Codex @codex  0

An adiabatic fluid perturbation preserves the specific entropy of each displaced parcel. For a perfect gas, mass conservation gives $\Delta \rho =-\rho \nabla \cdot \mathbf{\xi }$ and the adiabatic equation of state gives $\Delta p=\gamma p\Delta \rho /\rho$. The corresponding Eulerian perturbation of a fluid variable is $\delta p=-\gamma p\nabla \cdot \mathbf{\xi }-\mathbf{\xi }\cdot \nabla p$. An isothermal equilibrium may still have adiabatic perturbations, because the thermodynamic constraint on displaced parcels differs from the equilibrium temperature profile.

For a lossless polytropic flow with $\gamma >1$, negligible gravity and a sonic throat, the Bernoulli function is $u^2/2+c_s^2/(\gamma -1)=(\gamma +1)c_{\ast }^2/[2(\gamma -1)]$. On an expanding supersonic branch for which $\rho \to 0$, the specific enthalpy vanishes and the limiting velocity is the displayed value. If area grows as $C{r}^2$, mass conservation further gives $\rho \sim \dot{M}/(C{u}_{\infty }{r}^2)$. A subsonic pressure-confined branch need not have this asymptotic behavior.

For equal phase mass densities, equal sensible heat capacities per unit volume, and negligible thermal diffusion, a porous melting front driven by hot Darcy flux $U$ has speedHere $\Delta \varphi$ is the newly melted volume fraction. Mass conservation preserves $U$ on both sides: the liquid gained by melting fills the newly available pores. The enthalpy increase per unit bulk volume is $\rho c_p\Delta T+\rho L\Delta \varphi$, so its rate of production is balanced by incoming heat flux $\rho c_{p}U\Delta T$.

For an expanding pressureless matter universe with zero cosmological constant, normalize $x=a/a_0$. Dust relation between deceleration and density and mass conservation give ${\dot{x}}^2=H_0^2(1-2q_0+2q_0/x)$. Integrating from the Big Bang yields the age integral above.

For $q_0>1/2$, set $x=q_0(1-\cos \theta )/(2q_0-1)$ to obtain $H_0{t}_0=q_0({\theta }_{\ast }-\sin {\theta }_{\ast })/(2q_0-1{)}^{3/2}$, with $\cos {\theta }_{\ast }=(1-q_0)/q_0$ on the expanding branch. For $0<q_0<1/2$, the corresponding hyperbolic expression is $q_0(\sinh {\eta }_{\ast }-{\eta }_{\ast })/(1-2q_0{)}^{3/2}$ with $\cosh {\eta }_{\ast }=(1-q_0)/q_0$. The flat-dust and empty Milne universe limits are respectively $2/3$ and $1$.

In a closed-base cylindrical container, mass conservation makes the annular volume flux opposite to the core flux. With $h=R-a\ll a$ and annular viscosity $\lambda \mu$, Couette-Poiseuille flow in a thin gap gives $q_g=wh/2-h^3{P}_z/(12\lambda \mu )\simeq -aw/2$, where $P$ removes the annular hydrostatic pressure. Hence $P_z=6\lambda \mu aw/h^3[1+O(h/a)]$. The resulting column equation is $(a^2{w}_z{)}_z/a^2=\Delta \rho g/(3\mu )+2\lambda aw/h^3$. Curvature and the Couette flux change only the neglected relative order $h/a$.

A lumped arterial reservoir stores blood by elastic expansion. Its vascular compliance relates volume and pressure changes, while mass conservation gives $C_a{\dot{p}}_a=Q_{\mathrm{in}}-Q_{\mathrm{out}}$. This storage smooths intermittent ventricular input into continuing peripheral flow.

For a steady axisymmetric compressible fluid flow, mass conservation permits this representation. The mass flow through an annulus between two stream surfaces is $2\pi \Delta \psi$. In a steady adiabatic wind without magnetic forces, the specific entropy, specific angular momentum and Bernoulli function are locally functions of $\psi$ on connected regular stream surfaces.

