w = velocity m/s also ρ density also p = pressure and v dynamic viscosity
where:
wx- a particle velocity component of the boundary layer on the x-axis;
wy - a particle velocity component of the boundary layer on the oy-axis.
The x axis is oriented in the flow direction, and the oy axis is normal to the surface throughout which the flow occurs.
For a definitive answer one would need to see the boundary conditions and initial condition. At present I don't know if the surface at y=0 is moving or stationary, and the external flow field is presently unknown, but could be constant or accelerating (such as Falkner-Skan flow over a wedge). I will assume that the flow is of Blasius or Falkner-Skan type.
For ease of typing, I'll replace w_x by u and w_y by v.....
This is a parabolic system of PDEs, and the absence of the u_xx (streamwise diffusion) suggests to me that you have applied the boundary layer approximation. This tells me that your second equation, p_y=0, is incorrect. If one is applying standard boundary layer theory, then there must be a large parameter somewhere. Generally we can assume that x>>y, i.e. the boundary layer thickness is much smaller than the distance from the origin. For the two terms in the continuity equation to balance (which they must) it is essential that u>>v.
Now the strange bit..... p_y must be >> p_x, but v
Dear Malik, a simple consideration: integrate the first eq. respect to x so that p(x,y) = F(wx, wy); using 2nd eq. you have that F should depend only on x and wy could be a function only of x; from continuity eq. you have wx constant. Is this your hope? I think that 2nd eq. is a very strong condition for a 2D flow velocity field. Gianluca
Dear Malik
You should be aware of the classical boundary layer theory of Prandtl. You have to correct the first equation (momentum along x, see the consequence of the continuity equation on wy) and I suggest to have a look to the book of Kundu per the solution of the BL equations.
You may check Maple software. I strongly believe that you'll be able to use the inbuilt tools.
Regards.
I agree with previous comments. But you also need the boundary conditions and the equation of states for the pressure. Otherwise all techniques will be useless.
For a definitive answer one would need to see the boundary conditions and initial condition. At present I don't know if the surface at y=0 is moving or stationary, and the external flow field is presently unknown, but could be constant or accelerating (such as Falkner-Skan flow over a wedge). I will assume that the flow is of Blasius or Falkner-Skan type.
For ease of typing, I'll replace w_x by u and w_y by v.....
This is a parabolic system of PDEs, and the absence of the u_xx (streamwise diffusion) suggests to me that you have applied the boundary layer approximation. This tells me that your second equation, p_y=0, is incorrect. If one is applying standard boundary layer theory, then there must be a large parameter somewhere. Generally we can assume that x>>y, i.e. the boundary layer thickness is much smaller than the distance from the origin. For the two terms in the continuity equation to balance (which they must) it is essential that u>>v.
Now the strange bit..... p_y must be >> p_x, but v
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