Question

Consider a fully developed steady laminar flow of an incompressible fluid with viscosity $\mu$ through a circular pipe of radius R. Given that the velocity at a radial location of R/2 from the centerline of the pipe is $U_1$, the shear stress at the wall is $K_{\mu }{U}_1/R,$ where K is _____

• A. 2.66

Answer 1

The correct option is A 2.66

Shear stress at the wall,

${\tau }_0={\tau }_{max}=\frac{K\mu U_1}{R}$

We know that the velocity at any point for laminar flow through pipe,

$u=-\frac{1}{4\mu }\left(\frac{\partial p}{\partial x}\right)(R^2-r^2)$

$Atr=\frac{R}{2},u=U_1$

$\therefore U_1=-\frac{1}{4\mu }\left(\frac{\partial p}{\partial x}\right)(R^2-\frac{R^2}{4})$

$=-\frac{1}{4\mu }\left(\frac{\partial p}{\partial x}\right)\frac{3}{4}{R}^2$

$=-\frac{3}{16\mu }\left(\frac{\partial p}{\partial x}\right)R^2$

${\tau }_o=-\frac{K\mu }{R}\times \frac{3}{16\mu }\left(\frac{\partial p}{\partial x}\right)R^2$

$=-\frac{3}{16}K\left(\frac{\partial p}{\partial x}\right)R$

We know that maximum shear stress at wall,

${\tau }_o=-\frac{R}{2}\frac{\partial p}{\partial x}$

$\therefore -\frac{3}{16}K\left(\frac{\partial p}{\partial x}\right)R=-\frac{R}{2}\frac{\partial p}{\partial x}$

$orK=\frac{8}{3}=2.66$