Conjugate Heat Transfer (CHT) — Plate with Hollow Border and Channel Flow

\begin{tikzpicture}[line cap=round, line join=round, >=latex] \def\xL{-5} \def\xR{5} \def\yB{-3} \def\yT{3} \def\xs{-3} \def\xe{3} \def\ys{-2} \def\ye{2} \def\yin{1.8} \def\win{0.8}

\draw[very thick] (\xL,\yB) rectangle (\xR,\yT); \draw[very thick] (\xs,\ys) rectangle (\xe,\ye);

\node at (0,0) {$\Omega_s$}; \node at (0,-2.6) {$\Omega_f$};

\draw[line width=3pt,red] (\xL,\yin) -- (\xL,{\yin+\win}); \draw[line width=3pt,red] (\xL,{-\yin-\win}) -- (\xL,{-\yin}); \draw[line width=3pt,red] (\xR,\yin) -- (\xR,{\yin+\win}); \draw[line width=3pt,red] (\xR,{-\yin-\win}) -- (\xR,{-\yin});

\node[anchor=east] at (\xL,{\yin+0.4}) {$\Gamma_{\text{in}}^{(1)}$}; \node[anchor=east] at (\xL,{-\yin-0.4}) {$\Gamma_{\text{in}}^{(2)}$}; \node[anchor=west] at (\xR,{\yin+0.4}) {$\Gamma_{\text{out}}^{(1)}$}; \node[anchor=west] at (\xR,{-\yin-0.4}) {$\Gamma_{\text{out}}^{(2)}$};

\node at (0,3.3) {$\Gamma_w$}; \node at (0,-3.3) {$\Gamma_w$};

\node at (3.4,0) {$\Gamma_{fs}$}; \node at (-3.4,0) {$\Gamma_{fs}$}; \end{tikzpicture}% }

TL;DR
2D conjugate heat transfer (CHT) setup: an outer channel with two inlet/outlet “tabs” on each vertical wall and a centered solid plate (inner square).
This repo includes:

• A clean mathematical model (strong & weak forms).
• FreeFEM scripts to generate matching meshes for fluid and solid (shared interface nodes).
• A LaTeX project you can extend into a report.

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1) Context & Motivation

Conjugate heat transfer couples:

• Fluid mechanics (incompressible Navier–Stokes) in the fluid domain $\Omega_f$, and
• Heat conduction in the solid $\Omega_s$,

with thermal exchange across the common interface $\Gamma_{fs}$. This configuration models a plate cooled by channel flow: the plate acts as an internal boundary (a hole from the fluid’s point of view, and the solid itself for conduction).

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2) Geometry & Domains

• Outer rectangle (channel): $[x_L, x_R] \times [y_B, y_T] = [0,1.5]\times[-1,0]$, with two rectangular tabs on each vertical side (slots you can later label as inlets/outlets).
• Inner solid plate: centered square $[x_s, x_e]\times[y_s, y_e]=[0.10,1.40]\times[-0.90,-0.10]$.

Notation:

• $\Omega_f$: fluid domain (outer region minus inner square).
• $\Omega_s$: solid domain (the inner square).
• $\Gamma_{fs} = \overline{\Omega_f}\cap\overline{\Omega_s}$: fluid–solid interface (square boundary).
• $\Gamma_w$: remaining channel walls (outer boundary).
• Optionally: $\Gamma_{\text{in}}^{(1,2)}$, $\Gamma_{\text{out}}^{(1,2)}$ on tab segments.

Boundary labels in meshes

• 30 → $\Gamma_w$ (outer walls)
• 40 → $\Gamma_{fs}$ (fluid–solid interface)

Orientation (important for BAMG)

• Outer boundary: CCW.
• Inner boundary for the fluid (hole): CW.
• Inner boundary for the solid (filled): CCW.
• Same discretization count on $\Gamma_{fs}$ for fluid and solid ⇒ shared interface nodes.

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3) Governing Equations (Strong Form)

Unknowns (in $d=2$):

• Fluid: velocity $\mathbf{u}(x,t)$, pressure $p(x,t)$, and temperature $T_f(x,t)$ in $\Omega_f$.
• Solid: temperature $T_s(x,t)$ in $\Omega_s$.

Parameters:

• $\rho$: density; $\mu$: dynamic viscosity.
• $\kappa$: thermal diffusivity (solid); $\hat\kappa$: thermal diffusivity (fluid).
• $\alpha$: interface heat transfer coefficient (Robin coupling).

