Developing combined convection of non-Newtonian fluids in an eccentric annulus

Summary Laminar combined convection of non-Newtonian fluids in vertical eccentric annuli, in which the inner and outer walls are held at different constant temperatures is considered and a new economical method of solution for the three-dimensional flow in the annulus is developed. Assuming that the ratio of the radial to the vertical scale, , is small, as occurs frequently in many industrial applications, then the governing equations can be simplified by expanding all the variables in terms of . This simplification gives rise to the presence of a dominant cross-stream plane in which all the physical quantities change more rapidly than in the vertical direction. The solution trechnique consists of marching in the vertical streamwise direction using a finite-difference scheme and solving the resulting equations at each streamwise step by a novel technique incorporating the Finite Element Method. The process is continued until the velocity, pressure and temperature fields are fully developed, and results are presented for a range of the governing non-dimensional parameters, namely the Grashof, Prandtl, Reynolds and Bingham numbers.List of symbols Bn Bingham number, - d ^* difference between the radii of the outer and inner cylinders,r _o ^*–r_i ^* - e ^* distance between the axes of the inner and outer cylinders - e eccentricity,e ^*/d^* - F ^* external force acting on the fluid - g ^* acceleration due to gravity - g ^* gravitational vector, (0,0,g ^*) - Gr Grashof number, _m *² g**(T ₀*–T _e*)d*³/ _m *² - K ^* consistency of the fluid - L ^* height of the cylinders of the annulus - n flow behaviour index - p ^* dimensional pressure - P dimensionless pressure gradient - Pr Prandtl number, _m */ _m ** - r _i ^* radius of the inner cylinder of the annulus - r _o ^* radius of the outer cylinder of the annulus - r _T wall temperature difference ratio,(T _i ^*–T_e ^*)/(T_o ^*–T_e ^*) - Re Reynolds number, _m d*w _m */ _m * - T dimensionless temperature of the fluid,(T ^*–T_e ^*)/(T_o ^*–T_e ^*) - T _dif ^* temperature difference between the walls of the annulus - T _e ^* temperature at the fluid at the entrance of the annulus - T _i ^* temperature at the inner cylinder of the annulus - T _o ^* temperature at the outer cylinder of the annulus - u dimensionless transverse velocity in thex direction,u ^*/(w_m ^*) - U dimensionless transverse velocity in the annulus,Reu - u ^* fluid velocity vector, (u ^*, v^*, w^*) - v dimensionless transverse velocity in they direction,v ^*/(w_m ^*) - V dimensionless transverse velocity in the annulus,Rev - w dimensionless vertical velocity,w ^*/w_m ^* - w _m scaling used to non-dimensionalise the vertical velocity - x dimensionless transverse coordinate,x ^*/d^* - y dimensionless transverse coordinate,y ^*/d^* - z dimensionless vertical coordinate,z ^*/L^* - Z dimensionless vertical coordinate,z/Re - Z _r dimensionless distance in the vertical direction where the final wall temperatures are attained,Z _r ^*/L^* - * dimensional molecular thermal diffusivity - * coefficient of thermal expansion, - dimensional rate of strain tensor - dimensionless ratio of the length scales in the annulus,d ^*/L^* - * dimensional apparent non-Newtonian viscosity - _m * mean viscosity, - * dimensional fluid density - _m * dimensional reference fluid density - * dimensional stress tensor - yield stress