Chapter 3, Problem 3.12P

Interpretation Introduction

Interpretation: The shear rate and shear stress around a cylinder containing oil is to be calculated.

Concept introduction: The shear rate is the velocity gradient or the rate of shear and it is defined as,

Shear rate = $\frac{\text{Δu}}{\text{y}}\mathrm{.......}\text{(1)}$

The notations used are,

  $\begin{array}{l}\text{Δu}\text{=}\text{Change}\text{in}\text{velociy}\\ \text{y}\text{=}\text{Gap}\text{width}\text{between}\text{cylinders}\end{array}$

The gap width is given as,

  $\begin{array}{l}\text{y}\text{=}\frac{\left({\text{D}}_{\text{o}}{\text{-D}}_{\text{i}}\right)}{\text{2}}\mathrm{.......}\text{(2)}\\ {\text{D}}_{\text{o}}\text{=}\text{Outer}\text{diameter}\text{of}\text{cylinder}\\ {\text{D}}_{\text{i}}\text{=}\text{Inner}\text{diameter}\text{of}\text{cylinder}\end{array}$

The shear stress is Force per unit area and is given by Newton’s law of viscosity which states that shear stress is directly proportional to the rate of shear.

  $\begin{array}{l}\text{τ}\text{=}\text{μ}\frac{\text{du}}{\text{dy}}\mathrm{.......}\text{(3)}\\ \text{τ}\text{=}\text{Shear}\text{stress}\\ \text{μ}\text{=}\text{Viscosity}\text{of}\text{fluid}\end{array}$