Question

Consider the flow of a fluid with viscosity  through a circular pipe. The velocity profile in the pipe is given as u(r) = umax(1 - r n /R n ), where umax is the maximum flow velocity, which occurs at the centerline; r is the radial distance from the centerline; and u(r) is the flow velocity at any position r. If n=3, develop a relation for the drag force exerted on the pipe wall by the fluid in the flow direction per unit length of the pipe.

Answer 1

Answer:

$F = 6u_{max}\mu \pi \ N/M$

Explanation:

Given that

Viscosity of fluid   = μ

$u=u_{max}\left ( 1-\dfrac{r^n}{R^n} \right )$

We know that shear stress `

$\tau =-\mu \dfrac{du}{dr}$

F= τ A

A=2πRL

When n=3

$u=u_{max}\left ( 1-\dfrac{r^3}{R^3} \right )$

$\dfrac{du}{dr}=- 3u_{max}\dfrac{r^2}{R^3}$

$\tau = 3\mu u_{max}\dfrac{r^2}{R^3}$

$F = 3u_{max}\mu \dfrac{r^2}{R^3}\time 2\pi RL$

At pipe surface r=R

$F = 3u_{max}\mu \dfrac{R^2}{R^3}\time 2\pi RL$

$F = 6u_{max}\mu \pi L$

So force per unit length

$F = 6u_{max}\mu \pi \ N/M$