Fluid Dynamics

Derivation. Derives Definition 31.4. Let \(\vect{x}(t)\) be the path of a particle, so that \(\dd\vect{x}/\dd t=\vect{v}(\vect{x}(t),t)\). The chain rule in several variables (Proposition 7.104) gives

which is Equation (31.3). The two terms are physically distinct and it is worth naming them: \(\pp_{t}f\) is the change an instrument at a fixed point records, and \((\vect{v}\cdot\vect{\nabla})f\) is the change a particle experiences merely by being carried into a region where \(f\) is different. A steady flow, \(\pp_{t}f=0\) everywhere, can therefore accelerate a particle violently — which is what happens in the Venturi throat of Section 31.3.2, and the reason the acceleration in the equations below is not \(\pp_{t}\vect{v}\).

Derives Proposition 31.7. When \(\vect{v}\) does not depend on \(t\), the two ordinary differential equations of Definition 31.6 are the same autonomous system, and by uniqueness of its solutions (Theorem 9.8) the curve through a given point is the same in both cases. Every particle passing through a fixed point therefore follows the identical curve, so the streakline is that curve too. In unsteady flow the streamline pattern changes while a particle is traversing it and the three families separate: this is why a smoke photograph of an unsteady flow is not a picture of its instantaneous streamlines, and why the dye filament of Section 36.3 is read as a streakline.

Derives Proposition 31.9. Taylor's theorem gives \(\delta v_{i}=\left(\pp_{j}v_{i}\right)\delta x_{j}\) to first order. Split the gradient into symmetric and antisymmetric parts, \(\pp_{j}v_{i}=e_{ij}+\Omega_{ij}\) with \(\Omega_{ij}=\tfrac{1}{2}\left(\pp_{j}v_{i}-\pp_{i}v_{j}\right)\). Now

so that \(\Omega_{ij}\delta x_{j} =-\tfrac{1}{2}\epsilon_{ijk}\omega_{k}\delta x_{j} =\tfrac{1}{2}\epsilon_{ikj}\omega_{k}\delta x_{j} =\tfrac{1}{2}\left(\vect{\omega}\times\delta\vect{x}\right)_{i}\), which is Equation (31.8). The factor of one half is the whole content of the statement that vorticity is twice the local angular velocity, and it is a standing source of confusion: a fluid in rigid rotation at angular velocity \(\vect{\Omega}\) (Rigid Bodies and Rotating Frames) has \(\vect{\omega}=2\vect{\Omega}\), not \(\vect{\Omega}\).

Derives Lemma 31.11. In a time \(\dd t\) the boundary element \(\dd S\) with outward unit normal \(\vect{n}\) sweeps the volume \(\left(\vect{v}\cdot\vect{n}\right) \dd S\,\dd t\), so the integral changes for two reasons and only two: the integrand changes at fixed points, and the domain moves. Hence

and the divergence theorem (Theorem 7.133) turns the surface term into \(\int_{V}\vect{\nabla}\cdot(f\vect{v})\,\dd V\). Expanding \(\vect{\nabla}\cdot(f\vect{v}) =\vect{v}\cdot\vect{\nabla}f+f\,\vect{\nabla}\cdot\vect{v}\) and using Equation (31.3) gives the second form. The first term of the first form requires differentiating an integral with respect to a parameter under a fixed domain, which is elementary here because \(f\) is \(C^{1}\) and \(V\) is bounded.

Derives Theorem 31.12. The mass of a material volume is constant, since the volume contains the same particles at all times and mass is neither created nor destroyed. Setting \(f=\rho\) in Equation (31.9) and equating the result to zero,

for every material volume. A continuous function whose integral vanishes over every subvolume vanishes identically — if it were positive at a point it would be positive on a ball about that point, by continuity, and the integral over that ball would not vanish — which is Equation (31.10). The second form follows from the first by Equation (31.3). First written as a field equation by Euler [Euler:1757].

Derives Corollary 31.13. Immediate from the second form of Equation (31.10), since \(\rho>0\). Note what the statement does not say: it does not say the density is uniform. Salt water stratified by depth satisfies Equation (31.11) exactly while \(\rho\) varies from place to place, and that is precisely the configuration of the Rayleigh–Taylor and Kelvin–Helmholtz problems of Section 31.8.1.

Derives Proposition 31.16. By Definition 31.1 the traction on any internal surface of a fluid at rest is \(-p\vect{n}\). The force on a volume \(V\) is therefore \(-\oint_{\pp V}p\vect{n}\,\dd S=-\int_{V}\vect{\nabla}p\,\dd V\) by the divergence theorem (Theorem 7.133), and the body force is \(\int_{V}\rho\vect{g}\,\dd V\). Equilibrium requires the sum to vanish for every \(V\), hence Equation (31.13) pointwise. Taking the curl of Equation (31.13) with \(\vect{g}=-g\hat{\vect{z}}\) constant gives \(\vect{\nabla}\rho\times\vect{g}=\vect{0}\), so \(\vect{\nabla}\rho\) is vertical: a fluid at rest in gravity is necessarily stratified horizontally, which is why a tilted density interface cannot be at rest.

Derivation. Derives Phenomenon 31.17. Let the applied pressure change the boundary value from \(p\) to \(p+\Delta p\) with the fluid remaining at rest and its density unchanged. Both states satisfy Equation (31.13) with the same right-hand side, so the difference field satisfies \(\vect{\nabla}\left(\Delta p\right)=\vect{0}\) and \(\Delta p\) is the same constant throughout a connected body of liquid — which is Pascal's statement. The force on a piston of area \(A\) is then \(F=\Delta p\,A\) at each piston, giving Equation (31.14). Conservation of volume for an incompressible liquid, \(A_{1}d_{1}=A_{2}d_{2}\) for the displacements \(d_{i}\), gives \(F_{1}d_{1}=F_{2}d_{2}\): the press multiplies force and divides distance, and creates no energy. The idealization is the constancy of \(\rho\); a real liquid transmits the change at the sound speed of Section 31.10.1, not instantaneously.

Derivation. Derives Phenomenon 31.18. A fluid at rest carries no shear stress, so the traction on any internal surface is the normal pressure alone and the balance of forces on a fluid element reads \(\vect{\nabla}p=\rho_{\mathrm{f}}\vect{g}\). The force exerted on the immersed body is the integral of that pressure over its wetted surface \(S\), directed inwards along the outward normal \(\vect{n}\):

the second step by the divergence theorem (Theorem 7.133). The manoeuvre deserves a word: the pressure field is defined only outside the body, and what the divergence theorem is applied to is its hydrostatic continuation into \(V\). That is legitimate because the surface integral depends only on the boundary values, which are unchanged if the body is replaced by the same volume of fluid — and that replacement is in equilibrium, which is Archimedes' own argument. Taking moments the same way places the resultant at the centroid of \(V\), the centre of buoyancy. Since \(\vect{F}_{\mathrm{b}}\) involves the body only through \(V\), two bodies of equal volume are buoyed equally however they differ in material, and a body of mean density \(\rho_{\mathrm{b}}\) floats with the fraction \(\rho_{\mathrm{b}}/\rho_{\mathrm{f}}\) of its volume submerged.

