# Seminar1

## OBJECTIVES

At the end of this seminar, the reader should be able to able to derive Basic Governing Equations for Compressible/Incompressible equations which include continuity equation, Momentum equation and Energy equation.

## BASIC CONCEPTS

• A System is defined as an arbitrary quantity of mass of fixed identity. Everything external to system is called its surrounding and the system is separated from its surroundings by boundaries.
• A control volume is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference, it is a fixed volume in space through which the fluid flows. The surface enclosing the control volume is referred to as control surface.
• In continuum assumption, Fluid is considered as continuous such that fluid properties are continuous functions of spatial coordinates.
• MASS CONSERVATION: Mass of a system is conserved i.e it can neither be created nor destroyed.
• The rate of change of momentum equals the sum of the forces on a fluid particle. (Newton 's second law of motion)
• First Law of Thermodynamics:The increase in the internal energy of a system is equal to the amount of energy added by heating the system, minus the amount lost as a result of the work done by the system on its surroundings.
$dU=\delta Q-\delta w\,$
• Divergence of a Vector:Let x, y, z be a system of Cartesian coordinates on a 3-dimensional space, and let i, j, k be the corresponding unit vectors.

The divergence of a continuously differentiable vector field F = F1 i + F2 j + F3 k is defined as

$\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z}.$
• Substantive (Total) Derivative:
$\frac{D}{Dt}=\frac{\partial}{\partial t}+{\mathbf v}\cdot\nabla$

Where

• Reynold Transport theorem: The Reynolds transport theorem refers to any extensive property, N, of the fluid in a particular control volume. It is expressed in terms of a substantive derivative on the left-hand side.

## STUDY MATERIAL

For study material, See Reference1

## Back ground Study material

For better understanding of the derivations, It is recommended that before reading this material following topics must be read in detail
• Pressure force and equilibrium of a fluid (Reference 2)
• Reynold transport theorem (Reference 3)
• Conservation of Mass (Reference 4)
• The Linear Momentum Equation (Reference 5)
• The energy equation (Reference 6)
The Derivation of Differential equation for Energy is better explained in Reference 7.

Detailed FInite Volume Formulation of Naiver-stokes equations can be found in Reference 8.

## REFERENCES

1. Fluid Mechanics by FRANK M WHITE page 225-249
2. Fluid Mechanics by FRANK M WHITE page 65
3. Fluid Mechanics by FRANK M WHITE page 139
4. Fluid Mechanics by FRANK M WHITE page 147
5. Fluid Mechanics by FRANK M WHITE page 153
6. Fluid Mechanics by FRANK M WHITE page 172
7. Viscous Flow by FRANK M White page 69
8. An introduction to Computational Fluid Dynamic; The Finite Volume Approach by HK Vesteeg & W Malalasekera Chapter2