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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: [1] = +,
In reality, very close to the origin, the motion resembles a solid body rotation. The Rankine vortex model assumes a solid-body rotation inside a cylinder of radius and a potential vortex outside the cylinder. The radius is referred to as the vortex-core radius.
These flows correspond closely to real-life flows over the whole of fluid mechanics; in addition, many valuable insights arise when considering the deviation (often slight) between an observed flow and the corresponding potential flow. Potential flow finds many applications in fields such as aircraft design.
Example [ edit ] Consider a uniform irrotational flow f ( z ) = U z {\displaystyle f(z)=Uz} with velocity U {\displaystyle U} flowing in the positive x {\displaystyle x} direction and place an infinitely long cylinder of radius a {\displaystyle a} in the flow with the center of the cylinder at the origin.
As an example, consider the problem of determining the potential of a unit source located at (,,) inside a conducting cylindrical tube (e.g. an empty tin can) which is bounded above and below by the planes = and = and on the sides by the cylinder =. [3]
The topspinning cylinder "pulls" the airflow up and the air in turn pulls the cylinder down, as per Newton's Third Law. On a cylinder, the force due to rotation is an example of Kutta–Joukowski lift. It can be analysed in terms of the vortex produced by rotation.
Bessel functions describe the radial part of vibrations of a circular membrane.. Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, [1] are canonical solutions y(x) of Bessel's differential equation + + = for an arbitrary complex number, which represents the order of the Bessel function.
Setup of a Taylor–Couette system. In fluid dynamics, the Taylor–Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders. For low angular velocities, measured by the Reynolds number Re, the flow is steady and purely azimuthal.