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Biological fluid dynamics studies fluid flows generated by and acting on organisms, biological cells, tissues and biological transport networks. Examples include blood flow through microcirculation, cytoplasmic streaming within a biological cell, flagellum propulsion and bioconvection in swimming suspensions.

The Reynolds number $\mathrm{Re}=\rho UL/\mu$ compares inertia with viscous forces, using density $\rho$, dynamic viscosity $\mu$, speed $U$ and length $L$. A microscopic swimmer can operate in Stokes flow, where viscous forces dominate and motion responds to the instantaneous force balance. Larger swimming animals can instead transfer appreciable momentum to the surrounding fluid, as modeled by Lighthill elongated-body theory. ^[ref]

Transport introduces another comparison: the Péclet number $\mathrm{Pe}=UL/D$ measures advection against diffusion with diffusion coefficient $D$. Thus the same flow can move slowly enough to have negligible inertia, yet carry a slowly diffusing substance effectively. Choosing the appropriate dimensionless quantities connects biological scale to the governing mechanical and transport models.

At a micrometre scale, viscous forces normally dominate inertia. As an illustrative Newtonian estimate, $L=1\mu \mathrm{m}$, $U=10\mu \mathrm{m}{\mathrm{s}}^{-1}$, $\rho =10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $\eta =10^{-3}\mathrm{Pa}\mathrm{s}$ give a Reynolds number $\rho UL/\eta =10^{-5}$. The surrounding flow is approximately a Stokes flow, satisfying $-\nabla p+\eta {\nabla }^2\mathbf{u}=0$ and $\nabla \cdot \mathbf{u}=0$.

A body of radius $0.5\mu \mathrm{m}$ translated at that speed in an unbounded Newtonian liquid would need Stokes drag $6\pi \eta aU\simeq 9.4\times 10^{-14}\mathrm{N}$. It stops almost immediately when propulsion stops. The scallop theorem rules out propulsion by a single reciprocal stroke in this ideal regime; a rotating helix or a nonreciprocal cycle of shapes can propel it. Brownian motion also randomizes position and orientation: Stokes–Einstein relation scales as $a^{-1}$ and rotational diffusion as $a^{-3}$, so retaining heading becomes harder as size decreases.

The Newtonian calculation is a scale estimate, not a complete model of blood. Vessel walls alter drag, the imposed flow advects the body, and nearby red blood cells make the medium heterogeneous at this scale. Non-Newtonian fluid effects and deformable obstacles can matter; steering must be assessed against those effects rather than inertial coasting.

The Reynolds number compares inertial and viscous effects:where $\rho$ is fluid mass density, $U$ a characteristic swimming speed, $L$ a characteristic body length, and $\eta$ the dynamic viscosity. Both swimmers are in the same fluid, so the fluid properties cancel from the ratio:For an order-of-magnitude estimate, take $L_b=2\mu \mathrm{m}$, $U_b=20\mu \mathrm{m}{\mathrm{s}}^{-1}$, $L_w=10\mathrm{m}$ and $U_w=5\mathrm{m}{\mathrm{s}}^{-1}$. With water values $\rho \simeq 10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $\eta \simeq 10^{-3}\mathrm{Pa}\mathrm{s}$, these giveand henceThe dimensions and swimming speeds are estimates, so this ratio is not a universal species-independent constant. Its physical significance is robust: viscous forces dominate the bacterium's motion, whereas fluid inertia is crucial for the whale.

Resistive-force theory approximates the local viscous force on a moving slender elastic filament by separate drag coefficients along and across its tangent. Let $\mathbf{t}$ be the unit tangent and $\mathbf{v}$ the material velocity relative to the ambient fluid. The force per unit length on the elastic filament isThe positive parallel and perpendicular drag coefficients of a slender filament are defined per unit length; for a sufficiently slender isolated body, their leading logarithmic ratio is ${\xi }_{\perp }/{\xi }_{\parallel }\simeq 2$.

The directional difference can generate propulsion. For example, take a straight element with $\mathbf{t}=(\cos \theta ,\sin \theta )$ and impose transverse velocity $\mathbf{v}=V{\mathbf{e}}_y$. Direct substitution givesA transverse motion of a tilted element therefore has an axial force component. In a propagating flagellum deformation, the correlation of tilt and transverse velocity can produce nonzero mean thrust. For a freely swimming body, the unknown translation and rotation follow by imposing zero total external force and torque, including the resistance of the head if present.

This is a local Stokes flow approximation. It neglects the nonlocal hydrodynamic interactions retained by slender-body theory, so nearby boundary conditions, tightly spaced elastic filaments and detailed end effects can require a more complete model.