Please do part 3 this is a fluid dynamics problem The 2D flo

Please do part 3. this is a fluid dynamics problem

The 2D flowfield for water of depth H draining out of a small hole in the bottom of a large reservoir can be approximated in the neighborhood of the hole (i.e., r

Solution

1) above example is similar to emptying wash basin or free vortex

in which velocity is given as if er,eo,ez are unit vector along r,theta and altitude z as follows

V=Vr*er+Vo*eo+Vz*ez

where it is given that

Vr=0vz=0 and

Vo=rho*R2/r

2) where accelaration equation is obtain by taking substantial derivative with time

hence

a=DV/Dt

on differentiating and solving we get

radial accelaration as

ar=Vr*dVr/dr+Vo/r*dVr/do+vz*dVr/dz-Vo2/r+dVr/dt

where all term are zero except one hence

ar=-Vo^2/r

so we get

ar=-rho^2*R^4/r,

theta accelaration as

ao=Vr*dVo/dr+Vo/r*dVo/do+vz*dVo/dz+Vr*Vo/r+dVo/dt

here all term are zero

hence ao=0

where accelaration in z direction

az=ar=Vr*dVz/dr+Vo/r*dVz/do+vz*dVz/dz+dVz/dthere all term are zero hence

az=0

3) hence total accelaration as

a=ar*er+ao*eo+az*ez

a=ar*er

a=ar=-rho^2*R^4/r m/s2

4) where stream function is obtain by taking derivative of velocity in direction perpendicular to motion, hence taking derivative of Vo w.r.t r we get stream function psi as follows

d(psi)/dr=Vo=rho*R^2/r

on taking integration we get stream function on radial direction r

d(psi)=Vo*dr

d(psi)=rho*R^2/r*dr

on integrating we get

psi=rho*R^2*logr+C1

where C1=0

psi=rho*R^2*logr

as stream function exist hence flow is possible

4) where velocity potential function is obtain by taking negative derivative in same direction of motion as follows

Vo=-d(phi)/do

hence

d(phi)=-Vo*do

on integrating for zero constant of integration as

phi=ar=-rho*R^2/r*theta

here as velocity potential function exist hence flow is irrotational

5) to calculate the pressure distribution which means pressure variattion across the flow as follows

by bernoulies equation we get for free surface as

P/density*g+Vo2/2*g/z1=constant=c2

but c2 integration constant

if pressure is zero at surface then equation turns

vo2/2*g+Z1=c2

hence free surface altitude is

Z1=c2-(vo2/2*g)

Z1=c2-rho^2*R^4/r^2*2*g

which means as Z1 goes on decreasing velocity head goes on increasing reach to maximum at hole and pressure is negative at hole.