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A specification of the meaning of a mathematical term or notation. A structural definition specifies conditions, such as the group axioms; a definition by a universal property characterizes an object by its arrows. Such a characterization does not by itself prove that the object exists. For example, existence of an exponential object is an additional property of a category.

In Receptor sensitivity in bacterial chemotaxis by Victor Sourjik and others, Victor Sourjik and Howard C. Berg used a cyan fluorescent tag on CheZ as FRET donor and a yellow tag on CheY as FRET acceptor. Escherichia coli biological cells adhered to a polylysine-coated coverslip in a continuously perfused chamber. A fluorescence microscope excited them with a $442$-nanometre laser; split emission bands fed two photon-counting detectors. The yellow/cyan emission ratio fell when attractant was added and rose when it was removed.^[ref]

Saturating attractant increased cyan emission by an amount matching selective bleaching of the yellow tag: removing the FRET acceptor supported relief of FRET donor transfer. Strains carrying the tagged pair but lacking other chemotaxis machinery showed no stimulus-dependent emission ratio change. Weak illumination reduced bleaching drift.^[ref]

The inference is a change in the population of nearby tagged CheY/CheZ pairs associated with CheY phosphorylation. It neither measures each pair's separation nor directly observes motor motion. The readout also averages many biological cells, and perfusion takes finite time. The pathway interpretation belongs to the chemotaxis FRET assay; the general measurement lesson is that molecular association can alter FRET while the geometry of an individual associated pair stays the same.

In FRET photobleaching controls, selectively destroy the FRET acceptor's ability to absorb and test whether FRET donor fluorescence increases. Under unchanged excitation and detection, complete loss of transfer, and an intact FRET donor,This is a destructive comparison. Monitor acceptor loss, donor-only samples exposed to the bleaching illumination, and an unbleached reference region or time trace. Incomplete bleaching, donor damage, movement or label replenishment violate the comparison. During ordinary recordings, gradual unequal bleaching can instead produce an apparent change of FRET without a biological change.^[ref]

FRET sensitized emission is FRET acceptor fluorescence caused by transfer from an excited FRET donor. Excite predominantly the FRET donor and collect both emission bands. Reduced FRET donor emission and increased FRET acceptor emission are useful together, but an acceptor-to-donor emission ratio is not automatically the FRET efficiency: brightness, label abundance and channel collection also enter.^[ref]

FRET efficiency is the probability that an excitation of the FRET donor ends in transfer to the FRET acceptor. Write $k_D=k_{\mathrm{r}}+k_{\mathrm{nr}}$ for the total decay rate without a FRET acceptor, including radiative fluorescence and other decay, and $k_T$ for the transfer rate. For constant competing rates,The integral counts transfer events among all excitations in a population with exponential decay. Substitution gives the separation law above. This short rate argument assumes one FRET donor, one FRET acceptor, weak coupling, an available FRET acceptor, and fixed geometry during the decay; it is not a model for arbitrary interacting labels.^[ref]

A FRET acceptor receives the excited state energy. Its absorption must support FRET spectral overlap with the FRET donor. FRET sensitized emission requires a fluorescent FRET acceptor; measuring a shortened FRET donor lifetime does not.

A FRET donor supplies the initial excited state energy. Its fluorescence provides both an emission channel and a reference FRET donor lifetime when no FRET acceptor is present. Calling a label a FRET donor describes its role in a chosen pair, rather than a permanent classification of the molecule.

A wake is the disturbed fluid region trailing a body relative to the surrounding stream. It can contain a velocity deficit, rotation, shed vortices and mixing. Surface viscosity and flow separation help produce it, and its structure depends on geometry and Reynolds number.

In a steady towing experiment, the body's drag transfers momentum to the fluid. Comparing upstream and downstream momentum transport therefore provides a way to infer that force when the control-surface stresses and pressure terms are also accounted for. A visible dye streak only identifies part of this field; it does not by itself measure the entire momentum budget. A steady mean wake can coexist with unsteady vortex shedding, so instantaneous and averaged measurements reveal different features. Farther downstream, mixing spreads the disturbance and changes its width and peak deficit.

A plume is a localized buoyancy-driven flow generated by a source of heat or another density-changing substance. A plume can be laminar or turbulent; the transport regime is an additional condition.

