12 [NAVIER STOKES EQUATION] MOMENTUM EQUATION[NAVIER STOKES EQUATION]Momentum equation is derived from the fundamental physical principle ofNewton second lawFx = m a = Fg + Fp + FvFg is the gravity forceFp is the pressure forceFv is the viscous forceSince force is a vectar, all these forces will have three components.First we will go one component by next component thanwe will assemble all the components to get full Navier – Stokes Equation.
13 Fx – Inertial Force Inertial Force = Mass X Acceleration derivative. Fx – Inertial ForceInertial Force = Mass X Acceleration derivative.Inertial Force in x direction = m Xrepresents instantaneous time rate of change of velocity of the fluid element as it moves through point through space.
14 Is called Material derivative or Substantial derivative or Acceleration derivative‘u’ is variableInertial force per unit volume in x direction =
15 Inertial force / volume in x direction Inertial force / volume in y directionInertial force / volume in z direction
16 Body force per unit volume Body forces act directly on the volumetric mass of the fluid element.The examples for the body forces areEg: gravitationalElectricMagnetic forces.Body force =Body force in y directionBody force in z direction
17 Pressure forces per unit volume Pressure on left hand face of the elementPressure on right hand face of the elementNet pressure force in X direction isNet pressure force per unit volume in X direction
18 Net pressure force per unit volume in X direction Net pressure force per unit volume in Y directionNet pressure force per unit volume in Z directionNet pressure force in all directionNet pressure force in 3 direction
38 Having derived equations for inertial force per unit volume, pressure force per unit volume body force per unit volume, and viscous force per unit volume now it is time to assemble together the subcomponents.
39 Assembly of all the components X direction:-Y direction:-Z direction:-
44 Divergence of the product of scalar times a vector. CONVERTING NON CONSERVATION FORM ONN-S EQUATION TO CONSERVATION FORM Navier-stokes equation in the X direction is given byDivergence of the product of scalar times a vector.
45 Taking RHS of N-S Equation we have Taking RHS of N-S Equation we have
53 Energy Equation Energy is not a vector So we will be having only one energy equation which includes the energy in all the direction.The rate of Energy = Force X velocityEnergy equation can be got by multiplying the momentum equation with the corresponding component of velocity
54 dQ = dE + dWdE = dQ - dW = dQ + dW [Work done is negative] because work is done on the system.Work done is given by dot product of viscous force and velocity vector.for Xdirection