© pritchard introduction to fluid mechanics chapter 5 introduction to differential analysis of...

© Pritchard

Introduction to Fluid Mechanics

Chapter 5

Introduction to Differential Analysis of

Fluid Motion

© Pritchard

Main Topics

Conservation of MassMotion of a Fluid Particle (Kinematics)Momentum Equation

Introduction

In Chapter 4, integral equations for finite control volumes are derived, which reflect the overall balance over the entire control volume under consideration -- A top down approach.

However, only information related to the gross behavior of a flow field is available. Detailed point-by-point knowledge of the flow field is unknown.

Additionally, velocity and pressure distributions are often assumed to be known or uniform in Chapter 4. However, for a complete analysis, detailed distributions of velocity and pressure fields are required.

A bottom-up approach is needed.

© Pritchard

Conservation of Mass

Basic Law for a System

© Pritchard

Conservation of MassRectangular Coordinate System

Differential control volume herein vs. finite control volume in Chapter 4. The differential approach has the ability to attain field solutions. The basic equations from Chapter 4 are still applicable here but with infinitesimal CV in conjunction with coordinate system.

dydzdx

x

uudydzudydz

dydzdx

x

uudydzudydz

xdxx

xdxx

)2

()(

)())((

)2

()(

)())((

2/

2/

dxdydzx

u

udydzudydz dxxdxx

2/2/ ))(())((

The net mass flow rate out of the CV in x direction is:

© Pritchard


Rectangular Coordinate System

© Pritchard



“Continuity Equation”

© Pritchard



“Del” Operator

© Pritchard



Incompressible Fluid:

Steady Flow:

© Pritchard


Cylindrical Coordinate System

© Pritchard



“Del” Operator

© Pritchard



Incompressible Fluid:

Steady Flow:

Motion of a fluid element (Kinematics)

Motion of a fluid element

According to multiple-variable Taylor expansion series

Particle (system) acceleration is expressed in terms of a velocity field (space quantity)

© Pritchard

Motion of a Fluid Particle (Kinematics)

Fluid Translation: Acceleration of aFluid Particle in a Velocity Field

© Pritchard

Motion of a fluid element

© Pritchard

Motion of a Fluid Particle (Kinematics)

Fluid Translation: Acceleration of aFluid Particle in a Velocity Field (Cylindrical)

© Pritchard

Momentum Equations (Navier-Stokes Equations)

The force here is that acting on the control volume/surface occupied by the fluid element at time t

© Pritchard

Momentum Equation

Forces Acting on a Fluid Particle

To determine the surface force, the stress condition on the surfaces of the CV element occupied by the fluid element is considered

© Pritchard

Navier-Stokes Equations

where p is the local thermodynamic pressure, which is related to the density and temperature by the thermodynamic relation usually called the equation of state. Notice that when velocity is zero, all the shear stresses are zero and all the normal stresses reduce to pressure under hydrostatic condition.

c08f001

c08f004

© pritchard introduction to fluid mechanics chapter 5 introduction to differential analysis of...

Documents