The equation for the condition of motion is as follows:

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \overrightarrow{v}\right)={S}_{m}$$

1

…………….

In equation one it is in the case of general motion, but in Eq. 2 the equation is in the form of a direction, that is, it is in a special case, as it is given in the following form

$$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x}\left(\rho {v}_{x}\right)+\frac{\partial }{\partial r}\left(\rho {v}_{r}\right)+\frac{\rho {v}_{r}}{r}={S}_{m}$$

2

…………….

where x is the essential heading, r is the winding bearing, vx is the center speed, and vr is the extended speed.

Insurance of power in an inertial (non-accelerating)

$$\frac{\partial }{\partial t}\left(\rho \overrightarrow{v}\right)+\nabla \cdot \left(\rho \overrightarrow{v}\overrightarrow{v}\right)=-\nabla p+\nabla \cdot \left(\stackrel{-}{\stackrel{⃐}{\tau }}\right)+ \rho \overrightarrow{g}+\overrightarrow{F}$$

3

……….

where p is the static strain, \(\stackrel{-}{\stackrel{⃐}{\tau }}\) is the tension tensor (portrayed underneath), and \(\overrightarrow{{\rho }\text{g}}\) and\(\overrightarrow{F}\) are the gravitational body power and outside body powers (for example, that rise out of association with the dissipated stage), independently. \(\overrightarrow{F}\) in like manner contains other model-subordinate source terms, for instance, porous media and customer described sources. The strain tensor \(\stackrel{-}{\stackrel{⃐}{\tau }}\)is given by:

$$\stackrel{-}{\stackrel{⃐}{\tau }}=\mu \left[\left(\nabla \overrightarrow{v}+\nabla {\overrightarrow{v}}^{T}\right)-\frac{2}{3}\nabla \cdot \overrightarrow{v}I\right]$$

4

…………..

where µ is the nuclear consistency, I is the unit tensor, and the second term on the right hand side is the effect of volume extension.

For 2D axisymmetric computations, the center point and extended power assurance conditions are given by:

$$\begin{array}{rr}\frac{\partial }{\partial t}\left(\rho {v}_{x}\right)+\frac{1}{r}\frac{\partial }{\partial x}\left(r\rho {v}_{x}{v}_{x}\right)+\frac{1}{r}\frac{\partial }{\partial r}\left(r\rho {v}_{r}{v}_{x}\right)=& -\frac{\partial p}{\partial x}\\ & +\frac{1}{r}\frac{\partial }{\partial x}\left[r\mu \left(2\frac{\partial {v}_{x}}{\partial x}-\frac{2}{3}(\nabla \cdot \overrightarrow{v})\right)\right]\\ & +\frac{1}{r}\frac{\partial }{\partial r}\left[r\mu \left(\frac{\partial {v}_{x}}{\partial r}+\frac{\partial {v}_{r}}{\partial x}\right)\right]+{F}_{x}\end{array}$$

5

…

And

$$\begin{array}{rr}\frac{\partial }{\partial t}\left(\rho {v}_{r}\right)+\frac{1}{r}\frac{\partial }{\partial x}\left(r\rho {v}_{x}{v}_{r}\right)+\frac{1}{r}\frac{\partial }{\partial r}\left(r\rho {v}_{r}{v}_{r}\right)=& -\frac{\partial p}{\partial r}\\ & +\frac{1}{r}\frac{\partial }{\partial x}\left[r\mu \left(\frac{\partial {v}_{r}}{\partial x}+\frac{\partial {v}_{x}}{\partial r}\right)\right]\\ & +\frac{1}{r}\frac{\partial }{\partial r}\left[r\mu \left(2\frac{\partial {v}_{r}}{\partial r}-\frac{2}{3}(\nabla \cdot \overrightarrow{v})\right)\right]\\ & -2{\mu }_{{r}^{r}}^{{v}_{r}}+\frac{2}{3}\frac{\mu }{r}(\nabla \cdot \overrightarrow{v})+\rho \frac{{v}_{r}}{r}+{F}_{r}\end{array}$$

6

..

where

$$\nabla \cdot \overrightarrow{v}=\frac{\partial {v}_{x}}{\partial x}+\frac{\partial {v}_{r}}{\partial r}+\frac{{v}_{r}}{r}$$

7

………….

what's more \({v}_{z}\) is the whirl speed.