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.

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Netta00 Netta00 · Answer 1 · 2019-10-15T08:45:53+02:00

Answer:

[tex]F = 6u_{max}\mu \pi \ N/M[/tex]

Explanation:

Given that

Viscosity of fluid = μ

[tex]u=u_{max}\left ( 1-\dfrac{r^n}{R^n} \right )[/tex]

We know that shear stress `

[tex]\tau =-\mu \dfrac{du}{dr}[/tex]

F= τ A

A=2πRL

When n=3

[tex]u=u_{max}\left ( 1-\dfrac{r^3}{R^3} \right )[/tex]

[tex]\dfrac{du}{dr}=- 3u_{max}\dfrac{r^2}{R^3}[/tex]

[tex]\tau = 3\mu u_{max}\dfrac{r^2}{R^3}[/tex]

[tex]F = 3u_{max}\mu \dfrac{r^2}{R^3}\time 2\pi RL[/tex]

At pipe surface r=R

[tex]F = 3u_{max}\mu \dfrac{R^2}{R^3}\time 2\pi RL[/tex]

[tex]F = 6u_{max}\mu \pi L[/tex]

So force per unit length

[tex]F = 6u_{max}\mu \pi \ N/M[/tex]