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A planar strained shear layer is a Burgers vortex sheet. For strain $u_x=-\alpha x$, $u_z=\alpha z$ and transverse velocity $u_y=U\mathrm{erf}(x/{\delta }_s)$, viscous balance gives ${\delta }_s^2=2\nu /\alpha$. At fixed velocity jump its excess viscous dissipation per sheet area scales as $U^2\sqrt{\nu \alpha }$, which vanishes as kinematic viscosity tends to zero.

Coarse-grained energy dissipation is a spatial average of the local rate $2\nu S_{ij}{S}_{ij}$ over a region of size $r$. Its random value ${ϵ}_r$ has mean $ϵ$ in homogeneous turbulence. Refined similarity uses this local averaged viscous dissipation to scale velocity increments.

Dynamic pressure is the bulk kinetic energy per unit volume associated with fluid motion at speed $U$ in a specified reference frame:Here $\rho =m/V$ is mass density. The word "pressure" identifies its units and its use as a pressure scale; $q$ is not an extra isotropic stress to add to the local static pressure in every surface traction. A stationary fluid can have large static pressure with $q=0$. Conversely, knowing $U$ and $\rho$ fixes $q$ without fixing static pressure. The definition remains useful in viscous fluid flow and compressible flow, even when the simplest Bernoulli equation is inapplicable. ^[ref]

For stipulated mass density ${\rho }_{\infty }=1.20\mathrm{kg}{\mathrm{m}}^{-3}$ and reference speed $U_{\infty }=20\mathrm{m}{\mathrm{s}}^{-1}$, $q_{\infty }=240\mathrm{Pa}$. At $40\mathrm{m}{\mathrm{s}}^{-1}$ with the same mass density, $q_{\infty }=960\mathrm{Pa}$. This quadratic scaling makes dynamic pressure a useful normalization for the lift coefficient, drag coefficient and pressure coefficient; it predicts four times a particular force only if its corresponding coefficient remains unchanged.

The connection with stagnation pressure requires hypotheses. For steady flow, constant mass density, inviscid flow, and gravity as the only body force, the Bernoulli equation gives, along one streamline,Here $p$ is static pressure, $g$ is gravitational acceleration, and $z$ is height. Bring that flow to rest without losses at the same height and the limiting static pressure is the stagnation pressure $p_0$:There must be no intervening pump supplying work or other energy input. Constancy along one streamline does not assert equal $B$ on all streamlines; irrotational flow in a connected region supplies that additional conclusion. Changing height requires retaining $\rho gz$. In compressible flow the stopping pressure difference generally differs from $\rho U^2/2$, and viscous dissipation can reduce recovered stagnation pressure. ^[ref]

A Pitot-static tube turns this relation into a measurement: its upstream-facing opening senses stagnation pressure, while separate side openings sample static pressure. Connect these to the positive and negative ports of a differential pressure transducer or manometer. Under the hypotheses above, the measured difference $\Delta p$ givesThus the stipulated $240\mathrm{Pa}$ and $1.20\mathrm{kg}{\mathrm{m}}^{-3}$ imply $20\mathrm{m}{\mathrm{s}}^{-1}$, not an independently measured speed. Align the probe with the incoming flow and keep its passages clear: blockage and pinched tubing invalidate the comparison. At very low speed, the difference can be smaller than instrument error; at high Mach number, compressibility and possibly a shock wave require a different reduction. ^[ref]

A useful control is differential-pressure calibration. For a digital manometer, zeroing establishes the reading at equal port pressures; it does not establish the gain away from zero. The manufacturer's Series 475 Handheld Digital Manometer: Specifications - Installation and Operating Instructions by Dwyer Instruments, Inc. directs the user to vent both ports to the atmosphere before zeroing. Its stated accuracy is a fraction of full scale, not of the reading, and depends on instrument temperature. Select a suitable range and record the instrument variant and units; the manual covers several ranges.^[ref]

An independent physical check can use a liquid-column manometer: apply a stable pressure difference to both instruments and compare the digital reading with the hydrostatic pressure inferred from the vertical level difference. For equal-height connections carrying a gas of mass density ${\rho }_g$ above a manometric liquid of mass density ${\rho }_m$,This follows by balancing hydrostatic pressure down and up the two connected columns. Use vertical height, even if a tube is inclined. Known mass densities, measured height and gravitational acceleration set the comparison scale. Check several differences across the working range and reverse the port connections to check the sign. This is a proposed independent check, not a claim that the cited teaching experiment performed a full instrument calibration. Recheck zero after reconnecting the sealed tubing; a zero adjustment alone cannot detect a blocked passage or an incorrect scale.