Derivation. Derives Phenomenon 31.20. A curved interface supports a pressure difference, the Young–Laplace relation [Young:1805],

the pressure being higher on the concave side; \(R_{1}\) and \(R_{2}\) are the principal radii of curvature of the surface (Section 13.3.3). In a tube of radius \(r\) wetted at contact angle \(\theta\) the meniscus is a spherical cap of radius \(R=r/\cos\theta\), so both principal radii equal \(R\) and the pressure immediately below the meniscus is lower than the atmospheric pressure above it by \(2\gamma\cos\theta/r\). Hydrostatic balance (Equation (31.13)) along the liquid from that point down to the level of the flat outer surface, where the pressure is atmospheric, requires the meniscus to stand a height \(h\) above it with \(\rho g h=2\gamma\cos\theta/r\), which is Equation (31.17). Two checks are worth making. The rise is not a force exerted by the tube on the liquid — it is the pressure deficit under a curved surface — so it does not depend on the thickness or material of the wall except through \(\theta\). And the \(1/r\) dependence is the measured signature: a set of tubes of different bores gives a straight line through the origin when \(h\) is plotted against \(1/r\), from whose slope \(\gamma\) is obtained, which is the classical capillary measurement of surface tension.

Derives Theorem 31.22. Apply Lemma 31.11 to the momentum of a material volume, \(f=\rho v_{i}\), and use Equation (31.10):

the cross terms cancelling by Equation (31.10) exactly. The rate of change of momentum equals the applied force, which for an ideal fluid is the pressure traction plus the body force, \(-\oint_{\pp V}p n_{i}\,\dd S+\int_{V}\rho g_{i}\,\dd V =\int_{V}\left(-\pp_{i}p+\rho g_{i}\right)\dd V\) by Theorem 7.133. Equating the two for every material volume, and arguing from continuity as in Theorem 31.12, gives Equation (31.19). The result is Cauchy's momentum equation (Section 30.3.2) with the constitutive relation of Definition 31.21 inserted, so nothing new about mechanics has been assumed: the whole content of Equation (31.19) is Definition 31.21.

Derives Theorem 31.24. Divide Equation (31.19) by \(\rho\) and use Equation (31.4) on the advective term together with \(\vect{\nabla}p/\rho=\vect{\nabla}P\), valid exactly when \(\rho=\rho(p)\):

For steady flow the first term vanishes and \(\vect{\nabla}B=\vect{v}\times\vect{\omega}\). Contract with \(\vect{v}\): the right-hand side is orthogonal to \(\vect{v}\), so \(\vect{v}\cdot\vect{\nabla}B=0\), and \(B\) is constant along the integral curves of \(\vect{v}\) — the streamlines (Definition 31.6). If moreover \(\vect{\omega}=\vect{0}\) then \(\vect{\nabla}B=\vect{0}\) outright and \(B\) is one constant for the whole (connected) fluid. Finally, if \(\vect{v}=\vect{\nabla}\phi\) then \(\pp_{t}\vect{v}=\vect{\nabla}\pp_{t}\phi\) and \(\vect{v}\times\vect{\omega}=\vect{0}\), so Equation (31.22) says that the gradient of the left-hand side of Equation (31.21) vanishes, which is the claim. Note which hypothesis does which work: barotropy is what makes \(\vect{\nabla}p/\rho\) a gradient, steadiness is what removes \(\pp_{t}\vect{v}\), and irrotationality is what upgrades “along a streamline” to “everywhere”. Dropping any one of them leaves a weaker but still usable statement, and dropping viscosity is what restricts every application to regions outside boundary layers and wakes.

Derivation. Derives Phenomenon 31.25. For steady flow of a fluid of constant density in a uniform gravitational field the Euler equation reads \(\rho(\vect{v}\cdot\vect{\nabla})\vect{v} =-\vect{\nabla}p-\rho g\vect{\nabla}z\). Use the identity Equation (31.4) and divide by \(\rho\):

Project onto the local flow direction. The right-hand side is orthogonal to \(\vect{v}\) and its projection vanishes identically, so the directional derivative of the bracket along a streamline is zero and the bracket is constant on each streamline — which is Equation (31.23), the constant-density case of Equation (31.20). Two riders follow at once. If the flow is in addition irrotational, \(\vect{\nabla}\times\vect{v}=0\) everywhere and the constant is the same on every streamline. And at a stagnation point, where \(v=0\) at the same height, the pressure exceeds the free-stream value by exactly \(\tfrac{1}{2}\rho v^{2}\): that difference is what a Pitot tube reads [Pitot:1732]. The derivation used the inviscid equation, so the relation holds only where viscous stresses are negligible along the streamline followed — outside boundary layers and away from wakes.

Derives Proposition 31.28. Irrotationality is \(\vect{\nabla}\times\vect{v}=\vect{0}\); on a simply connected region Proposition 7.136 then supplies a potential \(\phi\) with \(\vect{v}=\vect{\nabla}\phi\). Substituting into Equation (31.11) gives \(\vect{\nabla}\cdot\vect{\nabla}\phi=\nabla^{2}\phi=0\), which is Equation (31.25), an elliptic equation with the full apparatus of Section 10.4.3 available: uniqueness up to a constant for the Neumann problem, the maximum principle (Theorem 10.77) applied to each velocity component, and separation of variables in the coordinate systems of Theorem 10.34. Ellipticity has a physical reading worth stating: the potential at each point depends on the boundary data everywhere, instantaneously, so an incompressible ideal fluid transmits the influence of a moving boundary with infinite speed. That is a property of the incompressible idealization, not of nature, and it is repaired by the finite sound speed of Section 31.10.1.

Derivation. Derives Phenomenon 31.30. Take the sphere of Example 31.29. The drag is the component of the pressure resultant along the stream,

and by Equation (31.28) the integrand is a combination of \(\cos\theta\sin\theta\) and \(\cos\theta\sin^{3}\theta\), both odd about \(\theta=\pi/2\) and both integrating to zero over \(\left(0,\pi\right)\). Hence \(F_{z}=0\) exactly. The mechanism is visible in Equation (31.28): the pressure distribution is symmetric fore and aft, so the high pressure at the rear stagnation point pushes the sphere forward exactly as hard as the high pressure at the front pushes it back. The general statement, for a body of arbitrary shape in three dimensions, follows from the same fore-and-aft structure of the far field and is left to the appendix.

Derivation. Derives Phenomenon 31.34. The derivation is molecular and belongs to Kinetic Theory of Gases, where it is carried out; the one-line reason is that momentum is carried across a shear layer by molecules crossing it, at a rate proportional to the number density \(n\), each carrying the momentum difference accumulated over one mean free path \(\lambda\propto1/n\), so that \(\mu\sim n\lambda\times(\text{molecular momentum})\) is independent of \(n\). What belongs here is the consequence: the continuum theory of this chapter takes \(\mu\) as an empirical coefficient and can neither predict this behaviour nor be embarrassed by it. That is the honest division of labour between Fluid Dynamics and Kinetic Theory of Gases, and it is why the value of \(\mu\) appears in Table 31.1 as a measurement.

