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Chapter 2 Reynolds Transport Theorem (RTT) 2.1 The Reynolds Transport Theorem 2.2 Continuity Equation 2.3 The Linear Momentum Equation 2.4 Conservation of Energy

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2.1 The Reynolds Transport Theorem (1)

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2.1 The Reynolds Transport Theorem (2)

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2.1 The Reynolds Transport Theorem (3) Special Case 1: Steady Flow Special Case 2: One-Dimensional Flow

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2.2 Continuity Equation (1) An Application: The Continuity Equation

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2.3 The Linear Momentum Equation (1) ..

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2.3 The Linear Momentum Equation (2)

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2.3 The Linear Momentum Equation (3) Special Cases

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2.3 The Linear Momentum Equation (4)

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2.4 Conservation of Energy

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Chapter 3 Flow Kinematics 3.1Conservation of Mass 3.2 Stream Function for Two-Dimensional Incompressible Flow 3.3 Fluid Kinematics 3.4 Momentum Equation

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3.1 Conservation of mass Rectangular coordinate system x y z dx dy dz o u v w

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x y z dx dy dz o u v w

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x y z dx dy dz o u v w

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dx dy dz o u v w x y z

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Net Rate of Mass Flux

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Net Rate of Mass Flux Rate of mass change inside the control volume

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Continuity Equation

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3.2 Stream Function for Two- Dimensional Incompressible Flow A single mathematical function (x,y,t) to represent the two velocity components, u(x,y,t) and (x,y,t). A continuous function (x,y,t) is defined such that The continuity equation is satisfied exactly

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Equation of Streamline Lines drawn in the flow field at a given instant that are tangent to the flow direction at every point in the flow field. Along a streamline

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Volume flow rate between streamlines u v x y Flow across AB Along AB, x = constant, and

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Volume flow rate between streamlines u v x y Flow across BC, Along BC, y = constant, and

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Stream Function for Flow in a Corner Consider a two-dimensional flow field

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Motion of a Fluid Element Translation x y z Rotation Angular deformation Linear deformation 3.3 Flow Kinematics

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Fluid Translation x y z Fluid particle path At t At t+dt

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Scalar component of fluid acceleration

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Fluid acceleration in cylindrical coordinates

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Fluid Rotation x y a a' b b' o xx yy

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a a' b b' o xx yy

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a a' b b' o xx yy Similarily, considering the rotation of pairs of perpendicular line segments in yz and xz planes, one can obtain

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Fluid particle angular velocity Vorticity: A measure of fluid element rotation Vorticity in cylindrical coordinates

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Fluid Circulation, c y x o b a Circulation around a close contour =Total vorticity enclosed Around the close contour oacb,

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Fluid Angular Deformation x y a a' b b' o xx yy

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Fluid Linear Deformation x y a a' b b' o xx yy

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a a' b b' o xx yy

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Rate of shearing strain (Angular deformation) Rate of Strain Rate of normal strain

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3.4 Momentum Equation

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x y z

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Forces acting on a fluid particle x y z x-direction + +

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Forces acting on a fluid particle x-direction + +

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Components of Forces acting on a fluid element x-direction y-direction z-direction

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Differential Momentum Equation

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Momentum Equation:Vector form is treated as a momentum flux

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Stress and Strain Relation for a Newtonian Fluid Newtonian fluid viscous stress rate of shearing strain

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Surface Forces

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Momentum Equation:Navier-Stokes Equations

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Navier-Stokes Equations For flow with =constant and =constant

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3.5 Conservation of Energy

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Summary of Basic Equations

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