# Drag force on cylinder in parallel flow

## Main Question or Discussion Point

Hi,
I'm looking for a correlation of the drag force on a cylinder in parallel creeping (stokes') flow (i.e. the flow is alongside the axis of the cylinder). My length-to-width ratio is such that assuming an infinitely long cylinder is perfectly okay.

Does anyone know where I can find such a correlation?

I've tried deriving it myself but I ran into something called the Stokes' paradox. I know this can be solved by an approximation method (taking a linear inertia term into account), but I do not want to dive in that deep, I just need the equation for a calculation.

## Answers and Replies

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I've found useful equations from Batchelor (1970) and Keller (1976) who use 'slender-body theory' in which the rod is approximated as a line of stokeslets (singular force terms)

Andy Resnick
Science Advisor
Education Advisor
Are you asking if the flow is parallel to or or perpendicular to the axis of the cylinder?

Stokes' paradox is, IIRC, for flow perpendicular to the axis of the cylinder. There's an exact solution in Lamb's 'hydrodynamics', which is at work. It involves the Euler constant (0.577...).

Stokes' paradox was first found for a cylinder perpendicular to the flow, but the same effect (and for the same reason) is present for flow parallel to the cylinder. The basic problem is not the direction of flow but rather the fact the effect of inertial as compared to viscous forces are no longer negligible at large distances from the body.

As for the flow solution: slender body theory provides results for both parallel and perpendicular flow for finite cylinders. Lamb's solution is for an infinite cylinder. Although I said that would be perfectly fine given my length to width ratio, the results for slender body theory apply to finite cylinders which is more useful to me. Moreover, lamb's solution is not exact either, it also involves assuming a linear inertia term in the navier stokes equation