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2006 Variational Approach to the Volume Viscosity of Fluids Allan J. Zuckerwar
Robert L. Ash Old Dominion University, [email protected]
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Repository Citation Zuckerwar, Allan J. and Ash, Robert L., "Variational Approach to the Volume Viscosity of Fluids" (2006). Mechanical & Aerospace Engineering Faculty Publications. 21. https://digitalcommons.odu.edu/mae_fac_pubs/21 Original Publication Citation Zuckerwar, A. J., & Ash, R. L. (2006). Variational approach to the volume viscosity of fluids. Physics of Fluids, 18(4), 047101. doi:10.1063/1.2180780
This Article is brought to you for free and open access by the Mechanical & Aerospace Engineering at ODU Digital Commons. It has been accepted for inclusion in Mechanical & Aerospace Engineering Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. PHYSICS OF FLUIDS 18, 047101 ͑2006͒
Variational approach to the volume viscosity of fluids ͒ Allan J. Zuckerwara NASA Langley Research Center, Mail Stop 238, Hampton, Virginia 23681 ͒ Robert L. Ashb Old Dominion University, Department of Aerospace Engineering, Norfolk, Virginia 23508 ͑Received 18 July 2005; accepted 2 February 2006; published online 4 April 2006͒ The variational principle of Hamilton is applied to develop an analytical formulation to describe the volume viscosity in fluids. The procedure described here differs from those used in the past in that a dissipative process is represented by the chemical affinity and progress variable ͑sometimes called “order parameter”͒ of a reacting species. These state variables appear in the variational integral in two places: first, in the expression for the internal energy, and second, in a subsidiary condition accounting for the conservation of the reacting species. As a result of the variational procedure, two dissipative terms appear in the Navier-Stokes equation. The first is the traditional volume viscosity term, proportional to the dilatational component of velocity; the second term is proportional to the material time derivative of the pressure gradient. Values of the respective volume viscosity coefficients are determined by applying the resulting volume-viscous Navier-Stokes equation to the case of acoustical propagation and then comparing expressions for the dispersion and absorption of sound. The formulation includes the special case of equilibration of the translational degrees of freedom. As examples, values are tabulated for dry and humid air, argon, and sea water. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2180780͔
I. INTRODUCTION be modeled numerically as discontinuities for many types of flow studies, thus eliminating the need for constitutive stress- The shear or dynamic viscosity of a fluid can be mea- strain rate coefficients. Strong shock waves in air and gases sured unambiguously in a variety of viscometric apparatuses. such as carbon dioxide and nitrogen result in significant de- By imposing a specific velocity gradient across a fluid and partures from thermodynamic equilibrium, requiring separate measuring the forces required to maintain that gradient, the models for the various molecular degrees of freedom and shear viscosity can be deduced directly. That is not the case making it extremely difficult to separate volume-viscous ef- for the volume viscosity. From the general form of isotropic fects from nonequilibrium thermodynamic effects. fourth-rank tensors, the second-rank stress tensor, when re- Frequency-dependent attenuation of sound using a con- lated linearly to rates of strain ͑Newtonian fluid͒, includes a tinuum constitutive model ͑i.e., shear and volume viscosi- volumetric dilatation term that alters the normal stresses. The ties͒ can result in frequency-dependent coefficients that vio- linear coefficient characterizing that effect is called the vol- late the so-called frame independence or material 1 ume viscosity. ͑Various algebraic representations of the vol- indifference requirement, where observers moving with dif- ume viscosity have been termed bulk viscosity and second ferent reference speeds would need to use different attenua- coefficient of viscosity, as will be mentioned in the text.