# Governing Equations

This section describes the definition and formulation of the equations governing the conservations of mass, momentum and energy in a fluid flow and obtaining a closed form solution from these equations.

The equations which govern the motion of a fluid are called the Navier-Stokes equations. These equations refer to the basic governing continuity equations for a compressible, viscous, heat conducting fluid.

They describe the transport of three conserved quantities for a local system: mass, momentum and energy and can be expressed in an integral form when applied to a finite region known as the control volume, or a differential form when applied at an infinitesimally small fluid element.

## Continuity Equation

The instantaneous mass conservation equation commonly termed as the continuity equation, is derived by applying mass conservation to a control volume for a general fluid.

Its differential form is written as:

- $\rho $ is the fluid density
- $t$ is time
- $\overrightarrow{u}$ is the flow velocity vector

The first term $\frac{\partial \rho}{\partial t}$ describes the rate of change of density with respect to time and the second term $\nabla .\left(\rho \overrightarrow{u}\right)$ describes the divergence of the vector field $\rho \overrightarrow{u}$ at a particular point fixed in space. The continuity equation can also be expressed using the substantial derivative and is written as:

## Momentum Equation

The momentum conservation equation is derived by applying Newton’s second law of motion to the fluid control volume, which applies as the rate of change of momentum of the fluid particle equals the sum of surface and body forces.

The surface forces comprise of viscous forces (Tensile and shear) and pressure (compressive) and the body (volume) forces can be gravity, centrifugal force, Coriolis force, electromagnetic force, and so on.

The momentum equation is written as:

- $p$ is the pressure
- $b$ is the source term (generally gravity)
- $\tau $ is the viscous stress tensor

If the fluid is Newtonian in nature, the viscous stresses are proportional to the time rate at which strain occurs. Stokes hypothesis for a Newtonian fluid simplifies the stress tensor to:

where $\mu $ is the dynamic viscosity of the fluid

The general form of the momentum equation can be seen to be comprised of the following terms:

Time derivative + Convection terms = Forcing (Source) terms + Diffusive terms

The momentum conservation equation for the flow velocity vector $\overrightarrow{u}$ can be written as three separate equations in an orthogonal system. These equations describe the conservation of momentum in x, y, z directions with flow velocity components u, v and w.

## Energy Equation

The energy conservation equation is derived by applying first law of thermodynamics to fluid control volume, which applies as rate of change of energy of a fluid particle equals the sum of rate of heat addition and rate of work done.

The energy of a fluid particle comprises of internal (thermal) energy and kinetic energy. Heat addition generally takes place in the form of conduction due to a temperature difference through sources such as chemical reactions, potential energy, and so on. Work is done on the fluid volume by the surface and body forces.

The energy equation is written as

- $E$ is the total energy (internal + kinetic)
- $k\nabla T$ is the heat flux given by Fourier’s Law
- $S$ is the heat source

There are different possible versions of the energy equation. The form written above is known as the total energy form. Another form which describes the convection of internal energy and more commonly used is written as:

- $e$ is the internal energy substituted from the relation $E=e+\frac{1}{2}{\left|\overrightarrow{u}\right|}^{2}$
- $T$ is the temperature
- $k$ is the thermal conductivity of the fluid
- $\nabla \overrightarrow{u}:\tau $ is the double dot product defined as $\frac{\partial {\text{u}}_{j}}{\partial {x}_{i}}{\tau}_{ij}$ describing the irreversible transfer of mechanical energy into heat

Another form known as the enthalpy form is written as:

- $h$ is the enthalpy substituted from the relation $h=e+\frac{p}{\rho}$
- $\frac{Dp}{Dt}$ is the energy transfer due to work done by compressive forces, that is, pressure work

The five equation sets described above, one continuity, three momentum and one energy equation contain unknowns: $\rho $ , $p$ , $T$ , $e$ , $u$ , $v$ , $w$ . In order to obtain a closed form solution for this system of differential equations, two more equations are needed. For a compressible flow these equations come in the form of equations of state. They are stated as:

and

For an ideal gas, these equations take the form

and

These relations give an additional equation for enthalpy given by $h={C}_{p}T$ . When substituted into the energy equation it simplifies to: