Similarity Solutions for the Two-Dimensional Diffusion–Advection Equation in Sharp-Corner Geometries

Problems involving corner geometries arise frequently in science and engineering, including in fluid mechanics. Low-Reynolds-number flow patterns near sharp corners, including Moffatt eddies, have been studied extensively in the past. Here, we focus on the effect of Moffatt eddies on concentration profiles near sharp corners. The Lie symmetry method is applied to the 2D diffusion-advection equation to obtain a similarity ansatz, reducing the equation to a time-independent second-order PDE. By modifying the ansatz, the problem can be recast as an eigenvalue problem. Exploiting elliptic regularity and the strong maximum principle, we invoke the Krein-Rutman theorem to establish the existence and domain-size monotonicity of the principal eigenvalue. We then investigate how varying domain length and corner angle affect the magnitude of the principal eigenvalue, observing an inverse relationship in both cases. The behavior of the eigenfunction is sensitive to small changes in the domain shape and size, so we truncate the domain along separating streamlines. This study adds to our understanding of the low-Reynolds-number transport dynamics near sharp corners from a pattern formation perspective, and can be extended to other systems.

Name thesis:degree_name

Master of Science

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Engineering Mechanics

Department dc:contributor.department

Mechanical Engineering

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2025

Author dc:creator

• Khattab, Mohamed Osama Mahmoud

Chair dc:contributor.committeechair

• Stremler, Mark A.

Committee members dc:contributor.committeemember

• Staples, Anne E.
• Paul, Mark R.

• Lie Symmetry
• Diffusion-advection
• Krein-Rutman theorem
• Similarity solutions
• Moffatt eddies

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/10919/137640

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/137640