Post on 29-Dec-2015
transcript
© Pritchard
Introduction to Fluid Mechanics
Chapter 5
Introduction to Differential Analysis of
Fluid Motion
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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.
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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:
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)
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Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field
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Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field
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Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field
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Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field (Cylindrical)
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Momentum Equations (Navier-Stokes Equations)
The force here is that acting on the control volume/surface occupied by the fluid element at time t
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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
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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.