diff --git a/docs/documentation/case.md b/docs/documentation/case.md
index 259b0bed47..130663767d 100644
--- a/docs/documentation/case.md
+++ b/docs/documentation/case.md
@@ -581,6 +581,9 @@ See @ref equations "Equations" for the mathematical models these parameters cont
 | `t_save`                   | Real    | Time duration between data output |
 | `t_stop`                   | Real    | Simulation stop time |
 | `surface_tension`          | Logical | Activate surface tension |
+| `sigma_model`              | Integer | Surface tension model: 0 constant, 1 linear in temperature |
+| `sigma_T_ref`              | Real    | Reference temperature of the linear sigma(T) model |
+| `sigma_dTdT`               | Real    | Slope dsigma/dT of the linear sigma(T) model |
 | `viscous`                  | Logical | Activate viscosity |
 | `hypoelasticity`           | Logical | Activate hypoelasticity* |
 | `riemann_hypo_ADC`         | Logical | Enable hypo anti-diffusion correction for HLLC/HLLD (default F) |
@@ -687,6 +690,8 @@ This option requires `weno_Re_flux` to be true because cell boundary values are
 
 - `surface_tension` activates surface tension when set to ``'T'``. Requires `sigma` to be set and `num_fluids`. The color function in each patch should be assigned such that `patch_icpp(i)%%cf_val = 1` in patches where `patch_icpp(i)%%alpha = 1 - eps` and `patch_icpp(i)%%cf_val = 0` in patches where `patch_icpp(i)%%alpha = eps`.
 
+- `sigma_model` selects the surface tension coefficient. With `sigma_model = 0` (default) it is the constant `sigma`. With `sigma_model = 1` it follows the linear thermal Marangoni model \f$\sigma(T) = \sigma + (\mathrm{d}\sigma/\mathrm{d}T)\,(T - T_\mathrm{ref})\f$, where `sigma_dTdT` (typically negative) and `sigma_T_ref` must be set. The temperature is the one Fourier heat conduction computes, so at least one fluid needs `fluid_pp(i)%%k_therm > 0` (and therefore `fluid_pp(i)%%cv > 0`). The tangential Marangoni force comes from the spatial variation of the capillary stress tensor; no separate source term is added. The model is only physical while \f$\sigma(T) > 0\f$, which holds whenever \f$|T - T_\mathrm{ref}| < \sigma/|\mathrm{d}\sigma/\mathrm{d}T|\f$.
+
 - `viscous` activates viscosity when set to ``'T'``. Requires `Re(1)` and `Re(2)` to be set.
 
