Consider two stationary plane walls with a constant volume flow rate is injected/sucked at the point of intersection of plane walls and let the angle subtended by two walls be . Take the cylindrical coordinate system with representing point of intersection and the centerline and are the corresponding velocity components. The resulting flow is two-dimensional if the plates are infinitely long in the axial 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., .
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.
Formulation
The first equation tells that is just function of , the function is defined as
Different authors defines the function differently, for example, Landau[8] defines the function with a factor . But following Whitham,[11]Rosenhead[12] the momentum equation becomes
Now letting
the and momentum equations reduce to
and substituting this into the previous equation(to eliminate pressure) results in
Multiplying by and integrating once,
where are constants to be determined from the boundary conditions. The above equation can be re-written conveniently with three other constants 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 .
The boundary conditions reduce to
where is the corresponding Reynolds number. The solution can be expressed in terms of elliptic functions. For convergent flow , the solution exists for all , but for the divergent flow , the solution exists only for a particular range of .
The equation takes the same form as an undamped nonlinear oscillator(with cubic potential) one can pretend that is time, is displacement and is velocity of a particle with unit mass, then the equation represents the energy equation(, where and ) with zero total energy, then it is easy to see that the potential energy is
where in motion. Since the particle starts at for and ends at for , there are two cases to be considered.
First case are complex conjugates and . The particle starts at with finite positive velocity and attains where its velocity is and acceleration is and returns to at final time. The particle motion represents pure outflow motion because and also it is symmetric about .
Second case , all constants are real. The motion from to to represents a pure symmetric outflow as in the previous case. And the motion to to with for all time() represents a pure symmetric inflow. But also, the particle may oscillate between , representing both inflow and outflow regions and the flow is no longer need to symmetric about .
The rich structure of this dynamical interpretation can be found in Rosenhead(1940).[7]
Pure outflow
For pure outflow, since at , integration of governing equation gives
and the boundary conditions becomes
The equations can be simplified by standard transformations given for example in Jeffreys.[14]
The limiting condition is obtained by noting that pure outflow is impossible when , which implies from the governing equation. Thus beyond this critical conditions, no solution exists. The critical angle is given by
As Reynolds number increases ( becomes larger), the flow tends to become uniform(thus approaching potential flow solution), except for boundary layers near the walls. Since is large and is given, it is clear from the solution that must be large, therefore . But when , , the solution becomes
It is clear that everywhere except in the boundary layer of thickness . The volume flux is so that and the boundary layers have classical thickness .
References
^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.
^Hamel, Georg. "Spiralförmige Bewegungen zäher Flüssigkeiten." Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1917): 34–60.
^von Kármán, and Levi-Civita. "Vorträge aus dem Gebiete der Hydro-und Aerodynamik." (1922)
^Walter Tollmien "Handbuch der Experimentalphysik, Vol. 4." (1931): 257.
^Fritz Noether "Handbuch der physikalischen und technischen Mechanik, Vol. 5." Leipzig, JA Barch (1931): 733.
^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.
^ abLouis Rosenhead "The steady two-dimensional radial flow of viscous fluid between two inclined plane walls." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 175. No. 963. The Royal Society, 1940.
^G.K. Batchelor. An introduction to fluid dynamics. Cambridge university press, 2000.
^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.
^Whitham, G. B. "Chapter III in Laminar Boundary Layers." (1963): 122.
^Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.
^Jeffreys, Harold, Bertha Swirles, and Philip M. Morse. "Methods of mathematical physics." (1956): 32–34.
