Introduction to Fluid Mechanics Chapter 5 Introduction to Differential Analysis of Fluid Motion

Main Topics Conservation of Mass Motion of a Fluid Particle (Kinematics) Momentum Equation

 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.

Conservation of Mass Basic Law for a System

Conservation of Mass Rectangular Coordinate System The net mass flow rate out of the CV in x direction is: Differential control volume herein vs. finite control volume in Chapter 4. The differential approach has the ability to attain field solutions. The basic equations from Chapter 4 are still applicable here but with infinitesimal CV in conjunction with coordinate system.

Conservation of Mass Rectangular Coordinate System

Conservation of Mass Rectangular Coordinate System “Continuity Equation”

Conservation of Mass Rectangular Coordinate System “Del” Operator

Conservation of Mass Rectangular Coordinate System Incompressible Fluid: Steady Flow:

Conservation of Mass Cylindrical Coordinate System

Conservation of Mass Cylindrical Coordinate System

Conservation of Mass Cylindrical Coordinate System “Del” Operator

Conservation of Mass Cylindrical Coordinate System Incompressible Fluid: Steady Flow:

Motion of a fluid element (Kinematics)

Motion of a fluid element According to multiple-variable Taylor expansion series Particle (system) acceleration is expressed in terms of a velocity field (space quantity)

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

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

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

Motion of a fluid element

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

Momentum Equations (Navier-Stokes Equations) The force here is that acting on the control volume/surface occupied by the fluid element at time t

Momentum Equation Forces Acting on a Fluid Particle To determine the surface force, the stress condition on the surfaces of the CV element occupied by the fluid element is considered

Momentum Equations  

Momentum Equations  

Momentum Equations  

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.

Navier-Stokes Equations

Momentum Equation (incompressible flow) Navier–Stokes Equations: Cylindrical coordinate

Momentum Equation Special Case: Euler’s Equation

c05u011 c05u011

c08f001 c08f001

c08f004 c08f004