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System of Navier-Stokes equations

    From the above equations we can obtain system of Navier-Stokes equations for incompressible, viscous fluid:

$\displaystyle \frac{\partial u}{\partial x} +\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}$$\displaystyle =$0(21.15)
$\displaystyle \rho \Big( \frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}+
v \frac{\partial u}{\partial y}+w \frac{\partial u}{\partial z}\Big)$$\displaystyle =$$\displaystyle - \frac{\partial p}{\partial x}
+\frac{\partial}{\partial x} \Big( \eta \frac{\partial
u}{\partial x} \Big)
+\frac{\partial}{\partial y} \Big( \eta
\frac{\partial u}{\partial y} \Big)
+\frac{\partial}{\partial z}
\Big( \eta \frac{\partial u}{\partial z} \Big)+f_1$ 
$\displaystyle \rho \Big( \frac{\partial v}{\partial t}+ u \frac{\partial v}{\partial x}+
v \frac{\partial v}{\partial y}+w \frac{\partial v}{\partial z}\Big)$$\displaystyle =$$\displaystyle - \frac{\partial p}{\partial y}
+\frac{\partial}{\partial x} \Big( \eta \frac{\partial
v}{\partial x} \Big)
+\frac{\partial}{\partial y} \Big( \eta
\frac{\partial v}{\partial y} \Big)
+\frac{\partial}{\partial z}
\Big( \eta \frac{\partial v}{\partial z} \Big)+f_2$ 
$\displaystyle \rho \Big( \frac{\partial w}{\partial t}+ u \frac{\partial w}{\partial x}+
v \frac{\partial w}{\partial y}+w \frac{\partial w}{\partial z}\Big)$$\displaystyle =$$\displaystyle - \frac{\partial p}{\partial z}
+\frac{\partial}{\partial x} \Big( \eta \frac{\partial
w}{\partial x} \Big)
+\frac{\partial}{\partial y} \Big( \eta
\frac{\partial w}{\partial y} \Big)
+\frac{\partial}{\partial z}
\Big( \eta \frac{\partial w}{\partial z} \Big)+f_3$ 


These equations create the complete system, we will solve in this work. For more simplicity we will consider the volume forces $ f_i$ to be equal $ 0$.