 - `hypoelasticity` activates elastic stress calculations for fluid-solid interactions. Requires `G` to be set in `fluid_pp`, and `fd_order` to be set (1, 2, or 4). Compatible with HLL (`riemann_solver = 1`), HLLC (`riemann_solver = 2`), and HLLD (`riemann_solver = 4`). The Riemann solver choice determines how the elastic stress source term \f$\mathbf{S}^e\f$ is discretized:
diff --git a/examples/2D_thermocapillary_migration/.gitignore b/examples/2D_thermocapillary_migration/.gitignore
new file mode 100644
index 0000000000..be4b366370
--- /dev/null
+++ b/examples/2D_thermocapillary_migration/.gitignore
@@ -0,0 +1,14 @@
+# Simulation output (large; reproducible via run.py)
+runs/
+__pycache__/
+*.pyc
+# In-place output of the quick-start run (./mfc.sh run .../case.py from this directory)
+restart_data/
+D/
+silo_hdf5/
+viz/
+*.inp
+*.dat
+MFC.out
+MFC.sh
+run_time.inf
diff --git a/examples/2D_thermocapillary_migration/README.md b/examples/2D_thermocapillary_migration/README.md
new file mode 100644
index 0000000000..0912a8c9d0
--- /dev/null
+++ b/examples/2D_thermocapillary_migration/README.md
@@ -0,0 +1,72 @@
+# Thermocapillary droplet migration — σ(T) validation against Samareh et al. (2014)
+
+Validation of temperature-dependent surface tension (`sigma_model = 1`, linear σ(T)) against the
+thermocapillary-migration benchmarks of
+
+> B. Samareh, J. Mostaghimi, C. Moreau, "Thermocapillary migration of a deformable droplet,"
+> *Int. J. Heat Mass Transfer* 73 (2014) 616–626.
+
+A drop in an imposed linear temperature field sees a surface-tension gradient along its interface
+(σ falls as T rises, dσ/dT < 0). The resulting tangential Marangoni stress drags interfacial fluid
+hot→cold and, by reaction, the drop migrates toward the hot wall.
+
+<img src="mechanism_schematic.png" height="MAX_HEIGHT"/>
+
+This example covers the two σ(T)-only Samareh cases (§4.1.1 and §4.1.2). The large-Marangoni case
+(§4.2) also needs temperature-dependent viscosity and is not included.
+
+| | Samareh scenario | Regime | Figure | Reference |
+|---|---|---|---|---|
+| TC1 (`case_Ma_0p001.py`) | Zero Marangoni number (§4.1.1) | Ma → 0 | Fig 5 (2D cylinder) | plateau v_t/v_YGB ≈ 0.80 |
+| TC2 (`case_Ma_20.py`) | Low Marangoni, Nas & Tryggvason (§4.1.2) | Re = 5, Ma = 20, Ca = 0.0167 | Fig 7 | peak U* ≈ 0.13 |
+
+## Setup
+
+`sigma_model = 1` evaluates σ(T) = `sigma` + `sigma_dTdT`·(T − `sigma_T_ref`) per cell and averages
+it to each face of the capillary flux, so the tangential variation of σ along the interface is the
+Marangoni stress. T is the conduction temperature, so the model requires `fluid_pp(i)%k_therm > 0`.
+
+MFC is compressible: at uniform pressure the EOS gives T = (p+p∞)/((γ−1)ρcv), so the linear T(y) is
+imposed through the density initial condition, ρ(y) ∝ 1/T(y). The drop is painted inside a single
+full-box analytic patch so T is continuous across the interface. Bulk conduction and isothermal
+slip walls (`bc_y = -15`, `isothermal_in/out`) then hold the imposed gradient against the drop's own
+advection.
+
+## Results
+
+Measured with `measure.py` at 64 cells across the box (Intel ifx, CPU, `--case-optimization`):
+
+| | Cells per D | Measured | Reference | Wall time |
+|---|---|---|---|---|
+| TC1 | 12.8 | plateau v_t/v_YGB = 0.857 at t/t_r = 0.6, 0.876 at t/t_r = 2 | ≈ 0.80 | 3.7 h on 6 ranks |
+| TC2 | 32 | peak U* = 0.134 at t* = 3.7 | ≈ 0.13 | 18.5 min on 8 ranks |
+
+TC1 settles 7% above the reference and then creeps up slowly (+0.005 per t_r over the last t_r);
+finer TC1 grids were not run for this table. In TC2 the drop rises toward the hot wall and U* decays
+steadily after the peak, reaching 0.071 at t* = 15 when the run ends; the late-time value is not
+converged.
+
+## Caveats
+
+- Compressible acoustics. The closed box rings (the unbalanced t = 0 Laplace jump excites the
+  vertical acoustic standing wave). Read the migration velocity as the mean of the oscillation.
+  Samareh's incompressible solver has no acoustics.
+- Small Ma is expensive. TC1 approaches Samareh's Ma = 0 (frozen-T) limit through a large thermal
+  diffusivity, so its time step is conduction-limited (dt ∝ Ma).
+- Distinct-fluid finite Ma is qualitative. In TC2 each fluid's absolute density stratifies as ~1/T
+  (a compressibility artifact absent in the incompressible reference), so the exact peak is
+  fidelity-limited; TC1's identical fluids avoid this.
+
+## Reproducing
+
+```bash
+# One case in place. Output lands beside the case files and measure.py reads every snapshot in
+# restart_data/, so delete restart_data/ before running the other case here.
+# --case-optimization is roughly 10x faster.
+./mfc.sh run examples/2D_thermocapillary_migration/case_Ma_20.py -n 8 --case-optimization
+python3 examples/2D_thermocapillary_migration/measure.py examples/2D_thermocapillary_migration fig7
+
+# Grid sweeps into runs/<case>/<grid>/, summarized in runs/*.json
+python3 examples/2D_thermocapillary_migration/run.py fig5   # TC1, 64/128/256 cells across
+python3 examples/2D_thermocapillary_migration/run.py fig7   # TC2, 64/128 cells across
+```
diff --git a/examples/2D_thermocapillary_migration/case_Ma_0p001.py b/examples/2D_thermocapillary_migration/case_Ma_0p001.py
new file mode 100644
index 0000000000..5e4ef29ea8
--- /dev/null
+++ b/examples/2D_thermocapillary_migration/case_Ma_0p001.py
@@ -0,0 +1,175 @@
+#!/usr/bin/env python3
+# Thermocapillary migration of a 2D drop in the near-zero-Marangoni limit -- Samareh, Mostaghimi &
+# Moreau, Int. J. Heat Mass Transfer 73 (2014) 616-626, Sec. 4.1.1 / Fig. 5. The temperature field is
+# imposed through density (rho = rho_coeff/T at uniform p, recovered from the EOS); a large thermal
+# conductivity (small Marangoni number Ma) + isothermal gradient walls then hold T near the imposed
+# linear profile against the drop's own advection. As Ma -> 0 this approaches Samareh's invariant-T
+# limit (v_t / v_YGB ~ 0.80). Ma is very small here (deep limit), so dt is conduction-limited
+# (dt ~ Ma) and the run is long -- coarse grids only are practical.
+#
+# MFC is compressible, so a temperature gradient IS a density gradient: at uniform pressure the EOS
+# gives T = (p+p_inf)/((gam-1)*rho*cv), so rho(y) = rho_coeff/T(y) encodes the imposed field. Both
+# fluids are identical, so one analytic patch (shared smooth circle eta(x,y) driving color, the
+# Laplace pressure jump, and density together) makes the recovered T exactly linear everywhere --
+# drop included. T is shifted up by T0 = 10 (Samareh's T0 = 0) to keep rho positive; only gradT and
+# sigma_T (which set v_YGB) are physical.
+
+import json
+
+# Marangoni number: the small-Ma realization of Samareh's Ma = 0 limit (smaller -> closer to invariant T)
+Ma = 0.001
+
+# Geometry: D = 1 drop, 5D wide x 7.5D tall box; drop 1.5D above the cold floor, rise axis = y
+D = 1.0
+r = D / 2.0
+W = 5.0 * D
+Ly = 7.5 * D
+y_drop = -Ly / 2.0 + 1.5 * D
+
+Nx = 64  # cells across the box width (Samareh used 64, 128, 256)
+dx = W / Nx
+Ny = round(Ly / dx)
+
+# Stiffened-gas EOS: two identical fluids (gamma = 2); low-Mach, c ~ 20 at rho = 0.2
+gam = 2.0
+p_inf, p0 = 32.0, 8.0
+mu = 0.1  # dynamic viscosity of both phases (MFC takes Re = 1/mu)
+
+# Imposed linear field T(y) = T0 + gradT*y, encoded as rho(y) = rho_coeff/T(y) at uniform p0
+T0 = 10.0
+gradT = 2.0 / 15.0  # |dT/dy| = 0.1333 (Samareh)
+sigma0 = 0.1
+sigma_T = -0.1  # dsigma/dT
+G = abs(sigma_T * gradT)  # Marangoni stress scale = 0.013333
+rho_drop = 0.2  # Samareh: rho_d = rho_b = 0.2 (density at the drop's reference temperature T0)
+rho_coeff = rho_drop * T0
+cv = (p0 + p_inf) / ((gam - 1.0) * rho_coeff)  # closes the EOS so rho(T0) = rho_drop
+T_ref = T0 + gradT * y_drop  # sigma = sigma0 at the drop's initial position
+
+# Conduction properties from Ma: smaller Ma = larger diffusivity = T held closer to the gradient.
+# k = alpha*rho*cp at the reference state; the local diffusivity k/(rho*cp) varies mildly with rho.
+alpha_b = G * r**2 / (mu * Ma)  # bulk thermal diffusivity
+cp = gam * cv
+k_b = alpha_b * rho_drop * cp  # bulk conductivity
+
+eps = 1.0e-9  # trace volume fraction of the (identical) second phase
+
+# One smooth circle profile eta(x,y) -- ~1 in the drop, ~0 outside -- drives the color field, the
+# Laplace pressure jump, AND the density together, so the EOS-recovered T is the imposed linear field
+# everywhere (drop included): rho tracks p so the (p+p_inf) factors cancel. A single patch => no
+# analytic-density leak across patches. Hardcode the drop center/radius as decimals (`r` is a reserved
+# IC-expression name).
+xc_d, yc_d, r_d = 0.0, y_drop, r
+w_if = 0.75 * dx  # interface half-width (~3-cell transition)
+laplace = sigma0 / r  # Laplace pressure jump sigma/r
+dist = f"sqrt((x - ({xc_d:.9f}))**2 + (y - ({yc_d:.9f}))**2)"
+eta = f"0.5*(1.0 - tanh(({dist} - {r_d:.9f})/{w_if:.9f}))"
+cf_expr = f"({eta})"
+pres_expr = f"{p0:.9f} + {laplace:.9f}*({eta})"
+rho_num = (1.0 - eps) * rho_coeff
+rho_expr = f"{rho_num:.9f}*({p0 + p_inf:.9f} + {laplace:.9f}*({eta}))/({p0 + p_inf:.9f}*({T0} + {gradT:.9f}*y))"
+
+
+def T_of_y(y):
+    return T0 + gradT * y
+
+
+# Time stepping: min(acoustic CFL, explicit-conduction limit). Small Ma => conduction-limited dt.
+rho_min = rho_coeff / (T0 + gradT * Ly / 2.0)  # hot wall: lowest density, max sound speed
+c_max = (gam * (p0 + p_inf) / rho_min) ** 0.5
+t_r = mu / G  # capillary-thermal time = 7.5
+mydt = 0.35 * dx / c_max
+mydt = min(mydt, 0.35 * dx**2 / (4.0 * alpha_b))  # 2D explicit diffusion number
+t_step_stop = round(2.0 * t_r / mydt)  # 2 capillary-thermal times (the conduction-sweep window)
+t_step_save = max(1, t_step_stop // 80)
+
+# Configuration case dictionary
+data = {
+    # Logistics
+    "run_time_info": "T",
+    # Computational Domain
+    "x_domain%beg": -W / 2,
+    "x_domain%end": W / 2,
+    "y_domain%beg": -Ly / 2,
+    "y_domain%end": Ly / 2,
+    "m": Nx - 1,
+    "n": Ny - 1,
+    "p": 0,
+    "cyl_coord": "F",
+    "dt": mydt,
+    "t_step_start": 0,
+    "t_step_stop": t_step_stop,
+    "t_step_save": t_step_save,
+    # Simulation Algorithm
+    "model_eqns": 3,
+    "alt_soundspeed": "F",
+    "mixture_err": "T",
+    "mpp_lim": "F",
+    "time_stepper": 3,
+    "weno_order": 5,
+    "weno_eps": 1e-16,
+    "mapped_weno": "T",
+    "null_weights": "F",
+    "mp_weno": "T",
+    "weno_avg": "T",
+    "weno_Re_flux": "T",
+    "riemann_solver": 2,
+    "wave_speeds": 1,
+    "avg_state": 2,
+    # Free-slip box (Samareh): reflective side walls, slip walls on y (isothermal walls require -15/-16)
+    "bc_x%beg": -2,
+    "bc_x%end": -2,
+    "bc_y%beg": -15,
+    "bc_y%end": -15,