Derives Theorem 31.36. By (i), \(\sigma_{ij}+p\delta_{ij}=A_{ijkl}\pp_{k}v_{l}\) for a constant fourth-rank tensor \(A\). By (ii) that tensor is isotropic, and every isotropic fourth-rank tensor in three dimensions is a combination of the three available products of Kronecker deltas,

a representation this chapter takes from the algebra of Cartesian tensors (Section 13.2.2). Symmetry of \(\sigma\) in \((i,j)\) forces \(\mu_{1}=\mu_{2}=:\mu\), and then

which involves only the symmetric part \(e_{ij}\) — so requirement (iv) is automatically met, since the antisymmetric part of the velocity gradient, which by Proposition 31.9 is the rigid rotation, has dropped out. Writing \(e_{ij}=\left(e_{ij}-\tfrac{1}{3}e_{kk}\delta_{ij}\right) +\tfrac{1}{3}e_{kk}\delta_{ij}\) and collecting the isotropic pieces gives Equation (31.30) with \(\zeta=\lambda+\tfrac{2}{3}\mu\). The split is the useful one because the first viscous term is trace free and responds only to shape change, while \(\zeta\) responds only to volume change and therefore cannot be measured in any incompressible flow.

Derives Theorem 31.37. Cauchy's momentum equation (Section 30.3.2), derived exactly as in Theorem 31.22 but keeping the full traction, is \(\rho\,Dv_{i}/Dt=\pp_{j}\sigma_{ij}+\rho g_{i}\). Insert Equation (31.30) with \(\mu\) and \(\zeta\) constant:

which collects to Equation (31.31). Under Equation (31.11) every term carrying \(\vect{\nabla}\cdot\vect{v}\) vanishes and division by \(\rho\) gives Equation (31.32). The bulk viscosity has disappeared, which is why it plays no part anywhere in this chapter except in the attenuation of sound.

The equations are second order in space, so they carry one more boundary condition than Equation (31.19) (Remark 31.23). That the extra condition is no slip rather than some finite slip is not a theorem: it is a physical hypothesis, and the one Navier's own derivation from a molecular model [Navier:1823] left open. It was settled by measurement, and the measurement is Poiseuille's — Section 36.2 turns the accuracy of Equation (31.35) into a bound on the slip length of a few tens of nanometres.

Derives Proposition 31.39. With \(\vect{v}=v(y)\hat{\vect{x}}\) the advective term vanishes, the flow is divergence free, and the steady \(x\) component of Equation (31.32) without a pressure gradient reduces to \(\mu\,\dd^{2}v/\dd y^{2}=0\). Hence \(v\) is linear in \(y\); no slip at both plates fixes \(v(0)=0\) and \(v(h)=U\), giving Equation (31.33). The stress follows from Equation (31.29). Two features are worth recording: the profile is independent of \(\mu\), so a Couette measurement determines \(\mu\) from the force and not from the shape of the profile, and the velocity gradient is uniform, so this is the one flow in which the constitutive relation Equation (31.29) is tested at a single, controlled shear rate. Couette's rotating-cylinder viscometer [Couette:1890] is the practical realization, with the small gap between the cylinders playing the part of \(h\).

Derives Proposition 31.40. Seek \(\vect{v}=v_{\varphi}(r)\hat{\vect{\varphi}}\). The flow is divergence free, and the advective term contributes only the centripetal \(-v_{\varphi}^{2}/r\) in the radial direction, which is balanced by the radial pressure gradient and does not enter the azimuthal equation. That equation is

whose general solution is \(v_{\varphi}=Ar+B/r\); the two no-slip conditions \(v_{\varphi}(R_{i})=\Omega_{i}R_{i}\) give the constants in Equation (31.34). The term \(Ar\) is rigid rotation and carries no shear; the shear stress is \(\sigma_{r\varphi}=\mu r\,\dd\left(v_{\varphi}/r\right)/\dd r =-2\mu B/r^{2}\), so the torque per unit length on a cylindrical surface of radius \(r\) is \(2\pi r^{2}\times2\mu B/r^{2}=4\pi\mu B\), the same at every radius — as it must be, since no torque accumulates in a steady state. This is the working equation of the Couette viscometer [Couette:1890]. The solution loses stability when the inner cylinder rotates fast enough, in the way computed and measured by Taylor [Taylor:1923] and described in Section 31.8.2.

Derivation. Derives Phenomenon 31.41. Seek a steady, axisymmetric, unidirectional solution \(\vect{v}=v(r)\,\hat{\vect{z}}\) of the incompressible Navier–Stokes equations in a tube of radius \(R\). Such a field is divergence-free automatically, and its inertial term vanishes identically: \((\vect{v}\cdot\vect{\nabla})\vect{v}=v\,\pp_{z}\bigl(v\, \hat{\vect{z}}\bigr)=0\), so the nonlinearity plays no part and the solution below is exact rather than approximate. What remains, using the cylindrical Laplacian Equation (7.130), is the axial component

in which the first term is independent of \(r\) and the second independent of \(z\), so both are the same constant: \(\dd p/\dd z=-\Delta p/L\). Integrating once and discarding the \(\ln r\) solution as singular on the axis, then integrating again and imposing the no-slip condition \(v(R)=0\) at the wall,

the parabolic profile. The flux is

which is Equation (31.35). The fourth power is the signature: two of its powers come from the cross-sectional area and two from the mean speed, which itself grows as \(R^{2}\) because the wall is the only brake on the flow. The agreement of the measured flux with Equation (31.35), with no fitted constant beyond the viscosity, is the quantitative laboratory test of the no-slip condition: a wall at which the liquid retained a finite tangential speed would deliver a larger flux than this, by the factor computed in Equation (36.8).

Derives Proposition 31.44. Equation (31.38) is linear and homogeneous in \(\left(\vect{v},p\right)\), so multiplying a solution by \(-1\) gives a solution, and the boundary data are multiplied by \(-1\) with it; uniqueness of the Stokes problem then identifies it as the solution. Nothing of the sort holds for Equation (31.32), whose advective term is quadratic and changes sign under \(\vect{v}\longmapsto-\vect{v}\) in a way the other terms do not. The consequence is observable and is the reason a swimming micro-organism cannot propel itself by any reciprocal stroke: a sequence of shapes retraced in reverse produces no net displacement whatever, however fast it is performed, because the speed cancels out of Equation (31.38) entirely.

Derivation. Derives Phenomenon 31.45. What can be settled in a few lines is the form of Equation (31.39), and it is worth separating that from the numerical factor because the form is what carries the physics. Equation (31.38) is linear in \(\vect{v}\) and the boundary data are linear in \(U\), so the solution, and with it the traction integrated over the sphere, is exactly proportional to \(U\): the drag is linear in the speed, not quadratic, and this is a theorem rather than a small-amplitude approximation. The quantities available are then \(\mu\), \(a\) and \(U\), whose dimensions \(\mathsf{M}\mathsf{L}^{-1}\mathsf{T}^{-1}\), \(\mathsf{L}\) and \(\mathsf{L}\mathsf{T}^{-1}\) admit exactly one combination with the dimensions of force (Theorem D.2), namely \(\mu aU\); note that \(\rho\) cannot appear, because it has been removed from the equations along with the inertial terms. Hence \(F=C\mu aU\) with \(C\) a pure number. Solving Equation (31.38) in full for the sphere and integrating the traction gives \(C=6\pi\) [Stokes:1851]; that calculation is longer than a page and is carried out in the appendix.

