Chapter 9, Problem 9.10P

Interpretation Introduction

Interpretation: The equations $9.17$ in component form for the Cartesian coordinates are to be written.

Concept introduction:

The equation $9.17$ in the component form for the Cartesian coordinates is written by simplifying the Navier-Stokes equations and differential expression of Newton’s second law of motion for a Newtonian fluid in $x,y,\text{and}z$ directions.

The Navier-Stokes equation differential expression of Newton’s second law of motion for a Newtonian fluid in $x$ direction is,

  $\rho \frac{D{v}_x}{Dt}=\rho g_x-\frac{\partial P}{\partial x}-\frac{\partial }{\partial x}\left(\frac{2}{3}\mu \nabla \cdot v\right)+\nabla \cdot \left(\mu \frac{\partial v}{\partial x}\right)+\nabla \cdot \left(\mu \nabla v_x\right)$

The Navier-Stokes equation differential expression of Newton’s second law of motion for a Newtonian fluid in $y$ direction is,

  $\rho \frac{D{v}_y}{Dt}=\rho g_y-\frac{\partial P}{\partial y}-\frac{\partial }{\partial y}\left(\frac{2}{3}\mu \nabla \cdot v\right)+\nabla \cdot \left(\mu \frac{\partial v}{\partial y}\right)+\nabla \cdot \left(\mu \nabla v_y\right)$

The Navier-Stokes equation differential expression of Newton’s second law of motion for a Newtonian fluid in $z$ direction is,

  $\rho \frac{D{v}_z}{Dt}=\rho g_z-\frac{\partial P}{\partial z}-\frac{\partial }{\partial z}\left(\frac{2}{3}\mu \nabla \cdot v\right)+\nabla \cdot \left(\mu \frac{\partial v}{\partial z}\right)+\nabla \cdot \left(\mu \nabla v_z\right)$