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The differential equation of linear momentum & Navier Stoke EquationEngr. Mahesh KumarIntegral Relation for Momentum

Moment Flux Term denoted by MomentumThe differential equation of linear momentumAs we Known Sides of cube are dx , dy and dz Therefore

Take component of Velocity in x,y,z direction as u,v and w

Momentum flux occur at all six faces, Picture shows mass flow in and out from all faces.

We can sayAs

Cancel row*dx.dy.dz----------(a)

There are two types of forces act on a body force and surface forceBody forces are due to external field . i-e due to gravity and magnetismFtotal = Fsur+ Fbody ------------------------ (1)Here body force

Stresses (sigma) are sum of hydrostatic pressure and Viscous stress (Tau)

As on face P and Viscous forces both act

In X direction

As we know Stresses (sigma) are sum of hydrostatic pressure and Viscous stress (Tau)dx.dy.dz is volume

Similarly do for y and z direction

We can say that-----------(2)

Sometime above equation is expressed in divergence form

As We Know Total Force in sum of Surface and Gravitational -------------------------------------------------------(3)The Value of dV/dt is given in equation (a)

Therefore equation (3) for x direction can be written as

Similarly for y and z direction

Navier Stoke EquationAs we know

Put in Last Equation of linear momentum

Navier Stoke Equation

Moody Chart