Derivation. Derives Phenomenon 31.49. Let \(L\) and \(U\) be the size of the body and the free-stream speed, and measure lengths, velocities, times and pressures in the units \(L\), \(U\), \(L/U\) and \(\rho U^{2}\), writing the resulting dimensionless variables with primes. The incompressible Navier–Stokes system

becomes, after dividing throughout by \(U^{2}/L\),

From Equations (31.32) and (31.42) (measuring lengths in \(L\), speeds in \(U\), times in \(L/U\) and pressures in \(\rho U^{2}\), then dividing throughout by \(U^{2}/L\)). with \(\mathrm{Re}\) as in Equation (31.42) the only parameter left. Geometric similarity of the bodies makes the dimensionless boundary conditions identical as well. Two flows agreeing in \(\mathrm{Re}\) therefore solve the same dimensionless problem, and if that problem has a unique solution their primed fields coincide; every dimensionless quantity built from them — the drag coefficient, the pressure coefficient, the Strouhal number of any shedding — is a function of \(\mathrm{Re}\) alone. The same count follows from the \(\Pi\) theorem (Theorem D.2): five dimensional quantities \(\rho,U,L,\mu\) and the force, in three independent dimensions, leave \(5-3=2\) dimensionless groups, one of which may be taken as the drag coefficient and the other as \(\mathrm{Re}\) [Buckingham:1914].

The uniqueness clause is load-bearing and is exactly where the argument stops: two flows at the same \(\mathrm{Re}\) solve the same problem, but if that problem admits more than one solution — as pipe flow above \(\mathrm{Re}\approx2000\) manifestly does — similarity says nothing about which one is realized.

Derivation. Derives Phenomenon 31.53. Three statements are contained in the phenomenon and they have three quite different standings. Separating them is the substance of this derivation, because two of the three follow and the third does not.

That the criterion, if there is one, is a condition on a pure number. A straight circular pipe fixes the geometry completely, so by Phenomenon 31.49 the dimensionless problem Equation (31.43) carries the single parameter \(\mathrm{Re}=\rho\bar{v}D/\mu\) and nothing else. Any criterion separating the two observed states is therefore a condition on \(\mathrm{Re}\) alone, and not on the diameter, the speed, the density or the viscosity separately. That is a prediction and it is the one Reynolds tested: he varied the diameter and, by warming the water, the viscosity, and located the change of state at one value of the combination rather than at one value of any factor in it [Reynolds:1883]. What the same argument declines to supply is the number. The uniqueness clause in the proof of Phenomenon 31.49 is exactly what fails here — above the threshold the dimensionless problem possesses two attracting states, and similarity says nothing whatever about which of them is realized.

That the threshold is not the onset of a linear instability. The laminar profile Equation (31.36) is parabolic, so its curvature keeps one sign across the whole section and it possesses no inflexion point. For the plane analogue Theorem 31.77 then forbids inviscid instability outright, and for the circular pipe the computed viscous spectrum is damped at every Reynolds number [Avila:2011]: the laminar solution does not lose stability at \(\mathrm{Re}\approx2000\), or anywhere else. Whatever the transition is, it is not a bifurcation of the laminar state, and no linearization about that state can produce the observed threshold. This is the exact complement of Phenomenon 31.78, where the same programme delivers a threshold and a length scale with no adjustable parameter.

That the resistance rises as the square of the flux. The momentum balance on the column of fluid between two sections a distance \(L\) apart is exact and assumes no model of the motion: the pressure force on the two faces is carried entirely by the shear on the wall, so

Below the threshold Equation (31.35) gives \(\Delta p/L=32\mu\bar{v}/D^{2}\) and hence \(\tau_{\mathrm{w}}=8\mu\bar{v}/D\), linear in the speed because the transverse transport of streamwise momentum is molecular. Above it that transport is carried instead by the Reynolds stress \(-\rho\avg{v_{i}'v_{j}'}\) of Equation (31.70), in which the viscosity does not appear at all. Grant that the wall stress becomes independent of \(\mu\) and the quantities left are \(\rho\), \(\bar{v}\) and \(D\), from which Theorem D.2 admits exactly one form, \(\tau_{\mathrm{w}}=\tfrac{1}{2}f\rho\bar{v}^{2}\) with \(f\) a pure number; then Equation (31.49) and \(Q=\pi D^{2}\bar{v}/4\) give

The exponent moves from one to two because the mechanism of transverse momentum transport moves from molecular to inertial, and that is all this argument establishes. It is not exact, and the discrepancy is measurable: \(f\) is not quite constant but falls slowly as \(\mathrm{Re}\) grows in a smooth pipe, so the observed exponent over Reynolds's own range is close to \(1.72\) rather than \(2\) [Reynolds:1883], and approaches \(2\) only when the wall is rough enough for the roughness height to replace the viscous scale.

What is not derived is the threshold itself, and Remark 31.54 states what stands in its place.

Derivation. Derives Phenomenon 31.55. Three separate statements are contained in the phenomenon, and they have very different standing.

That the ideal model gives zero is Phenomenon 31.30, proved above for the sphere. That the real drag is not zero, and does not tend to zero as \(\mu\longrightarrow0\), follows from the boundary-layer scaling of Section 31.7.2: the viscous stress at the wall is of order \(\mu U/\delta\) with \(\delta\sim L\,\mathrm{Re}^{-1/2}\), so it falls only as \(\mathrm{Re}^{-1/2}\) while the area over which it acts is fixed, and the separated wake contributes a pressure defect that does not fall at all. The limit of vanishing viscosity is not uniform, which is why setting \(\mu=0\) in the equation gives the wrong answer for any \(\mu\) however small.

That the drag is measurable from the wake alone is the quantitative statement, and it follows from momentum conservation. Take a control volume bounded by a large cylinder around the body, with plane faces far upstream and far downstream where the pressure has returned to \(p_{\infty}\). The flux of \(x\) momentum into the volume through the upstream face is \(\rho U^{2}A\); out through the downstream face it is \(\rho\int u^{2}\,\dd A\); mass conservation requires the deficit \(\int\left(U-u\right)\dd A\) to leave through the side, carrying \(x\) momentum \(\rho U\int\left(U-u\right)\dd A\) with it. The net force on the body is the imbalance,

From Equations (31.10) and (31.19) (balancing the flux of streamwise momentum through a large control surface, the mass deficit leaving through its sides carrying momentum \(\rho U\) per unit volume). the integral running over the wake, where alone \(u\neq U\). Two things follow: the drag is exactly the momentum deficit the body leaves behind, so a body with no wake has no drag — which is Phenomenon 31.30 again, read backwards — and the drag can be measured by traversing a Pitot tube across the wake, without touching the body at all.

What is not derived here is the size of the wake, which requires the boundary layer to be followed to its separation point, and with it both the value of the plateau coefficient and the location and depth of the drag crisis.

Derives Theorem 31.58. Write Equation (31.32) using Equation (31.4) and \(\vect{g}=-\vect{\nabla}\Phi\):

Take the curl. Every gradient dies by Equation (7.167), so the pressure and the body force disappear entirely — the first thing to notice about vorticity is that pressure cannot create it. Expanding \(\vect{\nabla}\times\left(\vect{v}\times\vect{\omega}\right)\) by Equation (31.5) and using \(\vect{\nabla}\cdot\vect{\omega}=0\) (a curl is solenoidal, Equation (31.6)) together with \(\vect{\nabla}\cdot\vect{v}=0\),

which rearranges to Equation (31.52). For non-uniform density the pressure term is \(-\vect{\nabla}\times\left(\vect{\nabla}p/\rho\right) =-\vect{\nabla}\left(1/\rho\right)\times\vect{\nabla}p =\left(\vect{\nabla}\rho\times\vect{\nabla}p\right)/\rho^{2}\), as stated.

