Is there a jump condition for this PDE? ( Brinkman model , piecewise constant permeability)

Question

The Brinkman equations for steady flow of an incompressible fluid through rigid porous solid are:

$-\dfrac{\mu_0}{k}\mathbf{v} - \mathrm{grad}p + \mu_0 \Delta \mathbf{v} =0$ and $\mathrm{div}(\mathbf{v}) = 0$ in $\Omega$

where $k > 0$ is the permeability of the solid, $\mu_0 >0$ is the viscosity of the fluid, and $p,\mathbf{v}$ are the pressure and velocity fields.

Suppose that the domain is divided into two disjoint regions and $k$ assumes the values $k_0$ and $k_1$ on them.

My question: Is there a jump condition for this problem due to the discontinuity of $k$ ?

Ans: $\mathrm{jump}(\mathbf{t_n})=0$ where $\mathbf{t_n} := -p\mathbf{n} + \mu_0\nabla \mathbf{v}\;\mathbf{n}$

Answer 1

There is a jump condition, but it is automatically satisfied by the solution (i.e., you don't need to do anything special). This is no different than solving $-\nabla \cdot k \nabla u = f$: there is a jump condition discussed in Jump condition for elliptic equation in standard finite element method but it is automatically satisfied.