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© Fox, McDonald & Pritchard
Introduction to Fluid Mechanics
Chapter 5
Introduction to Differential Analysis of
Fluid Motion
© Fox, McDonald & Pritchard
Main Topics
Conservation of MassStream Function for Two-Dimensional
Incompressible FlowMotion of a Fluid Particle (Kinematics)Momentum Equation
© Fox, McDonald & Pritchard
Conservation of Mass
Basic Law for a System
© Fox, McDonald & Pritchard
Conservation of Mass
Rectangular Coordinate System
© Fox, McDonald & Pritchard
Conservation of Mass
Rectangular Coordinate System
© Fox, McDonald & Pritchard
Conservation of Mass
Rectangular Coordinate System
“Continuity Equation”
© Fox, McDonald & Pritchard
Conservation of Mass
Rectangular Coordinate System
“Del” Operator
© Fox, McDonald & Pritchard
Conservation of Mass
Rectangular Coordinate System
© Fox, McDonald & Pritchard
Conservation of Mass
Rectangular Coordinate System
Incompressible Fluid:
Steady Flow:
© Fox, McDonald & Pritchard
Conservation of Mass
Cylindrical Coordinate System
© Fox, McDonald & Pritchard
Conservation of Mass
Cylindrical Coordinate System
© Fox, McDonald & Pritchard
Conservation of Mass
Cylindrical Coordinate System
“Del” Operator
© Fox, McDonald & Pritchard
Conservation of Mass
Cylindrical Coordinate System
© Fox, McDonald & Pritchard
Conservation of Mass
Cylindrical Coordinate System
Incompressible Fluid:
Steady Flow:
© Fox, McDonald & Pritchard
Stream Function forTwo-Dimensional
Incompressible Flow
Two-Dimensional Flow
Stream Function
© Fox, McDonald & Pritchard
Stream Function forTwo-Dimensional
Incompressible Flow
Cylindrical Coordinates
Stream Function (r,)
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field
Fluid RotationFluid Deformation
• Angular Deformation• Linear Deformation
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Translation: Acceleration of aFluid Particle in a Velocity Field (Cylindrical)
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Rotation
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Rotation
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Rotation
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Deformation:• Angular Deformation
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Deformation:• Angular Deformation
© Fox, McDonald & Pritchard
Motion of a Fluid Particle (Kinematics)
Fluid Deformation:• Linear Deformation
© Fox, McDonald & Pritchard
Momentum Equation
Newton’s Second Law
© Fox, McDonald & Pritchard
Momentum Equation
Forces Acting on a Fluid Particle
© Fox, McDonald & Pritchard
Momentum Equation
Forces Acting on a Fluid Particle
© Fox, McDonald & Pritchard
Momentum Equation
Differential Momentum Equation
© Fox, McDonald & Pritchard
Momentum Equation
Newtonian Fluid: Navier-Stokes Equations
© Fox, McDonald & Pritchard
Momentum Equation
Special Case: Euler’s Equation