+    "num_patches": 1,
+    "num_fluids": 2,
+    # Physics: viscosity + temperature-dependent surface tension sigma(T); k_therm > 0 enables conduction
+    "viscous": "T",
+    "surface_tension": "T",
+    "sigma": sigma0,
+    "sigma_model": 1,
+    "sigma_T_ref": T_ref,
+    "sigma_dTdT": sigma_T,
+    # Database Structure Parameters
+    "format": 1,
+    "precision": 2,
+    "prim_vars_wrt": "T",
+    "cons_vars_wrt": "T",
+    "cf_wrt": "T",
+    "parallel_io": "T",
+    # Fluid 1 (continuous phase)
+    "fluid_pp(1)%gamma": 1.0 / (gam - 1.0),
+    "fluid_pp(1)%pi_inf": gam * p_inf / (gam - 1.0),
+    "fluid_pp(1)%cv": cv,
+    "fluid_pp(1)%Re(1)": 1.0 / mu,
+    "fluid_pp(1)%k_therm": k_b,
+    # Fluid 2 (identical properties)
+    "fluid_pp(2)%gamma": 1.0 / (gam - 1.0),
+    "fluid_pp(2)%pi_inf": gam * p_inf / (gam - 1.0),
+    "fluid_pp(2)%cv": cv,
+    "fluid_pp(2)%Re(1)": 1.0 / mu,
+    "fluid_pp(2)%k_therm": k_b,
+    # Single patch over the whole box. The drop lives entirely in the analytic profiles: cf, pres,
+    # and alpha_rho all share the one eta(x,y) above, so rho tracks the (jump-carrying) pressure and
+    # the EOS-recovered T is exactly the imposed linear field everywhere -- drop included, no bump.
+    "patch_icpp(1)%geometry": 3,
+    "patch_icpp(1)%x_centroid": 0.0,
+    "patch_icpp(1)%y_centroid": 0.0,
+    "patch_icpp(1)%length_x": W,
+    "patch_icpp(1)%length_y": Ly,
+    "patch_icpp(1)%vel(1)": 0.0,
+    "patch_icpp(1)%vel(2)": 0.0,
+    "patch_icpp(1)%pres": pres_expr,
+    "patch_icpp(1)%alpha_rho(1)": rho_expr,
+    "patch_icpp(1)%alpha_rho(2)": eps,
+    "patch_icpp(1)%alpha(1)": 1.0 - eps,
+    "patch_icpp(1)%alpha(2)": eps,
+    "patch_icpp(1)%cf_val": cf_expr,
+    # Isothermal Dirichlet gradient walls pin T to the imposed gradient (cold floor / hot ceiling)
+    "bc_y%isothermal_in": "T",
+    "bc_y%isothermal_out": "T",
+    "bc_y%Twall_in": T_of_y(-Ly / 2),
+    "bc_y%Twall_out": T_of_y(Ly / 2),
+}
+
+print(json.dumps(data))
diff --git a/examples/2D_thermocapillary_migration/case_Ma_20.py b/examples/2D_thermocapillary_migration/case_Ma_20.py
new file mode 100644
index 0000000000..fea9235fbb
--- /dev/null
+++ b/examples/2D_thermocapillary_migration/case_Ma_20.py
@@ -0,0 +1,212 @@
+#!/usr/bin/env python3
+# Thermocapillary migration of a 2D drop at finite Marangoni number -- Samareh, Mostaghimi & Moreau,
+# Int. J. Heat Mass Transfer 73 (2014) 616-626, Sec. 4.1.2 / Fig. 7 (test case of Nas & Tryggvason).
+# A drop of diameter D = 1 starts 1D above the cold floor of a 2D x 4D slip-wall box. With Ma finite
+# the energy equation is coupled: the drop's motion distorts the temperature field, a thermal
+# boundary layer forms at the interface, and sigma responds to the evolving local T. Drop and bulk
+# are genuinely DIFFERENT fluids (all material properties at ratio 0.5). Non-dimensional targets:
+# Re = 5, Ma = 20, Ca = 0.01666; Samareh's migration velocity U* = U/U_r peaks near 0.13 at t*~5.
+#
+# TEMPERATURE WITH DISTINCT FLUIDS (compressible code, incompressible reference).
+# MFC is compressible: at the low migration Mach here, pressure is ~uniform and the EOS locks
+# T = (p+p_inf)/((gam-1)*rho*cv), so a temperature gradient is a density gradient. With two distinct
+# fluids we impose T(y) by a PER-FLUID density proxy: each pure-fluid density absorbs the field,
+#   rho_i(x,y) = (p + p_inf_i)/((gam-1)*cv_i*T(y)).
+# The fluid-to-fluid density RATIO is then height-independent (T(y) cancels), so the prescribed
+# Nas-Tryggvason ratio 0.5 is preserved by tuning the per-fluid stiffening p_inf_i; and the MIXTURE
+# EOS recovers exactly T_mix(y) = T(y) across the interface (the volume fractions cancel) -- no jump.
+# Bulk conduction + isothermal gradient walls then sustain the field against the drop's advection.
+# Honest caveat: each fluid's ABSOLUTE density stratifies as ~1/T (a compressibility artifact absent
+# in Samareh's incompressible solver; magnitude ~ dT/T, shrunk by the T_base offset). So the peak
+# matches Fig 7 only qualitatively; the overshoot-then-settle shape is the validation target.
+
+import json
+
+# Non-dimensional targets (Samareh Fig. 7)
+Re = 5.0
+Ma = 20.0
+Ca = 0.01666
+prop_ratio = 0.5  # droplet / bulk material-property ratio (rho, mu, cv, k all at 0.5)
+
+# Geometry: D = 1 drop in a 2D wide x 4D tall box; drop 1D above the cold floor, gradient axis = y
+D = 1.0
+r = D / 2.0
+Wx = 2.0 * D
+Hy = 4.0 * D
+y_bottom = -Hy / 2.0
+y_drop = y_bottom + 1.0 * D
+
+Nx = 64  # cells across the box width (Samareh used 64, 128)
+dx = Wx / Nx
+Ny = round(Hy / dx)
+
+# Bulk reference scales (arbitrary; only Re, Ma, Ca are physical)
+rho_b = 1.0
+mu_b = 0.02
+gam = 2.0
+
+# Invert the non-dimensional numbers for the physical surface-tension and conduction properties
+G = Re * mu_b**2 / (rho_b * r**2)  # |sigma_T*gradT| = 0.008
+gradT = 1.0 / Hy  # T runs 1 (cold) .. 2 (hot) across the box height = 0.25
+sigma_T = -G / gradT  # dsigma/dT < 0 = -0.032
+sigma0 = G * r / Ca  # surface tension at T_ref ~ 0.240
+alpha_b = G * r**2 / (mu_b * Ma)  # bulk thermal diffusivity = 0.005
+
+# Imposed field T(y) = T_base + gradT*(y - y_bottom). T_base > 0 keeps rho positive and the
+# isothermal-wall validator happy; a larger offset reduces the ~1/T compressibility stratification.
+T_base = 1.0
+
+
+def T_of_y(y):
+    return T_base + gradT * (y - y_bottom)
+
+
+T_ref = T_of_y(y_drop)  # sigma = sigma0 at the drop's initial position; T_ref = 1.25
+
+# Stiffened-gas EOS, bulk fluid. Pick a reference sound speed (low migration Mach ~ U_r/c) and
+# background pressure (well above the Laplace jump sigma0/r); the stiffening and cv then close the
+# EOS so the bulk density is rho_b at (p0, T_ref).
+c_ref = 15.0
+p0 = 5.0
+p_inf_b = c_ref**2 * rho_b / gam - p0  # bulk stiffening
+cv_b = (p0 + p_inf_b) / ((gam - 1.0) * rho_b * T_ref)
+cp_b = gam * cv_b
+k_b = alpha_b * rho_b * cp_b  # bulk conductivity
+
+# Droplet = prop_ratio * bulk for every material property. The EOS stiffening p_inf_d follows so the
+# drop density is rho_d at (p0, T_ref): p0+p_inf_d = prop_ratio^2*(p0+p_inf_b) -> ratio rho_d/rho_b = 0.5.
+rho_d = prop_ratio * rho_b
+mu_d = prop_ratio * mu_b
+cv_d = prop_ratio * cv_b
+k_d = prop_ratio * k_b
+p_inf_d = (gam - 1.0) * rho_d * cv_d * T_ref - p0
+
+U_r = G * r / mu_b  # Marangoni velocity scale = 0.2
+t_r = mu_b / G  # capillary-thermal time = 2.5
+
+# Time stepping: min(acoustic CFL, explicit-conduction limit). Density is lowest (sound speed
+# highest) at the hot wall, where rho ~ rho_ref*T_ref/T_hot.
+T_hot = T_of_y(Hy / 2.0)
+rho_hot_min = prop_ratio * rho_b * T_ref / T_hot  # drop fluid at the hot wall: global density min
+c_max = (gam * (p0 + p_inf_d) / rho_hot_min) ** 0.5
+alpha_d = k_d / (rho_d * cp_b * prop_ratio)  # = k_d/(rho_d*cp_d), the fastest-diffusing phase
+alpha_max = max(alpha_b, alpha_d)
+mydt = 0.35 * dx / c_max
+mydt = min(mydt, 0.35 * dx**2 / (4.0 * alpha_max))
+t_step_stop = round(15.0 * t_r / mydt)  # 15 capillary-thermal times
+t_step_save = max(1, t_step_stop // 100)
+
+# One analytic patch over the whole box (a second patch's analytic density would leak globally). The
+# smooth circle eta(x,y) ~ 1 in the drop / 0 outside drives the color, the volume-fraction split, the
+# Laplace pressure jump, and BOTH per-fluid densities together. Hardcode the drop center/radius as
+# decimals (`r` and `eps` are reserved IC-expression names).
+xc_d, yc_d, r_d = 0.0, y_drop, r
+w_if = 0.75 * dx  # interface half-width (~3-cell transition)
+laplace = sigma0 / r  # Laplace pressure jump sigma/r
+dist = f"sqrt((x - ({xc_d:.9f}))**2 + (y - ({yc_d:.9f}))**2)"
+eta = f"0.5*(1.0 - tanh(({dist} - {r_d:.9f})/{w_if:.9f}))"
+Texpr = f"({T_base:.9f} + {gradT:.9f}*(y - ({y_bottom:.9f})))"
+pres_expr = f"({p0:.9f} + {laplace:.9f}*({eta}))"
+# volume fractions: drop = eta, bulk = 1 - eta, both floored at a trace eps_vf so no phase is ever
+# exactly empty (pressure relaxation rejects a cell holding volume fraction but no mass)
+eps_vf = 1.0e-9
+alpha2_expr = f"({eps_vf!r} + (1.0 - 2.0*{eps_vf!r})*({eta}))"
+alpha1_expr = f"(1.0 - {alpha2_expr})"
+# partial densities: alpha_i * rho_i, with rho_i = (p + p_inf_i)/((gam-1)*cv_i*T(y))
+cb = (gam - 1.0) * cv_b
+cd = (gam - 1.0) * cv_d
+arho1_expr = f"{alpha1_expr}*({pres_expr} + {p_inf_b:.9f})/({cb:.9f}*{Texpr})"
+arho2_expr = f"{alpha2_expr}*({pres_expr} + {p_inf_d:.9f})/({cd:.9f}*{Texpr})"
+cf_expr = f"({eta})"
+
+# Configuration case dictionary
+data = {
+    # Logistics
+    "run_time_info": "T",
+    # Computational Domain
+    "x_domain%beg": -Wx / 2,
+    "x_domain%end": Wx / 2,
+    "y_domain%beg": -Hy / 2,
+    "y_domain%end": Hy / 2,
+    "m": Nx - 1,
+    "n": Ny - 1,
+    "p": 0,
+    "cyl_coord": "F",
+    "dt": mydt,
+    "t_step_start": 0,
+    "t_step_stop": t_step_stop,
+    "t_step_save": t_step_save,
+    # Simulation Algorithm
+    "model_eqns": 3,
+    "alt_soundspeed": "F",
+    "mixture_err": "T",
+    "mpp_lim": "T",
+    "time_stepper": 3,
+    "weno_order": 5,
+    "weno_eps": 1e-16,
+    "mapped_weno": "T",
+    "null_weights": "F",
+    "mp_weno": "T",
+    "weno_avg": "T",
+    "weno_Re_flux": "T",
+    "riemann_solver": 2,
+    "wave_speeds": 1,
+    "avg_state": 2,
+    # Closed free-slip box (Samareh / Nas & Tryggvason): reflective side walls, slip walls on y
+    # (isothermal walls require -15/-16)
+    "bc_x%beg": -2,
+    "bc_x%end": -2,
+    "bc_y%beg": -15,
+    "bc_y%end": -15,
+    "num_patches": 1,
+    "num_fluids": 2,
+    # Physics: viscosity + temperature-dependent surface tension sigma(T); k_therm > 0 enables conduction
+    "viscous": "T",
+    "surface_tension": "T",
+    "sigma": sigma0,
+    "sigma_model": 1,
+    "sigma_T_ref": T_ref,
+    "sigma_dTdT": sigma_T,
+    # Database Structure Parameters
+    "format": 1,
+    "precision": 2,
+    "prim_vars_wrt": "T",
+    "cons_vars_wrt": "T",
+    "cf_wrt": "T",
+    "parallel_io": "T",
+    # Fluid 1 (bulk)
+    "fluid_pp(1)%gamma": 1.0 / (gam - 1.0),
+    "fluid_pp(1)%pi_inf": gam * p_inf_b / (gam - 1.0),
+    "fluid_pp(1)%cv": cv_b,
+    "fluid_pp(1)%Re(1)": 1.0 / mu_b,
+    "fluid_pp(1)%k_therm": k_b,
+    # Fluid 2 (droplet), all material properties at prop_ratio * bulk; stiffening keeps rho_d/rho_b = 0.5
+    "fluid_pp(2)%gamma": 1.0 / (gam - 1.0),
+    "fluid_pp(2)%pi_inf": gam * p_inf_d / (gam - 1.0),
+    "fluid_pp(2)%cv": cv_d,
+    "fluid_pp(2)%Re(1)": 1.0 / mu_d,
+    "fluid_pp(2)%k_therm": k_d,
+    # Single analytic patch: the drop lives entirely in eta(x,y). Color, volume fractions, pressure,
+    # and both per-fluid densities all share it, so the mixture EOS recovers the linear T(y) exactly
