Lamb–Chaplygin dipole
The Lamb–Chaplygin dipole model is a mathematical description for a particular inviscid and steady dipolar vortex flow. It is a non-trivial solution to the two-dimensional Euler equations. The model is named after Horace Lamb and Sergey Alexeyevich Chaplygin, who independently discovered this flow structure. This dipole is the two-dimensional analogue of Hill's spherical vortex. The model. A two-dimensional (2D), solenoidal vector field formula_1 may be described by a scalar stream function formula_2, via formula_3, where formula_4 is the right-handed unit vector perpendicular to the 2D plane. By definition, the stream  function is related to the vorticity formula_5 via a Poisson equation: formula_6. The Lamb–Chaplygin model follows from demanding the following characteristics: The solution formula_2 in cylindrical coordinates (formula_14), in the co-moving frame of reference reads: formula_15 where formula_16 are the zeroth and first Bessel functions of the first kind, respectively. Further, the value of formula_17 is such that formula_18, the first non-trivial zero of the first Bessel function of the first kind. Usage and considerations. Since the seminal work of P. Orlandi, the Lamb–Chaplygin vortex model has been a popular choice for numerical studies on vortex -environment interactions. The fact that it does not deform make it a prime candidate for consistent flow initialization. A less favorable property is that the second derivative of the flow field at the dipole's edge is not continuous. Further, it serves a framework for stability analysis on dipolar-vortex structures.