On the compressible Navier–Stokes equation with Coriolis force and density-dependent viscosity: the Cartesian vector solutions and their properties

Abstract The Navier–Stokes (NS) equation with Coriolis force and density-dependent viscosity is an important physical model, which has been widely used to understand and analyze a wide array of phenomena, including behaviors of the Gulf stream, dynamics of hurricanes, operation of chemical reactors and functionality of rotating machines. In this paper, based on the matrix and curve integration techniques, we build a sufficient condition for the existence of Cartesian vector solutions u = b ( t ) + A ( t ) x for the N -dimensional NS equation, in which A satisfies appropriate matrix equations. Then, we discuss two special cases of A and thereby explicit analytical solutions are obtained. To shed light on these solutions, we give some illustrative examples. Among them, some examples form the generalization previously obtained by other authors and some examples are quite new. Finally, we analyze the properties of Cartesian vector solutions in a special case.