A dilute suspension of identical rigid spheres raises the dynamic viscosity by a fraction $5\varphi /2$, where $\varphi$ is the particle volume fraction. At imposed trace-free rate-of-strain tensor $E$, a sphere of radius $a$ adds viscous dissipation $20\pi \mu a^3(E:E)/3$. Multiplication by the number density $3\varphi /(4\pi a^3)$ gives $5\mu \varphi E:E$, which added to $2\mu E:E$ proves the formula. This is the leading noninteracting-sphere term; hydrodynamic interactions enter at higher concentration.

Enstrophy measures squared vorticity: a local density is ${\omega }^2/2$, and its spatial integral or ensemble mean gives a global measure. In homogeneous incompressible flow, viscous dissipation satisfies $ϵ=\nu ⟨{\omega }^2⟩$. Vortex stretching can produce enstrophy in three dimensions; kinematic viscosity destroys it.

The equilibrium range consists of scales below the energy-containing motions whose statistics can adjust rapidly to the local energy supply. It includes the inertial range and dissipative scales. Approximate equilibrium concerns mean transfer and viscous dissipation, not absence of intermittent spatial fluctuations.

The swirl viscous dissipation of a steady Burgers vortex, integrated per unit length and per unit density, is $\nu \int {\omega }_z^{2}dA=\alpha {\Gamma }^2/(8\pi )$. It remains finite as kinematic viscosity tends to zero at fixed strain and circulation. This is excess over the background strain: the uniform strain itself has infinite total viscous dissipation over an infinite cross-section.

At fixed outer boundary velocity, extending the inclusion velocity rigidly through its interior makes the particle flow an admissible competitor for the particle-free Minimum-dissipation theorem for Stokes flow. Its interior rate-of-strain tensor vanishes. For a linear background, the increase in viscous dissipation is force times relative translation, torque times relative rotation, plus the contraction of the particle stresslet tensor with background strain.

The Minimum-dissipation theorem for Stokes flow compares fields with fixed boundary velocities. Introducing a freely moving rigid inclusion can reduce velocity and total power at fixed applied force while increasing the resistance at fixed velocity. Extending the inclusion's rigid velocity through its interior adds zero viscous dissipation and gives a trial field in the domain without that inclusion, proving the fixed-velocity comparison.

Frazil ice consists of small suspended ice crystals formed in supercooled turbulent water. A layer of crystals can damp surface gravity waves through viscous dissipation, even when individual crystals scatter very little wave energy.

Let $\mathcal{R}=k/({\mu }_m{m}_H)$, take constant opacity $\kappa$, and prescribe the dynamic viscosity $\mu =\alpha p_g/\Omega =\alpha \mathcal{R}\rho T/\Omega$ in a Keplerian disk. The pressure supporting the disk may contain both gas pressure and radiation pressure. Combining viscous dissipation and radiative diffusion gives ${\partial }_m^2{T}^4+A\Omega T=0$ in the vertical mass coordinate of a disk, where $A=27\alpha \kappa \mathcal{R}/(16\sigma )$. Set $\zeta =2m/\Sigma$, $T=T_{\ast }t(\zeta )$ and $T_{\ast }^3=A\Omega {\Sigma }^2/4$. With a radiative-zero disk surface, $t$ satisfies $(t^4{)}^{\prime \prime }+t=0$, $t^{\prime }(0)=t(1)=0$. Writing $I_t={\int }_0^{1}td\zeta$, the density-weighted mean kinematic viscosity is $\bar{\nu }=\alpha \mathcal{R}{T}_{\ast }{I}_t/\Omega$, giving the displayed formula. This calculation does not assume gas-pressure domination: the closure depends on gas pressure while hydrostatic support can be mixed.

