# Weak Formulation of Fluid Equation

I am currently trying to obtain the weak formulation of a system of differential equations to program in Freefem++ a solver for the growth of plaque in human arteries. I am, however, failing spectacularly.

Let $\vec{u}$ be the velocity field in 3D, and $p$ the pressure. Then, our system of equations is:

$$\rho \chi \vec{u} - \nu \Delta\vec{u} +\nabla p = f \\ \nabla . \vec{u} = 0$$

And the boundary equations are

$$\vec{u}=U_{in}, in \space \space \Gamma_{in} \\ T(\vec{u},p).\vec{n}=-p_{out}\vec{n}, in \space \space \Gamma_{out} \\ \vec{u}.\vec{n}=L_p(p-q), in \space \space \Gamma_{end} \\ \vec{u}-(\vec{u}.\vec{n}).\vec{n}=0, in \space \space \Gamma_{end}$$

where $q$ and $p_{out}$ are known, $\vec{n}$ is the normal vector to the surface, and the various $\Gamma$'s are the boundary (i.e their union is the boundary of $\Omega$), and

$T(\vec{u},p)=\nu (\nabla\vec{u} + \nabla\vec{u^T})-pI$

where I is the 3x3 identity matrix

Now, I confess I am kind of lost. I tried to multiply by a test function $\vec{v}$ but when I integrate by parts the laplacian, I get a term that is dependent on the gradient of $\vec{u}$ on the various boundaries. But, my boundary conditions don't depend on the gradient, only on the function itself, and so I am completely stuck.

Also, the boundary term that has the T(u,v) - I don't have any idea how to use it. I have been looking online for weak formulations of the navier-stokes equations but either they work in another form of the equation (which I can't) or don't have what I'm looking for.

Any help would be appreciated :/

• You should be using u = 0 at the solid boundaries. Where does the plaque come into the equations? – Chet Miller May 6 '17 at 21:04
• Actually that's not true. The idea is that you have a cylinder (your artery, for instance), and the blood which travels in it has particles (biology stuff, lets call it particles A,B,C...) which can exchange particles with the media that surrounds it, so you actually have a non-tangential velocity at the boundary - u_t=0 makes sense, but u_n isn't. The plaque is studied in a further system of equations, but right now I'm just dealing with this first, and then I'll couple them all. – user166271 May 6 '17 at 22:12
• So, you have a seepage flow at the wall. What determines the rate of seepage? – Chet Miller May 7 '17 at 0:12
• The third boundary condition I put there, basically. It has to deal with a diffusion of particles, which has another equation, and which is couple with the third boundary condition I put there – user166271 May 7 '17 at 10:23
• Are you sure about that first boundary condition? Where is the viscous contribution to the normal stress? – Chet Miller May 7 '17 at 12:07