Regularized Formulations Sample Clauses

Regularized Formulations. In Section 1.3 we introduced the partial differential equations (1.2), (1.3) describing the flow of a Xxxxxxx material. We will refer to this formulation as the formulation in primitive variables. Notice that in the constitutive relation (1.3) the deformation tensor Du is piecewise defined in fluid and plug region. A major difficulty associated with solving the Xxxxxxx equations is that the fluid and plug regions are unknown a priori and finding them is part of the problem. Further notice that µˆ = 2µ + τs is |Du| singular in the plug region where the strain rate tensor vanishes. These difficulties can 11 √ be addressed by regularizing µˆ. A common regularization technique is the Xxxxxxxxx- Xxxxxxxx regularization [16], in which |Du| is replaced by |Du|ε = |Du|2 + ε2 for some ε > 0. In practice, regularization techniques replace the plug region by a flow region in which the fluid has a high viscosity. The regularized formulation of the Xxxxxxx flow, using the Xxxxxxxxx-Xxxxxxxx approach, reads  ρ[ ∂u + (u · ∇)u] − ∇ · (2µDu + τs Du ) + ∇p = f ∂t |Du|ε  ∇ · u = 0 (2.1) and is imposed in the entire domain Ω. In the case of the steady Xxxxxx type equations, (2.1) reduces to  −∇ · (2µDu + τs |Du|ε ) + ∇p = f Du ∇ · u = 0 (2.2) While the Xxxxxxxxx-Xxxxxxxx regularization perturbs |Du| by a constant param- eter ε independent of |Du|, other techniques exist taking into account the size of |Du|. The model of Papanastasiou [70] is such a technique, providing an exponential regularization of (1.1) by introducing a parameter m which controls the exponential growth of the stress tensor. The Papanastasiou variant of (1.1) reads τ = 2µ + τs Du |Du|