Jeffery–Hamel flow explained

In fluid dynamics Jeffery–Hamel flow is a flow created by a converging or diverging channel with a source or sink of fluid volume at the point of intersection of the two plane walls. It is named after George Barker Jeffery(1915)^[1] and Georg Hamel(1917),^[2] but it has subsequently been studied by many major scientists such as von Kármán and Levi-Civita,^[3] Walter Tollmien,^[4] F. Noether,^[5] W.R. Dean,^[6] Rosenhead,^[7] Landau,^[8] G.K. Batchelor^[9] etc. A complete set of solutions was described by Edward Fraenkel in 1962.^[10]

Flow description

Consider two stationary plane walls with a constant volume flow rate

Q is injected/sucked at the point of intersection of plane walls and let the angle subtended by two walls be 2 \alpha . Take the cylindrical coordinate (r,\theta,z) system with r =0 representing point of intersection and \theta =0 the centerline and (u,v,w) are the corresponding velocity components. The resulting flow is two-dimensional if the plates are infinitely long in the axial z direction, or the plates are longer but finite, if one were neglect edge effects and for the same reason the flow can be assumed to be entirely radial i.e., u = u(r,\theta),v =0 ,w =0 .

Then the continuity equation and the incompressible Navier–Stokes equations reduce to

\begin{align} (\partial (ru))/(\partial r) & =0 , \\[ 6 pt] u (\partial u)/(\partial r) & = - (1)/(\rho) (\partial p)/(\partial r) + \nu \left[ (1)/(r) (\partial)/(\left(r \partial r) (\partial u)/(\partial r) \right) + (1)/(r ^{2}) (\partial ^{2} u)/(\partial \theta ^{2}) - (u)/(r ^{2}) \right] \\[ 6 pt] 0 & = - (1)/(\rho r) (\partial p)/(\partial \theta) + (2 \nu)/(r ^{2}) (\partial u)/(\partial \theta) \end{align}

The boundary conditions are no-slip condition at both walls and the third condition is derived from the fact that the volume flux injected/sucked at the point of intersection is constant across a surface at any radius.

u(\pm \alpha) = 0 , Q = \int ^{\alpha} _{- \alpha} u r d\theta

Formulation

The first equation tells that

ru is just function of \theta , the function is defined asF(\theta) = (r u)/(\nu) .

Different authors defines the function differently, for example, Landau^[8] defines the function with a factor

6 . But following Whitham,^[11] Rosenhead^[12] the \theta momentum equation becomes(1)/(\rho) (\partial p)/(\partial\theta) = (2 \nu ^{2})/(r ^{2}) (dF)/(d\theta)

Now letting

(p - p _{infty})/(\rho) = (\nu ^{2})/(r ^{2}) P(\theta),

the

r and \theta momentum equations reduce toP =- (1)/(2) (F ^{2+F'')} P' = 2 F', ⇒ P = 2 F + C

and substituting this into the previous equation(to eliminate pressure) results in

F'' + F ^{2} + 4 F + 2 C =0

Multiplying by

F' and integrating once,(1)/(2) F' ^{2+} (1)/(3) F ^{3} +2 F ^{2} +2 CF = D, (1)/(2) F' ^{2+} (1)/(3) (F ^{3} +6 F ^{2} +6 CF -3 D) = 0

where

C,D are constants to be determined from the boundary conditions. The above equation can be re-written conveniently with three other constants a,b,c as roots of a cubic polynomial, with only two constants being arbitrary, the third constant is always obtained from other two because sum of the roots is a + b + c =-6 .(1)/(2) F' ^{2+ (1)/(3)} (F - a)(F - b)(F - c) =0 , (1)/(2) F' ^{2- (1)/(3)} (a - F)(F - b)(F - c) =0 .

The boundary conditions reduce to

F(\pm \alpha) = 0 , (Q)/(\nu) = \int ^{\alpha} _{- \alpha} F d\theta

where

Re = Q/\nu is the corresponding Reynolds number. The solution can be expressed in terms of elliptic functions. For convergent flow Q< 0 , the solution exists for all Re , but for the divergent flow Q> 0 , the solution exists only for a particular range of Re .

Dynamical interpretation

Source:^[13]

The equation takes the same form as an undamped nonlinear oscillator(with cubic potential) one can pretend that

\theta is time, F is displacement and F' is velocity of a particle with unit mass, then the equation represents the energy equation(K.E. + P.E. =0 , where K.E. = (1)/(2) F' ^{2} and P.E. = V(F) ) with zero total energy, then it is easy to see that the potential energy isV(F) =- (1)/(3) (a - F)(F - b)(F - c)

where

V\leq 0 in motion. Since the particle starts at F =0 for \theta =- \alpha and ends at F =0 for \theta = \alpha , there are two cases to be considered.

• First case

b,c are complex conjugates and a> 0 . The particle starts at F =0 with finite positive velocity and attains F = a where its velocity is F' =0 and acceleration is F'' =- dV/dF< 0 and returns to F =0 at final time. The particle motion 0 <F<a represents pure outflow motion because F> 0 and also it is symmetric about \theta =0 .

• Second case

c<b< 0 <a , all constants are real. The motion from F =0 to F = a to F =0 represents a pure symmetric outflow as in the previous case. And the motion F =0 to F = b to F =0 with F< 0 for all time(- \alpha\leq\theta\leq\alpha ) represents a pure symmetric inflow. But also, the particle may oscillate between b\leq F\leq a , representing both inflow and outflow regions and the flow is no longer need to symmetric about \theta =0 .The rich structure of this dynamical interpretation can be found in Rosenhead(1940).^[7]

Pure outflow

For pure outflow, since

F = a at \theta =0 , integration of governing equation gives\theta = \sqrt{ (3)/(2) } \int_F^a \frac

and the boundary conditions becomes

\alpha = \sqrt{ (3)/(2) } \int_0^a \frac, \quad Re = 2\sqrt \int_0^\alpha \frac.

