Slow viscous flow (MAGIC102)

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The Reynolds number gives the ratio of inertial to viscous effects in a fluid flow. When the Reynolds number is small, inertial effects are negligible and the Du/Dt term in the Navier–Stokes equations may be neglected. This simplifies the Navier-Stokes equations, making them linear and instantaneous. These simplifications make solving low-Reynolds-number flow problems much easier than high Reynolds number flows. This module will consider the circumstances under which the Reynolds number will be small and examine the basic properties of low-Reynolds-number flows. We shall present a number of solution techniques, and show how they can be applied to a range of problems. In the course of this, students will meet various useful applied mathematics methods, including solution by potentials, boundary integral methods, and asymptotic approximations.

Essential

• Vector Calculus (div, grad, curl, line,surface/volume integrals, divergence theorem)
• Differential Equations (methods for first-order ordinary differential equations)
• Basic Fluid Mechanics (introductory course in inviscid fluid mechanics)

Desirable / Complimentary

• Further Fluid mechanics (introductory course in viscous fluid mechanics)
• Tensors and the Einstein Summation Convention (some previous experience useful)
• Non-dimensionalisation / scaling analysis

• Introduction to low-Reynolds-number flow (3 lectures)
  The Stokes equations and boundary conditions. Basic properties, uniqueness theorem, reciprocal theorem, minimum dissipation theorem. Oscillating Couette flow and Poiseuille flow.
• Fundamental solutions and representation by potentials (4 lectures)
  Solution using potentials. Papkovich-Neuber potentials, flow past a rigid sphere. Boundary integrals and the multi-pole expansion.
• Slender-body theory (3 lectures)
  Basic derivation. Applications to sedimenting slender objects and swimming micro-organisms.

No assessment information is available yet.