A complex-valued first integral of Navier-Stokes equations: Unsteady Couette flow in a corrugated channel system

Marner, F.; Gaskell, P.H.; Scholle, M.

Abstract

For a two-dimensional incompressible viscous flow, a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady Navier-Stokes equations in terms of complex variables and the subsequent introduction of a complex potential field; complementary solid and free surface boundary conditions are formulated. The methodology is used to solve the challenging problem of unsteady Couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral. In the Stokes flow limit, the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature. Similarly, for unsteady flow, the results are in accord with related investigations, exploring material transfer between trapped eddies and the associated bulk flow, and vice versa. It is shown how the work relates to the classical complex variable method for solving the biharmonic problem and perspectives are provided as to how the first integral may be further utilised to investigate other fluid flow features.

Citation

Marner, F., Gaskell, P., & Scholle, M. (2017). A complex-valued first integral of Navier-Stokes equations: Unsteady Couette flow in a corrugated channel system. Journal of Mathematical Physics, 58(4), Article 043102. https://doi.org/10.1063/1.4980086

• Journal Article Type: Article
• Acceptance Date: Mar 30, 2017
• Online Publication Date: Apr 19, 2017
• Publication Date: Apr 19, 2017
• Deposit Date: Jun 30, 2017
• Publicly Available Date: Jun 30, 2017
• Journal: Journal of Mathematical Physics
• Print ISSN: 0022-2488
• Electronic ISSN: 1089-7658
• Publisher: American Institute of Physics
• Peer Reviewed: Peer Reviewed
• Volume: 58
• Issue: 4
• Article Number: 043102
• DOI: https://doi.org/10.1063/1.4980086
• Public URL: https://durham-repository.worktribe.com/output/1383985