Sakiadis Flow of Harris Fluids: a Series-Solution

Abstract

Boundary layer theory is without doubt one of the most successful approximations in the history of fluid mechanics. This is certainly true for Newtonian fluids but for non-Newtonian fluids, the theory is still regarded as incomplete. A major obstacle in extending the theory to non-Newtonian fluids is in the diversity of their constitutive behavior meaning that each fluid model should be treated separately. Furthermore, the nonlinearity introduced by their shear-dependent viscosity and/or elasticity often gives rise to a formidable mathematical task which cannot be solved, at times, even numerically. Understandably, the situation becomes much more complicated when the viscosity of the fluid is time-dependent (e.g., when the fluid is thixotropic). Such fluid systems are quite frequent in industrial applications (e.g., drilling muds) with the common effect being that their viscosity drops with the progress of time at any given shear rate. Due to the complexity of their rheological behavior, working with thixotropic fluids is not an easy task. A major problem is the lack of a robust and easy-to-use rheological model which can describe such behavior. Among different rheological models available to represent such fluid systems Harris model is without doubt one of the simplest ones, albeit admittedly not the best one. Interestingly, the model developed by Harris can also represent purely-viscous shearthinning fluids for certain set of parameter values. Harris tried this version of his rheological model to study Blasius flow. He relied on the technique of similarity solution and reduced the boundary layer equations into a single ODE. But, the equation so obtained was realized to be too formidable to render itself to an analytical or even numerical solution so that it remained unsolved until recently. As a matter of fact, in a recent work Sadeqi et al relied on a robust numerical method to tackle Blasius flow of shear-thinning fluids obeying Harris model. They also showed that Harris model can represent thixotropic fluids only for certain values of the model parameters. In the present work we would like to extend the work carried out in Ref. 15 to Sakiadis flow. Due to the strong nonlinearity of the governing equation, we have decided to rely on the homotopy analysis method (HAM) in the present work. Unlike perturbation techniques, HAM is independent of the smallness/largeness of any parameter involved in the problem. In addition, it provides a simple way to ensure the convergence of the series-solution so that one can always come up with a sufficiently accurate approximation to the solution (even for strongly non-linear problems). Furthermore, unlike all other analytical techniques, the homotopy analysis method provides great freedom in choosing the so-called Sakiadis Flow of Harris Fluids: a Series-Solution