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 MassBasic Law for a System

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Conservation of MassRectangular Coordinate SystemDifferential control volume herein vs. finite control volume in Chapter 4. The differential approach has the ability to attain field solutions.

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

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Conservation of MassRectangular Coordinate System

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Conservation of MassRectangular Coordinate SystemContinuity Equation

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Conservation of MassRectangular Coordinate SystemDel Operator

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Conservation of MassRectangular Coordinate SystemIncompressible Fluid:Steady Flow:

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Conservation of MassCylindrical Coordinate System

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Conservation of MassCylindrical Coordinate System

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Conservation of MassCylindrical Coordinate SystemDel Operator

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Conservation of MassCylindrical Coordinate SystemIncompressible Fluid:Steady Flow:

Motion of a fluid element

Motion of a fluid element

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Motion of a Fluid Particle (Kinematics)Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

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Motion of a Fluid Particle (Kinematics)Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

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Motion of a Fluid Particle (Kinematics)Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

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Motion of a fluid element

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

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Momentum Equations (Navier-Stokes Equations)

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Momentum EquationForces Acting on a Fluid Particle

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Momentum Equations

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Momentum Equations

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Momentum Equations

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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.

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Navier-Stokes Equations

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Momentum EquationForces Acting on a Fluid Particle

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Momentum EquationDifferential Momentum Equation

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Momentum EquationNewtonian Fluid: NavierStokes Equations

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Momentum EquationSpecial Case: Eulers Equation

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