Chapter 8, Problem 8.7P

Interpretation Introduction

Interpretation:

The ratio of the diameter of the cylinder for which the linear velocity profile assumption is accurate within 1 % of the actual velocity profile is to be determined.

Concept Introduction:

A control volume in a circular conduit is shown below as:

For this control volume, the expression for linear momentum in x direction is written as:

  ${\displaystyle \sum F_x}={P\left(2\pi r\Delta r\right)|}_x-{P\left(2\pi r\Delta r\right)|}_{x+\Delta x}+{{\tau }_{rx}\left(2\pi r\Delta x\right)|}_{r+\Delta r}-{{\tau }_{rx}\left(2\pi r\Delta x\right)|}_r$ .... (1)

Here, $P$ is the pressure, r is the radius of the pipe, and $\tau$ is the shear stress.

Newton’s law of viscosity is defined as:

  $\tau =\mu r\frac{d}{dr}\left(\frac{v_{\theta }}{r}\right)$ .... (2)

Here, $\tau$ is shear stress, $\mu$ is the viscosity of the fluid, r is the radius of the pipe and $v_{\theta }$ is the velocity of the fluid.