# Hauptseminar Active Matter SS 2017/Hydrodynamics of Newtonian Fluids

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Date
2017-04-11
Time
14:00
Topic
Hydrodynamics of Newtonian Fluids
Speaker
Cameron Stewart
Tutor
Paolo Malgaretti

## Contents

Swimmers, as their name suggests, operate within liquids. Therefore hydrodynamics is essential for understanding how swimmers move, how they interact with other swimmers or confining boundaries, or how they respond to external flows.

This topic aims at providing a concise overview of hydrodynamics of Newtonian viscous liquids, with a focus on the case of motion at low Reynolds numbers (creeping flow) . The starting point is the derivation of the continuity and the Navier-Stokes (NS) equations, which provide the most general description of a fluid in motion. This is followed by the introduction of the notion of a Newtonian fluid and of an incompressible flow [1,2]. Boundary conditions appropriate for fluid-solid interfaces and fluid-fluid interfaces are discussed. Finally, definitions are introduced for energy dissipation in flowing Newtonian fluid, and for force and torque acting on a body in an incompressible fluid .

A case of particular significance is that of axisymmetric flows, which leads to the notion of stream function. Application of this concept can considerably simplify the analytical approach to a hydrodynamics problem by reducing it to solving for a single scalar field, the stream function . The effects of thermal fluctuations on the fluid flow and the so-called "linear fluctuating hydrodynamics" are briefly introduced .

Introducing the Reynolds number to obtain the dimensionless NS equations, the (steady-state) Stokes equations, which are most relevant for micro-swimmers, are obtained in the limit of very small (vanishing) Reynolds numbers [1-3]. In analogy with the case of electrostatics or diffusion, the steady state Stokes equations, which are linear, admit fundamental solutions (Green's function), often denoted also as "singularities", describing the flows induced by point sources. A few of them (the point force and force-doublet, the source and source doublet) are derived [4,5].

## Literature

1. J. Happel and H. Brenner, Low Reynolds number hydrodynamics (Noordhoff Int. Pub., Leyden, The Netherlands, 1973), Ch. 1, 3.5, 4.1-4.7, 6.1-6.5, 7.
2. R.G. Winkler, Low Reynolds number hydrodynamics and mesoscale simulations, Eur. Phys. J. ST 225, 2079 (2016).
3. E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys. 72, 096601 (2009).
4. J.R. Blake and A.T. Chwang, Fundamental singularities of viscous flow. Part I, J. Eng. Math. 8, 23 (1974).