Figure 5 fluid element with the normal and shear stress applied
on it [6]
Net force in the x – direction in the right and left face is
\(\left[\sigma_{\text{xx}}+\frac{\partial\sigma_{\text{xx}}}{\partial x}\frac{\text{δx}}{2}-\left(\sigma_{\text{xx}}-\frac{\partial\sigma_{\text{xx}}}{\partial x}\frac{\text{δx}}{2}\right)\right]\text{δyδz}=\frac{\partial\sigma_{\text{xx}}}{\partial x}\text{δxδyδz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.21\right)\)
Net force in the x – direction in the top and bottom face is
\(\left[\tau_{\text{yx}}+\frac{\partial\tau_{\text{yx}}}{\partial y}\frac{\text{δy}}{2}-\left(\tau_{\text{yx}}-\frac{\partial\tau_{\text{yx}}}{\partial y}\frac{\text{δy}}{2}\right)\right]\text{δyδz}=\frac{\partial\tau_{\text{yx}}}{\partial x}\text{δxδyδz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.22\right)\)
Net force in the x – direction in the front and back face is
\(\left[\tau_{\text{zx}}+\frac{\partial\tau_{\text{zx}}}{\partial z}\frac{\text{δz}}{2}-\left(\tau_{\text{zx}}-\frac{\partial\tau_{\text{zx}}}{\partial z}\frac{\text{δz}}{2}\right)\right]\text{δyδz}=\frac{\partial\tau_{\text{zx}}}{\partial z}\text{δxδyδz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.23\right)\)
\(\text{δF}_{s}=\left(\frac{\partial\sigma_{\text{xx}}}{\partial x}+\frac{\partial\tau_{\text{yx}}}{\partial y}+\frac{\partial\tau_{\text{zx}}}{\partial
z}\right)\text{δxδyδz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.24\right)\)
Now it is the time to write down the first form of the equation of
motion in the x – direction by conjunction the body and surface forces
\(\sum F=\delta m\ a_{x}\) ,\(\sum F=\text{δF}_{s}+S_{M}\ \) ,\(\delta m=\rho\delta x\delta y\delta z\) , \(S_{M}=\rho g\)
\(a_{x}=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\)
\(\sum F=\delta m\ a_{x}\)
\(\left(\frac{\partial\sigma_{\text{xx}}}{\partial x}+\frac{\partial\tau_{\text{yx}}}{\partial y}+\frac{\partial\tau_{\text{zx}}}{\partial z}\right)\text{δxδyδz}+S_{M}=\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)\text{δxδyδz}\)
By canceling δxδyδz and substitute the source term\(S_{M}=\rho g_{x}\)
\(\left(\frac{\partial\sigma_{\text{xx}}}{\partial x}+\frac{\partial\tau_{\text{yx}}}{\partial y}+\frac{\partial\tau_{\text{zx}}}{\partial z}\right)+\rho g_{x}=\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)\)
Then, we can obtain the x – component of momentum equation
\(\left(\frac{\partial\sigma_{\text{xx}}}{\partial x}+\frac{\partial\tau_{\text{yx}}}{\partial y}+\frac{\partial\tau_{\text{zx}}}{\partial z}\right)+\rho g_{x}=\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.25\right)\)
Similarly we can obtain the y – component of momentum equation
\(\left(\frac{\partial\sigma_{\text{yy}}}{\partial x}+\frac{\partial\tau_{\text{xy}}}{\partial y}+\frac{\partial\tau_{\text{zy}}}{\partial z}\right)+\rho g_{y}=\rho\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.26\right)\)
Similarly we can obtain the z – component of momentum equation
\(\left(\frac{\partial\sigma_{\text{zz}}}{\partial x}+\frac{\partial\tau_{\text{xz}}}{\partial y}+\frac{\partial\tau_{\text{yz}}}{\partial z}\right)+\rho g_{z}=\rho\left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.27\right)\)
From fluid mechanics textbooks; The normal stresses [6];
\(\sigma_{\text{xx}}=-P+2\mu\left(\frac{\partial u}{\partial x}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.28\right)\)
\(\sigma_{\text{yy}}=-P+2\mu\left(\frac{\partial v}{\partial y}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.29\right)\)
\(\sigma_{\text{zz}}=-P+2\mu\left(\frac{\partial u}{\partial x}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.30\right)\)
The shearing stresses [6];
\(\tau_{\text{xy}}=\tau_{\text{yx}}=\mu\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.31\right)\)
\(\tau_{\text{xz}}=\tau_{\text{zx}}=\mu\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.32\right)\)
\(\tau_{\text{yz}}=\tau_{\text{zy}}=\mu\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(3.33\right)\)