Biological Fluid Dynamics: Viscosity, the Navier-Stokes Equation, and Life at Low Reynolds Number

Vocabulary practice flashcards covering fluid continuity, viscosity, Navier-Stokes derivations, Poiseuille flow, low Reynolds number dynamics, Stokes drag, Purcell's scallop theorem, and helical bacterial propulsion.

Continuum Hypothesis (Fluid Dynamics)

The physical idealization describing a fluid as continuous fields of density $\rho(\mathbf{r}, t)$, pressure $p(\mathbf{r}, t)$, and velocity $\mathbf{v}(\mathbf{r}, t)$ by averaging over fluid elements that are large compared to molecular dimensions ($\approx 0.3\,\text{nm}$) but small compared to flow gradients.

Incompressibility Condition

The mathematical requirement $\nabla \cdot \mathbf{v} = 0$, expressing conservation of mass for a fluid with constant density ($\rho \approx 10^3\,\text{kg/m}^3$ for water), dictating that volume inflow equals volume outflow across any closed surface.

Dynamic Viscosity ($\eta$)

The proportionality coefficient relating shear stress to the transverse velocity gradient (shear rate) between sliding fluid layers ($F/A = \eta (v/d)$), with SI units of $\text{Pa}\cdot\text{s}$ ($\approx 10^{-3}\,\text{Pa}\cdot\text{s}$ for water at room temperature).

Newtonian Fluid

A fluid in which the internal viscous shear stress is linearly proportional to the local velocity gradient at all shear rates, exhibiting internal friction only when neighboring layers move at different speeds.

Kinematic Viscosity ($\nu$)

The ratio of dynamic viscosity to fluid density, $\nu = \eta / \rho$, representing the diffusion constant of momentum ($\approx 10^{-6}\,\text{m}^2/\text{s} = 1\,\text{mm}^2/\text{s}$ for water).

Material Time Derivative ($D/Dt$)

The differential operator $\frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla$, representing the total acceleration experienced by an observer moving along with a fluid element, combining local time variation and spatial convection.

Convection (Fluid Dynamics)

The nonlinear acceleration term $(\mathbf{v} \cdot \nabla)\mathbf{v}$ in Newton's second law for fluids, representing the change in velocity that arises when a fluid element traverses regions of varying flow speed even in steady state.

Navier-Stokes Equation

The fundamental momentum equation for an incompressible Newtonian fluid, $\rho \frac{\partial \mathbf{v}}{\partial t} + \rho(\mathbf{v} \cdot \nabla)\mathbf{v} = -\nabla p + \eta \nabla^2 \mathbf{v}$, which balances local inertia and convection with pressure and viscous forces.

Poiseuille Flow

Steady, laminar, pressure-driven flow through a straight, rigid circular pipe under no-slip conditions at the walls, forming a parabolic velocity profile given by $v(r) = \frac{\Delta p}{4\eta l}\left(\frac{d^2}{4} - r^2\right)$.

Hagen-Poiseuille Law

The relation giving total volume flow rate through a circular pipe as $Q = \frac{\pi \Delta p d^4}{128\eta l}$, highlighting a strong fourth-power dependence on vessel diameter $d$.

Reynolds Number ($Re$)

The dimensionless ratio of inertial force to viscous force, $Re = \frac{\rho a v}{\eta} = \frac{a v}{\nu}$, which also equals the ratio of fluid kinetic energy ($KE \sim \rho a^3 v^2$) to work dissipated by viscous forces ($W \sim \eta v a^2$).

Stokes Equation

The linear, time-independent governing equation $\nabla p = \eta \nabla^2 \mathbf{v}$ obtained when $Re \ll 1$, where inertial terms vanish and flow is determined completely by instantaneous boundary conditions without memory.

Pitch-Drop Experiment

A long-term fluid mechanics demonstration begun in 1927 at the University of Queensland exhibiting funnel flow of pitch ($\eta \sim 10^8\,\text{Pa}\cdot\text{s}$, $Re \sim 10^{-19}$), serving as a macroscopic model of overdamped life at low Reynolds number.

Stokes Flow Past a Sphere

Creeping flow ($Re \ll 1$) past a sphere characterized by perfect fore-aft symmetric streamlines with no trailing wake, dragging fluid along with velocity disturbances decaying slowly as $a/r$.

Stokes Drag Force ($F_s$)

The drag force resisting a sphere of radius $R$ moving at constant speed $v$ in an unbounded viscous fluid at low Reynolds number, given by the linear relation $F_s = 6\pi \eta R v$.

Stopping Distance (Purcell's Estimate)

The distance a sphere coasts after propulsion ceases ($v_0 \tau = \frac{2\rho R^2 v_0}{9\eta}$), evaluating to less than a tenth of an ångström ($\approx 0.07\,\text{\AA}$) over $\tau \approx 0.2\,\mu\text{s}$ for an E. coli cell, confirming inertia is negligible.

Viscous Limit (Damped Harmonic Oscillator)

The physical regime $km/\gamma^2 \ll 1$ ($\sqrt{m/k} \gg m/\gamma$), where inertial effects vanish within a rapid transient $\tau = m/\gamma$, causing the position to relax as $x(t) \approx x_0 e^{-(k/\gamma)t}$ on the slow timescale $\gamma/k$.

Spirostomum ambiguum

A large single-celled protozoan ($L_0 = 1.1 \pm 0.2\,\text{mm}$) that swims at $Re_0 \approx 0.2$ but contracts to a fraction $f_c = 0.40 \pm 0.03$ of its length in $\tau_c = 4.64 \pm 0.15\,\text{ms}$, temporarily boosting its Reynolds number to $Re_T \approx 50\text{--}80$ and generating vortices.

Hydrodynamic Trigger Waves

Ultra-fast intercellular communication waves propagating through colonies of Spirostomum ambiguum, mediated by transient vortex flows from rapid cellular contractions that trigger neighboring cells in a fast chain reaction.

Scallop Theorem (Purcell)

The principle stating that at low Reynolds number, reciprocal body deformations that trace the exact same geometric path in reverse produce zero net translation over a complete cycle, regardless of speed or tempo.

Reciprocal Motion

A cyclic body deformation parameterized by a single degree of freedom $\theta(t)$ where $\theta(0) = \theta(T)$, which retraces identical intermediate shapes in reverse and generates zero net motion in Stokes flow.

Purcell's Simple Swimmer

A minimal low-Reynolds-number model swimmer consisting of three rigid links and two hinges ($\theta_1, \theta_2$) that achieves net propulsion by traversing a closed loop of non-zero area in shape space, breaking kinematic time-reversal symmetry.

Anisotropic Drag on a Slender Rod

The property of slender filaments at low Reynolds number wherein the drag coefficient perpendicular to the filament ($\gamma_\perp \approx 4\pi\eta l$) is approximately twice as large as the drag coefficient parallel to its length ($\gamma_\parallel \approx 2\pi\eta l$).

Bacterial Propulsive Thrust ($F_p$)

The axial propulsion force generated by a rotating helical flagellar bundle, $F_p = 2\pi \eta L v \cos(\theta) \sin(\theta)$, which arises directly from the anisotropic drag difference ($\gamma_\perp - \gamma_\parallel \approx 2\pi\eta l$) acting on inclined helical segments.