August 23, 2023
jeffery, hamel, flow, fluid, dynamics, flow, created, converging, diverging, channel, with, source, sink, fluid, volume, point, intersection, plane, walls, named, after, george, barker, jeffery, 1915, georg, hamel, 1917, subsequently, been, studied, many, majo. 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 Karman 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 Contents 1 Flow description 2 Formulation 3 Dynamical interpretation 13 4 Pure outflow 4 1 Limiting form 5 Pure inflow 5 1 Limiting form 6 ReferencesFlow description EditConsider two stationary plane walls with a constant volume flow rate Q displaystyle Q is injected sucked at the point of intersection of plane walls and let the angle subtended by two walls be 2 a displaystyle 2 alpha Take the cylindrical coordinate r 8 z displaystyle r theta z system with r 0 displaystyle r 0 representing point of intersection and 8 0 displaystyle theta 0 the centerline and u v w displaystyle u v w are the corresponding velocity components The resulting flow is two dimensional if the plates are infinitely long in the axial z displaystyle 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 8 v 0 w 0 displaystyle u u r theta v 0 w 0 Then the continuity equation and the incompressible Navier Stokes equations reduce to r u r 0 u u r 1 r p r n 1 r r r u r 1 r 2 2 u 8 2 u r 2 0 1 r r p 8 2 n r 2 u 8 displaystyle begin aligned frac partial ru partial r amp 0 6pt u frac partial u partial r amp frac 1 rho frac partial p partial r nu left frac 1 r frac partial partial r left r frac partial u partial r right frac 1 r 2 frac partial 2 u partial theta 2 frac u r 2 right 6pt 0 amp frac 1 rho r frac partial p partial theta frac 2 nu r 2 frac partial u partial theta end aligned 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 a 0 Q a a u r d 8 displaystyle u pm alpha 0 quad Q int alpha alpha ur d theta Formulation EditThe first equation tells that r u displaystyle ru is just function of 8 displaystyle theta the function is defined as F 8 r u n displaystyle F theta frac ru nu Different authors defines the function differently for example Landau 8 defines the function with a factor 6 displaystyle 6 But following Whitham 11 Rosenhead 12 the 8 displaystyle theta momentum equation becomes 1 r p 8 2 n 2 r 2 d F d 8 displaystyle frac 1 rho frac partial p partial theta frac 2 nu 2 r 2 frac dF d theta Now letting p p r n 2 r 2 P 8 displaystyle frac p p infty rho frac nu 2 r 2 P theta the r displaystyle r and 8 displaystyle theta momentum equations reduce to P 1 2 F 2 F displaystyle P frac 1 2 F 2 F P 2 F P 2 F C displaystyle P 2F quad Rightarrow quad P 2F C and substituting this into the previous equation to eliminate pressure results in F F 2 4 F 2 C 0 displaystyle F F 2 4F 2C 0 Multiplying by F displaystyle F and integrating once 1 2 F 2 1 3 F 3 2 F 2 2 C F D displaystyle frac 1 2 F 2 frac 1 3 F 3 2F 2 2CF D 1 2 F 2 1 3 F 3 6 F 2 6 C F 3 D 0 displaystyle frac 1 2 F 2 frac 1 3 F 3 6F 2 6CF 3D 0 where C D displaystyle 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 displaystyle 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 displaystyle a b c 6 1 2 F 2 1 3 F a F b F c 0 displaystyle frac 1 2 F 2 frac 1 3 F a F b F c 0 1 2 F 2 1 3 a F F b F c 0 displaystyle frac 1 2 F 2 frac 1 3 a F F b F c 0 The boundary conditions reduce to F a 0 Q n a a F d 8 displaystyle F pm alpha 0 quad frac Q nu int alpha alpha F d theta where R e Q n displaystyle Re Q nu is the corresponding Reynolds number The solution can be expressed in terms of elliptic functions For convergent flow Q lt 0 displaystyle Q lt 0 the solution exists for all R e displaystyle Re but for the divergent flow Q gt 0 displaystyle Q gt 0 the solution exists only for a particular range of R e displaystyle Re Dynamical interpretation 13 EditThe equation takes the same form as an undamped nonlinear oscillator with cubic potential one can pretend that 8 displaystyle theta is time F