Abstract
Singularities in the stress field of the stagnation-point flow of a viscoelastic fluid have been studied for various viscoelastic constitutive models. Analyzing the analytical solutions of these models is the most effective way to study this problem. In this paper, exact analytical solutions of two-dimensional steady wall-free stagnation-point flows for the generic Oldroyd 8-constant model are obtained for the stress field using different material parameter relations. For all solutions, compatibility with the conservation of momentum is considered in our analysis. The resulting solutions usually contain arbitrary functions, whose choice has a crucial effect on the stress distribution. The corresponding singularities are discussed in detail according to the choices of the arbitrary functions. The results can be used to analyze the stress distribution and singularity behavior of a wide spectrum of viscoelastic models derived from the Oldroyd 8-constant model. Many previous results obtained for simple viscoelastic models are reproduced as special cases. Some previous conclusions are amended and new conclusions are drawn. In particular, we find that all models have singularities near the stagnation point and most of them can be avoided by appropriately choosing the model parameters and free functions. In addition, the analytical solution for the stress tensor of a near-wall stagnation-point flow for the Oldroyd-B model is also obtained. Its compatibility with the momentum conservation is discussed and the parameters are identified, which allow for a non-singular solution.
Highlights
The working fluids encountered in practical applications and industry are often non-Newtonian, and research on this type of fluids has been conducted for decades
We obtained the analytical solutions of the stress distributions of a wall-free steady stagnation-point flow with the proposed velocity profile u = for the Oldroyd 8-constant model under different material relations
All solutions here are compatible with the momentum conservation equation, and their singularities are discussed in detail
Summary
The working fluids encountered in practical applications and industry are often non-Newtonian, and research on this type of fluids has been conducted for decades. Meleshko et al [11] extended the analysis of the stress distribution in a wall-free stagnation-point flow from the UCM model to the Johnson–Segalman model and took the momentum conservation equation into account Their solution demonstrates that the Johnson–Segalman model has a non-removable logarithmic singularity. It is interesting to investigate whether the other viscoelastic models have similar singularities as the Oldroyd-B, Maxwell-B and Johnson–Segalman models For this purpose, we will search for exact analytical solutions for the Oldroyd 8-constant constitutive framework of a wall-free stagnation-point flow satisfying the compatibility with the conservation of momentum by means of the method of characteristics and analyze the effect of various model parameters on the solutions with regard to their singularity. We focus on the Oldroyd-B model and analyze the compatibility of the solution with the momentum conservation equation
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