Fluid dynamics problem Can someone solve this sourcesink typ

Solution

We have a sink and a vortex at (0,0)

For a sink of strengh A , the velocity at radius \"r\" , Vr =is -A/(2pi*r) while tangential velocity Vtheta =0

To find the stream function shi, must integrate: ( we know v(radial) = 1/r (dshi/dtheta), Vtangential =V(theta)= d(shi)/dr

-d(shi)/dr = V(theta) implies shi = constant,*(theta) = A/(2pi)*theta

For a Vortex of strength C, there is no radial velocity, only tangential velocity

Vr=0, Vtheta = C/r = - Gamma/(2pi* r) where the factor 2pi is used after integrating around the circle containing the vortex( convention is to express this way, although straight C can be used)

Hence the superposed velocity field for the combination is

radial Velocity = -A/(2pi*r)

tangential velocity = -Gamma/(2pi*r) ( sign depends on the circulation direction of the vortex)

To find the stream function for this part of the velocity (vortex), use the same method of integrating back, knowing

(1/r)d(shi)/d(theta) = -V(radial) =0, hence constant for theta dependence

and -d(shi)/dr = V(theta)

from the foregoing he contribution of stream function for vortex is constant*ln(r)

contrbution from sink is constant* theta

where the constants can be found from the given conditions.

b) V(radial) = 1/r d(shi)/d(theta)

V(theta) = = d(shi)/dr

where Shi is the overall stream function obtained above

c) Combine radial inflow with a circular vortex the resulting flow is a spiralling inward pattern.

Regret Dont have MATLAB