Navier-Stokes simplification, velocity distribution

Navier-Stokes equation (3-D, incompressible, constant $\mu$)

$$\begin{aligned} \rho\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right) &= \rho g_x - \frac{\partial p}{\partial x} + \mu\nabla^2 u\\ \rho\left(\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}\right) &= \rho g_y - \frac{\partial p}{\partial y} + \mu\nabla^2 v\\ \rho\left(\frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}\right) &= \rho g_z - \frac{\partial p}{\partial z} + \mu\nabla^2 w \end{aligned}$$

with $\nabla^2 = \partial^2/\partial x^2 + \partial^2/\partial y^2 + \partial^2/\partial z^2$ and continuity $\partial u/\partial x + \partial v/\partial y + \partial w/\partial z = 0$.

Simplification

Assumptions: steady ($\partial/\partial t = 0$); no z-variation and $w = 0$; fully developed ($\partial/\partial x = 0$ for velocity). Continuity then gives $\partial v/\partial y = 0$, and with $v = 0$ at a wall, $v = 0$ everywhere. Take $x$ along the flow and gravity along $-y$ ($g_x = 0$, $g_y = -g$).

• Left-hand side: all inertia terms vanish ($u\,\partial u/\partial x = 0$, $v = 0$).
• x-momentum:

$$0 = -\frac{\partial p}{\partial x} + \mu\frac{d^2u}{dy^2}$$

• y-momentum:

$$0 = -\rho g - \frac{\partial p}{\partial y}$$

• z-momentum: $0 = -\partial p/\partial z$.

So the pressure is hydrostatic across the flow and $dp/dx$ is constant, giving the simplified model:

$$\mu\frac{d^2u}{dy^2} = \frac{dp}{dx}$$

Velocity distribution

Yes. Integrating twice:

$$\frac{du}{dy} = \frac{1}{\mu}\frac{dp}{dx}y + C_1,\qquad u = \frac{1}{2\mu}\frac{dp}{dx}y^2 + C_1y + C_2$$

For Couette-type flow between plates at $y = 0$ (fixed) and $y = h$ (moving at $U$), with $u(0) = 0$, $u(h) = U$:

$$u = \frac{U y}{h} + \frac{1}{2\mu}\frac{dp}{dx}\left(y^2 - hy\right)$$

If $dp/dx = 0$, this reduces to the linear profile $u = Uy/h$ (simple Couette flow). If both plates are fixed ($U=0$), it gives the parabolic Poiseuille profile $u = \dfrac{1}{2\mu}\dfrac{dp}{dx}(y^2 - hy)$.