page_title: Potential flow around a circular cylinder
section_title: Mathematical solution
hierarchical_section_title: Potential flow around a circular cylinder / Mathematical solution
caption_attribution_description: English: Flow past a cylinder. Pressure field and velocity vectors.
context_page_description: 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.
context_section_description: 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
  where U is a constant, and at the boundary of the cylinder
  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:
  Being incompressible, ∇ · V = 0, so φ must satisfy Laplace's equation:
  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):
  The solution that satisfies the boundary conditions is
  The velocity components in polar coordinates are obtained from the components of ∇φ in polar coordinates:
  and
  Being inviscid and irrotational, Bernoulli's equation allows the solution for pressure field to be obtained directly from the velocity field:
  where the constants U and p∞ appear so that p → p∞ far from the cylinder, where V = U. Using V² = V²
  ᵣ + V²
  θ,
  In the figures, the colorized field referred to as "pressure" is a plot of
  On the surface of the cylinder, or r = R, pressure varies from a maximum of 1 (shown in the diagram in red) at the stagnation points at θ = 0 and θ = π to a minimum of −3 (shown in blue) on the sides of the cylinder, at θ =  and θ = . Likewise, V varies from V = 0 at the stagnation points to V = 2U on the sides, in the low pressure.
caption: Colors: pressure field. Red is high and blue is low. Velocity vectors.
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