Chapter 7, Problem 7.19P

Interpretation Introduction

Interpretation: The value for the distance from the wall at which maximum shear work can be obtained is to be determined for a parabolic velocity profile.

Concept introduction: The shear stress on any object is given by the Newton’s law of viscosity,

  $\text{τ}\text{=}\text{μ}\frac{\text{dv}}{\text{dr}}\mathrm{.......}\text{ (1)}$

In the above equation, the notations used are,

  $\begin{array}{l}\text{τ}\text{= }\text{Shear}\text{stress}\\ \text{μ}\text{=}\text{Viscosity}\text{of}\text{fluid}\\ \frac{\text{dv}}{\text{dr}}\text{=}\text{Change}\text{in}\text{velocity}\text{with respect to the distance}\end{array}$

For the parabolic distribution the velocity is given as,

  $\text{v}\text{=}{\text{v}}_{\text{max}}\left[\text{1-}{\left(\frac{\text{r}}{\text{R}}\right)}^{\text{2}}\right]\mathrm{.......}\text{ (2)}$

In the above equation, the notations used are,

v is the velocity of flow of the fluid and ${\text{v}}_{\text{max}}$ is the maximum velocity of the fluid.

R is the radius of the tube and r is the distance from the center of the tube.