There are a number of exact solutions to the Navier-Stokes equations. How many exact solutions are currently known? Is it possible to enumerate all of the solutions to the Navier-Stokes equations?
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1$\begingroup$ One can only enumerate the exact solutions known at a certain point in time, and even that is quite tedious since exact solutions depend on the precise formulation of the problem (changing the shape of the section of a pipe has an important impact, for example), and researchers have found solutions through various methods at various moments, often ignoring each others contributions. But you can get a decent list. $\endgroup$– Christoph B.Commented Apr 10, 2013 at 8:36
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2$\begingroup$ This appears to be a "big list" type question, which is considered off-topic here. $\endgroup$– Kyle KanosCommented Dec 24, 2015 at 18:10
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Frank White's Viscous Fluid Flow book contains a good list of these "exact" solutions. I am not sure if it is complete though. I've provided links to a few of the solutions.
- Steady flow between a fixed and moving plate
- Axially moving concentric cylinders
- Flow between rotating concentric cylinders
- Hagan-Poiseuille flow
- Combined Couette-Poiseuille flow between plates
- Noncircular ducts -- fully developed flow
- Starting flow in a circular pipe
- Pipe flow due to an oscillating pressure gradient
- Suddenly accelerating plate
- Oscillating plate/oscillating freestream
- Steady Couette flow where the moving wall suddenly stops
- Unsteady Couette flow between a fixed and an oscillating plate
- Radial outflow from a porous cylinder
- Radial outflow between two circular plates
- Combined Poiseuille and Couette flow in a tube or annulus
- Gravity-driven thin fluid films
- Decay of a line Oseen-Lamb vortex
- The Taylor vortex profile
- Uniform suction on a plane
- Flow between plates with bottom injection and top suction
- Start up of wind driven surface water
- The Ekman Spiral
- Plane stagnation flow
- Axisymmetric stagnation flow
- Flow near an infinite rotating disk
- Jeffrey-Hamel flow in a wedge-shaped region
- Stokes' Solution for an Immersed Sphere -- Creeping Flow
- Creeping flow past a fluid sphere
- Blasius boundary layer
- Falkner-Skan-Cooke boundary layer
- Compressible self-similar boundary layer
- Free-shear flows
- Plane laminar wake -- far field
- Plane laminar jet
- Flat-plate with uniform wall-suction
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1$\begingroup$ Ah wonderful! I was wondering if such a compilation existed as well. Most fluids texts have some subset of the analytic solutions but they are usually scattered through some giant tome of a book. Thank you! $\endgroup$ Commented Apr 9, 2013 at 16:39
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1$\begingroup$ C. Y. Wangs' 1991 « Exact Solutions of the Steady-State Navier-Stokes Equation » paper published in Annual Review of Fluid Mechanics, gives an overview of exact solutions in the steady case as its name suggests: annualreviews.org/doi/abs/10.1146/annurev.fl.23.010191.001111 It doesn't go into great depths (and is over 20 years old, but White's book was first published in 1974 and then updated in the nineties), but it gives extensive references to other works. I guess one could extract a few additional examples to complete the above list, such as Burgers' vortex and Beltrami flows. $\endgroup$ Commented Apr 10, 2013 at 8:31
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1$\begingroup$ This helped me some, with my question: spinning fluid inside a sphere. I would just add that you can approximate many things from these canned solutions. I have done: - fluid pushed through a cone - fluid between two spheres coming together - fluid between a slowly varying surface. $\endgroup$ Commented Jun 4, 2014 at 3:58
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1$\begingroup$ P. G. Drazin has written a small book on exact solutions to the Navier-Stokes. Being published more recently its list is even more comprehensive than Wang's. $\endgroup$ Commented Apr 17, 2018 at 1:45