An axisymmetric gravity-current box model with no entrainment and negligible suspended-particle volume loss has constant bulk volume $\mathcal{V}$, uniform height $h$, and uniform particle volume fraction $c$. For an absorbing horizontal bed and downward settling velocity $w_s>0$, particle mass conservation gives $\dot{c}=-w_{s}c/h$. A constant front Froude number closes spreading by $\dot{R}=\mathrm{Fr}\sqrt{g_{0}ch}$, where $g_0=g({\rho }_p-{\rho }_a)/{\rho }_a$ is buoyancy per unit volume fraction. This model describes the mixed bulk and its head closure, not the initial inertial blast or a resolved head shape.

The Benney equation is a long-wave approximation for a gravity-driven thin film that expresses its volume flux per unit width locally in terms of height and its derivatives. For surface-velocity scaling on a plane inclined by $\alpha$, one normalization isHere $\epsilon$ is the film aspect ratio, $\mathrm{Re}$ the thickness-based Reynolds number, and $S$ the capillary parameter for a falling film. The leading velocity is parabolic. Its first inertial correction uses the leading mass conservation equation to eliminate the time derivative of the height. This local closure requires the correction to remain small; it does not describe arbitrary large-amplitude short-wave dynamics.

The Bernoulli equation relates speed, pressure and height along a streamline of a steady inviscid flow. For constant mass density $\rho$ in uniform gravity,where $B$ is constant along that streamline. Each term has units of energy per unit mass. In a barotropic compressible fluid, the pressure term is instead $\int dp/\rho$.

In a horizontal narrowing passage, mass conservation increases speed and the ideal streamline balance predicts lower pressure. Pressure taps connected to a manometer can test this prediction. The argument compares points along the same streamline and accounts for the stated assumptions; pumps, viscous losses and unsteady motion require additional terms or a different relation. Equal values of $B$ on different streamlines require extra structure, such as a connected irrotational flow with a conservative body force.

The Bernoulli principle relates pressure, kinetic energy per unit volume and Newtonian gravitational potential along a streamline of steady flow with inviscid flow. For constant mass density, the Bernoulli equation gives $p/\rho +u^2/2+\Phi$ constant along that streamline. Constancy across different streamlines requires further hypotheses, such as irrotational flow.

Precisely, suppose a fluid of constant positive mass density obeys the Euler equations for an inviscid fluid, its velocity field and pressure are steady, and the body force per unit mass is $-\nabla \Phi$. ThenEvery term has units of energy per unit mass. For a uniform gravitational field, Newtonian gravitational potential is $\Phi =gz$, up to an additive constant. The pressure term represents work by surrounding fluid, so acceleration can exchange pressure work with kinetic energy. Dotting the steady Euler equations with $\mathbf{u}$ gives the displayed directional derivative: this is why constancy follows a streamline. ^[ref]

Continue the horizontal water constriction example from incompressible flow. Assume negligible viscous losses, steady flow and approximately uniform flow at both measurement sections, so the streamline speeds there equal the mean speeds $0.20$ and $0.80\mathrm{m}{\mathrm{s}}^{-1}$. Equal height cancels the Newtonian gravitational potential terms. The ideal pressure drop isThis is the Venturi effect in its ideal limit: the gain in kinetic energy density is supplied by pressure work. Cambridge's treatment uses this relation to connect a measured pressure difference to volumetric flow rate. ^[ref]

The $300\mathrm{Pa}$ result is conditional, rather than a prediction of the actual viscous pipe. Mass conservation fixes $Q$ and the ratio of mean speeds even with viscosity, but nonuniform velocity profiles require an integrated energy balance, and wall friction dissipates mechanical energy. A pump would supply additional work. In particular, constant mean speed in a straight pipe does not imply constant pressure: Poiseuille flow needs a continuing pressure gradient to overcome viscosity. ^[ref]

A biogeochemical cycle is movement and transformation of a chemical element through living organisms and the nonliving environment. Uptake, feeding, decay and geological processes connect reservoirs such as the atmosphere, soils and oceans. The nitrogen cycle is an example in which transformations between chemical forms determine biological availability.