The two terms on the right of Equation (31.52) are the whole dynamics of the subject. The first, \(\left(\vect{\omega}\cdot\vect{\nabla}\right)\vect{v}\), is vortex stretching: a vortex line lying along a direction in which the fluid is being stretched has its vorticity amplified in proportion, because angular momentum is conserved while the moment of inertia falls. It is absent in any flow independent of one Cartesian direction, which is why such flows cannot produce the cascade of Section 31.9.2 and why turbulence is an intrinsically three-dimensional phenomenon. The second, \(\nu\nabla^{2} \vect{\omega}\), diffuses vorticity, with \(\nu\) as the diffusivity — which is the reading of the kinematic viscosity promised in Remark 31.33.

Derives Theorem 31.59. (1) \(\vect{\nabla}\cdot\vect{\omega}=0\) identically by Equation (31.6). Apply the divergence theorem (Theorem 7.133) to the piece of tube between two cross-sections: the side wall is made of vortex lines and carries no flux, so the two end fluxes are equal. If a tube ended in the interior its strength would have to fall to zero there, contradicting that equality.

(2) and (3) follow from Kelvin's theorem (Theorem 31.64), which is proved below independently. Take a material circuit lying in the surface of a vortex tube and not girdling it; the circulation around it is zero at the initial instant, hence zero for all time, so the circuit continues to lie in a surface of vortex lines — the tube surface is a material surface, and since this holds for every tube of every thickness the vortex lines themselves are material. Take next a material circuit girdling the tube once: its circulation is the strength, and Kelvin's theorem says the strength does not change. The three statements are Helmholtz's [Helmholtz:1858], and they are the reason a vortex ring is a persistent object: it cannot end, it cannot be cut, and its strength cannot decay, until viscosity is restored.

Derives Theorem 31.64. Let \(C(t)\) be a material circuit and parametrize it by a label \(s\) carried with the fluid, so that differentiating under the integral sign is legitimate:

The second integral is elementary: a material line element is stretched by the velocity difference across it, so \(D\left(\dd\vect{x}\right)/Dt=\dd\vect{v}\), and \(\oint\vect{v}\cdot\dd\vect{v}=\oint\dd\left(\tfrac{1}{2}v^{2}\right) =0\) because the circuit is closed. In the first integral substitute Equation (31.19) divided by \(\rho\): with \(\rho=\rho(p)\) the pressure term is \(-\vect{\nabla}P\) and with conservative forces the body force is \(-\vect{\nabla}\Phi\), so the integrand is an exact differential and its integral around a closed circuit vanishes by Equation (7.166). Hence Equation (31.56) [Thomson:1869].

Derives Corollary 31.65. By Theorem 7.132 the circulation around every reducible circuit in the region is zero initially, hence by Theorem 31.64 zero at all times, hence the vorticity flux through every small surface vanishes and \(\vect{\omega}=\vect{0}\). This is what justifies the potential-flow model of Section 31.3.3 for a body set in motion from rest in still fluid — and, read the other way, it is exactly the difficulty of the lift problem, since it forbids the circulation of Section 31.7.1 from ever appearing. The escape is that the theorem is about a material region, and the circuit that would have to be used around an aerofoil sweeps through the boundary layer and the wake, where the hypotheses fail.

Derives Lemma 31.67. For a counterclockwise contour the outward normal is \(\left(\dd y,-\dd x\right)/\abs{\dd z}\), so the pressure force on an element is \(\dd X=-p\,\dd y\) and \(\dd Y=p\,\dd x\), whence

By Equation (31.23) at constant height, \(p=B-\tfrac{1}{2}\rho q^{2}\) with \(q^{2}=\left(\dd w/\dd z\right)\overline{\left(\dd w/\dd z\right)}\); the constant \(B\) contributes \(-\ii B\oint\overline{\dd z}=0\) around a closed contour. Hence

The body surface is a streamline, so \(\psi\) is constant along \(C\) and \(\dd w=\dd\phi\) is real there: \(\overline{\dd w}=\dd w =\left(\dd w/\dd z\right)\dd z\). Substituting gives Equation (31.58). The integrand is holomorphic everywhere in the fluid, so by the deformation theorem (Corollary 8.15) the contour may be pushed out to any circuit enclosing the body — which is what makes the formula usable, since the far field is known even when the body's shape is not.

Derives Theorem 31.68. Far from the body the complex velocity is holomorphic in an annulus and has a Laurent expansion (Theorem 8.21); the constant term is the free stream and the \(1/z\) coefficient is fixed by the circulation, since \(\oint\left(\dd w/\dd z\right)\dd z=\Gamma+\ii Q\) with \(Q\) the volume flux out of the contour, zero for a solid body. Hence

Only the \(1/z\) term survives the contour integral, by the residue theorem (Theorem 8.24): \(\oint\left(\dd w/\dd z\right)^{2}\dd z =2\pi\ii\times\left(-\ii U\Gamma/\pi\right)=2U\Gamma\). Substituting in Equation (31.58),

which is Equation (31.59). The sign is worth checking physically rather than trusting: a counterclockwise circulation added to a rightward stream speeds the flow up underneath the body and slows it on top, so by Phenomenon 31.25 the pressure is lower underneath and the force is downward — which is what \(Y=-\rho U\Gamma<0\) says for \(\Gamma>0\). That \(X=0\) is Phenomenon 31.30 once more: circulation buys lift and not one newton of drag.

Derives Theorem 31.71. Within the layer the streamwise velocity rises from zero at the wall to \(U\) over the transverse distance \(\delta\), so transverse gradients exceed streamwise ones by the factor \(L/\delta\gg1\) and the dominant viscous term in Equation (31.32) is \(\nu\,\pp_{y}^{2}u\sim\nu U/\delta^{2}\). The streamwise inertia is \(u\,\pp_{x}u\sim U^{2}/L\). A layer in which viscosity acts at all is one where these balance, \(\nu U/\delta^{2}\sim U^{2}/L\), giving Equation (31.61). The same statement read as a diffusion problem is more transparent: momentum diffuses away from the wall a distance \(\sqrt{\nu t}\) in time \(t\), and the time available is the transit time \(L/U\), so \(\delta\sim\sqrt{\nu L/U}\), which is Equation (31.61) again — and is the estimate used for the entry length in Remark 31.42.

The scaling settles Phenomenon 31.30 as a matter of principle. The layer disappears as \(\mu\longrightarrow0\), which is why the outer potential solution describes the observed streamlines ever better; but the velocity gradient across it grows as \(U/\delta\sim\mathrm{Re}^{1/2}\), so the surface stress falls only as \(\mathrm{Re}^{-1/2}\) over a fixed area, and the separated wake it permits contributes a pressure drag that does not fall at all. A limit of that kind cannot be taken by setting the small parameter to zero in the equation, which is exactly what d'Alembert's argument does. Prandtl's water channel made the layer visible and, by removing it through a suction slit, showed that the whole flow around the body is controlled by it (Section 36.5).

