Burgers vortex
In fluid dynamics, the Burgers vortex or Burgers–Rott vortex is an exact solution to the Navier–Stokes equations governing viscous flow, named after Jan Burgers and Nicholas Rott. The Burgers vortex describes a stationary, self-similar flow. An inward, radial flow, tends to concentrate vorticity in a narrow column around the symmetry axis. At the same time, viscous diffusion tends to spread the vorticity. The stationary Burgers vortex arises when the two effects balance. The Burgers vortex, apart from serving as an illustration of the vortex stretching mechanism, may describe such flows as tornados, where the vorticity  is provided by continuous convection-driven vortex stretching. Flow field. The flow for the Burgers vortex is described in cylindrical formula_1 coordinates. Assuming axial symmetry (no formula_2-dependence), the flow field associated with the axisymmetric stagnation point flow is considered: where formula_6 (strain rate) and formula_7 (circulation) are constants. The flow satisfies the continuity equation by the two first of the above equations. The azimuthal momentum equation of the Navier-Stokes equations then reduces to where formula_9 is the kinematic viscosity of the fluid. The equation is integrated with the condition formula_10 so that at infinity the solution behaves like  a potential vortex, but at finite location, the flow is rotational. The choice formula_11 ensures formula_12 at the axis. The solution is The vorticity equation only gives a non-trivial component in the formula_14-direction, given by Intuitively the flow can be understood by looking at the three terms in the vorticity equation for formula_16, The first term on the right-hand side of the above equation corresponds to vortex stretching which intensifies the vorticity of the vortex core due to the axial-velocity component formula_18. The intensified vorticity tries to diffuse outwards radially due to the second term on  the right-hand side, but is prevented by radial vorticity convection due to formula_19 that emerges on the left-hand side of the above equation. The three-way balance establishes a steady solution. Burgers vortex layer. Burgers vortex layer or Burgers vortex sheet is a strained shear layer, which is a two-dimensional analogue of Burgers vortex. This is also an exact solution of the Navier-Stokes equations, first described by A. A. Townsend in 1951. The velocity field formula_20 expressed in the Cartesian coordinates are where formula_6 is the strain rate, formula_25 and formula_26. The value formula_27 is interpreted  as the vortex sheet strength. The vorticity equation only gives a non-trivial component in the formula_14-direction, given by Non-axisymmetric Burgers vortices. Non-axisymmetric Burgers' vortices emerge in non-axisymmetric strained flows. The theory for non-axisymmetric Burgers's vortex for small vortex Reynolds numbers formula_30 was developed by A. C. Robinson and Philip Saffman in 1984, whereas Keith Moffatt, S. Kida and K. Ohkitani has developed the theory for formula_31 in 1994. The structure of non-axisymmetric Burgers' vortices for arbitrary values of vortex Reynolds number can be discussed through numerical integrations. The velocity field takes the  form subjected to the condition formula_35. Without loss of generality, one assumes formula_6 and formula_37. The vortex cross-section lies in formula_38 plane, providing a non-zero vorticity component in the formula_14 direction The axisymmetric Burgers' vortex is recovered when formula_41 whereas the Burgers' vortex layer is recovered when formula_42 and formula_43. Burgers vortex in cylindrical stagnation surfaces. Explicit solution of the Navier-Stokes equations for the Burgers vortex in stretched cylindrical stagnation surfaces were discussed recently. The solution is expressed in the cylindrical coordinate system as follows where formula_6 is the strain rate, formula_48 is the radial location of  the cylindrical stagnation surface, formula_7 is the circulation and formula_50 is the regularized gamma function. This solution is nothing but the Burger's vortex in the presence of a line source with source strength formula_51. The vorticity equation only gives a non-trivial component in the formula_14-direction, given by where formula_54 in the above expression is the gamma function. As formula_55, the solution reduces to Burgers' vortex solution and as formula_56, the solution becomes the Burgers' vortex layer solution. Explicit solution for Sullivan vortex in cylindrical stagnation surface also exists. Sullivan vortex. In 1959, Roger D. Sullivan extended the  Burgers vortex solution by considering the solution of the form where formula_60. The functions formula_61 and formula_62 are given by For Burgers vortex formula_65, formula_66 and formula_67 are always positive, Sullivans result shows that formula_68 for formula_69 and formula_70 for formula_71. Thus Sullivan vortex resembles Burgers vortex for formula_72, but develops a two-cell structure near the axis due to the sign change of formula_73.