͒ tion constants. It was this dilemma that caused George Gab- However, because of the need to utilize thermodynamic pres- riel Stokes in 1845 to assume that the second coefficient of sure and associated equations of state, the volume viscosity viscosity constant was related linearly to the shear viscosity appears in the constitutive equation for a Newtonian fluid, ͑the so-called “Stokes hypothesis,” which will be introduced where it must either include or be added to the thermody- later͒, in order to exclude volume-viscous effects from his namic pressure. equations of motion. In order to develop our volume viscos- Consequently, it is not possible to model simple fluids ity approach, we will need to review the basic continuum using only shear viscosity when large axial velocity gradients model in order to synchronize our notation and illuminate the ͑ ͒ approach. compared with transverse velocity gradients exist in the 2 flow. That is the case for flows with shock waves and in the In the absence of rotational viscosity, the most general study of acoustics, where transverse velocity gradients can linear relationship between the second-order stress tensor ij be missing altogether and frequency-dependent attenuation and the rate of strain tensor ˙ ij for a simple, isotropic fluid is effects must be modeled. In the case of shock waves, the given by axial flow gradients are confined to very small spatial dis- ͑ ͒ ͓ ͔␦ ͑ ͒ tances on the order of a molecular mean free path and can ij = − P + ˙ kk ij +2 ˙ ij, 1
͒ where ␦ is called the Kronecker delta ͑with the value of a Electronic mail: [email protected] ij b͒ ͒ Author to whom correspondence should be addressed. Electronic mail: unity when i and j are the same indices, and zero otherwise [email protected] using index notation and the Einstein summation convention
1070-6631/2006/18͑4͒/047101/10/$23.0018, 047101-1 © 2006 American Institute of Physics 047101-2 A. J. Zuckerwar and R. L. Ash Phys. Fluids 18, 047101 ͑2006͒ vץ ץ ץ for repeated indices͒, and the rate of strain tensor is defined͑ + + j =0, ץ ץ as vj t xj xjץ v or, recognizing thatץ vץ 1 ˙ ϵ ͩ i + j ͪ, ץ ץ ij Dץ ץ xj xi 2 + ϵ , ͑3͒ ץ vj t xj Dtץ where vi represents the ith component of the velocity vector ͑ ͒ in an x1 ,x2 ,x3 Eulerian, Cartesian coordinate system, with where D/Dt represents the material time derivative of den- the linear coefficients and , defined as the second coeffi- sity, or the rate of change of density for an infinitesimal cient of viscosity and the dynamic viscosity, respectively. In volume of fluid moving through a spatial location in an Eu- agreement with solid mechanics, the diagonal stress tensor lerian coordinate system at a given instant. Hence, we see components are considered to be positive in tension; thus, that absolute pressure is a negative quantity. v 1 D D͑1/͒ץ Introductory fluid mechanics texts tabulate the dynamic ˙ = j =− = ץ jj or shear viscosity for common liquids and gases. In fact, the xj Dt Dt shear viscosity can be estimated for monatomic gases using and therefore, the general stress tensor relationship can actu- the Boltzmann equation with a restricted set of distribution ally be written functions that are slightly perturbed from Maxwellian form. vץ vץ On the other hand, the second coefficient of viscosity or 2 1 D =−ͫP + ͩ − ͪ ͬ␦ + ͩ i + j ͪ. xץ xץ volume viscosity, does not enjoy the same clarity. Firstly, we ij V 3 Dt ij note that when the fluid is in thermodynamic equilibrium, in j i the sense that an equation of state in the form P= P͑,T͒ can If one is interested primarily in fluid systems that have be used to relate temperature, pressure, and density, the trace relatively small variations in temperature, one can assume of the stress tensor kk must be equal to three times the that the volume viscosity V and dynamic viscosity are average normal stress, and thus should yield −3P ͑because constants throughout the volume. the trace of the Kronecker delta is 3͒, and we have the re- quirement that the average normal stress ¯, given by II. DISSIPATION OF MECHANICAL ENERGY IN vץ v 2ץ v 2ץ ¯ = kk = ͫ− P + k ͬ + k =−P + ͩ + ͪ k . FLUIDS ץ ץ ץ 3 xk 3 xk 3 xk There are four recognized classes of processes by which Stokes ͑1845͒1 