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## Chapter 2 Reynolds Transport Theorem (RTT)

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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**2.1The 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**2.3The Linear Momentum Equation (3)**• Special Cases**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**y**dy dz dx x v o u z w 3.1 Conservation of mass • Rectangular coordinate system**y**dy dz dx x v o u z w**y**dy dz dx x v o u z w**y**dy dz dx x v o u z w**Net Rate of Mass Flux**Rate of mass change inside the control volume**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**y**v u x • Volume flow rate between streamlines Flow across AB Along AB, x = constant, and**y**v u x • Volume flow rate between streamlines Flow across BC, Along BC, y = constant, and**Stream Function for Flow in a Corner**Consider a two-dimensional flow field**3.3 Flow Kinematics**Rotation Translation y x Linear deformation z Angular deformation • Motion of a Fluid Element**Fluid particle path**At t+dt At t y x z • Fluid Translation**b**y y o a a' b' x x • Fluid Rotation**b**y o a a' b' x**b**y o a a' b' x 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**y**b c a o x • Fluid Circulation, Around the close contour oacb, Circulation around a close contour =Total vorticity enclosed**y**b y o a a' x b' x • Fluid Angular Deformation**b**y b' y o a a' x x • Fluid Linear Deformation**b**b' y o a a' x**Rate of Strain**Rate of normal strain Rate of shearing strain(Angular deformation)**y**x z**y**x z Forces acting on a fluid particle x-direction + +**Forces acting on a fluid particle**x-direction + +**Components of Forces acting on a fluid element**x-direction y-direction z-direction**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**Navier-Stokes Equations**For flow with =constant and =constant