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 \frac{DV}{Dt}=(A²(x²-y²))/r; A=const; r-distance from the center of the corner

Homework Equations

Vx=U=\frac{∂ψ}{∂y} . . Vy=V=-\frac{∂ψ}{∂x}

a=\frac{∂V}{∂t}+U\frac{∂V}{∂x}+\frac{∂V}{∂y}

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\frac{A(x-y)}{∂x} - Ay\frac{A(x-y)}{∂y}

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.

a_x=u\frac{∂u}{∂x}+v\frac{∂u}{∂y}
a_y=u\frac{∂v}{∂x}+v\frac{∂v}{∂y}

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

Chet