On the Validity of Boundary Layer Theory for Simulating von Karman Flows of Bingham Fluids

Abstract

Exact solutions are rather rare in fluid mechanics, and this is particularly so for non-Newtonian fluids. This is perhaps why the field of non-Newtonian fluid mechanics relies so heavily on approximate theories such as boundary layer theory. Having said this, it should be conceded that while this theory has been very successful for Newtonian fluids, for nonNewtonian fluids its validity, in general, is in doubt. That is to say, there is always the danger that by dropping certain non-Newtonian terms from the governing equations, the nonNewtonian flavor of the flow is tarnished thereby affecting the physics of the problem. Also, it is by no means certain that the potential flow outside the boundary remains uninfluenced by the non-Newtonian behavior of the fluid―a point missed in virtually all boundary layer studies carried out in the past in relation to non-Newtonian fluids. Ironically, an exact solution is needed at the first place to assess the validity of boundary layer approximation for any given non-Newtonian fluid. In a recent work Ahmadpour and Sadeghy have shown that for Bingham fluids, an exact solution can be found in von Karman flow (i.e., the swirling flow generated by a rotating disk in an otherwise quiescent fluid). This exact solution provides us with a perfect tool to investigate the validity of boundary theory for Bingham fluids. With this in mind, it is the main objective of the present work to show that for Bingham fluids, the boundary layer theory is valid over a broad range of parameters. To achieve this goal, we will rely on the idea that a suitable similarity variable can be found [see Ref. 1] which transforms the governing partial differential equations into ordinary differential equations. (The idea, which was first introduced by von Karman while obtaining a self-similar exact solution for Newtonian fluids above a rotating disk [see, also, Refs. 3-5], has been shown to be valid for a variety of non-Newtonian fluids comprising shear-thinning fluid, viscoelastic fluids and viscoplastic fluids.) The work is organized as follows: we start with presenting the governing equations in its most general form before simplifying them using the boundary layer approximation. The Bingham model will be introduced next as the rheological model of interest. We will then proceed with transforming the set of governing PDEs into ODEs using an appropriate similarity variable. The numerical method of solution used to solve the governing ODEs will be described next. Numerical results are presented showing the validity of boundary layer approximation in von Karman flow of Bingham fluids. The work is concluded by highlighting its major findings. On the Validity of Boundary Layer Theory for Simulating von Karman Flows of Bingham Fluids