displaystyle F is displacement and F displaystyle F is velocity of a particle with unit mass then the equation represents the energy equation K E P E 0 displaystyle K E P E 0 where K E 1 2 F 2 displaystyle K E frac 1 2 F 2 and P E V F displaystyle P E V F with zero total energy then it is easy to see that the potential energy is V F 1 3 a F F b F c displaystyle V F frac 1 3 a F F b F c where V 0 displaystyle V leq 0 in motion Since the particle starts at F 0 displaystyle F 0 for 8 a displaystyle theta alpha and ends at F 0 displaystyle F 0 for 8 a displaystyle theta alpha there are two cases to be considered First case b c displaystyle b c are complex conjugates and a gt 0 displaystyle a gt 0 The particle starts at F 0 displaystyle F 0 with finite positive velocity and attains F a displaystyle F a where its velocity is F 0 displaystyle F 0 and acceleration is F d V d F lt 0 displaystyle F dV dF lt 0 and returns to F 0 displaystyle F 0 at final time The particle motion 0 lt F lt a displaystyle 0 lt F lt a represents pure outflow motion because F gt 0 displaystyle F gt 0 and also it is symmetric about 8 0 displaystyle theta 0 Second case c lt b lt 0 lt a displaystyle c lt b lt 0 lt a all constants are real The motion from F 0 displaystyle F 0 to F a displaystyle F a to F 0 displaystyle F 0 represents a pure symmetric outflow as in the previous case And the motion F 0 displaystyle F 0 to F b displaystyle F b to F 0 displaystyle F 0 with F lt 0 displaystyle F lt 0 for all time a 8 a displaystyle alpha leq theta leq alpha represents a pure symmetric inflow But also the particle may oscillate between b F a displaystyle b leq F leq a representing both inflow and outflow regions and the flow is no longer need to symmetric about 8 0 displaystyle theta 0 The rich structure of this dynamical interpretation can be found in Rosenhead 1940 7 Pure outflow EditFor pure outflow since F a displaystyle F a at 8 0 displaystyle theta 0 integration of governing equation gives 8 3 2 F a d F a F F b F c displaystyle theta sqrt frac 3 2 int F a frac dF sqrt a F F b F c and the boundary conditions becomes a 3 2 0 a d F a F F b F c R e 2 3 2 0 a F d F a F F b F c displaystyle alpha sqrt frac 3 2 int 0 a frac dF sqrt a F F b F c quad Re 2 sqrt frac 3 2 int 0 alpha frac FdF sqrt a F F b F c The equations can be simplified by standard transformations given for example in Jeffreys 14 First case b c displaystyle b c are complex conjugates and a gt 0 displaystyle a gt 0 leads toF 8 a 3 M 2 2 1 cn M 8 k 1 cn M 8 k displaystyle F theta a frac 3M 2 2 frac 1 operatorname cn M theta kappa 1 operatorname cn M theta kappa M 2 2 3 a b a c k 2 1 2 a 2 2 M 2 displaystyle M 2 frac 2 3 sqrt a b a c quad kappa 2 frac 1 2 frac a 2 2M 2 dd where sn cn displaystyle operatorname sn operatorname cn are Jacobi elliptic functions Second case c lt b lt 0 lt a displaystyle c lt b lt 0 lt a leads toF 8 a 6 k 2 m 2 sn 2 m 8 k displaystyle F theta a 6k 2 m 2 operatorname sn 2 m theta k m 2 1 6 a c k 2 a b a c displaystyle m 2 frac 1 6 a c quad k 2 frac a b a c dd Limiting form Edit The limiting condition is obtained by noting that pure outflow is impossible when F a 0 displaystyle F pm alpha 0 which implies b 0 displaystyle b 0 from the governing equation Thus beyond this critical conditions no solution exists The critical angle a c displaystyle alpha c is given by a c 3 2 0 a d F F a F F a 6 3 2 a 0 1 d t t 1 t 1 1 6 a t K k 2 m 2 displaystyle begin aligned alpha c amp sqrt frac 3 2 int 0 a frac dF sqrt F a F F a 6 amp sqrt frac 3 2a int 0 1 frac dt sqrt t 1 t 1 1 6 a t amp frac K k 2 m 2 end aligned where m 2 3 a 3 k 2 1 2 a 3 a displaystyle m 2 frac 3 a 3 quad k 2 frac 1 2 left frac a 3 a right where K k 2 displaystyle K k 2 is the complete elliptic integral of the first kind For large values of a displaystyle a the critical angle becomes a c 3 a K 1 2 3 211 a displaystyle alpha c sqrt frac 3 a K left frac 1 2 right frac 3 211 sqrt a The corresponding