We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids, also called yield-stress fluids. In contrast to the conventional approaches relying on the non-linear viscosity concept of the Navier-Stokes theory and representation of the solid state as an infinitely rigid non-deformable solid, the solid state in our theory is deformable and the fluid state is considered rather as a “melted” solid via a certain procedure of relaxation of tangential stresses similar to Maxwell’s visco-elasticity theory. The model is formulated as a system of first-order hyperbolic partial differential equations with possibly stiff non-linear relaxation source terms. The computational strategy is based on a staggered semi-implicit scheme which can be applied in particular to low-Mach number flows as usually required for flows of non-Newtonian fluids. The applicability of the model and numerical scheme is demonstrated on a few standard benchmark test cases such as Couette, Hagen-Poiseuille, and lid-driven cavity flows. The numerical solution is compared with analytical or numerical solutions of the Navier-Stokes theory with the Herschel-Bulkley constitutive model for nonlinear viscosity.

我们讨论了统一的双曲模型对于连续流体和固体力学对非牛顿流建模，尤其是对粘塑性流体（也称为屈服应力流体）中的应力驱动的固体流体转换建模的适用性。与依靠Navier-Stokes理论的非线性粘度概念和将固态表示为无限刚性的不可变形固体的常规方法相反，我们理论中的固态是可变形的，而流体状态则被认为是可变形的。作为“融化通过某种类似于麦克斯韦粘弹性理论的切线应力松弛程序来固结。该模型被公式化为一阶双曲偏微分方程组，其中可能包含刚性非线性松弛源项。该计算策略基于交错的半隐式方案，该方案可特别应用于非牛顿流体流动通常需要的低马赫数流量。在一些标准基准测试案例（例如Couette，Hagen-Poiseuille和盖驱动的空腔流）上证明了该模型和数值方案的适用性。将该数值解与具有非线性粘度的Herschel-Bulkley本构模型的Navier-Stokes理论的解析解或数值解进行比较。