A particle suspension is a dispersed population of solid particles carried by a fluid. Its dynamics couple particle transport, drag force, gravity, and the compensating motion of the carrier fluid.

Drag is the component of fluid force opposing an object's motion relative to the surrounding fluid. Surface pressure and viscous shear stress both contribute; their relative importance depends on shape and flow regime. A body can therefore experience drag even if most resistance comes from an uneven pressure field rather than direct tangential rubbing.

At low Reynolds number, the Stokes drag law gives $F=6\pi \mu aU$ for a sphere under its creeping-flow assumptions. In many inertial regimes, a useful representation is $F=C_D\rho A{U}^2/2$, with a measured or modeled drag coefficient $C_D$. That coefficient can vary with Reynolds number, surface condition and geometry. Towing a body while measuring force at several speeds tests which regime applies. The supplied towing power is $FU$ for steady motion along the drag direction.

Mass density is mass per unit volume, $\rho =m/V$ for a uniform sample, with SI units $\mathrm{kg}{\mathrm{m}}^{-3}$. For a nonuniform continuum, integrating the local field gives the total mass, $M={\int }_V\rho (\mathbf{x})dV$. The averaging volume must be specified when unresolved grains, pores or multiple phases are present.

For a fixed mass, expansion increases volume and lowers mass density. A material element obeying mass conservation satisfies $D\rho /Dt=-\rho \nabla \cdot \mathbf{v}$, linking that change to the velocity field. Density enters both weight and inertia, but it is different from number density and from relative density. Measuring a sample's mass and external volume can give a different value from measuring the density of its solid grains alone if the sample volume includes void space.

A buoyant plume is a localized rising or sinking flow driven by a density difference from its surroundings. Heating fluid can lower its mass density and produce upward buoyancy; a denser injected fluid can produce a descending counterpart. The sign of the density anomaly and the direction of gravity specify the motion being described.

As a plume travels, fluid entrainment incorporates ambient fluid, increasing its volume flux and diluting the source anomaly. A warm narrow stream can therefore carry a growing amount of air while its mean excess temperature decreases. The buoyancy flux has a different balance from the volume transport, especially when the ambient fluid is stratified or heat is lost. Dye or temperature measurements across successive heights reveal widening and dilution; their interpretation requires the cross-sectional velocity as well as anomaly size to determine a transported flux.

A fluid is a material that flows under sustained shear stress and cannot maintain a static shear stress in its fluid state. Liquids and gases are the usual examples. A Newtonian fluid has deviatoric stress proportional to its rate-of-strain tensor, whereas more general fluid constitutive equations can be nonlinear or have memory. In a porous medium, the fluid occupies connected voids within a load-bearing solid framework. The important distinction is the response to a continuing tangential force: an ordinary elastic solid can settle into a fixed deformation, while an ordinary fluid continues to deform. A fluid can nevertheless be at rest when no unbalanced shear stress drives it. ^[ref]

For a worked example, place a Newtonian fluid between parallel plates separated by $h=1\mathrm{mm}$, keep the lower plate fixed and move the upper one at $U=0.10\mathrm{m}{\mathrm{s}}^{-1}$. With no-slip boundary conditions, steady flow, no imposed pressure gradient and negligible edge effects, Couette flow has $u(y)=Uy/h$. For dynamic viscosity $\mu =0.001\mathrm{Pa}\mathrm{s}$, the shear stress magnitude isThe velocity gradient describes neighbouring layers sliding at different speeds. Doubling $U$ doubles the required shear stress; maintaining the same velocity keeps producing deformation, rather than fixing a final shape. This gives viscosity a concrete meaning: resistance to a rate of motion, not a prohibition on motion.

Physiological fluid dynamics studies the motion of fluids in biological systems, including blood flow in elastic arteries, flow through deformable vessels, and liquid layers lining airways. Wall compliance couples pressure to geometry, so wave propagation and mechanical feedback are as important as viscosity.

Hydraulic resistance relates the pressure drop across a resistive fluid element to its volumetric flow rate. In a linear model, $\Delta p=RQ$ with $R>0$, so dissipated mechanical power is $\Delta pQ=R{Q}^2$. A peripheral circulatory bed can be represented by such a resistance when its detailed dynamics are not retained. This terminal pressure-to-flow resistance differs from the area-dependent friction rate multiplying velocity in a distributed tube momentum equation.