critical Reynolds number or volume flux is given by R e c Q c n 2 0 a c a 6 k 2 m 2 sn 2 m 8 d 8 12 k 2 1 2 k 2 0 K cn 2 t d t 12 1 2 k 2 E k 2 1 k 2 K k 2 displaystyle begin aligned Re c frac Q c nu amp 2 int 0 alpha c a 6k 2 m 2 operatorname sn 2 m theta d theta amp frac 12k 2 sqrt 1 2k 2 int 0 K operatorname cn 2 tdt amp frac 12 sqrt 1 2k 2 E k 2 1 k 2 K k 2 end aligned where E k 2 displaystyle E k 2 is the complete elliptic integral of the second kind For large values of a k 2 1 2 3 2 a displaystyle a left k 2 sim frac 1 2 frac 3 2a right the critical Reynolds number or volume flux becomes R e c Q c n 12 a 3 E 1 2 1 2 K 1 2 2 934 a displaystyle Re c frac Q c nu 12 sqrt frac a 3 left E left frac 1 2 right frac 1 2 K left frac 1 2 right right 2 934 sqrt a Pure inflow EditFor pure inflow the implicit solution is given by 8 3 2 b F d F a F F b F c displaystyle theta sqrt frac 3 2 int b F frac dF sqrt a F F b F c and the boundary conditions becomes a 3 2 b 0 d F a F F b F c R e 2 3 2 a 0 F d F a F F b F c displaystyle alpha sqrt frac 3 2 int b 0 frac dF sqrt a F F b F c quad Re 2 sqrt frac 3 2 int alpha 0 frac FdF sqrt a F F b F c Pure inflow is possible only when all constants are real c lt b lt 0 lt a displaystyle c lt b lt 0 lt a and the solution is given by F 8 a 6 k 2 m 2 sn 2 K m 8 k b 6 k 2 m 2 cn 2 K m 8 k displaystyle F theta a 6k 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 displaystyle m 2 frac 1 6 a c quad k 2 frac a b a c where K k 2 displaystyle K k 2 is the complete elliptic integral of the first kind Limiting form Edit As Reynolds number increases b displaystyle b becomes larger the flow tends to become uniform thus approaching potential flow solution except for boundary layers near the walls Since m displaystyle m is large and a displaystyle alpha is given it is clear from the solution that K displaystyle K must be large therefore k 1 displaystyle k sim 1 But when k 1 displaystyle k approx 1 sn t tanh t c b a 2 b displaystyle operatorname sn t approx tanh t c approx b a approx 2b the solution becomes F 8 b 3 tanh 2 b 2 a 8 tanh 1 2 3 2 displaystyle F theta b left 3 tanh 2 left sqrt frac b 2 alpha theta tanh 1 sqrt frac 2 3 right 2 right It is clear that F b displaystyle F approx b everywhere except in the boundary layer of thickness O b 2 displaystyle O left sqrt frac b 2 right The volume flux is Q n 2 a b displaystyle Q nu approx 2 alpha b so that R e O b displaystyle Re O b and the boundary layers have classical thickness O R e 1 2 displaystyle O left Re 1 2 right References Edit 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 Hamel Georg Spiralformige Bewegungen zaher Flussigkeiten Jahresbericht der Deutschen Mathematiker Vereinigung 25 1917 34 60 von Karman and Levi Civita Vortrage aus dem Gebiete der Hydro und Aerodynamik 1922 Walter Tollmien Handbuch der Experimentalphysik Vol 4 1931 257 Fritz Noether Handbuch der physikalischen und technischen Mechanik Vol 5 Leipzig JA Barch 1931 733 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 a b Louis Rosenhead The steady two dimensional radial flow of viscous fluid between two inclined plane walls Proceedings of the Royal Society of London A Mathematical Physical and Engineering Sciences Vol 175 No 963 The Royal Society 1940 a b Lev Landau and E M Lifshitz Fluid Mechanics Pergamon New York 61 1959 G K Batchelor An introduction to fluid dynamics Cambridge university press 2000 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 Whitham G B Chapter III in Laminar Boundary Layers 1963 122 Rosenhead Louis ed Laminar boundary layers Clarendon Press 1963 Drazin Philip G and Norman Riley The Navier Stokes equations a classification of flows and exact solutions No 334 Cambridge University Press 2006 Jeffreys Harold Bertha Swirles and Philip M Morse Methods of mathematical physics 1956 32 34 Retrieved from https en wikipedia org w index php title Jeffery Hamel flow amp oldid 1169191201, wikipedia, wiki, book, books, library,