+    # (drop included) while the distinct fluid-to-fluid density ratio (0.5) is preserved.
+    "patch_icpp(1)%geometry": 3,
+    "patch_icpp(1)%x_centroid": 0.0,
+    "patch_icpp(1)%y_centroid": 0.0,
+    "patch_icpp(1)%length_x": Wx,
+    "patch_icpp(1)%length_y": Hy,
+    "patch_icpp(1)%vel(1)": 0.0,
+    "patch_icpp(1)%vel(2)": 0.0,
+    "patch_icpp(1)%pres": pres_expr,
+    "patch_icpp(1)%alpha_rho(1)": arho1_expr,
+    "patch_icpp(1)%alpha_rho(2)": arho2_expr,
+    "patch_icpp(1)%alpha(1)": alpha1_expr,
+    "patch_icpp(1)%alpha(2)": alpha2_expr,
+    "patch_icpp(1)%cf_val": cf_expr,
+    # Isothermal Dirichlet gradient walls pin T to the imposed gradient (cold floor / hot ceiling)
+    "bc_y%isothermal_in": "T",
+    "bc_y%isothermal_out": "T",
+    "bc_y%Twall_in": T_of_y(-Hy / 2),
+    "bc_y%Twall_out": T_of_y(Hy / 2),
+}
+
+print(json.dumps(data))
diff --git a/examples/2D_thermocapillary_migration/measure.py b/examples/2D_thermocapillary_migration/measure.py
new file mode 100644
index 0000000000..66593d483f
--- /dev/null
+++ b/examples/2D_thermocapillary_migration/measure.py
@@ -0,0 +1,273 @@
+#!/usr/bin/env python3
+"""Measure a thermocapillary drop's rise from MFC restart data, for both validation cases.
+
+One tool, two reporting modes (auto-detected from the domain, or forced with a 2nd argument):
+
+  fig5  (case_Ma_0p001.py, Samareh Sec 4.1.1 / Fig 5) -- v/v_YGB vs t/t_r. The drop rises in +y and is
+        tracked by its color function. In the slip-wall box (Samareh's geometry, default) the box is
+        the rest frame, so the lab-frame rise velocity is the measure; in an open box we report the
+        drift-corrected U = u_drop - u_far. Reports the terminal velocity (post-overshoot minimum
+        of the smoothed curve), the overshoot peak, the endpoint, and the late-time drift slope.
+        Compare against Samareh 2D ~ 0.80.
+  fig7  (case_Ma_20.py, Sec 4.1.2 / Fig 7, Nas & Tryggvason) -- U*=U/U_r vs t*=t/t_r, with
+        U_r=|sigma_T gradT| r0/mu_b and t_r=mu_b/|sigma_T gradT|. Reports the overshoot peak and the
+        terminal (final-t_r-window) value. Compare the peak against ~0.13.
+
+Mode auto-detection uses the rise-axis extent Ly: fig5 is Ly=7.5D, fig7 is Ly=4D. Pass
+fig5|fig7 as the 2nd argument to override.
+
+All run-dependent constants come from simulation.inp / pre_process.inp so this can't silently
+disagree with the data. The conserved-variable layout is (model_eqns=3, num_fluids=2,
+surface_tension): index 0,1 = partial densities, 3 = y-momentum, and the color function is the last
+conserved variable.
+
+Usage:  python3 measure.py [case_dir] [fig5|fig7]
+Writes: <case_dir>/viz/<mode>*.png  and prints a JSON summary line (tag: RESULT_JSON).
+"""
+
+import glob
+import json
+import os
+import re
+import sys
+
+import matplotlib
+
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt
+import numpy as np
+
+case_dir = sys.argv[1] if len(sys.argv) > 1 else os.path.dirname(os.path.abspath(__file__))
+mode_arg = sys.argv[2].lower() if len(sys.argv) > 2 else None
+
+R = 0.5  # droplet radius (D = 1)
+
+
+def read_namelist(path):
+    """Parse a Fortran namelist file's plain "name = value" lines into a dict (lowercase keys)."""
+    out = {}
+    with open(path) as fh:
+        for line in fh:
+            if "=" in line:
+                name, value = line.split("=", 1)
+                out[name.strip().lower()] = value.strip().rstrip(",")
+    return out
+
+
+params = read_namelist(os.path.join(case_dir, "simulation.inp"))
+patches = read_namelist(os.path.join(case_dir, "pre_process.inp"))
+
+
+def param(name, src=params):
+    """Look up a parameter by name and return it as a float."""
+    return float(src[name.lower()])
+
+
+nx = int(param("m")) + 1  # box-width cells (x)
+ny = int(param("n")) + 1  # rise-axis cells (y)
+nz = int(param("p")) + 1  # short-axis cells (z); 1 in 2D
+dt = param("dt")
+Wx = param("x_domain%end", patches) - param("x_domain%beg", patches)  # box width
+Ly = param("y_domain%end", patches) - param("y_domain%beg", patches)  # rise-axis extent
+dim = 3 if int(param("p")) > 0 else 2
+
+# Mode: explicit 2nd arg, else auto-detect from the rise-axis extent.
+if mode_arg in ("fig5", "fig7"):
+    mode = mode_arg
+elif Ly > 6.0:
+    mode = "fig5"  # 7.5D box
+else:
+    mode = "fig7"  # 4D box
+
+restart_dir = os.path.join(case_dir, "restart_data")
+# Cell-center y positions from the boundary file (last ny+1 boundaries are the interior).
+yb = np.fromfile(os.path.join(restart_dir, "lustre_y_cb.dat"), np.float64)[-(ny + 1) :]
+y = 0.5 * (yb[:-1] + yb[1:])  # shape (ny,)
+
+steps = sorted(int(m.group(1)) for f in glob.glob(os.path.join(restart_dir, "lustre_*.dat")) if (m := re.search(r"lustre_(\d+)\.dat$", f)))
+if not steps:
+    sys.exit(f"no lustre_<step>.dat snapshots in {restart_dir!r} -- run the case first")
+
+cells = nx * ny * nz
+nvars = np.fromfile(os.path.join(restart_dir, f"lustre_{steps[0]}.dat"), np.float64).size // cells
+c_idx = nvars - 1  # color function is the last conserved variable
+
+
+def field(snapshot, i):
+    """Variable i of a snapshot, reshaped to (nz, ny, nx) -- x is MFC's fast (innermost) index."""
+    return snapshot[i * cells : (i + 1) * cells].reshape(nz, ny, nx)
+
+
+# Shared core: walk the snapshots, building color-weighted time series. Identical math in 2D/3D.
+yb3 = y[None, :, None]  # broadcast y over (nz, ny, nx)
+times, y_centroid, u_lab, u_far, rho_drop_t = [], [], [], [], []
+for s in steps:
+    snap = np.fromfile(os.path.join(restart_dir, f"lustre_{s}.dat"), np.float64)
+    rho = field(snap, 0) + field(snap, 1)  # total density (sum of partial densities)
+    vy = field(snap, 3) / rho  # rise (y) velocity
+    c = np.clip(field(snap, c_idx), 0.0, 1.0)  # color function (1 inside the drop)
+    csum = c.sum()
+    times.append(s * dt)
+    y_centroid.append((c * yb3).sum() / csum)  # color-weighted y-centroid
+    u_lab.append((c * vy).sum() / csum)  # color-weighted drop rise velocity
+    rho_drop_t.append((c * rho).sum() / csum)  # color-weighted drop density
+times, y_centroid, u_lab, rho_drop_t = map(np.array, (times, y_centroid, u_lab, rho_drop_t))
+
+viz_dir = os.path.join(case_dir, "viz")
+os.makedirs(viz_dir, exist_ok=True)
+
+
+def reference_scales():
+    """Marangoni velocity/time scales U_r, t_r and the YGB speed, from the run's constants."""
+    mu = 1.0 / param("fluid_pp(1)%re(1)")  # MFC takes Re = 1/mu
+    dsigma_dT = param("sigma_dtdt")  # sigma(T) slope
+    # gradT: prefer the imposed wall temperatures (conduction cases); else the IC construction 1/Ly.
+    if "bc_y%twall_out" in params and "bc_y%twall_in" in params:
+        gradT = abs(param("bc_y%twall_out") - param("bc_y%twall_in")) / Ly
+    else:
+        gradT = 2.0 / 15.0 if mode == "fig5" else 1.0 / Ly  # fig5 uses 1/(7.5D)=2/15 exactly
+    G = abs(dsigma_dT * gradT)
+    return mu, dsigma_dT, gradT, G, G * R / mu, mu / G  # mu, sigma_T, gradT, G, U_r, t_r
+
+
+if mode == "fig5":
+    mu, dsigma_dT, gradT, _, _, t_r = reference_scales()
+    wall = int(param("bc_y%beg")) in (-2, -15, -16)  # walled box (Samareh) vs open box
+    far_y = 0.75 * (Ly / 2.0)
+    is_far = np.abs(y) > far_y  # a cell is "far field" once |y| exceeds this (open-box drift)
+    # far-field y-velocity per snapshot (open-box return-flow drift correction)
+    u_far = []
+    for s in steps:
+        snap = np.fromfile(os.path.join(restart_dir, f"lustre_{s}.dat"), np.float64)
+        u_far.append((field(snap, 3) / (field(snap, 0) + field(snap, 1)))[:, is_far, :].mean())
+    u_far = np.array(u_far)
+    U = u_lab if wall else u_lab - u_far
+    v_YGB = (2.0 / 15.0) * (-dsigma_dT) * gradT * R / mu  # YGB terminal speed (mu* = k* = 1)
+    rho_drop = float(rho_drop_t[0])
+    tau = rho_drop * R**2 / mu  # viscous time
+    samareh_ratio = 0.80
+
+    # Terminal velocity = post-overshoot minimum of the curve smoothed over ~one viscous time
+    # (the comparison to Samareh's invariant-T terminal velocity; the endpoint drifts as T distorts).
+    ratio_t = U / v_YGB
+    dt_snap = times[1] - times[0] if len(times) > 1 else tau
+    smooth_w = max(1, int(round(tau / dt_snap)))
+    if smooth_w > 1:  # edge-corrected boxcar so the ends aren't depressed by zero-padding
+        k = np.ones(smooth_w)
+        sm = np.convolve(ratio_t, k, mode="same") / np.convolve(np.ones_like(ratio_t), k, mode="same")
+    else:
+        sm = ratio_t
+    peak_i = int(np.argmax(np.where(times < 0.6 * t_r, sm, -np.inf)))
+    plateau_i = peak_i + int(np.argmin(sm[peak_i:]))
+    ratio_plateau, t_plateau_tr, overshoot, ratio_final = float(sm[plateau_i]), float(times[plateau_i] / t_r), float(sm[peak_i]), float(ratio_t[-1])
+    in_tail = times >= times[-1] - tau
+    slope_per_tr = float(np.polyfit(times[in_tail] / t_r, ratio_t[in_tail], 1)[0]) if in_tail.sum() > 1 else 0.0
+
+    mode_label = "slip-wall box (lab-frame U)" if wall else "open box (drift-corrected U)"
+    print(f"[fig5] dim={dim}D  nx={nx} ny={ny} nz={nz}  cells={cells}  nvars={nvars}  {mode_label}")
+    print(f"rho_drop={rho_drop:.4f}  tau={tau:.3f}  t_r={t_r:.3f}  v_YGB={v_YGB:.6f}  run length={times[-1] / t_r:.2f} t_r\n")
+    print(f"{'step':>7} {'t':>8} {'t/t_r':>6} {'y_drop':>9} {'u_lab':>9} {'u_far':>9} {'U':>9} {'U/vYGB':>8}")
+    for i, t in enumerate(times):
+        print(f"{steps[i]:>7} {t:>8.4f} {t / t_r:>6.2f} {y_centroid[i]:>9.5f} {u_lab[i]:>9.5f} {u_far[i]:>9.5f} {U[i]:>9.5f} {U[i] / v_YGB:>+8.2f}")
+    print(f"\nrising (+y, toward hot top): {y_centroid[-1] > y_centroid[0]}")
+    print(f"terminal velocity  v_t/v_YGB = {ratio_plateau:+.3f}  (at t/t_r = {t_plateau_tr:.2f})   [Samareh {dim}D ~ {samareh_ratio:.2f}]")
+    print(f"overshoot peak = {overshoot:+.3f}   endpoint = {ratio_final:+.3f}   late-time drift = {slope_per_tr:+.3f}/t_r")
+    if abs(slope_per_tr) > 0.05:
+        print(f"  NOTE: endpoint is still drifting ({slope_per_tr:+.3f}/t_r) -- the plateau, not the endpoint, is the Samareh comparison")
+
+    fig, ax = plt.subplots(figsize=(7.2, 4.6))
+    ax.plot(times / t_r, ratio_t, "o-", color="C0", ms=3.0, lw=1.0, label="MFC " + ("(lab frame)" if wall else "(drift-corrected)"))
+    ax.axhline(1.0, ls=":", color="0.45", lw=1.3)
+    ax.text(0.02, 1.0, r" $v_{\mathrm{YGB}}$ (zero-Ma Stokes, sphere)", va="bottom", ha="left", color="0.4", fontsize=9)
+    ax.axhline(samareh_ratio, ls="--", color="C3", lw=1.3, label=rf"Samareh {dim}D $\approx$ {samareh_ratio:.2f}")
+    ax.plot(t_plateau_tr, ratio_plateau, "*", color="k", ms=14, zorder=5, label=rf"terminal velocity = {ratio_plateau:.2f}")
+    ax.set_xlabel(r"$t / t_r$  ($t_r = \mu / |\sigma_T \nabla T|$, Samareh time scale)")
