## Governing Equations

The main structure of thermo-fluids examinations is directed by governing equations that are based on the conservation law of fluid’s physical properties. The basic equations are the three physics laws of conservation10,11

${}^{10,11}$
:

**1. Conservation of Mass: Continuity Equation**

**2. Conservation of Momentum: Momentum Equation of Newton’s Second Law**

**3. Conservation of Energy: First Law of Thermodynamics or Energy Equation**

These principles state that mass, momentum and energy are stable constants within a closed system. Basically: What comes in, must also go out somewhere else.

The investigation of fluid flow with thermal changes relies on certain physical properties. The three unknowns which must be obtained simultaneously from these three basic conservation equations are the velocity v⃗

$\overrightarrow{v}$
, pressure p

$p$
and the absolute temperature T

$T$
. Yet p

$p$
and T

$T$
are considered the two required independent thermodynamics variables. The final form of the conservation equations also contains four other thermodynamics variables; density ρ

$\rho $
, the enthalpy h

$h$
as well as viscosity μ

$\mu $
and thermal conductivity k

$k$
; the last two are also transport properties. Since p

$p$
and T

$T$
are considered two required independent thermodynamics variables, these four properties are uniquely determined by the value of p

$p$
and T

$T$
. Fluid flow should be analyzed to know vecv

$vecv$
, p

$p$
and T

$T$
throughout every point of the flow regime. This is most important before designing any product which involves fluid flow. Furthermore, the method of fluid flow observation based on kinematic properties is a fundamental issue. Movement of fluid can be investigated with either Lagrangian or Eulerian methods. Lagrangian description of fluid motion is based on the theory to follow a fluid particle which is large enough to detect properties. Initial coordinates at time t0

${t}_{0}$
and coordinates of the same particle at time t1

${t}_{1}$
have to be examined. To follow millions of separate particles through the path is almost impossible. In the Eulerian method, any specific particle across the path is not followed, instead, the velocity field as a function of time and position is examined. This missile example precisely fits to emphasize the methods.