Let $\mathcal{R}=k/({\mu }_m{m}_p)$ and choose $h^6=3\alpha \kappa {\Sigma }^2{\mathcal{R}}^4/(16\sigma {\Omega }^5)$. Set $z=h\zeta$, $\rho =(\Sigma /h)D$, $p=\Sigma {\Omega }^{2}hP$, $T=({\Omega }^2{h}^2/\mathcal{R})\Theta$ and $F=\alpha \Sigma {\Omega }^3{h}^{2}Q$. Vertical hydrostatic equilibrium, viscous dissipation, radiative diffusion and the ideal gas law reduce to $P^{\prime }=-D\zeta$, $Q^{\prime }=9P/4$, ${\Theta }^{\prime }=-DQ/{\Theta }^3$, $P=D\Theta$. Midplane symmetry gives $Q(0)=0$; a regular radiative-zero surface has $P=\Theta =0$ at a universal ${\zeta }_s$, with $2{\int }_0^{{\zeta }_s}Dd\zeta =1$. All physical parameters have disappeared. Uniqueness of this normalized boundary-value problem makes every physical profile a rescaling of one dimensionless solution.

// Target: astrophysics.bigb

The inertial range lies between the energy-containing and viscous scales. Direct forcing and viscous dissipation are small there, and nonlinear transfer carries an approximately constant mean energy flux in a three-dimensional forward cascade. Scale similarity and universality are additional statistical hypotheses.

Integral-scale intermittency is modulation of the energy-containing motions and the energy supply on large lengths or long turnover times. Averaging velocity statistics over regions or realizations with differing viscous dissipation rates can invalidate universal coefficients based only on the global mean viscous dissipation.

Internal intermittency is the fluctuating concentration of gradients, transfer and viscous dissipation inside turbulent fluid. It is distinct from an interface between turbulent and non-turbulent regions intermittently passing an observer. Its scale dependence is reflected in non-Gaussian increments and coarse-grained viscous dissipation moments.

For a fixed geometry, the steady Stokes equation maps $(\mathbf{u},p^{\prime },\mathbf{f})$ to $(-\mathbf{u},-p^{\prime },-\mathbf{f})$ when all imposed boundary velocities and disturbance forces are reversed; $p^{\prime }$ is pressure relative to any unchanged hydrostatic pressure. Linearity of Stokes flow and Uniqueness of Stokes flow therefore reverse the entire velocity field. If the driving schedule and evolving geometry are retraced, fluid elements retrace their paths too. Reversing just one drive while leaving another applied force unchanged does not meet these conditions.

This reversal concerns the paths, while viscous dissipation remains positive: for a sphere, the required power $6\pi \mu a{U}^2$ is unchanged by $U\mapsto -U$. A reversed experiment must supply fresh energy. Inertia, Brownian motion and irreversible contact can prevent perfect retracing. Edward M. Purcell illustrated the mechanism with a viscous fluid flow demonstration in Life at low Reynolds number, American Journal of Physics 45, 3–11 (1977).

Kolmogorov 1941 theory assumes locally universal small-scale velocity statistics set by mean viscous dissipation and kinematic viscosity at large Reynolds number. In its inertial range dimensional form, $S_p(r)\sim {ϵ}^{p/3}{r}^{p/3}$. Large-scale modulation and internal intermittency require qualifying this unconditional scaling.

In the locally isotropic equilibrium range, the small-scale velocity increment statistics depend on the separation, mean viscous dissipation and kinematic viscosity, independently of large-scale forcing details. Dimensional analysis leaves the Kolmogorov microscales as normalization scales and gives the displayed second-order longitudinal structure function. Universality of $F$ is part of the hypothesis, not something proved by dimensional analysis.

Mixing turbulent populations with differing viscous dissipation gives $S_p={\beta }_p{r}^{p/3}⟨{ϵ}_a^{p/3}⟩$. Relative to the same power of the mean viscous dissipation, the coefficient changes with the intensity distribution except at $p=3$. Thus universality of all unconditional coefficients is incompatible with arbitrary integral-scale intermittency.

For a periodic thin mucus gap $h(x)$ translating as a backward wave relative to a force-free foot, put $I_j={\int }_0^L{h}^{-j}dx$. In the wave frame, the plane and foot have horizontal velocities $V-U$ and $V$. A Couette-Poiseuille flow with periodic pressure has flux $q=(V-U/2)I_2/I_3$. Zero net plane force gives the displayed crawling speed. The leading viscous dissipation per wavelength is $\Phi =2\mu UV{I}_1$.