+    ax.set_ylabel(r"normalized rise velocity  $v / v_{\mathrm{YGB}}$")
+    ax.set_xlim(left=0.0)
+    ax.set_ylim(0.0, max(1.1, 1.05 * ratio_t.max()))
+    ax.set_title(f"{dim}D thermocapillary rise ({'Samareh slip-wall box' if wall else 'open box'})")
+    ax.grid(alpha=0.3)
+    ax.legend(loc="lower right", fontsize=9)
+    fig.tight_layout()
+    out_png = os.path.join(viz_dir, "rise_velocity.png")
+    fig.savefig(out_png, dpi=150)
+    print(f"saved figure -> {out_png}")
+    summary = {
+        "mode": "fig5",
+        "dim": dim,
+        "nx_width": nx,
+        "cells_per_D": nx / 5.0,
+        "cells": cells,
+        "mu": mu,
+        "rho_drop": rho_drop,
+        "dsigma_dT": dsigma_dT,
+        "gradT": gradT,
+        "r": R,
+        "v_YGB": v_YGB,
+        "tau": tau,
+        "t_r": t_r,
+        "wall": wall,
+        "drift_corrected": not wall,
+        "samareh_ratio": samareh_ratio,
+        "ratio_plateau": ratio_plateau,
+        "t_plateau_tr": t_plateau_tr,
+        "overshoot": overshoot,
+        "ratio_final": ratio_final,
+        "slope_per_tr": slope_per_tr,
+        "u_lab_final": float(u_lab[-1]),
+        "u_far_final": float(u_far[-1]),
+        "rises": bool(y_centroid[-1] > y_centroid[0]),
+        "t_end": float(times[-1]),
+        "t_end_tr": float(times[-1] / t_r),
+    }
+
+else:  # fig7
+    mu_b, sigma_T, gradT, _, U_r, t_r = reference_scales()
+    NAS_TRYGGVASON_PEAK = 0.13
+    t_star, U_star = times / t_r, u_lab / U_r
+    peak_i = int(np.argmax(U_star))
+    peak, t_peak = float(U_star[peak_i]), float(t_star[peak_i])
+    in_tail = times >= times[-1] - t_r
+    terminal = float(U_star[in_tail].mean())
+    print(f"[fig7] nx={nx} ny={ny} nz={nz}  cells={cells}  cells/D={nx / (Wx / 1.0):.1f}  nvars={nvars}  color idx={c_idx}")
+    print(f"mu_b={mu_b:.4f}  sigma_T={sigma_T:.5f}  gradT={gradT:.5f}  U_r={U_r:.5f}  t_r={t_r:.4f}  run={t_star[-1]:.2f} t_r\n")
+    print(f"{'step':>8} {'t*':>7} {'y_drop':>9} {'U':>10} {'U*':>9}")
+    for i in range(len(times)):
+        print(f"{steps[i]:>8} {t_star[i]:>7.2f} {y_centroid[i]:>9.5f} {u_lab[i]:>10.6f} {U_star[i]:>9.4f}")
+    print(f"\nrising (+y, toward hot top): {y_centroid[-1] > y_centroid[0]}")
+    print(f"overshoot peak  U* = {peak:.4f}  at t* = {t_peak:.2f}   [Nas & Tryggvason / Samareh Fig 7 peak ~ {NAS_TRYGGVASON_PEAK:.2f}]")
+    print(f"terminal (final t_r window)  U* = {terminal:.4f}")
+    fig, ax = plt.subplots(figsize=(7.2, 4.6))
+    ax.plot(t_star, U_star, "o-", color="C0", ms=3.0, lw=1.2, label=f"MFC ({nx}/box width, {nx / (Wx / 1.0):.0f}/$D$)")
+    ax.axhline(NAS_TRYGGVASON_PEAK, ls="--", color="C3", lw=1.3, label=rf"Nas & Tryggvason peak $\approx$ {NAS_TRYGGVASON_PEAK:.2f}")
+    ax.plot(t_peak, peak, "*", color="k", ms=14, zorder=5, label=rf"MFC peak = {peak:.3f}")
+    ax.set_xlabel(r"$t^* = t / t_r$  ($t_r = \mu_b / |\sigma_T \nabla T|$)")
+    ax.set_ylabel(r"$U^* = U / U_r$  ($U_r = |\sigma_T \nabla T|\, r_0 / \mu_b$)")
+    ax.set_title("Fig 7: finite-Ma thermocapillary migration (Re=5, Ma=20, Ca=0.01666)")
+    ax.set_xlim(left=0.0)
+    ax.set_ylim(bottom=min(0.0, 1.05 * U_star.min()))
+    ax.grid(alpha=0.3)
+    ax.legend(loc="lower right", fontsize=9)
+    fig.tight_layout()
+    out_png = os.path.join(viz_dir, "fig7_migration.png")
+    fig.savefig(out_png, dpi=150)
+    print(f"saved figure -> {out_png}")
+    summary = {
+        "mode": "fig7",
+        "nx_width": nx,
+        "cells_per_D": nx / (Wx / 1.0),
+        "cells": cells,
+        "mu_b": mu_b,
+        "sigma_T": sigma_T,
+        "gradT": gradT,
+        "U_r": U_r,
+        "t_r": t_r,
+        "peak": peak,
+        "t_peak_tr": t_peak,
+        "terminal": terminal,
+        "nas_tryggvason_peak": NAS_TRYGGVASON_PEAK,
+        "rises": bool(y_centroid[-1] > y_centroid[0]),
+        "t_end_tr": float(t_star[-1]),
+    }
+
+print("RESULT_JSON " + json.dumps(summary))
diff --git a/examples/2D_thermocapillary_migration/mechanism_schematic.png b/examples/2D_thermocapillary_migration/mechanism_schematic.png
new file mode 100644
index 0000000000..6980739b92
Binary files /dev/null and b/examples/2D_thermocapillary_migration/mechanism_schematic.png differ
diff --git a/examples/2D_thermocapillary_migration/run.py b/examples/2D_thermocapillary_migration/run.py
new file mode 100644
index 0000000000..62d525ef62
--- /dev/null
+++ b/examples/2D_thermocapillary_migration/run.py
@@ -0,0 +1,149 @@
+#!/usr/bin/env python3
+"""Run the thermocapillary validation cases and measure them.
+
+Each grid variant runs in its own runs/<name>/ dir (MFC writes output next to the case
+file), is measured by measure.py, and its result is collected into runs/<summary>.json.
+
+Targets:
+  fig5  case_Ma_0p001.py -- 2D zero-Ma rise, grid convergence (v/v_YGB -> ~0.80)
+  fig7  case_Ma_20.py    -- 2D low-Ma migration (U*/U_r, peak ~0.13)
+  all   fig5 + fig7
+
+Usage (invokes mpirun -- run from a normal shell):
+    python3 run.py <fig5|fig7|all> [run|remeasure]   (default: fig5 run)
+
+  run        run each variant, then measure it
+  remeasure  re-measure existing runs/ WITHOUT running any simulation
+
+The canonical case files have no grid knob, so a variant's grid is set by copying the case
+into its run dir and rewriting its `Nx = <n>` line; the committed case is never touched.
+"""
+
+import json
+import os
+import re
+import shutil
+import subprocess
+import sys
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+REPO = os.path.abspath(os.path.join(HERE, "..", ".."))
+RUNS = os.path.join(HERE, "runs")
+
+# Per target: the case file, the measure.py mode, the summary file, and the grid variants to run.
+# Each variant is (run-dir name, MPI ranks, Nx override or None for the case default).
+TARGETS = {
+    "fig5": dict(
+        case="case_Ma_0p001.py",
+        mode="fig5",
+        summary="summary.json",
+        variants=[
+            ("tc1/w064", 6, 64),
+            ("tc1/w128", 16, 128),
+            ("tc1/w256", 64, 256),  # Ma=0.001 is conduction-dt-limited; w256 is very expensive
+        ],
+    ),
+    "fig7": dict(
+        case="case_Ma_20.py",
+        mode="fig7",
+        summary="fig7_summary.json",
+        variants=[
+            ("tc2/w064", 8, 64),
+            ("tc2/w128", 16, 128),
+        ],
+    ),
+}
+
+# Headline numbers to show per mode in the end-of-run table.
+TABLE_KEYS = {
+    "fig5": ("cells_per_D", "ratio_plateau", "overshoot", "ratio_final"),
+    "fig7": ("cells_per_D", "peak", "t_peak_tr", "terminal"),
+}
+
+
+def run_variant(case_file, name, ranks, nx):
+    """Run one variant in a fresh runs/<name>/ dir. Returns True on success.
+
+    The case is copied in; if nx is given, the copy's `Nx = <n>` line is rewritten. The
+    committed case file is never touched.
+    """
+    wd = os.path.join(RUNS, name)
+    if os.path.isdir(wd):
+        shutil.rmtree(wd)
+    os.makedirs(wd)
+
+    dst = os.path.join(wd, os.path.basename(case_file))
+    shutil.copy(os.path.join(HERE, case_file), dst)
+    if nx is not None:
+        text, hits = re.subn(r"(?m)^Nx = \d+", f"Nx = {nx}", open(dst).read(), count=1)
+        if hits != 1:
+            print(f"  WARNING: no `Nx = <int>` line in {name} -- using case default")
+        open(dst, "w").write(text)
+
+    print(f"\n>>> {name}: {os.path.basename(case_file)} ranks={ranks} Nx={nx or 'default'}", flush=True)
+    rel = os.path.relpath(dst, REPO)
+    p = subprocess.run(
+        ["./mfc.sh", "run", rel, "-n", str(ranks), "--case-optimization"],
+        cwd=REPO,
+        capture_output=True,
+        text=True,
+        check=False,
+    )
+    if p.returncode != 0:
+        print(f"  RUN FAILED (exit {p.returncode}):")
+        print("\n".join((p.stdout + p.stderr).splitlines()[-25:]))
+        return False
+    return True
+
+
+def measure(wd, mode):
+    """Run measure.py <wd> <mode> and return its RESULT_JSON dict (or None on failure)."""
+    m = subprocess.run([sys.executable, os.path.join(HERE, "measure.py"), wd, mode], capture_output=True, text=True, check=False)
+    print("\n".join(m.stdout.splitlines()[-4:]))
+    if m.returncode != 0:
+        print(f"  MEASURE FAILED: {m.stderr[-500:]}")
+        return None
+    for line in m.stdout.splitlines():
+        if line.startswith("RESULT_JSON "):
+            return json.loads(line[len("RESULT_JSON ") :])
+    return None
+
+
+def process_target(target, run_mode):
+    """Run (or just re-measure) every variant of one target and update its summary file."""
+    cfg = TARGETS[target]
+    spath = os.path.join(RUNS, cfg["summary"])
+    summary = json.load(open(spath)) if os.path.isfile(spath) else {}
+
+    for name, ranks, nx in cfg["variants"]:
+        wd = os.path.join(RUNS, name)
+        if run_mode == "run":
+            if not run_variant(cfg["case"], name, ranks, nx):
+                continue
+        elif not os.path.isfile(os.path.join(wd, "simulation.inp")):
+            continue  # remeasure: skip variants that were never run
+        res = measure(wd, cfg["mode"])
+        if res is not None:
+            summary[name] = res
+            json.dump(summary, open(spath, "w"), indent=2)  # checkpoint after each variant
+
+    keys = TABLE_KEYS[target]
+    print(f"\n=== {target} summary ({len(summary)} runs) -> {spath} ===")
+    for name, res in sorted(summary.items()):
+        print(f"  {name:>16}: " + "  ".join(f"{k}={res[k]}" for k in keys if k in res))
+
+
+def main():
+    target = sys.argv[1] if len(sys.argv) > 1 else "fig5"
+    run_mode = sys.argv[2] if len(sys.argv) > 2 else "run"
+    targets = ["fig5", "fig7"] if target == "all" else [target]
+    if any(t not in TARGETS for t in targets):
+        sys.exit(f"unknown target {target!r}; choose from {', '.join(TARGETS)}, all")
+
+    os.makedirs(RUNS, exist_ok=True)
+    for t in targets:
+        process_target(t, run_mode)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/src/simulation/m_global_parameters.fpp b/src/simulation/m_global_parameters.fpp
index 6157db87ef..92ee078763 100644
--- a/src/simulation/m_global_parameters.fpp
+++ b/src/simulation/m_global_parameters.fpp
@@ -389,6 +389,9 @@ contains
         adv_src_mode = adv_src_mode_unset
         use_nc_iface_vel = .false.
         low_Mach = 0
+        sigma_model = 0
+        sigma_T_ref = dflt_real
+        sigma_dTdT = 0._wp
         wave_speeds = dflt_int
         null_weights = .false.
         precision = 2
diff --git a/src/simulation/m_rhs.fpp b/src/simulation/m_rhs.fpp
index 7e74b33ed2..a36c5abbca 100644
--- a/src/simulation/m_rhs.fpp
+++ b/src/simulation/m_rhs.fpp
@@ -669,7 +669,7 @@ contains
 