The equations can be simplified by standard transformations given for example in Jeffreys.^[14]

• First case

b,c are complex conjugates and a> 0 leads toF(\theta) = a - (3 M ^{2})/(2) (1- \operatorname{cn)/((M\theta,\kappa)}{ 1+ \operatorname{cn}(M\theta,\kappa)}) M ^{2} = (2)/(3) \sqrt{(a - b)(a - c)}, \kappa ^{2= (1)/(2)} + (a +2)/(2 M ^{2})

where

\operatorname{sn}, \operatorname{cn} are Jacobi elliptic functions.

• Second case

c<b< 0 <a leads toF(\theta) = a -6 k ^{2} m ^{2\operatorname{sn}} ^{2(m\theta,k)} m ^{2} = (1)/(6) (a - c), k ^{2= (a - b)/(a - c)} .

Limiting form

The limiting condition is obtained by noting that pure outflow is impossible when

F'(\pm\alpha) =0 , which implies b =0 from the governing equation. Thus beyond this critical conditions, no solution exists. The critical angle \alpha _{c} is given by\begin{align} \alpha _{c} & = \sqrt{ (3)/(2) } \int_0^a \frac,\\ &= \sqrt \int_0^1 \frac,\\ &= \frac\end

where

m ^{2} = (3+ a)/(3) , k ^{2} = (1)/(\left( 2) (a)/(3+ a) \right)

where

K(k ^{2)} is the complete elliptic integral of the first kind. For large values of a , the critical angle becomes \alpha _{c} = \sqrt{ (3)/(a) }K\left(\frac\right)=\frac.

The corresponding critical Reynolds number or volume flux is given by

\begin{align} Re _{c} = (Q _{c})/(\nu) & = 2 \int ^{\alpha _{c}} _{0} (a -6 k ^{2} m ^{2\operatorname{sn}} ^{2} m\theta) d\theta,\\ & = (12 k ^{2})/(\sqrt{ 1-2 k ^{2}) } \int_0^K \operatorname^2 t dt,\\ &= \frac[E(k^2) -(1-k^2)K(k^2)] \end

where

E(k ^{2)} is the complete elliptic integral of the second kind. For large values of a, \left( k ^{2\sim} (1)/(- 2) (3)/(2 a) \right) , the critical Reynolds number or volume flux becomes Re _{c = (Q _{c})/(\nu)} = 12 \sqrt{ (a)/(3) } \left[E\left(\frac{1}{2}\right)-\frac{1}{2}K\left(\frac{1}{2}\right) \right]=2.934 \sqrt.

Pure inflow

For pure inflow, the implicit solution is given by

\theta = \sqrt{ (3)/(2) } \int_b^F \frac

and the boundary conditions becomes

\alpha = \sqrt{ (3)/(2) } \int_b^0 \frac, \quad Re = 2\sqrt \int_\alpha^0 \frac.

Pure inflow is possible only when all constants are real

c<b< 0 <a and the solution is given byF(\theta) = a -6 k ^{2} m ^{2\operatorname{sn}} ^{2(K-m\theta,k)=b+6k} ^{2} m ^{2} \operatorname{cn} ^{2(K-m\theta,k)} m ^{2} = (1)/(6) (a - c), k ^{2= (a - b)/(a - c)}

where

K(k ^{2)} is the complete elliptic integral of the first kind.

Limiting form

As Reynolds number increases (

- b becomes larger), the flow tends to become uniform(thus approaching potential flow solution), except for boundary layers near the walls. Since m is large and \alpha is given, it is clear from the solution that K must be large, therefore k\sim 1 . But when k ≈ 1 , \operatorname{sn} t ≈ \tanh t, c ≈ b, a ≈ -2 b , the solution becomesF(\theta) = b\left\{ 3 \tanh ^{2} \left[\sqrt{ - (b)/(2) }(\alpha-\theta)+\tanh^\sqrt\right]-2\right\}.

It is clear that

F ≈ b everywhere except in the boundary layer of thickness O\left(\sqrt{ - (b)/(2) }\right). The volume flux is Q/\nu ≈ 2 \alpha b so that |Re| = O(|b|) and the boundary layers have classical thickness O\left(|Re| ^{1/2} \right) .

Notes and References

1. Jeffery, G. B. "L. The two-dimensional steady motion of a viscous fluid." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 29.172 (1915): 455–465.
2. Hamel, Georg. "Spiralförmige Bewegungen zäher Flüssigkeiten." Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1917): 34–60.
3. [Theodore von Kármán|von Kármán]
4. [Walter Tollmien]
5. [Fritz Noether]
6. Dean, W. R. "LXXII. Note on the divergent flow of fluid." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 18.121 (1934): 759–777.
7. [Louis Rosenhead]
8. [Lev Landau]
9. [George Keith Batchelor|G.K. Batchelor]
10. Fraenkel, L. E. (1962). Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 267(1328), 119-138.
11. [Gerald B. Whitham|Whitham, G. B.]
12. Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.
13. [Philip Drazin|Drazin, Philip G.]
14. Jeffreys, Harold, Bertha Swirles, and Philip M. Morse. "Methods of mathematical physics." (1956): 32–34.