An incompressible fluid flows steadily in a twodimensional c

An incompressible fluid flows steadily in a two-dimensional channel of height it h = 25cm and width w = 100cm. Find the uniform velocity U_1 at the entrance for a constant flow rate of Q = 10m^3/s. The velocity distribution downstream at section (2) is u/u_max = 1 - (y/h)^2 Evaluate the maximum velocity at this section. Calculate the pressure drop that would exist in the channel if viscous friction at the walls could be neglected and rho = 1000kg/m^3.

Solution

Given,

= 1000 kg/m^3

h = 25 cm = 0.25 m

w = 100 cm = 1 m

Q = 10 m^3/s

Steady state flow

Two dimension flow

U / U_max = 1 – (y/h)^2

Here,

Q = U₁*A₁ = U₁* (2h*w) = U₁* 0.5*1 = 10

Therefore, U₁ = 20 m/s

Since,

Q = 10 = U₁*A₁ = U₂*A₂ where U₂ is average velocity.

And A₁ = A₂

Hence, U₂ = 20 m/s

U_max*(1 – (y/h)^2)*w*dh = U₂*w*dh

Integrating both side, we get

U_max*w*2*2h/3 = U₂*w*2h

U₂ = 2*U_max/3

So, U_max = 3*U₂/2 = 30 m/s

By applying Bernoulli equation on section 1 & 2, we get

There is not pressure drop from section 1 to 2 as mass flow rate, height & average velocity are same at both section.