Derivation of thermal boundary conditions at fluid-fluid interface?

I am solving a problem in thermoelectric magnetohydrodynamics (TEMHD). I am currently going through the process of understanding the boundary conditions at a fluid-fluid interface. I understand that in electromagnetism, there are the boundary conditions

\begin{align*} \left[\varepsilon\boldsymbol{E}\cdot\hat{\boldsymbol{n}}\right]_{-}^{+} & =\xi\\ \left[\boldsymbol{E}\times\hat{\boldsymbol{n}}\right]_{-}^{+} & =0\\ \left[\boldsymbol{B}\cdot\hat{\boldsymbol{n}}\right]_{-}^{+} & =0\\ \left[\frac{1}{\mu}\boldsymbol{B}\times\hat{\boldsymbol{n}}\right]_{-}^{+} & =0\\ \left[\boldsymbol{J}\cdot\hat{\boldsymbol{n}}\right]_{-}^{+} & =0 \end{align*}

where $$\hat{\boldsymbol{n}}$$ is the unit normal vector pointing from medium $$-$$ into medium $$+$$ at an interface.

I am also aware that for incompressible flow, it is possible to show (from the incompressibility condition) that the fluid velocity $$\boldsymbol{u}$$ is continuous at such an interface, so that $$\left[\boldsymbol{u}\right]_{-}^{+}=0$$. It is also possible to show from the Navier-Stokes equations that boundary conditions on the stress tensor $$\boldsymbol{\underline{\tau}}$$ hold, that is, \begin{align*} \left[\left(\underline{\boldsymbol{\tau}}\cdot\hat{\boldsymbol{n}}\right)\cdot\hat{\boldsymbol{n}}\right]_{-}^{+} & =\gamma\varkappa\\ \left[\left(\underline{\boldsymbol{\tau}}\cdot\hat{\boldsymbol{n}}\right)\times\hat{\boldsymbol{n}}\right]_{-}^{+} & =\hat{\boldsymbol{n}}\times\boldsymbol{\nabla}_{S}\gamma \end{align*} where $$\boldsymbol{\nabla}_{S}:=\boldsymbol{\nabla}-\hat{\boldsymbol{n}}\left(\hat{\boldsymbol{n}}\cdot \boldsymbol{\nabla}\right)$$ is the surface gradient operator, $$\gamma$$ is the surface tension and $$\varkappa=\boldsymbol{\nabla}\cdot\hat{\boldsymbol{n}}$$ is the surface curvature.

My understanding is that all these boundary conditions can be derived by integrating Maxwell's equations and the incompressible Navier-Stokes equations by integrating them over a small cylinder or a "Gaussian pillbox" enclosing a small patch of the interface, taking the limit of the height of the cylinder to zero and spotting which terms remain dominant (see here for example).

I have been assured that the boundary conditions for temperature $$T$$ are that temperature is continuous at an interface, so that $$\left[T\right]^+_-=0$$, and that the normal components of heat flux $$\boldsymbol{q}$$ are also continuous, so that $$\left[\boldsymbol{q}\cdot\hat{\boldsymbol{n}}\right]^+_-=0$$ - provided that there are no sources or sinks of heat at the interface.

I cannot seem to find any resources that use the heat equation (or conservation of energy) and a similar approach to the above to prove these boundary conditions. I can still perform my modelling duties without proving these boundary conditions, but my understanding feels incomplete. Could anybody please signpost me in the right direction?

Incidentally, does anyone know where I can find a proof of the boundary condition on $$\boldsymbol{u}$$ where the assumption of incompressible flow is relaxed?

• I agree that the normal component of the heat flux must be continuous at the interface. Commented Dec 27, 2022 at 22:31

The temperature $$T$$ spatiotemporal distribution is governed by the heat equation. For isotropic homogeneous media, it reduces to a parabolic PDE, namely $$\partial_t T-a\nabla^2T= S(t,\mathbf{r})$$, where $$a$$ is a positive constant and $$S$$ is an external source. The boundary conditions for interfaces can be generically derived in the same way you do for the Navier-Stokes and Maxwell's equation -- by properly integrating the PDE at the surface. There is no magic.
If for example $$S=0$$, both $$T$$ and $$\nabla T$$ (which is proportional to the heat flux $$\mathbf{q}$$ according to Fourier's law) must be continuous across the interface. Likewise, they could be discontinuous if there was a heat source at the interface -- in the very same way you have the stress condition associated to the surface tension.
Indeed, the general procedure of integration-at-interfaces is valid, in principle, for every PDE. For an incompressible fluid, the condition $$\nabla\cdot \mathbf u=0$$ is obtained from the continuity equation for mass, $$\partial_t \rho+\nabla\cdot(\rho \mathbf{u})=0$$, assuming the mass density $$\rho$$ is constant. If that is not the case, constructing the appropriate surface integral at the interface (analogous to the one used in Gauss' law in your first equation) leads to the boundary condition at equilibrium, in your notation, $$[\rho \mathbf{u}]^{+}_{-}=0$$, assuming there is no source of mass at the surface.