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A non-Newtonian fluid has a stress--strain-rate relation with nonlinear, time-dependent, history-dependent, or microstructure-dependent behavior.

A Bingham plastic is a yield-stress fluid, and hence a non-Newtonian fluid. It is also an instantaneous generalized-Newtonian model after yielding. It cannot describe memory-dependent viscoelasticity, including stress relaxation, elastic recoil, and time-dependent normal-stress effects.

For scalar shear stress $\tau$ and shear rate $\dot{\gamma }$, its complete ideal constitutive law isEquivalently, in yielded material,

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.

Viscous fluid flow includes the transfer of momentum by internal viscous stresses. For an incompressible Newtonian fluid with constant dynamic viscosity $\mu$, the extra stress is proportional to the rate-of-strain tensor:Combining this relation with momentum conservation gives the Navier-Stokes equation. Other constitutive equations, including those for non-Newtonian fluids, give different stress responses.

A simple example is Couette flow between plates separated by $h$, with one fixed and the other moving at speed $U$. A steady linear profile $u_x=Uy/h$ produces uniform shear stress ${\tau }_{xy}=\mu U/h$. The moving plate supplies power per area $\mu U^2/h$, which becomes viscous dissipation. ^[ref]

The Reynolds number $\mathrm{Re}=\rho UL/\mu$ compares advective momentum transport with viscous transport. Small values motivate Stokes flow; large values can still require thin viscous boundary layers near solid walls. Viscous flow therefore includes both smooth laminar flow and turbulent flow.