Quaternionic operator methods in fluid dynamics. (English) Zbl 1320.76096

Sprössig, Wolfgang

Adv. Appl. Clifford Algebr. 18, No. 3-4, 963-978 (2008).

Summary: The paper is devoted to the consideration of an initial-boundary value fluid flow problem of Oseen’s type. A brief historical introduction to fluid dynamics is given. A special time-discretisation method conveys the problem to a quaternionic Yukawa equation for the \((k + 1)\)-th time step of the fluid velocity in dependence on earlier times. Yukawa’s problem can be decomposed into a product of two associated operators of Dirac’s type with pure complex potential. With the help of a generalized Bergman projection the solution can be found by an iteration principle. The quaternionic operator calculus ensures that initial values and boundary values are fulfilled. Approximation and stability prove the convergence of the iteration process.

MSC:

76M25	Other numerical methods (fluid mechanics) (MSC2010)
30G35	Functions of hypercomplex variables and generalized variables

Keywords:

quaternionic analysis; semi-discretisation method; Yukawa equation; fluid flow problems