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Stokes' law states that force on slow moving sphere (i.e. $Re\ll1$) in liquid is $$ F_d = 6 \pi \mu R V $$

In two dimensions we are in trouble (flow around disk in 2d or around cylinder in 3d), because there is no solution to the Stokes' problem (known as Stokes' paradox), but from dimensional analysis we can still conclude that

$$ F_d = C \mu V $$

I did some numerical tests of Navier-Stokes equations for small Reynolds numbers and found that $F_d$ really does not depend on $R$ and $C\approx 4\pi$.

I find it quite counter-intuitive that the force in 2D does not depend on the disk radius. Have I done something wrong? Or it really does not depend on radius of the disk?

Only thing which depends on the disk radius is the admissible range of input velocities. If you increase $R$ than you have to lower the max $V$ to ensure the condition $Re \ll 1$.

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  • $\begingroup$ Intuitively, I would say that the contact area for the sphere scales with $R^2$ and for the rod with $R$, so in that sense I can understand that for the rod the order in $R$ is one lower. $\endgroup$
    – Bernhard
    Sep 26, 2015 at 9:13
  • $\begingroup$ It is impressive that you have modelled the situation. It would be helpful if you provided more details of your model and algorithm. Of course drag per unit length depends on the radius of the cylinder. Your error is likely to be in your algorithm. $\endgroup$ Jun 3, 2018 at 0:03
  • $\begingroup$ A have used FEniCS library to do the simulation and followed this tutorial karlin.mff.cuni.cz/~hron/fenics-tutorial/stokes/doc.html $\endgroup$
    – Tom
    Jun 3, 2018 at 5:10

2 Answers 2

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Making conclusion based on dimensional analysis without testing the underlying assumptions is dangerous.

The paradox occurs because the validity of the Stokes' equations rely on the Reynolds number being small. This is not the case in 2D as inertia cannot be disregarded in the far-field and therefore a solely viscous dependent force is not possible. Instead a pertubation analysis using the Oseen equations (known as Oseen's approximation) is required leading to a form of the Stokes' drag multiplied by some corrective factor which depends on the Reynolds number.

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  • $\begingroup$ "...Reynolds number being small. This is not the case in 2D..." I do not understand this. I can have perfectly normal 2d flow with small Raynolds number. $\endgroup$
    – Tom
    Sep 25, 2015 at 12:39
  • $\begingroup$ @Tom - please see the revision to my answer, i hope that clarifies things $\endgroup$
    – nluigi
    Sep 25, 2015 at 13:10
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Viscosity has different size dependence in two and three dimensions. Am example is the Enskog values of viscosity for hard spheres and hard disks gases (Gass 1970s article). These scale with diameter of the particles in different ways, since cinematic viscosity (LENGTH*LENGTH/TIME) es viscosity/density and density es mass/area in two dimensions but mass/volumen in 3D.

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  • $\begingroup$ Welcome to Physics. I'm not sure how this answers the question at the top of the page; the OP is not asking about the dependence of viscosity on the microscopic material properties, but rather about the drag force on an object for a given viscosity. Can you edit your answer to show how this addresses the original question? $\endgroup$ Feb 8, 2022 at 16:47

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