Linear Momentum Equation

• 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 linearmomentum or just the momentum of the body.

• In fluid mechanics, Newton’s second law is usually referred to as thelinear momentum equation.

Linear Momentum Equation

• conservation of momentum principle: The momentum of a systemremains constant when the net force acting on it is zero, and thus themomentum of such systems is conserved.

• Momentum: vector quantityNewton’s second law for rotating rigid bodies

Linear Momentum Equation

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

Linear Momentum Equation

• 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 andviscous forces and reaction forces at points of contact).

Forces Acting On a Control Volume

Linear Momentum Equation

• Surface forces consist of:– Normal and tangential components.

• 2nd order tensor calledthe stress tensor describethe surface stresses at a pointin the fluid.

Linear Momentum Equation

• 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.

Linear Momentum Equation

• We should choose the control volume such that forces that are not ofinterest remain internal, and thus they do not complicate the analysis.

• subtract the atmospheric pressure and work with gage pressures

Linear Momentum Equation

Linear Momentum Equation

Linear Momentum Equation

• Which can be stated as:

Linear Momentum Equation

• momentum equation is a vector equation, and thus each term should betreated as a vector.

• Force consists of weights, pressure forces, and reaction forces. Themomentum equation is commonly used to calculate the forces (usuallyon support systems or connectors) induced by the flow.

In many practical applications, fluid crosses the boundaries of the controlvolume at one or more inlets and one or more outlets, and carries with itsome momentum into or out of the control volume.

Linear Momentum Equation

• Momentum flow rate across a uniform inlet or outlet:

Special Cases: Steady Flow

Linear Momentum Equation

Linear Momentum Equation

• Unfortunately, the velocity across most inlets and outlets of practicalengineering interest is not uniform. Nevertheless, it turns out that wecan still convert the control surface integral into algebraic form, but adimensionless correction factor, called the momentum-flux correctionfactor, is required,

Momentum-Flux Correction Factor

Linear Momentum Equation

• For the case in which density is uniform over the inlet or outlet and V isin the same direction as Vavg over the inlet or outlet,

Linear Momentum Equation

Linear Momentum Equation

Linear Momentum Equation

Linear Momentum Equation

Linear Momentum Equation

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

Linear Momentum Equation

• single-stream systems:

Steady Flow with One Inlet and One Outlet

Linear Momentum Equation

• V: average velocities across the inlet and outlet

Linear Momentum Equation

• An interesting situation arises when there are no external forces such asweight, pressure, and reaction forces acting on the body in the directionof motion—a common situation for space vehicles and satellites.

• When the mass m of the control volume remains nearly constant:

Flow with No External Forces

Linear Momentum Equation

• the control volume in this case can be treated as a solid body, with a netforce or thrust of:

• This approach can be used to determine the linear acceleration of spacevehicles when a rocket is fired.

Angular Momentum Equation

• The linear momentum equation is useful in determining the relationshipbetween the linear momentum of flow streams and the resultant forces.

Angular Momentum Equation

Angular Momentum Equation

Analogy between corresponding linear and angular quantities

Angular Momentum Equation

The moment of a force F abouta point O:

Angular Momentum Equation

Angular Momentum Equation

• Angular momentum equation for a general control volume (stationary ormoving, fixed shape or distorting) is obtained to be:

Angular Momentum Equation

• For a fixed control volume:

• an approximate form of the angular momentum equation in terms ofaverage properties at inlets and outlets becomes:

Special Case1: Steady State

Angular Momentum Equation

• If the flow is steady:

• It states that the net torque acting on the control volume during steadyflow is equal to the difference between the outgoing and incomingangular momentum flow rates.

• In many problems, all the significant forces and momentum flows are inthe same plane, and thus all give rise to moments in the same plane andabout the same axis.

Angular Momentum Equation

• r: average normal distance between the point about which moments aretaken and the line of action of the force or velocity,

• Provided that the sign convention for the moments is observed.

• That is, all moments in the counterclockwise direction are positive, andall moments in the clockwise direction are negative.

Angular Momentum Equation

• conservation of angular momentum principle, which can be stated as inthe absence of external moments,

• the rate of change of the angular momentum of a control volume is equalto the difference between the incoming and outgoing angularmomentum fluxes.

Special Case2: Flow with No External Moments