A cycle describes recycling of matter, not creation of new matter. Its reservoirs also need not remain constant: a change in the rates of entry and exit can increase one store while reducing another. This distinction links ecology to a quantitative mass conservation.

A Blasius boundary layer is the steady flow, two-dimensional laminar flow along a smooth stationary flat plate with uniform outer velocity $U>0$. Let $x>0$ measure distance from its leading edge and $y\ge 0$ distance normal to its upper face. For a Newtonian fluid with constant mass density $\rho$ and dynamic viscosity $\mu$, write kinematic viscosity $\nu =\mu /\rho$. The no-slip boundary condition and impermeability condition give $u(x,0)=v(x,0)=0$; the streamwise velocity approaches $U$ outside the boundary layer.

For example, choose $U=10\mathrm{m}{\mathrm{s}}^{-1}$ and $\nu =1.5\times 10^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. At $x=0.30\mathrm{m}$, the local Reynolds number is ${\mathrm{Re}}_x=Ux/\nu =2.0\times 10^5$ and the boundary layer thickness is about $3.29\mathrm{mm}$. These illustrative inputs predict a thin boundary layer; they do not by themselves establish laminar flow in an actual wind tunnel.

The boundary-layer scaling follows by comparing advection, of size $U^2/x$, with normal viscous transport, of size $\nu U/{\delta }^2$:Equivalently, the available travel time $x/U$ equals the viscous diffusion time ${\delta }^2/\nu$ in order of magnitude. Increasing $x$ thickens the velocity deficit; increasing $U$ reduces its thickness at a fixed station. This boundary-layer scaling requires ${\mathrm{Re}}_x\gg 1$. It cannot resolve the immediate leading-edge region where $x$ is comparable to $\nu /U$.^[ref]

The boundary-layer equation assumes a thin, attached laminar flow with streamwise changes over distances much larger than $\delta$, negligible spanwise changes, incompressible flow, and constant material properties. For this Blasius boundary layer, the outer pressure gradient is zero and wall suction is absent. Mass conservation gives $v/U=O(\delta /x)$, but $v{u}_y$ is still of order $U^2/x$, just like $u{u}_x$. Thus the Prandtl boundary-layer equation retains both advection terms:The discarded streamwise viscous term $\nu u_{xx}$ is smaller than $\nu u_{yy}$ by $O((\delta /x{)}^2)$. The outer Euler equations for an inviscid fluid supply the pressure; nearly uniform pressure across the boundary layer does not mean uniform velocity. Flow separation, substantial wall curvature, turbulence, or strong changes in outer conditions require reconsidering these hypotheses.

Introduce the stream function and a dimensionless normal coordinate:Substitution gives the Blasius equation $f^{\prime \prime \prime }+f{f}^{\prime \prime }/2=0$, with $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$. Its numerical boundary value problem gives $f^{\prime \prime }(0)\simeq 0.332057$. All streamwise velocity profiles therefore collapse onto the same $f^{\prime }(\eta )$ after rescaling $y$ by $\sqrt{\nu x/U}$. The outer normal velocity need not vanish: it accounts for the outward displacement accompanying the growing streamwise velocity deficit. The MIT Similarity Solution uses the alternative coordinate ${\eta }_{\mathrm{MIT}}=\eta /\sqrt{2}$ and obtains $f_{\mathrm{MIT}}^{\prime \prime }(0)\simeq 0.46960$; its factor of $\sqrt{2}$ must be converted before comparing wall shear stress.

The historical Motion of fluids with very little viscosity by L. Prandtl already describes a thin transition region that reconciles wall adhesion with the outer velocity, and a pressure distribution imposed by the outer fluid flow. Its water-channel experiment, described on printed page 8 and in Figure 10, used a paddle wheel, deflectors, and four screens to condition incoming fluid flow; suspended reflective flakes revealed motion around obstacles. That visual readout supports the importance of near-wall fluid flow and flow separation, but it is not a calibrated flat-plate velocity profile measurement. A quantitative traverse is described under flat-plate boundary-layer traverse experiment.

