Figure 3 Schematic representation of fluid element [5].
P.S. The fluid thermophysical properties will be as a function of space and time and they write as\(\rho=\rho\left(x,y,z,t\right),T=T\left(x,y,z,t\right),P=P\left(x,y,z,t\right),u=u\left(x,y,z,t\right)\)for density, temperature, pressure and velocity of fluid.
P.S. Taylors series forward and backward will be used up to first two terms measure from the central point. For example;
forward Taylors series:\(u\left(x+x\right)=u\left(x\right)+\frac{\text{δu}}{\text{δx}}x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3.1\right)\text{\ \ }\)
backward Taylors series
\(u\left(x-x\right)=u\left(x\right)-\frac{\text{δu}}{\text{δx}}x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3.2\right)\)
In this way, if we select the pressure for example;
forward Taylors series
\(P_{2}=P+\frac{\text{δP}}{\text{δx}}\frac{\text{δx}}{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.3\right)\)
backward Taylors series
\(P_{2}=P-\frac{\text{δP}}{\text{δx}}\frac{\text{δx}}{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.4\right)\)
Conservation of Mass [5]
\(\frac{\text{DM}_{\text{sys}}}{\text{Dt}}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3.5\right)\ \)
\(\frac{\partial}{\partial t}\int_{\text{cv}}{\text{ρd}V}+\int_{\text{cs}}{\rho V\bullet\mathbf{n}\text{\ dA}}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(3.6\right)\)
Rate of increases of mass in fluid element can be calculated from the equation inserted below;
\(\frac{\partial}{\partial t}\int_{\text{cv}}{\text{ρd}V}=\frac{\partial\rho}{\partial t}\text{δxδyδz}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.7\right)\)
Regarding the net rate of mass flow rate through each face of the fluid element which is the density multiplied by the velocity components perpendicular to the face by the cross sectional area as illustrated in Figure 4. The analysis is inserted below;