# Simulating Stokes flow with an obstacle

I was asked to compute the Stokes flow (i.e. a low Reynolds fluid) near and obstacle. This is the first time I face a fluid and I am lost. What reference/general ideas/big theorems can you recommend me?

More details: given and obstacle (lets say, a bounded simply connected region $D\subset\mathbb{R}^2$ with $C^\infty$ boundary and all necessary stuff) and a steady flow $W\hat{x}$ from $-\infty\hat{x}$, I want to calculate the resultanting Stokes flow near $D$. For $D=\text{circle}$, the solution is well known (see, for example, page 4: http://web.mit.edu/2.21/www/Lec-notes/chap2_slow/2-5Stokes.pdf). In this reference, the formulas are derived from the continuity equation $\nabla\cdot q =0$ with boundary conditions $\text{at infinity}$. My idea was to use this approach, but I don't even know if it is possible to solve numerically a PDE with this kind of boundary conditions.

Thanks!

• This question is pretty confusing. (Though I am a big Better off Ted fan.) The first tool I'd recommend for flow around arbitrary objects might be the method of Regularized Stokeslets - this is very simple to code, and well-explained in Cortez et al. 2005, doi.org/10.1063/1.1830486 and the original Cortez paper (cited in that) – AJK Jan 5 '18 at 3:42
• You can't, in general, numerically solve problems that are posed on an infinite domain. But you can solve on a sequence or larger and larger domains, and then extrapolate the result to an infinite domain. – Wolfgang Bangerth Jan 7 '18 at 22:41

## 1 Answer

As previously mentioned, you can't technically solve a problem on an infinite domain. However, you can solve a problem over a domain that is sufficiently large enough to approximate the conditions you would find at infinity- that is the spacial gradient at your boundary is numerically zero (Dirichlet boundary condition). This is known as the "farfield" boundary. Since you are just starting out, just add more space away from your point of interest and see how your solution changes.

• This is the more simple approach. I was expecting a more sophisticated solution. I don't know the field, so another solution maybe even don't exists – Veridian Dynamics Jan 24 '18 at 18:46
• Truthfully, you never know the field before you solve for it, you only impose boundaries and initial conditions at the onset. Remember that the point of numerical methods is to walk the razors edge between computation cost and solution accuracy. By adding space away from your cylinder of interest, you'll be able to see that the flow is only disturbed relatively close to the body, and you can make a best guess determination on what your bounds should be. – tdso222 Jan 25 '18 at 6:10
• For solving this problem, first initialize stream function value over your domain, then use some constant value of stream function at your boundary to provide information to the rest of the domain. Since this problem is elliptic, and you're just starting out, I'd recommend using an explicit solution method such as the Point Gauss-Seidel method. – tdso222 Jan 25 '18 at 6:27