Flow between two parallel plates with a distance $h$
Governing equations:
$$\frac{\partial U}{\partial x}+\frac{\partial V}{\partial y}+\frac{\partial W}{\partial z}=0$$
$$\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial x}+V\frac{\partial U}{\partial y}+W\frac{\partial U}{\partial z}=-\frac{1}{\rho} \frac{\partial p}{\partial x}+\frac{\mu}{\rho}\left ( \frac{\partial^2 U}{\partial x^2}+\frac{\partial^2 U}{\partial y^2}+\frac{\partial^2 U}{\partial z^2} \right )$$
$$\frac{\partial V}{\partial t}+U\frac{\partial V}{\partial x}+V\frac{\partial V}{\partial y}+W\frac{\partial V}{\partial z}=-\frac{1}{\rho} \frac{\partial p}{\partial y}+\frac{\mu}{\rho}\left ( \frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}+\frac{\partial^2 V}{\partial z^2} \right )$$
$$\frac{\partial W}{\partial t}+U\frac{\partial W}{\partial x}+V\frac{\partial W}{\partial y}+W\frac{\partial W}{\partial z}=-\frac{1}{\rho} \frac{\partial p}{\partial z}+\frac{\mu}{\rho}\left ( \frac{\partial^2 W}{\partial x^2}+\frac{\partial^2 W}{\partial y^2}+\frac{\partial^2 W}{\partial z^2} \right )$$
Considering a bidimensional fully developed fluid flow in a steady state,
$$\frac{\partial V}{\partial y}=0 \Rightarrow V(y)=C_1$$
As $V(0)=0$ at the lower boundary, $C_1=0$ so we conclude that:
$$V(y)=0$$
And
$$0=-\frac{1}{\rho} \frac{\partial p}{\partial x}+\frac{\mu}{\rho}\frac{\partial^2 U}{\partial y^2} \Leftrightarrow \frac{\partial^2 U}{\partial y^2}=\frac{1}{\mu} \frac{\partial p}{\partial x}$$
$$0=-\frac{1}{\rho} \frac{\partial p}{\partial y} \Leftrightarrow \frac{\partial p}{\partial y}=0 $$
Integrating the first equation two times:
$$U(y)=\int \int \frac{1}{\mu} \frac{\partial p}{\partial x}dydy=\int \frac{1}{\mu} \frac{\partial p}{\partial x}y+C_2 dy=\frac{1}{2\mu} \frac{\partial p}{\partial x}y^2+C_2y+C_3$$
Since at the plates, $y=0$ and $y=h$, we have the no-slip condition:
$$\left\{\begin{matrix}
U(0)=0\\
U(h)=0
\end{matrix}\right.
\Leftrightarrow
\left\{\begin{matrix}
0=\frac{1}{2\mu} \frac{\partial p}{\partial x}0^2+C_20+C_3\\
0=\frac{1}{2\mu} \frac{\partial p}{\partial x}h^2+C_2h+C_3\end{matrix}\right.
\Leftrightarrow
\left\{\begin{matrix}
C_3=0\\
C_2=-\frac{1}{2\mu} \frac{\partial p}{\partial x}h\end{matrix}\right.$$
The velocity profile is then:
$$U(y)=\frac{1}{2\mu} \frac{\partial p}{\partial x}y^2-\frac{1}{2\mu} \frac{\partial p}{\partial x}hy$$
$$U(y)=\frac{1}{2\mu} \frac{\partial p}{\partial x}(y^2-hy)$$