Derives Proposition 31.73. Evaluate the streamwise momentum equation of the layer, \(\rho\left(u\pp_{x}u+v\pp_{y}u\right) =-\dd p/\dd x+\mu\pp_{y}^{2}u\), at the wall, where no slip makes \(u=v=0\): the entire left-hand side vanishes and Equation (31.62) remains. If \(\dd p/\dd x<0\) the profile is concave towards the wall throughout and \(\pp_{y}u\) cannot vanish inside the layer, so the flow cannot reverse; if \(\dd p/\dd x>0\) the profile has an inflexion and reversal becomes possible. The criterion is necessary and not sufficient — how far into a rising pressure the layer can go before separating depends on its history — but its necessity already explains the observations of Section 36.5: separation occurs on the rear of a bluff body and in a diverging channel, both of them regions of rising pressure, and never on the accelerating forward face.

The drag crisis follows from the same criterion applied to a layer that has itself become turbulent. A turbulent layer carries fast-moving fluid closer to the wall, so it withstands more adverse pressure before reversing, separates further back, leaves a narrower wake and by Equation (31.51) a smaller drag. That is why roughening a sphere can reduce its resistance [Eiffel:1912], and it is Prandtl's explanation of Eiffel's measurement.

Derives Theorem 31.74. Each layer is irrotational in the undisturbed state and remains so by Corollary 31.65, so write \(\vect{v}_{j}=U_{j}\hat{\vect{x}} +\vect{\nabla}\phi_{j}\) with \(\phi_{j}\) harmonic (Proposition 31.28) and small. Separable harmonic solutions periodic in \(x\) with wavenumber \(k\) and satisfying \(\pp_{y}\phi=0\) on the rigid boundaries are

and let the interface be \(y=\eta=HE\). Write \(s_{j}:=\ii\left(kU_{j}-\omega\right)\), so that \(\left(\pp_{t}+U_{j}\pp_{x}\right)\) acting on anything proportional to \(E\) multiplies it by \(s_{j}\).

Kinematic condition. A particle on the interface stays on it, so to first order \(\left(\pp_{t}+U_{j}\pp_{x}\right)\eta =\pp_{y}\phi_{j}\) at \(y=0\) for each fluid. This gives \(s_{1}H=-Ak\sinh\left(kh_{1}\right)\) and \(s_{2}H=Bk\sinh\left(kh_{2}\right)\), whence

Dynamic condition. The unsteady Bernoulli relation Equation (31.21) in each layer, linearized about the uniform stream, gives the pressure at the interface as \(p_{j}=-\rho_{j}\left(\pp_{t}+U_{j}\pp_{x}\right)\phi_{j} -\rho_{j}g\eta+\text{const}\), the constants being equal because the undisturbed interface is in equilibrium. The Young–Laplace relation Equation (31.18) for a surface of small slope reads \(p_{2}-p_{1}=-\gamma\,\pp_{x}^{2}\eta=\gamma k^{2}\eta\). Substituting,

and inserting the two expressions for \(\phi_{j}(0)\) together with \(s_{j}^{2}=-\left(kU_{j}-\omega\right)^{2}\) gives Equation (31.63) after multiplying through by \(-k/H\). The disturbance grows exponentially when \(\omega\) has a positive imaginary part, and since the two roots of a real quadratic are complex conjugates, instability at a given \(k\) is equivalent to the discriminant being negative.

Derives Corollary 31.75. Set \(U_{j}=0\) and \(\coth\left(kh_{j}\right)\longrightarrow1\) in Equation (31.63) and solve for \(\omega^{2}\), giving Equation (31.64). With \(\rho_{1}>\rho_{2}\) the bracket is negative for \(k<k_{\mathrm{c}}\), so \(\omega^{2}<0\), \(\omega\) is purely imaginary, and one root grows. Neglecting \(\gamma\) gives \(\abs{\omega}=\sqrt{\mathcal{A}gk}\). The cut-off is the physically important part: surface tension stabilizes short wavelengths, so a water surface held above air is stable in a tube narrower than \(\lambda_{\mathrm{c}}\), which for water is a few centimetres — the reason a glass of water inverted over a fine mesh does not empty.

Derives Corollary 31.76. Write Equation (31.63) with \(\coth\longrightarrow1\) as a quadratic in the phase speed \(c=\omega/k\): \(\left(\rho_{1}+\rho_{2}\right)c^{2} -2\left(\rho_{1}U_{1}+\rho_{2}U_{2}\right)c +\rho_{1}U_{1}^{2}+\rho_{2}U_{2}^{2} -\gamma k-\left(\rho_{2}-\rho_{1}\right)g/k=0\). Its discriminant, divided by four, is

and the first two terms combine to \(-\rho_{1}\rho_{2}\left(U_{1}-U_{2}\right)^{2}\) identically. The discriminant is negative exactly under Equation (31.66). Setting \(\rho_{1}=\rho_{2}=\rho\) and \(g=\gamma=0\) leaves \(c=\tfrac{1}{2}\left(U_{1}+U_{2}\right) \pm\tfrac{\ii}{2}\abs{U_{1}-U_{2}}\), unstable at every \(k\) with growth rate \(k\abs{U_{1}-U_{2}}/2\), which grows without bound as \(k\longrightarrow\infty\) — the vortex sheet is an idealization whose short-wave behaviour is regularized in a real fluid by the finite thickness of the shear layer and by viscosity.

Two consequences of Equation (31.66) are worth recording because both are observed. Stable stratification does not confer immunity: with the denser fluid below, the right-hand side is positive but has a minimum over \(k\), so a large enough velocity difference destabilizes some band of wavelengths — which is how billow clouds form on a temperature inversion, and how wind raises waves on water. And with \(g\) and \(\gamma\) present the instability sets in first at a definite wavelength, so the billows appear with a selected spacing rather than at all scales at once.

Derives Theorem 31.77. Linearizing Equation (31.19) about the parallel flow and eliminating the pressure gives Rayleigh's equation,

Suppose \(\operatorname{Im}c\neq0\). Then \(U-c\) never vanishes and the equation may be divided by it. Multiply by the complex conjugate \(\varphi^{\ast}\) and integrate across the channel, integrating the first term by parts and using the boundary conditions:

The first integral is real. Taking the imaginary part of the second, with \(c=c_{\mathrm{r}}+\ii c_{\mathrm{i}}\) and \(1/(U-c)=\left(U-c_{\mathrm{r}}+\ii c_{\mathrm{i}}\right)/ \abs{U-c}^{2}\), leaves

Since \(c_{\mathrm{i}}\neq0\) by hypothesis and the remaining weight \(\abs{\varphi}^{2}/\abs{U-c}^{2}\) is non-negative and not identically zero, \(U''\) must change sign, hence vanish, somewhere in the interval.

The criterion is necessary and not sufficient, and it is worth seeing what it does and does not settle. A free shear layer or a jet has an inflexion point and is violently unstable, which is Corollary 31.76 in a smoothed form. The parabolic pipe and channel profiles of Equation (31.36) have none, so they are stable in the inviscid theory — and they are observed to break down anyway (Phenomenon 31.53), which is the first hint that the pipe transition is not a linear instability at all.

