Aerodynamics HW - fluid particles acceleration based on stream functio

[leonida]

Homework Statement

prove that with flow in a corner, with stream function ψ=Axy, particles are accelerating per [itex]\frac{DV}{Dt}[/itex]=(A²(x²-y²))/r; A=const; r-distance from the center of the corner

Homework Equations

Vx=U=[itex]\frac{∂ψ}{∂y}[/itex] . . Vy=V=-[itex]\frac{∂ψ}{∂x}[/itex]

a=[itex]\frac{∂V}{∂t}[/itex]+U[itex]\frac{∂V}{∂x}[/itex]+[itex]\frac{∂V}{∂y}[/itex]

The Attempt at a Solution

As per above equations i get velocity components as
U=Ax and V=-Ay


then since local acc is 0 acceleration is:

a=Ax[itex]\frac{A(x-y)}{∂x}[/itex] - Ay[itex]\frac{A(x-y)}{∂y}[/itex]

finally, as per my calcs, accelerations is:

a=A²(x+y)

where did this r come from and also (x²-y²). i was thinking using r²=x²+y², and using to multiply the whole acceleration expression with r²/(x²+y²), but i am getting nowhere.

help please

 

[Chestermiller]

Your equation does not give the acceleration. The acceleration is a vector, and you need to find its components first, before determining its magnitude.

[tex]a_x=u\frac{∂u}{∂x}+v\frac{∂u}{∂y}[/tex]
[tex]a_y=u\frac{∂v}{∂x}+v\frac{∂v}{∂y}[/tex]

Even with this, you still don't match the answer you are trying to prove.

Chet