Axisymmetric Finite Element Solution of Non-isothermal Parallel-plate Flow.pdfVIP

Axisymmetric Finite Element Solution of Non-isothermal Parallel-plate Flow Shangyou Zhang and David O. Olagunju? Department of Mathematical Sciences University of Delaware Newark, DE 19716. USA November 19, 2004 Abstract Steady non-isothermal parallel-plate flow of a Newtonian fluid with a temperature dependent viscosity is considered. The viscosity is modelled by a Nahme type law. We apply axisymmetric Qk finite elements to the coupled nonlinear system to obtain numerical solutions for a wide range of parameters. Keywords: Parallel-plate flow, viscous heating, axisymmetric finite elements. AMS subject classification: 76D07, 76M10, 65F10 ?corresponding author: olagunju@math.udel.edu 1 1 Introduction It is a well known fact that heating effects due to viscous dissipation could cause appreciable deviation from isothermal theory in shear flow. If not properly accounted for, this could lead to errors in viscometric applications [1, 2, 3, 4, 5]. Viscous heating has also been shown to significantly alter the onset and nature of instabilities in shear flows [7, 8, 9]. Due to the strong coupling between the velocity and temperature fields, no exact solution of the non-isothermal flow between finite coaxial parallel plates exist. Consequently, approximate techniques have been used to reduce the equations to more tractable forms. For example, in [2] Bird and Turian assumed a velocity profile of the form vθ = rg(z), which satisfies the momentum equation only if the flow is isothermal. Regular perturbation methods have also been used. If the aspect ratio α ≡ H/a, where a is the disk radius and H is the gap, an exact solution exists for the limiting case α = 0 [3, 6, 10]. In this limit the problem reduces to that of plane Couette flow. However, the plane Couette solution does not satisfy the boundary conditions at the fluid- air interface since the problem is singular in the limit α → 0. In [3], Turian and Bird noted that errors up to 15% existed between the theoretical results