I would like to know whether exact analytical solutions exist for the fluid flow between two planes with a linearly increasing height, when the convective term is zero (please see the attached image). The flow is pressure driven flowing in the x direction (the gravity effect is neglected). Unlike a plane Poiseuille flow, the component w pointing toward the wall does not vanish. The geometry is infinite along the y direction. That is to solve for the two unknown velocities u and w, both are functions of x and z. Any hints / references would be highly appreciated. Thank you.
Hi, there are analytical solutions, but your gonna have to do some numerical works in the way. for example you can see in the following book(Iain G. Currie Fundamental Mechanics Of Fluids) that this problem ends up with an integral which you cannot solve right ahead and you need to do it numerically, like error functions. and ofcourse there are quasi-analytical ways, i have been triying to solve the same problem numerically.
Thanks Ehsan for your quick reply. I think that the book you mentioned treats the problem of fluid flows (not necessary Stokes flow, see the section 7.8 page 276) in a convergent / divergent channel of a circular cross-section. Here I am interested in the problem of plane channel flows of infinite width. Thank you.
i have read it many times, i know why you would think that its circular, maybe because of the cylindrical coordinates. you have to look again and concentrate on the solution. its a channel with infinite width.
Can help?
http://www.m-hikari.com/ams/ams-2011/ams-53-56-2011/nofalAMS53-56-2011.pdf
Franco
Yes there should be!
Start from the general streamfunction solution for the Stokes equation in cylindrical polars and then take the appropriate derivatives to get the generalised velocity field. The constants are determined by applying the boundary conditions at the edges of the wedge.
See the last section here for the general solution:
http://www.maths.manchester.ac.uk/~mheil/Lectures/Fluids/Material/Chapter8.pdf
The similar problem is Jeffery-Hamel problem (~1917). They consider 2D flow between corner sides goes from (or to) its' vertex, in polar coordinates, without the angular component of the velocity (only radial flow).
The solution for the convergent flow (it is opposite to the flow in your image) does not given in closed form but it is an elliptic integral can be calculated numerically. The radial flow is always unidirectional. It is only convergent at any Reynolds (in radial-flow approximation, of course).
The divergent flow can be presented in the same form but it has some additional features. At some critical Re, the counter-flow occurs in the diffuser even in pure radial flow approximation.
I think these results may be applied in your problem despite you have not a point fluid source or sink here.
You can find a very good brief description of the Hamel problem in Fluid Mechanics by L.D. Landau (L.D. Landau & E.M. Lifshitz Fluid Mechanics, Volume 6 of A Course of Theoretical Physics / Butterworth-Heinemann 1987 or Pergamon Press 1959, or, maybe, there is some newer edition).
Your little problem is a lot to think about. To give you something you may think about, Poiseuille may do something for you at a point where the plates are very close together so that the viscous term control the motion. Actually, maybe not. The pressure gradient can not be constant along any distance as the pressure loss varies (and have to) along the direction of the fluid. Solving exactly seems unlikely. Remember that in the essence you are trying to compute shear stress across an expansion of a given geometry so that you have to decide on your geometry model and how the stresses are computed across that geometry up to the boundary at the wall. Thinking of it in the per square meter basis may help. So think of 1 meter perpendicular to the paper and 1 m of surface (boundary wall). Your geometry model may do something for you and assume that if the space between walls get to be about a couple of milimeters wide (in water) your relationship is meaningless and need to start assuming that you need to include the momentum terms. This is my 2 cents.
This is not to say that I know the answer, it is to say, where the math could aim to try to solve the problem. Good luck though.
May be you can check this:
P.G. Drazin, N. Riley; The Navier-Stokes equation, a classifications of flows and exact solutions, Cambridge University press
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