         if (surface_tension) then
             call nvtxStartRange("RHS-SURFACE-TENSION")
-            call s_get_capillary(q_prim_qp%vf, bc_type)
+            call s_get_capillary(q_prim_qp%vf, q_T_sf, bc_type)
             call nvtxEndRange
         end if
 
diff --git a/src/simulation/m_start_up.fpp b/src/simulation/m_start_up.fpp
index c4aa060a93..c32dc01ed3 100644
--- a/src/simulation/m_start_up.fpp
+++ b/src/simulation/m_start_up.fpp
@@ -1112,7 +1112,7 @@ contains
         $:GPU_UPDATE(device='[adv_n, adap_dt, adap_dt_tol, adap_dt_max_iters, pi_fac, low_Mach]')
 
         $:GPU_UPDATE(device='[acoustic_source, num_source]')
-        $:GPU_UPDATE(device='[sigma, surface_tension]')
+        $:GPU_UPDATE(device='[sigma, surface_tension, sigma_model, sigma_T_ref, sigma_dTdT]')
 
         $:GPU_UPDATE(device='[dx, dy, dz, x_cb, x_cc, y_cb, y_cc, z_cb, z_cc]')
         $:GPU_UPDATE(device='[bc_x%beg, bc_x%end, bc_y%beg, bc_y%end, bc_z%beg, bc_z%end]')
diff --git a/src/simulation/m_surface_tension.fpp b/src/simulation/m_surface_tension.fpp
index 2fbe62261b..d98a81cc29 100644
--- a/src/simulation/m_surface_tension.fpp
+++ b/src/simulation/m_surface_tension.fpp
@@ -34,6 +34,10 @@ module m_surface_tension
     !> @}
     $:GPU_DECLARE(create='[gL_x, gR_x]')
 
+    !> Cell-centered surface tension coefficient for sigma_model = 1
+    real(wp), allocatable, dimension(:,:,:) :: c_sigma
+    $:GPU_DECLARE(create='[c_sigma]')
+
     type(int_bounds_info) :: is1, is2, is3, iv
     $:GPU_DECLARE(create='[is1, is2, is3, iv]')
 
@@ -54,6 +58,12 @@ contains
         @:ALLOCATE(gL_x(idwbuff(1)%beg:idwbuff(1)%end, idwbuff(2)%beg:idwbuff(2)%end, idwbuff(3)%beg:idwbuff(3)%end, num_dims + 1))
         @:ALLOCATE(gR_x(idwbuff(1)%beg:idwbuff(1)%end, idwbuff(2)%beg:idwbuff(2)%end, idwbuff(3)%beg:idwbuff(3)%end, num_dims + 1))
 
+        if (sigma_model == 1) then
+            @:ALLOCATE(c_sigma(idwbuff(1)%beg:idwbuff(1)%end, idwbuff(2)%beg:idwbuff(2)%end, idwbuff(3)%beg:idwbuff(3)%end))
+        else
+            @:ALLOCATE(c_sigma(1:1, 1:1, 1:1))
+        end if
+
     end subroutine s_initialize_surface_tension_module
 
     !> Compute the capillary source flux from reconstructed color-gradient fields
@@ -105,11 +115,12 @@ contains
             real(wp), dimension(num_dims, num_dims) :: Omega
         #:endif
         real(wp) :: w1L, w1R, w2L, w2R, w3L, w3R, w1, w2, w3
-        real(wp) :: normWL, normWR, normW
+        real(wp) :: normWL, normWR, normW, sigma_face
         integer  :: j, k, l, i
 
         if (id == 1) then
-            $:GPU_PARALLEL_LOOP(collapse=3, private='[Omega, w1L, w2L, w3L, w1R, w2R, w3R, w1, w2, w3, normWL, normWR, normW]')
+            $:GPU_PARALLEL_LOOP(collapse=3, &
+                                & private='[Omega, w1L, w2L, w3L, w1R, w2R, w3R, w1, w2, w3, normWL, normWR, normW, sigma_face]')
             do l = isz%beg, isz%end
                 do k = isy%beg, isy%end
                     do j = isx%beg, isx%end
@@ -132,7 +143,9 @@ contains
                         normW = (normWL + normWR)/2._wp
 
                         if (normW > capillary_cutoff) then
-                            call s_compute_capillary_stress_tensor(sigma, w1, w2, w3, normW, Omega)
+                            sigma_face = sigma
+                            if (sigma_model == 1) sigma_face = (c_sigma(j, k, l) + c_sigma(j + 1, k, l))/2._wp
+                            call s_compute_capillary_stress_tensor(sigma_face, w1, w2, w3, normW, Omega)
 
                             do i = 1, num_dims
                                 flux_src_vf(eqn_idx%mom%beg + i - 1)%sf(j, k, l) = flux_src_vf(eqn_idx%mom%beg + i - 1)%sf(j, k, &
@@ -144,7 +157,7 @@ contains
 
                             ! Continuum surface force capillary stress, Schmidmayer et al. JCP (2017)
                             flux_src_vf(eqn_idx%E)%sf(j, k, l) = flux_src_vf(eqn_idx%E)%sf(j, k, &
-                                        & l) + sigma*c_divs(num_dims + 1)%sf(j, k, l)*vSrc_rsx_vf(j, k, l, 1)
+                                        & l) + sigma_face*c_divs(num_dims + 1)%sf(j, k, l)*vSrc_rsx_vf(j, k, l, 1)
                         end if
                     end do
                 end do
@@ -152,7 +165,8 @@ contains
             $:END_GPU_PARALLEL_LOOP()
         else if (id == 2) then
             #:if not MFC_CASE_OPTIMIZATION or num_dims > 1
-                $:GPU_PARALLEL_LOOP(collapse=3, private='[Omega, w1L, w2L, w3L, w1R, w2R, w3R, w1, w2, w3, normWL, normWR, normW]')
+                $:GPU_PARALLEL_LOOP(collapse=3, &
+                                    & private='[Omega, w1L, w2L, w3L, w1R, w2R, w3R, w1, w2, w3, normWL, normWR, normW, sigma_face]')
                 do l = isz%beg, isz%end
                     do k = isy%beg, isy%end
                         do j = isx%beg, isx%end
@@ -175,7 +189,9 @@ contains
                             normW = (normWL + normWR)/2._wp
 
                             if (normW > capillary_cutoff) then
-                                call s_compute_capillary_stress_tensor(sigma, w1, w2, w3, normW, Omega)
+                                sigma_face = sigma
+                                if (sigma_model == 1) sigma_face = (c_sigma(j, k, l) + c_sigma(j, k + 1, l))/2._wp
+                                call s_compute_capillary_stress_tensor(sigma_face, w1, w2, w3, normW, Omega)
 
                                 do i = 1, num_dims
                                     flux_src_vf(eqn_idx%mom%beg + i - 1)%sf(j, k, l) = flux_src_vf(eqn_idx%mom%beg + i - 1)%sf(j, &
@@ -186,7 +202,7 @@ contains
                                 end do
 
                                 flux_src_vf(eqn_idx%E)%sf(j, k, l) = flux_src_vf(eqn_idx%E)%sf(j, k, &
-                                            & l) + sigma*c_divs(num_dims + 1)%sf(j, k, l)*vSrc_rsx_vf(j, k, l, 2)
+                                            & l) + sigma_face*c_divs(num_dims + 1)%sf(j, k, l)*vSrc_rsx_vf(j, k, l, 2)
                             end if
                         end do
                     end do
@@ -195,7 +211,8 @@ contains
             #:endif
         else if (id == 3) then
             #:if not MFC_CASE_OPTIMIZATION or num_dims > 2
-                $:GPU_PARALLEL_LOOP(collapse=3, private='[Omega, w1L, w2L, w3L, w1R, w2R, w3R, w1, w2, w3, normWL, normWR, normW]')
+                $:GPU_PARALLEL_LOOP(collapse=3, &
+                                    & private='[Omega, w1L, w2L, w3L, w1R, w2R, w3R, w1, w2, w3, normWL, normWR, normW, sigma_face]')
                 do l = isz%beg, isz%end
                     do k = isy%beg, isy%end
                         do j = isx%beg, isx%end
@@ -218,7 +235,9 @@ contains
                             normW = (normWL + normWR)/2._wp
 
                             if (normW > capillary_cutoff) then
-                                call s_compute_capillary_stress_tensor(sigma, w1, w2, w3, normW, Omega)
+                                sigma_face = sigma
+                                if (sigma_model == 1) sigma_face = (c_sigma(j, k, l) + c_sigma(j, k, l + 1))/2._wp
+                                call s_compute_capillary_stress_tensor(sigma_face, w1, w2, w3, normW, Omega)
 
                                 do i = 1, num_dims
                                     flux_src_vf(eqn_idx%mom%beg + i - 1)%sf(j, k, l) = flux_src_vf(eqn_idx%mom%beg + i - 1)%sf(j, &
@@ -229,7 +248,7 @@ contains
                                 end do
 
                                 flux_src_vf(eqn_idx%E)%sf(j, k, l) = flux_src_vf(eqn_idx%E)%sf(j, k, &
-                                            & l) + sigma*c_divs(num_dims + 1)%sf(j, k, l)*vSrc_rsx_vf(j, k, l, 3)
+                                            & l) + sigma_face*c_divs(num_dims + 1)%sf(j, k, l)*vSrc_rsx_vf(j, k, l, 3)
                             end if
                         end do
                     end do
@@ -241,9 +260,10 @@ contains
     end subroutine s_compute_capillary_source_flux
 
     !> Compute color-function gradients and reconstruct them at cell boundaries
-    impure subroutine s_get_capillary(q_prim_vf, bc_type)
+    impure subroutine s_get_capillary(q_prim_vf, q_T_sf, bc_type)
 
         type(scalar_field), dimension(sys_size), intent(in)        :: q_prim_vf
+        type(scalar_field), intent(in)                             :: q_T_sf
         type(integer_field), dimension(1:num_dims,1:2), intent(in) :: bc_type
         type(int_bounds_info)                                      :: isx, isy, isz
         integer                                                    :: j, k, l, i
@@ -313,6 +333,19 @@ contains
         ! reconstruct gradient components at cell boundaries
         call s_reconstruct_cell_boundary_values_capillary(c_divs, gL_x, gR_x, i)
 
+        if (sigma_model == 1) then
+            ! q_T_sf is the conduction temperature, already filled in the ghost cells
+            $:GPU_PARALLEL_LOOP(collapse=3)
+            do l = idwbuff(3)%beg, idwbuff(3)%end
+                do k = idwbuff(2)%beg, idwbuff(2)%end
+                    do j = idwbuff(1)%beg, idwbuff(1)%end
+                        c_sigma(j, k, l) = sigma + sigma_dTdT*(q_T_sf%sf(j, k, l) - sigma_T_ref)
+                    end do
+                end do
+            end do
+            $:END_GPU_PARALLEL_LOOP()
+        end if
+
     end subroutine s_get_capillary
 
     !> Reconstruct left and right cell-boundary values of capillary variables
@@ -353,6 +386,7 @@ contains
         @:DEALLOCATE(c_divs)
 