postulated that setting the second coefficient nonrandom mechanical energy can be dissipated as random of viscosity equal to −͑2/3͒ resulted in the desired rela- energy ͑heat͒ in matter: viscous, relaxation, resonance, and 3 tionship between pressure and average normal stress. How- hysteretic. The latter two have not been observed in simple ever, it could be argued that ͑measured͒ pressure and average fluids and will not be considered further, although hysteretic 4 normal stress are equivalent only when the measurement is effects have been found in liquids in the glassy state. Viscous dissipation occurs in a fluid subjected to a shear ץ ץ͑ made in the absence of any volumetric dilatation vk / xk =0͒. This requirement is actually enforced when fluid pres- stress. The relationship between the shear stress and the ͑ ͒ sure or fluid temperature is measured using a solid sensing transverse velocity gradient is found from Eq. 1 for the ͑ ͒ element, since the no-slip boundary condition on any solid case i j. The process represented by Eq. 1 is characterized surface creates a “no volumetric dilatation” condition on that by a single constitutive constant , the absolute or dynamic surface. On the other hand, when optical techniques are used viscosity of the fluid. This constant can be determined by a to measure local temperatures and pressures in a flowing wide variety of experiments, can be derived from fundamen- ͑ ͒ fluid, nontrivial volumetric dilatation conditions can exist. tal physical laws e.g., Boltzmann transport equation , and is The departures of the second coefficient of viscosity from well tabulated in handbooks as a function of temperature, ͑ ͒ pressure, and fluid composition.5 Noteworthy is the fact that − 2/3 have been defined as the volume viscosity V, where the shear viscosity is a transport property, rather than an equilibrium property of the fluid. ϵ ͑ ͒ A relaxation process, on the other hand, is the response V + 2/3 . of a fluid to a dilatational disturbance. It can manifest itself The volume-viscous contribution to the average normal as ͑1͒ a transport process or ͑2͒ an equilibrium process, the stress is then latter rendering the return of the disturbed state of the fluid toward equilibrium without molecular transport. The trans- ͑ v port process, known as “translational relaxation” classicalץ k ͑ ͒ ¯ V = V . 2 heat conduction͒, involves the redistribution of excess ki- xץ k netic energy among all the velocity components. This will be Part of the volume viscosity measurement difficulty re- discussed in more detail below. The second process involves sides in the fact that for nonrelativistic systems, conservation the transition of a molecule or molecules Mi from one equi- of mass requires that librium state to another, as described by a reaction: 047101-3 Variational approach to the volume viscosity of fluids Phys. Fluids 18, 047101 ͑2006͒
͚ M ͚ ЈMЈ ͑i ͒ ͑ ͒ d i i i i , =1,2, ... , 4 = LA, ͑8͒ i i dt Ј where i and i are the stoichiometric coefficients of the where L is a constant and t is time. For small changes about Ј reactants Mi and products Mi , respectively. For example, Eq. equilibrium, the affinity A can be expanded as follows: ͑4͒ may represent ͑1͒ the excitation of a vibrational, rota- Aץ Aץ tional, or electronic degree of freedom of a fluid molecule, A = A͑,X,Y͒ = ͩ ͪ ͑ − ͒ + ͩ ͪ ͑X − X ͒ 0 ץ 0ץ X,Y X ,Y M0 + M M1 + M, Aץ where M and M represent the molecule in the ground and + ͩ ͪ ͑Y − Y ͒, 0 ץ 1 0 first excited levels, respectively; ͑2͒ the association of a Y X, dimer from a pair of monomers, as occurs in nitrogen 6 7 where X and Y are nonconjugate state variables, and sub- tetraoxide or acetic acid script 0 refers to the value at equilibrium. A relaxation time 2M M2, can be then defined as Aץ where M represents the dimer; ͑3͒ the recombination of a 1 2 =−Lͩ ͪ , ͑9͒ ץ 8 pair of ions to form a neutral electrolyte, XY X,Y + − M + N MN; which, when substituted into Eq. ͑8͒, leads to the familiar ͑4͒ the transition between two states of a rotational isomer relaxation equation: 9 like 1,1,2-trichloroethane, d − =− 0 . M MЈ; dt XY or ͑5͒ the redistribution of the populations of structural com- Noteworthy is the fact that the relaxation time depends upon plexes in a liquid, like water,10 the relaxation path. Meixner has demonstrated the generality of the formula- M + M + ЈM + ЈM + . 