Quaternionic Functions and Their Applications in a Viscous Fluid Flow

Abstract

We used a quaternion function method for the Moisil–Theodoresco system (MTS). Solutions of the MTS are (left-) regular quaternion functions $$f(\mathbf{r})=f_0(\mathbf{r})+\mathbf{f}(\mathbf{r})= f_0(x,y,z)+\mathbf{i}f_x(x,y,z)+\mathbf{j}f_y(x,y,z)+\mathbf{r}f_z(x,y,z)$$ of a reduced quaternion variable $$\mathbf{r}=\mathbf{i}x + \mathbf{j}y + \mathbf{k}z$$ . Here we present the quaternion three-dimensional representation of a general solution of the Stokes system for the slow flow of a viscous fluid in star-shaped domain. It is shown that in particular cases of plane and axially symmetric flows this representation goes into the representations by means of analytical and generalized analytical functions of complex variables. As applications the main problems the Stokes flow in a ball are solved.