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

2.1 The Reynolds Transport Theorem (1)

2.1 The Reynolds Transport Theorem (2)

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

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

2.3 The Linear Momentum Equation (1) ..

2.3 The Linear Momentum Equation (2)

2.3 The Linear Momentum Equation (3)  Special Cases

2.3 The Linear Momentum Equation (4)

2.4 Conservation of Energy

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

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

x y z dx dy dz o u v w

x y z dx dy dz o u v w

dx dy dz o u v w x y z

Net Rate of Mass Flux

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

Continuity Equation

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

 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

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

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

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

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

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

 Scalar component of fluid acceleration

 Fluid acceleration in cylindrical coordinates

 Fluid Rotation x y a a' b b' o xx yy

a a' b b' o xx yy

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

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

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

 Fluid Angular Deformation x y a a' b b' o xx yy 

 Fluid Linear Deformation x y a a' b b' o xx yy

a a' b b' o xx yy

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

3.4 Momentum Equation

x y z

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

Forces acting on a fluid particle x-direction + +

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

Differential Momentum Equation

Momentum Equation:Vector form is treated as a momentum flux

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

Surface Forces

Momentum Equation:Navier-Stokes Equations

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

3.5 Conservation of Energy

 Summary of Basic Equations