Stochastic shear thickening fluids: Strong convergence of the Galerkin approximation and the energy equality

• Published in: arXiv.org
• Main Author: Yoshida, Nobuo
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 08.10.2012
• Subjects: Fluid dynamics; Fluid flow; Galerkin method; Incompressible flow; Mathematical analysis; Mathematics - Probability; Newtonian fluids; Non Newtonian fluids; Partial differential equations; Polynomials; Shear thickening (liquids); Tensors; Thickening; Velocity distribution
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1009.2136

We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here, the extra stress tensor of the fluid is given by a polynomial of degree p-1 of the rate of strain tensor, while the colored noise is considered as a random force. We focus on the shear thickening case, more precisely, on the case \(p\in [1+{\frac{d}{2}},{\frac{2d}{d-2}})\), where d is the dimension of the space. We prove that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.