# Jeffery-Hamel flow

## Introduction

The flow between two planes that meet at an angle was first analyzed by Jeffery (1915) and Hamel (1916). Under suitable assumptions, the problem can be reduced to the solution of an ordinary differential equation. This ODE can be readily solved using various numerical techniques. This case is analyzed in several of the standard texts, including Batchelor and Landau and Lifshitz.

The flow geometry is shown in the figure below. The two plates confining the flow meet at an angle of $2\alpha$, and the flow can be either diverging or converging (shown in the figure).

The volume flow rate may be computed from

$Q = \int_{-\alpha}^\alpha Rv_r(R,\theta)d\theta$

at a specified radius $R$. For $Q$ positive the flow is diverging, while for $Q$ negative the flow is converging. The converging-flow case is of more interest as a test case since the diverging case may not be as stable (or steady) as the converging case. The appropriate Reynolds number is usually taken to be a function of the volume flow rate. Here, we will follow Rosenhead and use

$Re = \frac{|Q|}{\nu}$

## Similarity Solution

The flow is assumed to be purely radial and steady, or that $v_\theta = v_z = 0$ and $v_r = v_r(r,\theta)$ with no-slip conditions at $\theta=\pm\alpha$.

The Navier-Stokes equations in cylindrical coordinates then reduce to

$v_r\frac{\partial v_r}{\partial r} = -\frac{1}{\rho}\frac{\partial p}{\partial r} + \nu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial v_r}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 v_r}{\partial \theta^2} - \frac{v_r}{r^2}\right]$

$-\frac{1}{\rho r} \frac{\partial p}{\partial \theta} + \nu \frac{2}{r^2}\frac{\partial v_r}{\partial \theta} = 0$

$\frac{\partial p}{\partial z} = 0$

The continuity equation reduces to

$\frac{\partial}{\partial r}\left(r v_r\right) = 0,$

which requires that $r v_r$ be independent of $r$. This leads us to look for solutions of the form

$v_r = \frac{\nu}{r}f(\theta),$

where $f$ is a function of $\theta$ only. Substituting into the azimuthal momentum equation and integrating once, we obtain

$\frac{p}{\rho} = \frac{2\nu^2}{r^2}f+g$

where $g$ is a function of $r$ only. Substituting into the radial momentum equation, we find

$f'' + f^2 + 4f = \frac{g'r^3}{\nu^2}.$

Since the left hand side is a function of $\theta$ only and the right-hand side is a function of $r$ only, both must be constant. This gives an ODE for $f$:

$f'' + f^2 + 4f = C,$

and, after an integration, an expression for pressure:

$\frac{p}{\rho} = \frac{2\nu^2}{r^2}f - \frac{C\nu^2}{2 r^2} + \frac{p_0}{\rho},$

where $p_0$ is a constant of integration.

The constant $C$ must be chosen before a solution can be found. It should be noted that various authors have used cosmetically different definitions for $f$, which leads to cosmetically different ODE's.

The Reynolds number for this case is usually based upon the volume flow rate $Q$, which can now be written as

$Q = \nu\int_{-\alpha}^\alpha fd\theta.$

The Reynolds number is then

$Re = \frac{|Q|}{\nu} = \left|\int_{-\alpha}^\alpha fd\theta\right|.$

## Solution

The elliptic function solutions of Rosenhead and others are not very convenient for comparison purposes, so it is easier to compute a numerical solution to the similarity ODE. There problem can be solved as a boundary value problem (as it is), or it can be converted into an initial value problem and solved via a shooting method. Both approaches should produce acceptable results provided that sufficient care is taken. Shown below are the results for three converging-flow calculation using a Newton iteration technique for three values of $C$.

## References

Jeffery, G. B. (1915), "The Two-Dimensional Steady Motion of a Viscous Fluid", Philosophical magazine, Vol. 29, pp. 455-465.

Hamel, G. (1916), "Spiralfömrige Bewegungen zäher Flüssigheiten", Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 25, pp. 34-60.

Rosenhead, L. (1940), "The Steady Two-Dimensional Radial Flow of Viscous Fluid between Two Inclined Plane Walls", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 175, No. 963, pp. 436-467.