Laminar flow between fixed parallel plates

Assumptions

Steady, laminar, fully developed flow of a Newtonian incompressible fluid between two fixed wide plates, gap $h$. Flow is in the $x$-direction; $y$ is measured from the lower plate. There is no body force effect other than hydrostatic, and $dp/dx$ is constant.

   y ^ ---------------------------  upper plate (u = 0)
     |        ______
     |  h   /        \   u(y)
     |      \________/
     +---------------------------> x   lower plate (u = 0)

Force balance on an element

Consider a fluid element of length $dx$, thickness $dy$ and unit width. Net force in $x$ is zero:

$$p\,dy - \left(p + \frac{dp}{dx}dx\right)dy - \tau\,dx + \left(\tau + \frac{d\tau}{dy}dy\right)dx = 0$$ $$\frac{d\tau}{dy} = \frac{dp}{dx}$$

With $\tau = \mu\,du/dy$:

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

Velocity distribution

Integrating twice, $u = \dfrac{1}{2\mu}\dfrac{dp}{dx}y^2 + C_1y + C_2$. Boundary conditions $u = 0$ at $y = 0$ and at $y = h$ give $C_2 = 0$ and $C_1 = -\dfrac{h}{2\mu}\dfrac{dp}{dx}$.

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

The profile is parabolic. The pressure falls in the flow direction, so $dp/dx < 0$ and $u > 0$.

Shear stress distribution

$$\tau = \mu\frac{du}{dy} = -\frac{dp}{dx}\left(\frac{h}{2} - y\right)$$

It is zero at the centre line and maximum at the walls: $\tau_w = \dfrac{h}{2}\left|\dfrac{dp}{dx}\right|$ (linear variation).

Maximum velocity

At $y = h/2$:

$$u_{max} = -\frac{h^2}{8\mu}\frac{dp}{dx}$$

Discharge and average velocity

$$q = \int_0^h u\,dy = -\frac{1}{2\mu}\frac{dp}{dx}\left[\frac{hy^2}{2} - \frac{y^3}{3}\right]_0^h = -\frac{h^3}{12\mu}\frac{dp}{dx}$$ $$\bar u = \frac{q}{h} = -\frac{h^2}{12\mu}\frac{dp}{dx} = \frac23 u_{max}$$

Pressure drop

For a length $L$, $dp/dx = -\Delta p/L$:

$$\Delta p = \frac{12\mu \bar u L}{h^2} = \frac{12\mu q L}{h^3}$$

The head loss is $h_f = \Delta p/\rho g = \dfrac{12\mu\bar u L}{\rho g h^2}$.