         @:DEALLOCATE(gL_x, gR_x)
+        @:DEALLOCATE(c_sigma)
 
     end subroutine s_finalize_surface_tension_module
 
diff --git a/tests/E39EFF77/golden-metadata.txt b/tests/E39EFF77/golden-metadata.txt
new file mode 100644
index 0000000000..cc330e206b
--- /dev/null
+++ b/tests/E39EFF77/golden-metadata.txt
@@ -0,0 +1,196 @@
+This file was created on 2026-10-23 12:39:01.271819.
+
+mfc.sh:
+
+    Invocation: test --generate --only E39EFF77 -j 8 --no-debug
+    Lock:       mpi=Yes & gpu=No & debug=No & reldebug=No & gcov=No & unified=No & single=No & mixed=No & fastmath=No
+    Git:        3ef5e1ee9bbecbfcdaef18cfb151fb304e438290 on pr/variable-sigma-v2 (dirty)
+
+pre_process:
+
+    CMake Configuration:
+    
+        CMake v3.31.6 on ubuntu
+    
+        C       : IntelLLVM v2026.1.1 (/opt/intel/oneapi/compiler/2026.1/bin/icx)
+        Fortran : IntelLLVM v2026.1.1 (/opt/intel/oneapi/compiler/2026.1/bin/ifx)
+    
+        PRE_PROCESS   : ON
+        SIMULATION    : OFF
+        POST_PROCESS  : OFF
+        SYSCHECK      : OFF
+        DOCUMENTATION : OFF
+        ALL           : OFF
+    
+        MPI     : ON
+        OpenACC : OFF
+        OpenMP  : OFF
+    
+        Fypp    : /home/daveygravy/repos/MFC/.claude/worktrees/up-sigma/build/venv/bin/fypp
+        Doxygen : 
+    
+        Build Type : Release
+    
+    Configuration Environment:
+    
+        CC       : /opt/intel/oneapi/compiler/2026.1/bin/icx
+        CXX      : /opt/intel/oneapi/compiler/2026.1/bin/icpx
+        FC       : /opt/intel/oneapi/compiler/2026.1/bin/ifx
+        OMPI_CC  : 
+        OMPI_CXX : 
+        OMPI_FC  : 
+
+post_process:
+
+    CMake Configuration:
+    
+        CMake v3.31.6 on ubuntu
+    
+        C       : IntelLLVM v2026.1.1 (/opt/intel/oneapi/compiler/2026.1/bin/icx)
+        Fortran : IntelLLVM v2026.1.1 (/opt/intel/oneapi/compiler/2026.1/bin/ifx)
+    
+        PRE_PROCESS   : OFF
+        SIMULATION    : OFF
+        POST_PROCESS  : ON
+        SYSCHECK      : OFF
+        DOCUMENTATION : OFF
+        ALL           : OFF
+    
+        MPI     : ON
+        OpenACC : OFF
+        OpenMP  : OFF
+    
+        Fypp    : /home/daveygravy/repos/MFC/.claude/worktrees/up-sigma/build/venv/bin/fypp
+        Doxygen : 
+    
+        Build Type : Release
+    
+    Configuration Environment:
+    
+        CC       : /opt/intel/oneapi/compiler/2026.1/bin/icx
+        CXX      : /opt/intel/oneapi/compiler/2026.1/bin/icpx
+        FC       : /opt/intel/oneapi/compiler/2026.1/bin/ifx
+        OMPI_CC  : 
+        OMPI_CXX : 
+        OMPI_FC  : 
+
+simulation:
+
+    CMake Configuration:
+    
+        CMake v3.31.6 on ubuntu
+    
+        C       : IntelLLVM v2026.1.1 (/opt/intel/oneapi/compiler/2026.1/bin/icx)
+        Fortran : IntelLLVM v2026.1.1 (/opt/intel/oneapi/compiler/2026.1/bin/ifx)
+    
+        PRE_PROCESS   : OFF
+        SIMULATION    : ON
+        POST_PROCESS  : OFF
+        SYSCHECK      : OFF
+        DOCUMENTATION : OFF
+        ALL           : OFF
+    
+        MPI     : ON
+        OpenACC : OFF
+        OpenMP  : OFF
+    
+        Fypp    : /home/daveygravy/repos/MFC/.claude/worktrees/up-sigma/build/venv/bin/fypp
+        Doxygen : 
+    
+        Build Type : Release
+    
+    Configuration Environment:
+    
+        CC       : /opt/intel/oneapi/compiler/2026.1/bin/icx
+        CXX      : /opt/intel/oneapi/compiler/2026.1/bin/icpx
+        FC       : /opt/intel/oneapi/compiler/2026.1/bin/ifx
+        OMPI_CC  : 
+        OMPI_CXX : 
+        OMPI_FC  : 
+
+syscheck:
+
+    CMake Configuration:
+    
+        CMake v3.31.6 on ubuntu
+    
+        C       : IntelLLVM v2026.1.1 (/opt/intel/oneapi/compiler/2026.1/bin/icx)
+        Fortran : IntelLLVM v2026.1.1 (/opt/intel/oneapi/compiler/2026.1/bin/ifx)
+    
+        PRE_PROCESS   : OFF
+        SIMULATION    : OFF
+        POST_PROCESS  : OFF
+        SYSCHECK      : ON
+        DOCUMENTATION : OFF
+        ALL           : OFF
+    
+        MPI     : ON
+        OpenACC : OFF
+        OpenMP  : OFF
+    
+        Fypp    : /home/daveygravy/repos/MFC/.claude/worktrees/up-sigma/build/venv/bin/fypp
+        Doxygen : 
+    
+        Build Type : Release
+    
+    Configuration Environment:
+    
+        CC       : /opt/intel/oneapi/compiler/2026.1/bin/icx
+        CXX      : /opt/intel/oneapi/compiler/2026.1/bin/icpx
+        FC       : /opt/intel/oneapi/compiler/2026.1/bin/ifx
+        OMPI_CC  : 
+        OMPI_CXX : 
+        OMPI_FC  : 
+
+CPU:
+
+    CPU Info:
+    From lscpu
+    Architecture:                            x86_64
+    CPU op-mode(s):                          32-bit, 64-bit
+    Address sizes:                           48 bits physical, 48 bits virtual
+    Byte Order:                              Little Endian
+    CPU(s):                                  256
+    On-line CPU(s) list:                     0-255
+    Vendor ID:                               AuthenticAMD
+    Model name:                              AMD EPYC 7773X 64-Core Processor
+    CPU family:                              25
+    Model:                                   1
+    Thread(s) per core:                      2
+    Core(s) per socket:                      64
+    Socket(s):                               2
+    Stepping:                                2
+    Frequency boost:                         enabled
+    CPU(s) scaling MHz:                      88%
+    CPU max MHz:                             3529.9060
+    CPU min MHz:                             401.4410
+    BogoMIPS:                                4400.14
+    Flags:                                   fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt pdpe1gb rdtscp lm constant_tsc rep_good nopl xtopology nonstop_tsc cpuid extd_apicid aperfmperf rapl pni pclmulqdq monitor ssse3 fma cx16 pcid sse4_1 sse4_2 x2apic movbe popcnt aes xsave avx f16c rdrand lahf_lm cmp_legacy svm extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch osvw ibs skinit wdt tce topoext perfctr_core perfctr_nb bpext perfctr_llc mwaitx cpb cat_l3 cdp_l3 hw_pstate ssbd mba ibrs ibpb stibp vmmcall fsgsbase bmi1 avx2 smep bmi2 invpcid cqm rdt_a rdseed adx smap clflushopt clwb sha_ni xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local user_shstk clzero irperf xsaveerptr rdpru wbnoinvd amd_ppin brs arat npt lbrv svm_lock nrip_save tsc_scale vmcb_clean flushbyasid decodeassists pausefilter pfthreshold v_vmsave_vmload vgif v_spec_ctrl umip pku ospke vaes vpclmulqdq rdpid overflow_recov succor smca debug_swap
+    Virtualization:                          AMD-V
+    L1d cache:                               4 MiB (128 instances)
+    L1i cache:                               4 MiB (128 instances)
+    L2 cache:                                64 MiB (128 instances)
+    L3 cache:                                1.5 GiB (16 instances)
+    NUMA node(s):                            2
+    NUMA node0 CPU(s):                       0-63,128-191
+    NUMA node1 CPU(s):                       64-127,192-255
+    Vulnerability Gather data sampling:      Not affected
+    Vulnerability Ghostwrite:                Not affected
+    Vulnerability Indirect target selection: Not affected
+    Vulnerability Itlb multihit:             Not affected
+    Vulnerability L1tf:                      Not affected
+    Vulnerability Mds:                       Not affected
+    Vulnerability Meltdown:                  Not affected
+    Vulnerability Mmio stale data:           Not affected
+    Vulnerability Old microcode:             Not affected
+    Vulnerability Reg file data sampling:    Not affected
+    Vulnerability Retbleed:                  Not affected
+    Vulnerability Spec rstack overflow:      Vulnerable
+    Vulnerability Spec store bypass:         Vulnerable
+    Vulnerability Spectre v1:                Vulnerable: __user pointer sanitization and usercopy barriers only; no swapgs barriers
+    Vulnerability Spectre v2:                Vulnerable; IBPB: disabled; STIBP: disabled; PBRSB-eIBRS: Not affected; BHI: Not affected
+    Vulnerability Srbds:                     Not affected
+    Vulnerability Tsa:                       Vulnerable
+    Vulnerability Tsx async abort:           Not affected
+    Vulnerability Vmscape:                   Vulnerable
+
diff --git a/tests/E39EFF77/golden.txt b/tests/E39EFF77/golden.txt
new file mode 100644
index 0000000000..f3081e3662
--- /dev/null
+++ b/tests/E39EFF77/golden.txt
@@ -0,0 +1,20 @@
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0.56230648317897 0.45381568413614 0.15311605240012 0.42282880374853 0.47034233556485 0.46290956847265 0.41318125456531 0.29411709456526 0.23055494281443 0.20883532634676 0.20187300059645 0.24712464818755 0.23493094364803 0.2107561855786 0.14909040267079 0.80184019130762 0.918281429607 0.79469591683757 0.69224937503118 0.63999044882381 0.62740634888359 0.62532282676021 0.62503698951773 0.62500369991795 0.62500032836772 0.62500002618898 0.62500000199006 0.62500000010494 0.62499999999875 0.62499999999885 0.62500000000221 0.62499999967979 0.62499999566704 0.62499994560821 0.62499937482387 0.62499359770904 0.62494248799979 0.62455573906972 0.62212353043748 0.60987301588672 0.56230648317897 0.45381568413614 0.15311605240012 0.42282880374853 0.47034233556485 0.46290956847265 0.41318125456531 0.29411709456526 0.23055494281443 0.20883532634676 0.20187300059645 0.24712464818755 0.23493094364803 0.2107561855786 0.14909040267079 0.80184019130762 0.918281429607 0.79469591683757 0.69224937503118 0.63999044882381 0.62740634888359 0.62532282676021 0.62503698951773 0.62500369991795 0.62500032836772 0.62500002618898 0.62500000199006 0.62500000010494 0.62499999999875 0.62499999999885 0.62500000000221 0.62499999967979 0.62499999566704 0.62499994560821 0.62499937482387 0.62499359770904 0.62494248799979 0.62455573906972 0.62212353043748 0.60987301588672 0.56230648317897 0.45381568413614 0.15311605240012 0.42282880374853 0.47034233556485 0.46290956847265 0.41318125456531 0.29411709456526 0.23055494281443 0.20883532634676 0.20187300059645 0.24712464818755 0.23493094364803 0.2107561855786 0.14909040267079 0.80184019130762 0.918281429607 0.79469591683757 0.69224937503118 0.63999044882381 0.62740634888359 0.62532282676021 0.62503698951773 0.62500369991795 0.62500032836772 0.62500002618898 0.62500000199006 0.62500000010494 0.62499999999875 0.62499999999885 0.62500000000221 0.62499999967979 0.62499999566704 0.62499994560821 0.62499937482387 0.62499359770904 0.62494248799979 0.62455573906972 0.62212353043748 0.60987301588672 0.56230648317897 0.45381568413614 0.15311605240012 0.42282880374853 0.47034233556485 0.46290956847265 0.41318125456531 0.29411709456526 0.23055494281443 0.20883532634676 0.20187300059645 0.24712464818755 0.23493094364803 0.2107561855786 0.14909040267079 0.80184019130761 0.918281429607 0.79469591683757 0.69224937503118 0.63999044882381 0.62740634888359 0.62532282676021 0.62503698951773 0.62500369991795 0.62500032836772 0.62500002618898 0.62500000199006 0.62500000010494 0.62499999999875 0.62499999999885 0.62500000000221 0.62499999967979 0.62499999566704 0.62499994560821 0.62499937482387 0.62499359770904 0.62494248799979 0.62455573906972 0.62212353043748 0.60987301588672 0.56230648317897 0.45381568413614 0.15311605240012 0.42282880374853 0.47034233556485 0.46290956847265 0.41318125456531 0.29411709456526 0.23055494281443 0.20883532634676 0.20187300059645 0.24712464818755 0.23493094364803 0.2107561855786 0.14909040267079 0.80184019130762 0.918281429607 0.79469591683757 0.69224937503118 0.63999044882381 0.62740634888359 0.62532282676021 0.62503698951773 0.62500369991795 0.62500032836772 0.62500002618898 0.62500000199006 0.62500000010494 0.62499999999875 0.62499999999885 0.62500000000221 0.62499999967979 0.62499999566704 0.62499994560821 0.62499937482387 0.62499359770904 0.62494248799979 0.62455573906972 0.62212353043748 0.60987301588672 0.56230648317897 0.45381568413614 0.15311605240012 0.42282880374853 0.47034233556485 0.46290956847265 0.41318125456531 0.29411709456526 0.23055494281443 0.20883532634676 0.20187300059645 0.24712464818755 0.23493094364803 0.2107561855786 0.14909040267079 0.80184019130762 0.918281429607 0.79469591683757 0.69224937503118 0.63999044882381 0.62740634888359 0.62532282676021 0.62503698951773 0.62500369991795 0.62500032836772 0.62500002618898 0.62500000199006 0.62500000010494 0.62499999999875 0.62499999999885 0.62500000000221 0.62499999967979 0.62499999566704 0.62499994560821 0.62499937482387 0.62499359770904 0.62494248799979 0.62455573906972 0.62212353043748 0.60987301588672 0.56230648317897 0.45381568413614 0.15311605240012 0.42282880374853 0.47034233556485 0.46290956847265 0.41318125456531 0.29411709456526 0.23055494281443 0.20883532634676 0.20187300059645 0.24712464818755 0.23493094364803 0.2107561855786 0.14909040267079 0.80184019130762 0.918281429607 0.79469591683757 0.69224937503118 0.63999044882381 0.62740634888359 0.62532282676021 0.62503698951773 0.62500369991795 0.62500032836772 0.62500002618898 0.62500000199006 0.62500000010494 0.62499999999875 0.62499999999885 0.62500000000221 0.62499999967979 0.62499999566704 0.62499994560821 0.62499937482387 0.62499359770904 0.62494248799979 0.62455573906972 0.62212353043748 0.60987301588672 0.56230648317897 0.45381568413614 0.15311605240012 0.42282880374853 0.47034233556485 0.46290956847265 0.41318125456531 0.29411709456526 0.23055494281443 0.20883532634676 0.20187300059645
\ No newline at end of file
diff --git a/toolchain/mfc/case_validator.py b/toolchain/mfc/case_validator.py
index 476009fbd0..5dc98e0393 100644
--- a/toolchain/mfc/case_validator.py
+++ b/toolchain/mfc/case_validator.py
@@ -234,7 +234,10 @@
     "check_surface_tension": {
         "title": "Surface Tension",
         "category": "Feature Compatibility",
-        "explanation": "Requires model_eqns 2 or 3, num_fluids = 2.",
+        "explanation": (
+            "Requires model_eqns 2 or 3, num_fluids = 2. sigma_model = 1 (linear sigma(T)) reads the "
+            "temperature that Fourier conduction computes, so it requires fluid_pp(i)%k_therm > 0 for at least one fluid."
+        ),
     },
     "check_hypoelasticity": {
         "title": "Hypoelasticity",
@@ -1215,10 +1218,13 @@ def check_surface_tension(self):
         """Checks constraints on surface tension"""
         surface_tension = self.get("surface_tension", "F") == "T"
         sigma = self.get("sigma")
+        sigma_model = self.get("sigma_model")
+        sigma_T_ref = self.get("sigma_T_ref")
+        sigma_dTdT = self.get("sigma_dTdT")
         model_eqns = self.get("model_eqns")
         num_fluids = self.get("num_fluids")
 
