Problems with system of PDE's

Questions
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1

I’m working with a system of partial differential equations with Mathematica--the two linearized momentum equations below--and I’m having trouble figuring out how exactly to solve them.

PDE1.jpg
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I’ve tried using DSolve and NDSolve, but I keep running into errors. u and w (velocities) are functions of [t,x] and [t,z] respectively. The problem I’m getting is that Fx, Fz, and p include powers of u and w terms in their definitions and DSolve and NDSolve don’t seem to like that. The following definitions are straight from the above paper, consider just the u and x equation:

Fx:=(\[Rho]*B*Cx*u[t,x]*Abs[u[t,x]])/2
Cx:=0.4+(4600/Renx)^2.9
Ren:=(u[t,x]*L)/v
p:=(\[Rho]*u[t,x]^2)/2

Where Rho, B, L, and v are constants:

\[Rho]:=1000
B:=200
L:=.019
v:=1.005*10^-6

Is there a better way to solve this? Ideally I’d like to include the boundary conditions:

u[t,0]==t
u[0,x]==1.3*E^(x)

I’ve tried to find advice online and in person, but I’ve been unsuccessful. Any advice or code would be wicked much appreciated!

2

Your definitions of Fx, C, Ren, p don’t include their dependence on x and t. That may be the problem.

3

I appologize for not clarifying about that. u[t,x] is a velocity that is dependent on functions of t and x.

u[t,x]==T[t]*X[x]
T[t]=t
X[x]=E^x

So a stripped-down example Dsolve looks like:

DSolve[
{D[u[t,x],t]+2*u[t,x]D[u[t,x],x]==Fx,
u[t,x]==T[t]X[x]},u[t,x],{t,x}]
4

The definition of Cx contains Renx which isn’t defined. Ren is defined.

5

Also, your boundary conditions are inconsistent. u[0,0] is 0 from one and 1.3 from the other.