# Continuity Equation. Continuity Equation Continuity Equation Net outflow in x direction.

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Continuity Equation

Continuity Equation Net outflow in x direction

Continuity Equation net out flow in y direction,

Continuity Equation Net out flow in z direction

Net mass flow out of the element

Continuity Equation Time rate of mass decrease in the element
Net mass flow out of the element = Time rate of mass decrease in the control volume

The above equation is a partial differential equation form of the continuity equation. Since the element is fixed in space, this form of equation is called conservation form.

If the density is constant

This is the continuity equation for incompressible fluid

[NAVIER STOKES EQUATION]
MOMENTUM EQUATION [NAVIER STOKES EQUATION] Momentum equation is derived from the fundamental physical principle of Newton second law Fx = m a = Fg + Fp + Fv Fg is the gravity force Fp is the pressure force Fv is the viscous force Since force is a vectar, all these forces will have three components.    First we will go one component by next component than we will assemble all the components to get full Navier – Stokes Equation.

Fx – Inertial Force Inertial Force = Mass X Acceleration derivative.
Fx – Inertial Force Inertial Force = Mass X Acceleration derivative. Inertial Force in x direction = m X represents instantaneous time rate of change of velocity of the fluid element as it moves through point through space.

Is called Material derivative or Substantial derivative or
Acceleration derivative ‘u’ is variable Inertial force per unit volume in x direction =

Inertial force / volume in x direction
Inertial force / volume in y direction Inertial force / volume in z direction

Body force per unit volume
Body forces act directly on the volumetric mass of the fluid element. The examples for the body forces are Eg: gravitational Electric Magnetic forces. Body force = Body force in y direction Body force in z direction

Pressure forces per unit volume
Pressure on left hand face of the element Pressure on right hand face of the element Net pressure force in X direction is Net pressure force per unit volume in X direction

Net pressure force per unit volume in X direction
Net pressure force per unit volume in Y direction Net pressure force per unit volume in Z direction Net pressure force in all direction Net pressure force in 3 direction

Viscous forces

Resolving in the X direction
Net viscous forces

Net viscous force per unit volume in X direction
Net viscous force per unit volume in Y direction Net viscous force per unit volume in Z direction

UNDERSTANDING VISCOUS STRESSES

LINEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE

Linear strain in X direction
Volumetric strain

Three dimensional form of Newton’s law of viscosity for compressible flows involves two constants of proportionality. 1. dynamic viscosity. 2. relate stresses to volumetric deformation.

In this the second component is negligible
[ Effect of viscosity ‘ ’ is small in practice. For gases a good working approximation can be obtained taking Liquids are incompressible. div V = 0]

SHEAR STRESSES = ELASTIC
CONSTANT X STRAIN RATE

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.

Assembly of all the components
X direction:- Y direction:- Z direction:-

X direction:-

Y direction:-

Z direction:-

+

Divergence of the product of scalar times a vector.
CONVERTING NON CONSERVATION FORM ON N-S EQUATION TO CONSERVATION FORM  Navier-stokes equation in the X direction is given by Divergence of the product of scalar times a vector.

Taking RHS of N-S Equation we have
Taking RHS of N-S Equation we have

since Is equal to zero

CONSERVATION FORM:-

SIMPLICATION OF NAVIER STOKES EQUATION
If is constant

For Incompressible flow

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 velocity Energy equation can be got by multiplying the momentum equation with the corresponding component of velocity

dQ = dE + dW dE = 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

for Y direction for Z direction

Body force is given by

Total work done Net Heat flux into element = Volumetric Heating + Heat transfer across surface. Volumetric heating

Heat transfer in X direction
= Heating of fluid element

dQ = B = dQ = B

Energy Equation Nonconservation form

Non conservation:-

Conservation:-

Momentum Equation Non conservation form X direction Y direction
X direction Y direction Z direction

Momentum Equation Conservation form
X direction Y direction Z direction

Energy Equation    Non conservation form

Energy equation Conservation form

FORMS OF THE GOVERNING EQUATIONS PARTICULARLY SUITED FOR CFD

Solution vectar

Variation in x direction

Variation in y direction

Variation in z direction

Source vectar

Time marching Types of time marching 1. Implicite time marching
2. Explicite time marching

Explicit FDM

Implicit FDM

Crank-Nicolson FDM

Space marching

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