Pressure-driven flow in thin domains

Article type: Research Article

Authors: Fabricius, John^{a; *} | Miroshnikova, Elena^a | Tsandzana, Afonso^b | Wall, Peter^a

Affiliations: [a] Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden. E-mails: john.fabricius@ltu.se, elena.miroshnikova@ltu.se, peter.wall@ltu.se | [b] Department of Mathematics and Informatics, Eduardo Mondlane University, Praça 25 de Junho, 257 C. P. 257, Maputo, Mozambique. E-mail: tsandzana69@gmail.com

Correspondence: [*] Corresponding author. E-mail: john.fabricius@ltu.se.

Abstract: We study the asymptotic behavior of pressure-driven Stokes flow in a thin domain. By letting the thickness of the domain tend to zero we derive a generalized form of the classical Reynolds–Poiseuille law, i.e. the limit velocity field is a linear function of the pressure gradient. By prescribing the external pressure as a normal stress condition, we recover a Dirichlet condition for the limit pressure. In contrast, a Dirichlet condition for the velocity yields a Neumann condition for the limit pressure.

Keywords: Stokes equation, pressure boundary condition, two-scale convergence, thin domain, Bogovskii operator, Korn inequality

DOI: 10.3233/ASY-191535

Journal: Asymptotic Analysis, vol. 116, no. 1, pp. 1-26, 2020