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The age is the proper cosmic time elapsed from the scale-factor singularity $a=0$ to the present epoch.

The age of the universe is the cosmological quantity; an FLRW model computes it by integrating proper cosmic time from its initial scale-factor singularity.

Energy conservation in FLRW gives $\dot{\rho }=-3H(\rho +P)$ when $\rho$ and $P$ are energy density and pressure.

The FLRW metric is the homogeneous and isotropic spacetime metric with scale factor $a(t)$ and constant spatial curvature $k$.

Luminosity distance is defined by $F=L/(4\pi d_L^2)$ and includes geometric dilution, photon redshift, and arrival-rate dilation.

A fluid with $w<-1$ grows in density during expansion and can produce a finite-time Big Rip singularity.

For pressureless subhorizon matter perturbations, a Fourier mode in conformal time obeyswhere $\mathcal{H}=a^{\prime }/a$.

These equations describe the small-amplitude phase of structure formation; nonlinear growth requires a more general treatment.

Cosmic inflation is an early period of accelerated expansion, often modeled by a scalar field whose potential energy dominates its kinetic energy.

A Big Crunch is a future finite-time singularity in which a recollapsing universe reaches vanishing scale factor.

Cosmological radiation has equation of state $P_r={\rho }_r/3$ and density scaling ${\rho }_r\propto a^{-4}$.

In a decelerating hot Big Bang model, widely separated regions of the observable universe have nearly equal conditions despite having no shared past light cone. Inflation supplies an earlier era with a shrinking comoving Hubble radius, allowing a formerly causal patch to grow beyond the later horizon.

At Hubble parameter $H$, the critical energy density is

This is the density threshold in the Friedmann equation for spatial flatness; thermodynamic and plasma critical densities are different concepts.

For a homogeneous fluid of energy density $\rho$, pressure $P$, and scale factor $a(t)$, adiabatic work in a comoving volume gives

A transfer function packages the response of a linear time-invariant system into a rule for turning a plant input into a plant output. Instead of solving a differential equation separately for every signal, take a Laplace transform, multiply by $G(s)$, then transform back. This also makes it possible to compare slow and fast oscillations through the continuous-time frequency response.

Here consider a continuous-time, causal, finite-dimensional linear time-invariant system with one plant input $u(t)$ and one plant output $y(t)$. With every initial state set to zero, its transfer function is defined byThe ratio $G(s)=Y(s)/U(s)$ is used where $U(s)\ne 0$ in a common Laplace convergence region; $G$ describes the system, independently of that particular plant input. Zero initial state matters: stored motion or another nonzero initial condition can contribute to the plant output even when the plant input vanishes. ^[ref]

The first-order low-pass filter below shows the same transfer function predicting both a step and a sustained sinusoid. Its pole gives a decay rate, its value at zero gives the eventual gain, and its values along the imaginary axis give the continuous-time frequency response. These descriptions make a useful bridge between a physical relaxation model and control theory calculations.

A Burgers vortex balances axial vortex stretching against radial viscous diffusion, producing a Gaussian vorticity profile.

For harmonic acoustic amplitudes, $⟨\mathbf{I}⟩=\frac{1}{2}\mathrm{Re}(p{\mathbf{u}}^{\ast })$.

The instantaneous acoustic energy flux is $p\mathbf{u}$ for linear acoustics. Its time average is the sound intensity.

For $U=A{x}^m$, $\psi =\sqrt{\nu xU}f(\eta )$ giveswith $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$.

This equation and its boundary conditions determine the Falkner-Skan boundary layer.

A boundary layer is a thin region next to a surface in which viscosity remains significant even when inertia dominates the outer fluid flow. Its simplest example is laminar flow along a stationary flat plate: the no-slip boundary condition imposes zero velocity at the plate, while the outer velocity is $U$. The resulting thin velocity profile contains a large normal velocity gradient, so small dynamic viscosity can still produce appreciable wall shear stress. The Blasius boundary layer makes this competition quantitative and explains how a plate aligned with the incoming fluid flow acquires drag.

The same balance appears in a singular perturbation of a differential equation: large derivatives in a thin region can keep a term with a small coefficient significant. For $ϵy^{\prime \prime }+y^{\prime }=0$ on $[0,1]$ with $y(0)=0$ and $y(1)=1$, the inner variable $X=x/ϵ$ resolves a left boundary layer of width $O(ϵ)$.

The Rayleigh-Taylor instability occurs when a denser fluid lies above a lighter fluid in a gravitational field, causing interface perturbations to grow.

The material derivative follows a moving continuum parcel:

Mass conservation states that matter is neither created nor destroyed. For mass density $\rho$ and velocity $\mathbf{v}$, its local form is ${\partial }_t\rho +\nabla \cdot (\rho \mathbf{v})=0$.