Potential flow around a circular cylinder

Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox.

[1] The flow being incompressible, a stream function can be found such that It follows from this definition, using vector identities, Therefore, a contour of a constant value of ψ will also be a streamline, a line tangent to V. For the flow past a cylinder, we find: Laplace's equation is linear, and is one of the most elementary partial differential equations.

This simple equation yields the entire solution for both V and p because of the constraint of irrotationality and incompressibility.

Thus we find the maximum speed in the flow, V = 2U, in the low pressure on the sides of the cylinder.

Unlike an ideal inviscid fluid, a viscous flow past a cylinder, no matter how small the viscosity, will acquire a thin boundary layer adjacent to the surface of the cylinder.

Boundary layer separation will occur, and a trailing wake will exist in the flow behind the cylinder.

For more detailed analyses and discussions, readers are referred to Milton Van Dyke's 1975 book Perturbation Methods in Fluid Mechanics.

Then the solution to first-order approximation is Here the radius of the cylinder varies with time slightly so r = a(1 + ε f(t)).

Then the solution to first-order approximation is In general, the free-stream velocity U is uniform, in other words ψ = Uy, but here a small vorticity is imposed in the outer flow.

The governing equation is Then the solution to first-order approximation is Here a parabolic shear in the outer velocity is introduced.

Therefore, Then the solution to the first-order approximation is If the cylinder has variable radius in the axial direction, the z-axis, r = a (1 + ε sin ⁠z/b⁠), then the solution to the first-order approximation in terms of the three-dimensional velocity potential is where K1(⁠r/b⁠) is the modified Bessel function of the first kind of order one.