Linear Momentum Equation, Study notes of Fluid Mechanics

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• For a rigid body of mass m, Newton’s second law is expressed as:
• The product of the mass and the velocity of a body is called the linear momentum or just the momentum of the body.
• In fluid mechanics, Newton’s second law is usually referred to as the linear momentum equation.

• Conservation of angular momentum principle: total angular momentum of a rotating body remains constant when the net torque acting on it is zero, and thus the angular momentum of such systems is conserved.

• body forces that act throughout the entire body of the control volume (such as gravity, electric, and magnetic forces)
• and surface forces that act on the control surface (such as pressure and viscous forces and reaction forces at points of contact). Forces Acting On a Control Volume

• The diagonal components of the stress tensor: Normal Stresses: Pressure: which always acts inwardly normal.
• The off-diagonal components: Shear Stresses: Viscous Stresses
• : is defined as the stress (force per unit area) in the j-direction acting on a face whose normal is in the i-direction.

• We should choose the control volume such that forces that are not of interest remain internal, and thus they do not complicate the analysis.
• subtract the atmospheric pressure and work with gage pressures

• Which can be stated as:

• momentum equation is a vector equation, and thus each term should be treated as a vector.
• Force consists of weights, pressure forces, and reaction forces. The momentum equation is commonly used to calculate the forces (usually on support systems or connectors) induced by the flow. In many practical applications, fluid crosses the boundaries of the control volume at one or more inlets and one or more outlets, and carries with it some momentum into or out of the control volume.

• Unfortunately, the velocity across most inlets and outlets of practical engineering interest is not uniform. Nevertheless, it turns out that we can still convert the control surface integral into algebraic form, but a dimensionless correction factor, called the momentum-flux correction factor, is required, Momentum-Flux Correction Factor

• the net force acting on the control volume during steady flow is equal to the difference between the rates of outgoing and incoming momentum flows.