Chapter 9, Problem 9.18P

Interpretation Introduction

Interpretation:

The problems 8.13 is to be solved using Navier-Stokes equation.

Concept Introduction:

Navier-stokes describes the motion of a fluid having constant density and viscosity.

In x direction:

  $\rho \left(\frac{\partial v_x}{\partial t}+v_x\frac{\partial v_x}{\partial x}+v_y\frac{\partial v_x}{\partial y}+v_z\frac{\partial v_x}{\partial z}\right)=\mu \left(\frac{{\partial }^2{v}_x}{\partial x^2}+\frac{{\partial }^2{v}_x}{\partial y^2}+\frac{{\partial }^2{v}_x}{\partial z^2}\right)-\frac{\partial p}{\partial x}+\rho g_x$ ................ (1)

In y direction:

  $\rho \left(\frac{\partial v_y}{\partial t}+v_x\frac{\partial v_y}{\partial x}+v_y\frac{\partial v_y}{\partial y}+v_z\frac{\partial v_y}{\partial z}\right)=\mu \left(\frac{{\partial }^2{v}_y}{\partial x^2}+\frac{{\partial }^2{v}_y}{\partial y^2}+\frac{{\partial }^2{v}_y}{\partial z^2}\right)-\frac{\partial p}{\partial y}+\rho g_y$ ................ (2)

In z direction:

  $\rho \left(\frac{\partial v_z}{\partial t}+v_x\frac{\partial v_z}{\partial x}+v_y\frac{\partial v_z}{\partial y}+v_z\frac{\partial v_z}{\partial z}\right)=\mu \left(\frac{{\partial }^2{v}_z}{\partial x^2}+\frac{{\partial }^2{v}_z}{\partial y^2}+\frac{{\partial }^2{v}_z}{\partial z^2}\right)-\frac{\partial p}{\partial z}+\rho g_z$ ................ (3)

Here,

  $\begin{array}{c}\rho =\text{ density of the fluid}\\ \mu =\text{viscosity of the fluid}\\ p=\text{pressure of the system}\\ v_x=\text{velocity of fluid in }x\text{ direction}\\ v_y=\text{velocity of fluid in }y\text{ direction}\\ v_z=\text{velocity of fluid in }z\text{ direction}\\ t=\text{time}\\ g_x=\text{acceleration due to gravity in }x\text{ direction}\\ g_y=\text{acceleration due to gravity in }y\text{ direction}\\ g_z=\text{acceleration due to gravity in }z\text{ direction}\end{array}$