Derivation. Derives Phenomenon 31.78. The reduction of the linearized Boussinesq system to the marginal relation \(\mathrm{Ra}(a)=\left(\pi^{2}+a^{2}\right)^{3}/a^{2}\) between Rayleigh number and dimensionless horizontal wavenumber, and the minimization of that curve which yields Equation (31.68), are carried out in Section 36.4, where they are set beside the measurement. The point to be taken here is the contrast with Phenomenon 31.53: the same programme — linearize about the base state, seek normal modes, find the parameter value at which the growth rate first reaches zero — delivers a threshold and a length scale in this problem and delivers nothing in the pipe. The failure there is therefore a property of that particular flow, not of the method, and Theorem 31.77 says why: the parabolic profile has no inflexion point, so its instability, if it is one, cannot be of the kind linear theory finds.

Derives Proposition 31.80. Take two thin coaxial rings of equal volume, and hence in an incompressible fluid of equal mass \(m\), at radii \(r_{1}<r_{2}\), carrying angular momenta per unit mass \(L_{1}\) and \(L_{2}\). Interchange them and leave the rest of the fluid undisturbed. In an ideal axisymmetric motion nothing exerts a torque about the axis on a ring — viscous stresses are absent by hypothesis and the pressure force on a ring is directed through the axis — so each ring carries its own \(L\) to its new radius. The two rings have equal volume, so the ambient pressure field does no net work on the pair. The kinetic energy of the azimuthal motion of a ring is \(\tfrac{1}{2}mv_{\varphi}^{2}=\tfrac{1}{2}mL^{2}/r^{2}\), so the interchange costs

The second bracket is positive because \(r_{1}<r_{2}\). So the interchange must be paid for from outside exactly when \(L^{2}\) increases outward, and it releases kinetic energy — and will therefore proceed of itself, which is instability — exactly when \(L^{2}\) decreases outward. Applied to neighbouring radii this is the sign of \(\dd L^{2}/\dd r\), and dividing by the positive \(r^{3}\) to make a quantity with the dimensions of the square of a frequency gives Equation (31.69).

For Equation (31.34), \(L=r\left(Ar+B/r\right)=Ar^{2}+B\), so \(\dd L^{2}/\dd r=2L\cdot2Ar=4Ar^{2}v_{\varphi}\) and \(\Phi=4Av_{\varphi}/r\). With the outer cylinder at rest, \(A=-\Omega_{1}R_{1}^{2}/\left(R_{2}^{2}-R_{1}^{2}\right)<0\) while \(v_{\varphi}\) falls from \(\Omega_{1}R_{1}\) to zero across the gap, so \(\Phi<0\) throughout the interior of the gap and the inviscid flow is unstable for any \(\Omega_{1}>0\). Rotating the outer cylinder instead, with the inner one at rest, gives \(A>0\) and \(\Phi>0\): that flow is stable to axisymmetric disturbances at every speed. That asymmetry between driving from inside and driving from outside is what the experiment shows, and it is why Couette's viscometer turns the outer cylinder and measures the torque on the inner one [Couette:1890].

Derives Theorem 31.82. Average Equation (31.32) written in conservation form, \(\rho\pp_{t}v_{i}+\rho\pp_{j}\left(v_{i}v_{j}\right) =-\pp_{i}p+\mu\nabla^{2}v_{i}+\rho g_{i}\), which is equivalent to it by Equation (31.11). Every term is linear in the fields except the advective one, and for that

the cross terms vanishing because \(\avg{v'}=0\) and \(V\) is already an average. Moving the correlation to the right-hand side and using the averaged continuity equation gives Equation (31.70). Nothing has been approximated: Equation (31.70) is exact.

Derivation. Derives Phenomenon 31.84. The derivation is dimensional, and rests on two hypotheses that are themselves not derived from the Navier–Stokes equations but assumed and then tested: that at scales far below the stirring scale the statistics are locally isotropic and independent of how the turbulence was made, and that in the intermediate range they are independent of the viscosity as well, depending only on the rate \(\varepsilon\) at which energy is passed down the scales [Kolmogorov:1941]. Granted these, \(E(k)\) can depend only on \(\varepsilon\) and \(k\). In the base dimensions of length \(\mathsf{L}\) and time \(\mathsf{T}\), the spectrum is an energy per unit mass per unit wavenumber and so carries \([\,E\,]=\mathsf{L}^{3}\mathsf{T}^{-2}\), while \(\varepsilon\) is an energy per unit mass per unit time, \([\,\varepsilon\,]=\mathsf{L}^{2}\mathsf{T}^{-3}\), and \([\,k\,]=\mathsf{L}^{-1}\). Setting \(E=C\varepsilon^{a}k^{b}\) and matching both dimensions,

so \(a=2/3\) and \(b=-5/3\), which is Equation (31.71); the exponents are fixed uniquely by Theorem D.2 and the only freedom left is the dimensionless \(C\). The same argument locates the lower end of the range: the only length that can be built from \(\varepsilon\) and the kinematic viscosity \(\nu\) is \(\eta=(\nu^{3}/\varepsilon)^{1/4}\), below which the second hypothesis must fail because viscosity is precisely what acts there. What the argument does not give is any justification of the hypotheses, and measurement has since found small but systematic departures from \(-5/3\) at high order — the intermittency corrections — so Equation (31.71) is a leading approximation confirmed to within those, not an identity [Grant:1962].

Derives Proposition 31.85. The dimensions are \([\,\nu\,]=\mathsf{L}^{2}\mathsf{T}^{-1}\) and \([\,\varepsilon\,]=\mathsf{L}^{2}\mathsf{T}^{-3}\), and eliminating \(\mathsf{T}\) between them gives Equation (31.72) uniquely (Theorem D.2). Then \(u_{\eta}\eta/\nu=\left(\nu\varepsilon\right)^{1/4} \left(\nu^{3}/\varepsilon\right)^{1/4}/\nu=1\). With \(\varepsilon\sim U^{3}/L\) — the statement that the energy \(U^{2}\) is passed down in the eddy turnover time \(L/U\), which is the cascade hypothesis and not a theorem — \(L/\eta=L\left(\varepsilon/\nu^{3}\right)^{1/4} \sim\left(UL/\nu\right)^{3/4}\), and the number of independent degrees of freedom needed to represent the field is the number of boxes of side \(\eta\) in the volume.

The last estimate is the practical face of the closure problem. At \(\mathrm{Re}=10^{6}\), a modest laboratory value, Equation (31.73) demands of order \(10^{13}\) degrees of freedom per stirring volume for a computation resolving every scale. For concreteness, in the atmospheric surface layer \(\varepsilon\approx10^{-2}\,\mathrm{W}/\mathrm{kg}\) and, with air from Table 31.1, Equation (31.72) gives \(\eta\approx0.8\,\mathrm{mm}\) — so the smallest structures in a kilometre-deep flow are under a millimetre across, a scale separation of six decades.

Derivation. Derives Phenomenon 31.88. Write \(\rho=\rho_{0}+\rho'\), \(p=p_{0}+p'\) with the primed quantities and the velocity small, and keep first-order terms in the continuity and Euler equations of a fluid at rest:

Take \(\pp_{t}\) of the first and the divergence of the second and subtract: \(\pp_{t}^{2}\rho'=\nabla^{2}p'\). If the disturbance follows some definite thermodynamic path, \(p'=(\dd p/\dd\rho)\rho'\) along it, and the result is the wave equation \(\pp_{t}^{2}\rho'=c^{2}\nabla^{2}\rho'\) with \(c^{2}=\dd p/\dd\rho\) — non-dispersive, as observed, because \(c\) carries no reference to the wavelength.

