Chapter 9, Problem 9.8P

Interpretation Introduction

Interpretation: The expression for the rate of change of density with falling velocity is to be determined along with the value at a point for the same.

Concept introduction:

The substantial derivative equation for the rate of change of density with respect to time is given as,

  $\frac{\text{Dρ}}{\text{Dt}}\text{=}\frac{\partial \text{ρ}}{\partial \text{t}}\text{+}{\text{v}}_{\text{y}}\frac{\partial \text{ρ}}{\partial \text{y}}$  ........ (1)

In the above equation, the notations used are,

D is used for substantial derivative while $\partial$ is used for partial derivative.

  ${\text{v}}_{\text{y}}$ = Velocity of the body

The expression $\frac{\partial \text{ρ}}{\partial \text{t}}$ is the local change in density with respect to time while the other term ${\text{v}}_{\text{y}}\frac{\partial \text{ρ}}{\partial \text{y}}$ is the rate of change in density due to motion of the body.