-        if not surface_tension and sigma is None:
+        if not surface_tension and all(v is None for v in [sigma, sigma_model, sigma_T_ref, sigma_dTdT]):
             return
 
         self.prohibit(surface_tension and sigma is None, "sigma must be set if surface_tension is enabled")
@@ -1226,6 +1232,17 @@ def check_surface_tension(self):
         self.prohibit(sigma is not None and not surface_tension, "sigma is set but surface_tension is not enabled")
         self.prohibit(surface_tension and model_eqns not in [2, 3], "The surface tension model requires model_eqns = 2 or model_eqns = 3")
         self.prohibit(surface_tension and num_fluids != 2, "The surface tension model requires num_fluids = 2")
+        self.prohibit(sigma_model is not None and not surface_tension, "sigma_model is set but surface_tension is not enabled")
+        self.prohibit(
+            sigma_model != 1 and (sigma_T_ref is not None or sigma_dTdT is not None),
+            "sigma_T_ref and sigma_dTdT only apply with sigma_model = 1",
+        )
+        if sigma_model == 1:
+            heat_conduction = any((self.get(f"fluid_pp({i})%k_therm") or 0) > 0 for i in range(1, (num_fluids or 1) + 1))
+            self.prohibit(
+                not heat_conduction,
+                "sigma_model = 1 takes the temperature from Fourier heat conduction: set fluid_pp(i)%k_therm > 0 for at least one fluid",
+            )
 
     def check_mhd(self):
         """Checks constraints on MHD parameters"""
diff --git a/toolchain/mfc/params/definitions.py b/toolchain/mfc/params/definitions.py
index ba5613db24..88f6b9fe1c 100644
--- a/toolchain/mfc/params/definitions.py
+++ b/toolchain/mfc/params/definitions.py
@@ -385,6 +385,10 @@ def get_value_label(param_name: str, value: int) -> str:
     # Physics (must be non-negative)
     "R0ref": {"min": 0},
     "sigma": {"min": 0},
+    "sigma_model": {
+        "choices": [0, 1],
+        "value_labels": {0: "constant", 1: "linear in temperature"},
+    },
     # Counts (must be positive)
     "num_fluids": {"min": 1, "max": NF},
     "num_patches": {"min": 0, "max": NUM_PATCHES_MAX},
@@ -511,6 +515,11 @@ def get_value_label(param_name: str, value: int) -> str:
             "requires": ["sigma"],
         }
     },
+    "sigma_model": {
+        "when_value": {
+            1: {"requires": ["sigma_T_ref", "sigma_dTdT"]},
+        }
+    },
     "relativity": {
         "when_true": {
             "requires": ["mhd"],
@@ -666,6 +675,9 @@ def _load():
     # Surface tension
     _r("sigma", REAL, {"surface_tension"}, math=r"\f$\sigma\f$")
     _r("surface_tension", LOG, {"surface_tension"})
+    _r("sigma_model", INT, {"surface_tension"})
+    _r("sigma_T_ref", REAL, {"surface_tension"}, math=r"\f$T_{\mathrm{ref}}\f$")
+    _r("sigma_dTdT", REAL, {"surface_tension"}, math=r"\f$d\sigma/dT\f$")
 
     # Chemistry
     _r("cantera_file", STR, {"chemistry"})
@@ -1402,6 +1414,9 @@ def _decl(targets: set, *names: str) -> None:
     "hypo_hll_interface_rhs",
     "wave_speeds",
     "low_Mach",
+    "sigma_model",
+    "sigma_T_ref",
+    "sigma_dTdT",
     "hyper_cleaning_speed",
     "hyper_cleaning_tau",
     "run_time_info",
diff --git a/toolchain/mfc/params/descriptions.py b/toolchain/mfc/params/descriptions.py
index 7dfb46f3ea..cf580ba3aa 100644
--- a/toolchain/mfc/params/descriptions.py
+++ b/toolchain/mfc/params/descriptions.py
@@ -167,6 +167,9 @@
     "igr_order": "Implicit gradient reconstruction order",
     "poly_sigma": "Polydisperse distribution standard deviation",
     "sigma": "Surface tension coefficient",
+    "sigma_model": "Surface tension model: 0 constant, 1 linear in temperature",
+    "sigma_T_ref": "Reference temperature of the linear sigma(T) model",
+    "sigma_dTdT": "Slope dsigma/dT of the linear sigma(T) model",
     "Bx0": "Background magnetic field in x-direction",
     "relax": "Enable relaxation terms",
     "adv_n": "Enable advection of number density",
diff --git a/toolchain/mfc/params/generators/fortran_gen.py b/toolchain/mfc/params/generators/fortran_gen.py
index 605c97a9fa..1c2c12c251 100644
--- a/toolchain/mfc/params/generators/fortran_gen.py
+++ b/toolchain/mfc/params/generators/fortran_gen.py
@@ -105,6 +105,9 @@
     "relax_model",
     "riemann_hypo_ADC",
     "sigma",
+    "sigma_T_ref",
+    "sigma_dTdT",
+    "sigma_model",
     "surface_tension",
     "synth_U_inf",
     "synthetic_turbulence",
diff --git a/toolchain/mfc/test/cases.py b/toolchain/mfc/test/cases.py
index d169be52bc..8d2ae928c1 100644
--- a/toolchain/mfc/test/cases.py
+++ b/toolchain/mfc/test/cases.py
@@ -503,9 +503,16 @@ def alter_grcbc(dimInfo):
             cases.append(define_case_d(stack, ["grcbc z"], {}))
             stack.pop()
 
-    def alter_capillary():
+    def alter_capillary(dimInfo):
         stack.push("", {"patch_icpp(1)%cf_val": 1, "patch_icpp(2)%cf_val": 0, "patch_icpp(3)%cf_val": 1, "sigma": 1, "model_eqns": 3, "surface_tension": "T"})
         cases.append(define_case_d(stack, ["capillary=T", "model_eqns=3"], {}))
+        if len(dimInfo[0]) == 2:
+            # sigma(T) reads the conduction temperature; the patches differ in T, so sigma varies across them.
+            sigma_T = {"sigma_model": 1, "sigma_T_ref": 2.0, "sigma_dTdT": -0.1}
+            for fluid, k_therm, cv in [(1, 1.0e-3, 1.0), (2, 4.0e-3, 2.0)]:
+                sigma_T[f"fluid_pp({fluid})%k_therm"] = k_therm
+                sigma_T[f"fluid_pp({fluid})%cv"] = cv
+            cases.append(define_case_d(stack, ["capillary=T", "model_eqns=3", "sigma_model=1"], sigma_T))
         stack.pop()
 
     def alter_weno(dimInfo):
@@ -834,7 +841,7 @@ def alter_num_fluids(dimInfo):
                     )
 
                 if len(dimInfo[0]) > 1:
-                    alter_capillary()
+                    alter_capillary(dimInfo)
 
             alter_riemann_solvers(num_fluids)
             if num_fluids == 2 and len(dimInfo[0]) > 1: