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I'm a canoist, so I know that when I go with my kayak behind a rock in a river, I feel a current that is opposite to the river current.

I'm also I student mathematician, so I would like to see this phenomenon from equations. But I don't know anything about the equations that govern fluid dynamic.

So my problem is: let be $W = \mathbb{R} \times [-2, 2]$ the river and $R = B_1(0)$ the rock (a ball inside the river). Let be $M = W \backslash R$ the river without the rock, i.e. where the water could flow. Let be $\phi: \mathbb{R} \times M \to M$ a function such that $\phi(t, x)$ is the position at time $t$ of the fluid particle that at time $t = 0$ was in position $x$. Let be the water flow from left to right. My question is: what differential equation for $\phi$ I have to solve? And does this equation predict the correct flow behind the rock?

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The simplest model that fits is potential flow around a cylinder (or a circle in 2D). This assumes an an inviscid, incompressible fluid with no vorticity, which is too simple to model the backflow. The backflow occurs because of viscosity produces boundary layer separation.

I think the second simplest model possible would be to solve the steady-state Navier Stokes equations for an incompressible fluid. Then it is just $$(\mathbf{v}\cdot \nabla)\mathbf{v}-\nu\nabla^2\mathbf{v}=-\nabla w$$ $$\nabla\cdot \mathrm{v}=0$$ where $\nu$ is the viscosity and $\nabla w=(\nabla p)/\rho$ is the pressure term (or other forces expressed as hydraulic head). Since you are interested in the 2D case it can also be turned into an equation for stream functions that leaves out the pressure term.

The backflow starts at a Reynolds number of about 40. While there are no doubt some analytic solutions for this geometry and low velocities, as you approach higher flow rates/Reynolds numbers they become unstable and you have to rely on numerical models.

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The equations that describe the flow of fluids like water are the Navier-Stokes equations.

These are notoriously intractable. So much so that they are currently the subject of one of the Millenium prizes in mathematics. If you're interested in finding out more about this I recommend Terence Tao's article on the subject.

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  • $\begingroup$ Which role has the shape of the domain $M$ in these equations? $\endgroup$ – Marco Francesco Nervo Apr 1 '19 at 9:41
  • $\begingroup$ @MarcoNervo that would determine the boundary conditions $\endgroup$ – John Rennie Apr 1 '19 at 9:45
  • $\begingroup$ While it is true that the full NS equations apply, for this particular case one can likely make use of much simpler restricted versions, e.g. incompressible fluid, laminar flow. I doubt one can go all the way to the irrotational fluid that would turn it into Laplace's equation, but there are simpler models that are used here. $\endgroup$ – Anders Sandberg Apr 1 '19 at 9:56
  • $\begingroup$ Your attempt at a mathematical description of the problem leaves out the important fact that the flow is NOT two dimensional, even if you assume the river bed is a horizontal plane and the rock is a vertical cylinder. The surface of the water is an unknown shape which has to be solved as part of the problem. The boundary condition is that the pressure at the water surface is constant. $\endgroup$ – alephzero Apr 1 '19 at 10:30
  • $\begingroup$ @alephzero where did I say the flow was two dimensional? $\endgroup$ – John Rennie Apr 1 '19 at 10:31

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