Question

The stream function for a certain incompressible flow field is given by the expression ψ = −Ur sinθ+qθ=2π. Obtain an expres- sion for the velocity field. Find the stagnation point(s) where jV ! j=0, and show that ψ =0 there.

Answer 1

Answer:

See attachment for step by step approach to get answers.

Explanation:

Given that;

The stream function for a certain incompressible flow field is given by the expression ψ = −Ur sinθ+qθ=2π. Obtain an expres- sion for the velocity field. Find the stagnation point(s) where jV ! j=0, and show that ψ =0 there.

See attachment.

Answer 2

Answer/Explanation:

In a cylindrical coordinates,

The components of the velocity field are given by:

Vr = (1/r)partial derivative of ψ with respect to θ = -U cos θ + q/(2πr)

Vθ = - partial derivative of ψ with respect to r = U sin θ

At stagnation points, the magnitude of velocity is equal to zero /V/ = 0, which means the each velocity components is equal to zero.

- U cos θ + q/2πr = 0

U sin θ = 0

Since θ E [0, 2π]

-U + q/2πr = 0, since cos 0 = 1

Therefore, q/2πr = U, and the r coordinate of the stagnation is given by r = q/2πU

Putting θ = 0 and r = q/2πU into the stream function, we obtain

ψ(r, θ) = -Ur sin θ + qθ/2π = -0 + 0 = 0