Solid (heat conduction) in $\Omega_s$ $$ \partial_t T_s - \kappa \Delta T_s = 0 \quad \text{in } \Omega_s\times(0,T). $$

Fluid (incompressible Navier–Stokes) in $\Omega_f$

$$ \begin{aligned} \rho(\partial_t \mathbf{u} + \mathbf{u}\cdot\nabla \mathbf{u}) &= -\nabla p + \mu \Delta \mathbf{u} + T_f,\mathbf{f}_{ext} \quad \text{in } \Omega_f\times(0,T),\\ \nabla\cdot \mathbf{u} &= 0 \quad \text{in } \Omega_f\times(0,T). \end{aligned} $$

with no-slip $\mathbf{u}=\mathbf{0}$ on $\Gamma_w\cup\Gamma_{fs}$.
On inlet/outlet windows you can use do-nothing tractions or prescribe profiles.

Fluid temperature (advection–diffusion) in $\Omega_f$

$$ \partial_t T_f - \hat\kappa \Delta T_f + \mathbf{u}\cdot\nabla T_f = 0 \quad \text{in } \Omega_f\times(0,T). $$

Coupling and BCs

• Interface (Robin–Robin) on $\Gamma_{fs}$:

$$ \partial_{\mathbf{n}}T_s = \alpha (T_f - T_s), \qquad \partial_{\mathbf{n}}T_f = \alpha (T_s - T_f). $$

• Walls $\Gamma_w$: $\mathbf{u}=\mathbf{0}$, $\partial_{\mathbf{n}}T_f=0$.
• Inlet/Outlet tabs: optional temperature Dirichlet $T_f=T_{\text{in/out}}$ or natural.
• Initial data: $\mathbf{u}(x,0)=\mathbf{u}0(x)$, $T_f(x,0)=T{f,0}(x)$, $T_s(x,0)=T_{s,0}(x)$.

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4) Weak Formulation (Variational)

Spaces:

$$ \begin{aligned} V &:= {\mathbf{v}\in H^1(\Omega_f)^d:\ \mathbf{v}=\mathbf{0}\ \text{on }\Gamma_w\cup\Gamma_{fs}},\\ Q &:= L^2_0(\Omega_f) = {q\in L^2(\Omega_f): \int_{\Omega_f} q = 0},\\ W_s &:= H^1(\Omega_s),\qquad W_f := {\phi\in H^1(\Omega_f): \phi=0 \text{ on Dirichlet parts}}. \end{aligned} $$

Con $D\mathbf{u}=\tfrac12(\nabla\mathbf{u}+\nabla\mathbf{u}^\top)$ y tests $\mathbf{v}\in V$, $q\in Q$, $\phi_s\in W_s$, $\phi_f\in W_f$:

Solid heat

$$ \int_{\Omega_s}\partial_t T_s,\phi_s +\kappa\int_{\Omega_s}\nabla T_s\cdot\nabla\phi_s +\kappa\alpha\int_{\Gamma_{fs}} T_s,\phi_s = \kappa\alpha\int_{\Gamma_{fs}} T_f,\phi_s. $$

Fluid temperature

$$ \int_{\Omega_f}\partial_t T_f,\phi_f +\int_{\Omega_f}(\mathbf{u}\cdot\nabla T_f),\phi_f +\hat\kappa\int_{\Omega_f}\nabla T_f\cdot\nabla\phi_f +\hat\kappa\alpha\int_{\Gamma_{fs}} T_f,\phi_f = \hat\kappa\alpha\int_{\Gamma_{fs}} T_s,\phi_f. $$

Navier–Stokes

$$ \begin{aligned} &\rho\int_{\Omega_f}\partial_t\mathbf{u}\cdot\mathbf{v} +\rho\int_{\Omega_f}(\mathbf{u}\cdot\nabla)\mathbf{u}\cdot\mathbf{v} +2\mu\int_{\Omega_f} D\mathbf{u}:D\mathbf{v} -\int_{\Omega_f} p,\nabla\cdot\mathbf{v} = \int_{\Omega_f} T_f,\mathbf{f}_{ext}\cdot\mathbf{v},\\ &\int_{\Omega_f} q,\nabla\cdot\mathbf{u}=0. \end{aligned} $$

This matches the LaTeX derivation included in the project.