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

restart;

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

SetCoordinates(cartesian[x,y,z]):

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

v[1]:=(x,z,t)->diff(f(z),z)*exp(I*k*x+(-1)*I*omega*t);

v[2]:=(x,z,t)->0;

v[3]:=(x,z,t)->(-1)*I*k*f(z)*exp(I*k*x+(-1)*I*omega*t);

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.

NSE.mw

Dear experts

I have a complex equation (eq=0) which is a function of K, Q, p and s. K and Q can be complex variables and pa and s are parameters.

equation is

eq:= (K*( Q*sinh(K)*cosh(Q)-K*cosh(K)*sinh(Q))*(1+s*K^2)
   +p*(-4*K^2*Q*(K^2+Q^2)
       +Q*(Q^4+2*K^2*Q^2+5*K^4)*cosh(K)*cosh(Q)
       -K*(Q^4+6*K^2*Q^2+K^4)*sinh(K)*sinh(Q)))/(K^2*Q*cosh(Q)) =0 :

and we know that Q:= sqrt(K^2-I*omega); where omega is complex variable

i) How can I study and plot K vs p if Q is 1) real and 2) pure imaginary. 3) complex variable and vice versa

ii) How can I study and plot p vs s

Dear experts

I am using Maple to solve a complex equation. My idea is to separate real and imaginary parts and then solve a set of the equation when both real and imaginary parts are zero. the following are the equation and the way I made real and imaginary parts;

eq:= (K*( Q*sinh(K)*cosh(Q)-K*cosh(K)*sinh(Q))*(1+s*K^2)+p*(-4*K^2*Q*(K^2+Q^2)+Q*(Q^4+2*K^2*Q^2+5*K^4)*cosh(K)*cosh(Q)-K*(Q^4+6*K^2*Q^2+K^4)*sinh(K)*sinh(Q)))/(K^2*Q*cosh(Q)):

so the K and the Q are both complex variables and p and s are constant.

K:=Kr+I*Kim;

Q:= sqrt(K^2-I*h^2*2*Pi/1e-6);

therefore the real and imaginary parts of the equation are 

finally, I tried to solve it as following

sys:={eval(A,[p=nu^2/g/h^3,s=sigma/rho/g/h^2])=0,eval(B,[p=nu^2/g/h^3,s=sigma/rho/g/h^2])=0}:

sol2:=fsolve(sys,{Kr=0..5,Kim=0..5},maxsols=5);

the problem is that Maple can not solve it and returns the command. I  know that there is solutions. How can I solve this equation?

the maple file is attached.mapleprime.mw

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I guess Maple use Newton method to solve equation or system of equations. Is there an alternative? I mean what are the possible methods?