Linear Momentum Equation and its Applications

The 25th week
Linear Momentum Equation and its Applications

Linear Momentum
Definition: Linear momentum of a mass m moving with
velocity V ?
N ?=mV ?
Where:
N ?=momentum vector, kg.m/sec.
m= mass of fluid, kg.
V ?= velocity vector, m/sec.

Total momentum of several masses: m1 with velocity v1 , m2 with velocity v2, etc..
(N_tot ) ?=??(N_i ) ? =(N_1 ) ?+(N_2 ) ?+?= m_1 (V_1 ) ?+m_2 (V_2 ) ?+?.

Conservation of Momentum:
You can never create or destroy momentum; all we can do is transfer momentum from one object to another. Therefore, the total momentum of a system of masses isolated from external forces (forces from outside the system) is constant in time. Similar to Conservation of Energy – always true, no exceptions.
There is a relation between the forces acting on the fluid flow and the change in momentum. These forces are:
Body force
Hydrostatic forces
Pressure forces
Shear forces
The total force = sum of the acting forces.
The equations that show this relation are called momentum equations can be derived using Newton s second law of motion.
Basic Equation:
In fluid mechanics Newton’s law is called the linear-momentum relation
In fluid mechanics Newton’s law is called the linear-momentum relationMomentum Equation for two- and three- dimensional flow
The force in the x- direction:


Newton’s second law states that the mass will begin to accelerate:
F= ma=mdV/dt=d(mV/t)=??F
Note that it is a vector law which implies the three scalar equations:
Fx = max
a_x=(?V_x)/?t
m= ?Q?t

???F_x =?Q?t (?V_x)/?t=?Q(V_2x-V_1x ) (1)
Eq (1) is the momentum equation in x- direction.
In the same way the momentum equation in y- direction and z- direction:
??F_y =?Q?t (?V_y)/?t=?Q(V_2y-V_1y )
??F_z =?Q?t (?V_z)/?t=?Q(V_2z-V_1z )