Will you help me solve for the solution to problem 2 posted

Will you help me solve for the solution to problem 2 posted below

An incompressible fluid of negligible viscosity is pumped at a total volume flow rate Q through a porous surface into a small gap located between two closely spaced parallel plates. The flow in the gap can be considered one-dimensional (i.e., velocity is constant along any vertical cross-section) and only has one velocity component along the x-direction. Perform a control volume analysis on the small volume of fluid of length dx located at a distance x to obtain an expression for the pressure variation dp/dx.

Solution

Fluid is considered as Incompressible fluid.velocity is considered as constant along vertical cross section.

If we take control volume as an Open system then,we know

Fluid energy at Entry = Fluid energy at Exit

(By Bernoulli\'s theorem)

Now, if we consider change in pressure and velocity I.e.

P to P+DP and V to V+ dV.

Flow rate, Q=Area *Velocity

So, applying cons of mass to control volume

d/dT p.dV + u.dA = 0

Also,d/dt dV = 0 (steady flow at control volume)

Flow between flat plates is

u (y) = a^2/2u(DP/DX)(y^2/a^2-y/a)

Since viscosity is negligible,u =0

Q= Integral of udA constant.

Where, u = velocity.

We know, irrespective of flow pressure gradient drops linearly with x.

By Cons of Momentum,

pdu/dt = dP/DX pg = dp/DX.