In this paper, the novel Tensor Diffusion approach for the numerical simulation of viscoelastic fluids is introduced based on the idea, that the extra-stress tensor in the momentum equation of the flow model can be replaced by the product of the strain-rate tensor and a (nonsymmetric) tensor-valued viscosity. As potential advantage (which can be demonstrated for fully developed channel, resp., pipe flows), this approach offers the possibility to reduce the full nonlinear viscoelastic three-field model to a generalized Tensor Stokes problem, avoiding the need of considering a separate stress tensor in the solution process. Moreover, an (artificial) diffusive operator is introduced into the momentum equation in the case of “no solvent” viscoelastic fluids, which is related to the physics of the problem leading to an improved numerical behaviour w.r.t. approximation and convergence properties of the iterative solvers. After validating the Tensor Diffusion approach for fully developed channel flow configurations, it is evaluated in the context of more general two-dimensional flow settings of benchmarking character, too, taking into account a symmetrized formulation. Numerical simulations for several prototypical flow configurations illustrate the mathematical and numerical properties of this new approach and can be viewed as proof-of-concept for further development.

在此基础上，本文提出了一种新颖的张量扩散方法，用于粘弹性流体的数值模拟，即可以用应变率张量的乘积和公式的乘积代替流动模型动量方程中的超应力张量。 （非对称）张量值粘度。作为潜在的优势（可以为充分开发的通道，管道和管道流动证明），这种方法提供了将完整的非线性粘弹性三场模型简化为广义的Tensor Stokes问题的可能性，而无需考虑单独的应力解过程中的张量。此外，在“无溶剂”粘弹性流体的情况下，将（人工）扩散算子引入到动量方程中，这与导致改进的数值行为wr的问题的物理性质有关。t。迭代求解器的逼近和收敛性质。在针对完全发达的通道流量配置验证了Tensor扩散方法之后，还要在考虑对称公式的情况下，在更通用的基准特征二维流量设置的背景下对它进行评估。几种原型流动配置的数值模拟说明了这种新方法的数学和数值特性，可以视为进一步开发的概念验证。