Applications of Mathematics, first online, pp. 1-22
On modeling flow between adjacent surfaces where the fluid is governed by implicit algebraic constitutive relations
Andreas Almqvist, Evgeniya Burtseva, Kumbakonam R. Rajagopal, Peter Wall
Received June 19, 2024. Published online November 12, 2024. OPEN ACCESS
Abstract: We consider pressure-driven flow between adjacent surfaces, where the fluid is assumed to have constant density. The main novelty lies in using implicit algebraic constitutive relations to describe the fluid's response to external stimuli, enabling the modeling of fluids whose responses cannot be accurately captured by conventional methods. When the implicit algebraic constitutive relations cannot be solved for the Cauchy stress in terms of the symmetric part of the velocity gradient, the traditional approach of inserting the expression for the Cauchy stress into the equation for the balance of linear momentum to derive the governing equation for the velocity becomes inapplicable. Instead, a non-standard system of first-order equations governs the flow. This system is highly complex, making it important to develop simplified models. Our primary contribution is the development of a framework for achieving this. Additionally, we apply our findings to a fluid that exhibits an S-shaped curve in the shear stress versus shear rate plot, as observed in some colloidal solutions.
Keywords: implicit algebraic constitutive relation; flow between adjacent surfaces
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Affiliations: Andreas Almqvist, Evgeniya Burtseva, Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden, e-mail: andreas.almqvist@ltu.se, evgeniya.burtseva@ltu.se, Kumbakonam R. Rajagopal, Department of Mechanical Engineering, Texas A&M University, 3123 College Station, TX 77843-3123, USA, e-mail: krajagopal@tamu.edu; Peter Wall (corresponding author), Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden, e-mail: peter.wall@ltu.se