1 1 2 2 ¯ 1 1 2 2 ¯ tion given by Eqs. ͑4͒–͑9͒, and has shown that it does not In equilibrium, the forward rate of Eq. ͑4͒ equals the back- depend upon the molecular details of any specific ward rate, in which case the equilibrium concentrations of processes.11 It is the contention here that the processes gov- the reactants and products are determined by the law of mass erned by these equations are exhaustive. In other words, all action. In the event of a disturbance, the reaction proceeds to known processes for the volume-dissipation of mechanical the right or to the left to strive toward a new equilibrium. energy in fluids, with the exception of translational ͑classi- The time delay between the excitation and response is the cal͒ relaxation processes, are subject to and described by the cause of mechanical dissipation in fluids. physical laws outlined in the above equations. The force that drives the reaction toward the new equi- For convenience, choose ͑P,S,͒ to be the independent librium is the affinity A, as defined by Gibbs. The affinity state variables. One may then write A=A͑P,S,͒ In the re- represents the reaction in all thermodynamic potential func- maining analysis, it will be assumed for simplicity that the tions; for example applicable flows are adiabatic, for mixed flows ͑allowing heat conduction through the boundaries͒ depend upon the P 12 dU = TdS + d − Ad, ͑5͒ boundary conditions and add a level of complexity that will 2 obscure the physical principles leading to the derivation of where U, T, S, , P, and are, respectively, the internal the volume viscosity. The isobaric, isentropic relaxation time energy, temperature, entropy, molar density, pressure, and then becomes ͑ ͒ Aץ progress variable sometimes called “ordering parameter” of 1 ͑ ͒ =−Lͩ ͪ , ͑10͒ ץ the fluid. From Eq. 5 , one finds PS P,S Uץ A =−ͩ ͪ . ͑6͒ ͑ ͒ and the isochoric incompressible , isentropic relaxation timeץ S, is given by Pץ Aץ Aץ Aץ Upon the attainment of equilibrium, the affinity vanishes. 1 The progress variable, a measure of how far the reaction =−Lͩ ͪ =−Lͫͩ ͪ + ͩ ͪ ͩ ͪ ͬ ץ ץ ץ ץ has proceeded toward the new equilibrium, is defined as fol- VS V,S P,S P ,S V,S Pץ Vץ Aץ :lows =−Lͫͩ ͪ − ͩ ͪ ͩ ͪ ͬ, ͑11͒ ץ ץ ץ dn P,S P,S V,S d = i , ͑7͒ ⌬ i which uses the Maxwell relation Vץ Aץ where ni and i are, respectively, the molar concentration and stoichiometric coefficient of the ith species of the fluid. The ͩ ͪ =−ͩ ͪ . ץ P ץ foundation of the theory of irreversible thermodynamics is ,S P,S based on the hypothesis: Now, 047101-4 A. J. Zuckerwar and R. L. Ash Phys. Fluids 18, 047101 ͑2006͒
19 V Wang Chang, Uhlenbeck, and de Boer started with theץ ͩ ͪ = ⌬V ͑12͒ Boltzmann transport equation for a gas with internal degreesץ P S , of freedom and ultimately arrived at the following: and P 2 Cץ ͩ ͪ = ⌬P ͑13͒ i ͑ ͒ V = P 0 , 17ץ V,S 3 CV where ⌬V and ⌬P are the isentropic changes in volume and pressure per unit change of progress variable . Equation in which the relaxation time is related to the equilibration ͑11͒ can then be written of the translational degrees of freedom. Their derivation is based on the erroneous supposition that the relaxation time A 1ץ 1 ͫͩ ͪ ͑⌬ ͒͑⌬ ͒ͬ ͑⌬ ͒͑⌬ ͒ =−L − V P = + L V P . applies to both the translational and internal processes. ץ VS P,S PS The rigor of their development was lost when they, without ͑14͒ justification, substituted the “relaxation time introduced in the theory of the dispersion of sound by Herzfeld and Rice” If the flow is incompressible ͑⌬V=0͒ or isobaric ͑⌬P=0͒, in place of . then = , and volume dissipation can be described by a PS VS Hirshfelder, Curtiss, and Bird20 also started with a “gen- single constitutive constant. In the general case of compress- eralized Boltzmann equation,” from which they claimed to ible flow with a pressure gradient, however, the two relax- have derived the following: ation times are not equal, and volume dissipation accordingly requires two independent constitutive constants. This is the fallacy of relying on Eq. ͑2͒ alone to describe the volume RC = P i , ͑18͒ dissipation in fluids. V ