Some problems of nonisothermal steady flow of a viscous fluid

Abstract

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation μ=μ0 exp (−βT) (μ0, β=const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated. Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. μ=μm 1/1+α2(T−Tm. Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. μ=μ0 e −βT (μ0, β=const), (0.1) was studied in [7, 8]. Couette flow with a general μ(T) relation was studied in (9). Forced flow in a plane conduit and in a circular tube with a general μ(T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value. In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential μ(T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal “explosion” and the temperature and velocity profiles were calculated. In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.