Question: Navier stokes Equation and BCs

Dear experts

I am trying to study linear version of the Navier-Stokes (NS) equation. I define NS equations in a way suggested in this page


with(PDEtools): with(Student[VectorCalculus]):


V:= VectorField(< v[1](x,z,t),0,v[3](x,z,t)>);

NavierStokes:= diff(V,t) - nu*Laplacian(V) +1/rho* Gradient(p(x,z,t)) - VectorField(< 0,0,-g>)=0;

the velocity components are intriduced as following




So the problem is 2D model so that v[1] and v[3] satisfy incompresibility condition. It is clear that substituting velocity components into NS leads to differential equations for f(z). The general form of f(z) is

f(z) = a1*cosh(k*z)+a2*sinh(k*z)+a3*cosh(kappa*z)+a4*sinh(kappa*z).

my problems are

1) apply v[1], v[3] and find f(z) with Maple. then apply boundary conditions to find coeffs a[i] with Maple

2) In addition, Substituting the f(z) into BCs leads to a linear homogeneous system of equations for the coefficients. The vanishing of the determinant of this system which is the condition for the existence of a nontrivial solution is interested. How can I do that?

3) f(z) is for the case that the bottom is z=0 and surface is z=h. in my case bottom is z=-h and surface is z=0. how to do this in the solution. I dont know how to apply it.

4) calculate pressure

this file is what I did.

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