A Numerical Study of Taylor–Couette Problem for a Rarefied Gas: Effect of Rotation of the Outer Cylinder

The Taylor–Couette problem for a rarefied gas is studied numerically by the direct simulation Monte Carlo method. The gas is supposed to be contained in an annular domain, bounded by two coaxial rotating cylinders and top and bottom (specularly reflecting) boundaries, and the flow is assumed to be steady and axisymmetric. Special attention is focused on the effect of rotation of the outer cylinder on the type of the induced Taylor-vortex flow. It is shown that different types of flow can coexist stably in a wide range of speeds of rotation of the inner and outer cylinders unless the outer cylinder is rotating fast in the opposite direction to the inner. INTRODUCTION The Taylor–Couette problem, the instability of the cylindrical Couette flow and the occurrence of the secondary flow with vortices (Taylor vortex flow) in a fluid between two coaxial rotating cylinders, is a classical problem in fluid dynamics (e.g., Ref. [1, 2, 3]). However, the study of this problem for a rarefied gas started much later, and several numerical studies with different emphases [4, 5, 6, 7, 8] have been carried out by using the direct simulation Monte Carlo (DSMC) method [9]. For instance, the effect of the temperature difference between the two cylinders is investigated in detail in Ref. [8]. Lately, interest has been expanded to more complicated topics, such as the bifurcation of flows when evaporation and condensation take place on the cylinders [10] and the ghost effect when the rotation speeds of the cylinders as well as the Knudsen number are infinitesimal [11]. In the previous DSMC studies, the case where only the inner cylinder is rotating is mainly discussed. In the present paper, we investigate the same problem again by the DSMC method, restricting ourselves to the steady and axisymmetric flows. But our attention is focused on systematic understanding of the effect of rotation of the outer cylinder on the type of induced Taylor vortex flow and the parameter range in which different types of flow may exist. PROBLEM AND ASSUMPTIONS Let us consider a rarefied gas between two rotating coaxial circular cylinders with a finite length. To describe the problem, we use the cylindrical coordinate system (r, θ , z) with the z axis taken along the common axis of the cylinders. The radius of the inner cylinder is LI , that of the outer cylinder is LII , and the bottom and top ends of the cylinders are covered with plates located at z = 0 and Lz, respectively. Thus, we consider an annular domain LI ≤ r ≤ LII , 0 ≤ θ < 2π , and 0 ≤ z ≤ Lz. The inner and outer cylinders are rotating around the z-axis at surface velocities VI and VII in the θ direction, respectively, and both cylinders are kept at a common temperature T0. We investigate the steady behavior of the gas numerically on the basis of kinetic theory under the following assumptions: (i) the flow field is axisymmetric; (ii) the gas molecules are hard spheres; (iii) the gas molecules undergo diffuse reflection on the surface of the cylinders (r = LI and LII) and specular reflection on the bottom and top boundaries (z = 0 and Lz). In the present study, special attention is focused on the effect of the rotation of the outer cylinder on the flow patterns. Since the basic equations are essentially the same as those of Ref. [8] and the space is limited in the present proceedings, we omit them for brevity. We only note that, by introducing appropriate dimensionless variables, one finds that the present problem is characterized by the following five dimensionless parameters: LII/LI , Lz/LI , VI/(2RT0), VII/(2RT0), and Kn. Here, Kn= `0/LI is the Knudsen number, where `0 is the mean free path of the gas molecules