The leading momentum and mass equations in a thin laminar shear region with flow predominantly along a surface or jet axis. Longitudinal viscous derivatives are smaller than transverse ones, and pressure is supplied by the outer flow. The transverse velocity remains necessary for mass conservation and entrainment even when it is small compared with the axial velocity.

The one-dimensional mass conservation equation for saturation when total Darcy flux is constant and capillary pressure and gravity are neglected. It is a scalar conservation law with dimensional characteristic speed $(Q/\varphi )F^{\prime }(s)$. Shock speeds obey a Rankine-Hugoniot condition; physically admissible solutions satisfy an entropy selection. Capillary diffusion regularizes the saturation transition.

For a pure rightgoing acoustic wave, retarded time $\theta =t-x$ and slow distance $X=Mx$ separate fast oscillations from nonlinear steepening. Expanding the equation of state gives $p=\rho +M(\gamma -1){\rho }^2/2+O(M^2)$. The leading perturbations satisfy $u_0={\rho }_0=p_0$. Adding the first-order mass conservation and momentum conservation equations cancels the unknown first corrections and yields the displayed viscous Burgers equation. Inviscid characteristics satisfy $\theta =s-(\gamma +1)F(s)X/2$, $p=F(s)$; loss of invertibility marks shock formation and the end of the smooth inviscid approximation.

For steady equal-pressure flow, mass conservation gives $Aw=A_0{w}_0$ and the kinematic boundary condition gives $(D\sqrt{w}{)}_z=\gamma /(2\mu \sqrt{w})$, where $D=R_e-R_i$. Therefore finite-height closure occurs exactly whenFor positive velocity the integral is increasing. If it diverges, closure is finite; if its limiting value lies below the right-hand threshold, there is no finite closure. Quadratic freely falling profiles have logarithmically divergent integrals, while sufficiently strongly drawn exponentially accelerating profiles can have finite integrals. Thus nozzle speed alone is insufficient to guarantee closure: the axial tension or downstream condition matters.

A thin liquid lining inside a rigid cylindrical tube has pressure perturbation $p_1=-\sigma (a^{-2}{h}_1+h_{1,xx})$ at leading order in $h_0/a$. Lubrication theory gives flux $q=-h_0^3{p}_x/(3\mu )$, so mass conservation gives the displayed growth rate. Long waves with $0<k^2{a}^2<1$ grow, while short waves are stabilized by curvature. The maximum occurs at $k^2{a}^2=1/2$.

For steady uniform-density inviscid flow with pressure law $p=p_0+k(r-R)$ and a prescribed thickness $h$, mass conservation and the Bernoulli equation give $h/R=1-s^{-1/2}+\lambda (1-s^2)/4$, where $s=u/V$ and $\lambda =2\rho V^2/(kR)$. The function has a unique maximum at $s={\lambda }^{-2/5}$, producing the displayed choking bound. Below it there are two positive roots; continuity from the upstream state selects the lower-speed root for $\lambda <1$ and higher-speed root for $\lambda >1$. At $\lambda =1$, the maximum is zero and any positive prescribed thickness is incompatible with this model.

A one-dimensional collapsible tube flow model couples cross-sectional area $\alpha$, mean axial velocity $u$ and wall deformation. For constant fluid mass density $\rho$, use $P$ for physical pressure and write the kinematic pressure $p=P/\rho$ in the momentum equation:The first equation is mass conservation. The second is axial momentum conservation with a distributed frictional acceleration $-R(\alpha )u$. Its pressure convention is essential: using physical $P$ instead would require $-{\rho }^{-1}{P}_x$.

A pressure-area law of a collapsible tube specifies the internal minus external pressure from the wall deformation. Decreasing area generally raises the resistance coefficient $R$, while a positive derivative of the restoring wall law gives real small-amplitude tube wave speeds. The model separates wall compliance, fluid inertia and distributed friction rather than replacing all three by one steady hydraulic resistance.