Chapter 4, Problem 4.2P

(a)

Interpretation Introduction

Interpretation:

The minimum velocity of the plate moving upward is to be calculated so that all the fluid moves in upward direction.

Concept Introduction:

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

In x direction:

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

In y direction:

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

In z direction:

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

Here,

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

(b)

Interpretation Introduction

Interpretation:

The fluid velocity at the midway between the plates is to be calculated if $v_0$ is set the minimum velocity calculated in part (a).

Concept Introduction:

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

In x direction:

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

In y direction:

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

In z direction:

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

Here,

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

(c)

Interpretation Introduction

Interpretation:

The shear rate in the fluid at the stationary place, at the moving plate, and midway between them are to be calculated.

Concept Introduction:

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

In x direction:

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

In y direction:

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

In z direction:

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

Here,

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