In mathematics, potential flow around a circular cylinder is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a potential flow. Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox.
Mathematical solution
A cylinder (or disk) of radius R is placed in a two-dimensional, incompressible, inviscid flow. The goal is to find the steady velocity vector V and pressure p in a plane, subject to the condition that far from the cylinder the velocity vector (relative to unit vectors i and j) is:
V = U i + 0 j , {\displaystyle \mathbf {V} =U\mathbf {i} +0\mathbf {j} \,,}
where U is a constant, and at the boundary of the cylinder
V ⋅ n ^ = 0 , {\displaystyle \mathbf {V} \cdot \mathbf {\hat {n}} =0\,,}
where n̂ is the vector normal to the cylinder surface. The upstream flow is uniform and has no vorticity. The flow is inviscid, incompressible and has constant mass density ρ. The flow therefore remains without vorticity, or is said to be irrotational, with ∇ × V = 0 everywhere. Being irrotational, there must exist a velocity potential φ:
V = ∇ ϕ . {\displaystyle \mathbf {V} =\nabla \phi \,.}
Being incompressible, ∇ · V = 0, so φ must satisfy Laplace's equation:
∇ 2 ϕ = 0 . {\displaystyle \nabla ^{2}\phi =0\,.}
The solution for φ is obtained most easily in polar coordinates r and θ, related to conventional Cartesian coordinates by x = r cos θ and y = r sin θ. In polar coordinates, Laplace's equation is (see Del in cylindrical and spherical coordinates):
1 r ∂ ∂ r ( r ∂ ϕ ∂ r ) + 1 r 2 ∂ 2 ϕ ∂ θ 2 = 0 . {\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial \phi }{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\phi }{\partial \theta ^{2}}}=0\,.}
The solution that satisfies the boundary conditions is
ϕ ( r , θ ) = U r ( 1 + R 2 r 2 ) cos θ . {\displaystyle \phi (r,\theta )=Ur\left(1+{\frac {R^{2}}{r^{2}}}\right)\cos \theta \,.}
The velocity components in polar coordinates are obtained from the components of ∇φ in polar coordinates:
V r = ∂ ϕ ∂ r = U ( 1 − R 2 r 2 ) cos θ {\displaystyle V_{r}={\frac {\partial \phi }{\partial r}}=U\left(1-{\frac {R^{2}}{r^{2}}}\right)\cos \theta }
and
V θ = 1 r ∂ ϕ ∂ θ = − U ( 1 + R 2 r 2 ) sin θ . {\displaystyle V_{\theta }={\frac {1}{r}}{\frac {\partial \phi }{\partial \theta }}=-U\left(1+{\frac {R^{2}}{r^{2}}}\right)\sin \theta \,.}
Being inviscid and irrotational, Bernoulli's equation allows the solution for the pressure field to be obtained directly from the velocity field:
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