Which path? Newton took the compressions to be isothermal, giving \(p\propto\rho\) and \(c^{2}=p/\rho\) [Newton:1687]. The justification fails on comparison of two times: heat must diffuse between a compression and the adjacent rarefaction, half a wavelength away, within half a period, and for air at audible frequencies the thermal diffusion length in that time is smaller than the wavelength by orders of magnitude. The compressions are therefore adiabatic, not isothermal. For an ideal gas the adiabat is \(p\propto\rho^{\gamma}\), whence \((\pp p/\pp\rho)_{s}=\gamma p/\rho\) and Equation (31.75) [Laplace:1816]. Numerically, for air at \(20\,\mathrm{^\circ\mathrm{C}}\) treated as an ideal gas with specific gas constant \(287\,\mathrm{J}/\mathrm{kg}/\mathrm{K}\) and \(\gamma=7/5\), Equation (31.75) gives \(c=343\,\mathrm{m}/\mathrm{s}\) against Newton's \(290\,\mathrm{m}/\mathrm{s}\), and the measured value [Batchelor:1967] is the former. The two answers differ by the factor \(\sqrt{\gamma}=\sqrt{7/5}\approx1.18\) for a diatomic gas, and the measured speed in air selects the adiabatic one. The episode is a clean instance of the pattern this treatise follows throughout: a derivation whose logic is sound fails numerically, and the failure localizes the one assumption — here the thermodynamic path — that has to be replaced.

Derives Theorem 31.91. Work in the frame in which the discontinuity is stationary and the flow is steady and one-dimensional. Each of mass, momentum and energy obeys a conservation law of the form \(\pp_{t}q+\pp_{x}F=0\), which in the steady state reduces to \(\dd F/\dd x=0\): the flux is the same on both sides however violently \(q\) varies between them. Integrating each across a pillbox straddling the surface and shrinking its thickness to zero, only the flux difference survives, and the three fluxes are \(\rho u\), \(p+\rho u^{2}\) and \(\rho u\left(h+\tfrac{1}{2}u^{2}\right)\) — the last dividing through by the mass flux, which is common to both sides by the first relation. That gives Equation (31.76). The viscous and conducting terms of Equation (31.31) contribute nothing to the integrated balance because they are gradients of quantities that are equal on the two sides, where the flow is uniform; that is why the result is independent of \(\mu\), \(\zeta\) and the thermal conductivity, although those are exactly what determine the shock's internal structure and thickness.

Derivation. Derives Phenomenon 31.93. Each point of the trajectory emits a disturbance that spreads as a sphere of radius \(ct\) about the point of emission, while the body advances \(Ut\). For \(U>c\) the spheres are contained within the envelope that touches all of them, a cone whose apex is the body and whose half-angle satisfies \(\sin\mu=ct/Ut\), which is Equation (31.77). For \(U<c\) no envelope exists and the disturbance eventually fills the whole space, which is Remark 31.89 restated geometrically. The cone is the locus at which the fluid first learns of the body, so it is where the state jumps, and Theorem 31.91 applies to the component of the velocity normal to it. For a body of finite thickness the attached cone becomes a detached bow shock at speeds close to the sound speed, which is what the photographs of [Mach:1887] also show.

Derives Theorem 31.94. This is Equation (31.63) with the upper fluid removed, \(\rho_{1}=0\) and \(U_{1}=U_{2}=0\), the lower fluid identified as the liquid, \(\rho_{2}=\rho\) and \(h_{2}=h\): \(\rho\coth\left(kh\right)\omega^{2} =k\left(\rho g+\gamma k^{2}\right)\), which rearranges to Equation (31.78). Setting \(\gamma=0\) recovers the classical gravity-wave relation of [Lamb:1932]. The hypotheses inherited from Theorem 31.74 are worth restating because each is visible in the failures of the formula: ideal fluid, so no attenuation is predicted although real waves damp; irrotational motion, so no mean drift; and small amplitude, so no steepening or breaking.

Derivation. Derives Phenomenon 31.95. From Equation (31.78) the phase speed is \(c=\omega/k=\sqrt{\left(g/k+\gamma k/\rho\right)\tanh\left(kh\right)}\). For \(kh\gg1\), \(\tanh\longrightarrow1\); with \(\gamma\) negligible, \(c=\sqrt{g/k}\), which decreases with \(k\) — long waves are faster — and the group velocity is

For \(kh\ll1\), \(\tanh\left(kh\right)\approx kh\) and \(\omega^{2}\approx gh\,k^{2}\), so \(c=c_{\mathrm{g}}=\sqrt{gh}\) independent of \(k\): no dispersion, and the shallow-water speed. For the minimum, the deep-water phase speed \(c^{2}=g/k+\gamma k/\rho\) is a sum of a decreasing and an increasing term, minimized where their derivatives cancel, \(-g/k^{2}+\gamma/\rho=0\), i.e. at \(k_{\min}=\sqrt{\rho g/\gamma}\), giving \(c_{\min}^{2}=2\sqrt{g\gamma/\rho}\) and hence Equation (31.79). With \(\gamma=7.3\times 10^{-2}\,\mathrm{N}/\mathrm{m}\) and \(\rho=998\,\mathrm{kg}/\mathrm{m}^{3}\) these evaluate to \(0.23\,\mathrm{m}/\mathrm{s}\) and \(1.7\,\mathrm{cm}\), both directly observable on a still pond. The factor of one half between energy and crest speed is likewise observable: a group of deep-water waves advances with individual crests appearing at its rear, travelling through it and vanishing at its front. The same factor fixes the half-angle of a ship's wake at \(\arcsin\left(1/3\right) \approx19.5^\circ\), independent of the speed of the ship, which is the most easily checked prediction in this section.

Derivation. Derives Phenomenon 31.96. Long waves of small but finite amplitude on shallow water obey, to leading order in both small parameters \(a/h\) and \(\left(h/\ell\right)^{2}\) with \(\ell\) the length of the disturbance, the Korteweg–de Vries equation [Korteweg:1895]

whose two non-trivial terms are exactly the two effects named above: the amplitude-dependent speed \(c_{0}\left(1+3\eta/2h\right)\), which steepens the front, and the third derivative, which is the first correction to \(\omega=c_{0}k\) obtained by expanding \(\tanh\left(kh\right)\) in Equation (31.78). Seek a solution travelling without change of form, \(\eta=\eta(x-ct)\). Substituting and integrating twice, with \(\eta\) and its derivatives vanishing at infinity, gives a first-order equation whose solution is

Two of the three relations are testable and both were tested by Russell. The speed agrees with Equation (31.80) to first order, since \(\sqrt{g\left(h+a\right)}=c_{0}\sqrt{1+a/h} \approx c_{0}\left(1+a/2h\right)\), and the expansion is consistent only for \(a\ll h\), which is Russell's regime. And the width is inversely related to the amplitude, \(\ell\propto a^{-1/2}\): a taller solitary wave is narrower and faster, which is the counter-intuitive prediction, and it is what the wave-tank measurements show. The derivation of Equation (31.81) itself from the free-surface equations is longer than a page and is left to the appendix; the modern reading of Equation (31.82) as a soliton, with the collision properties that follow, belongs to Nonlinear Dynamics and Chaos.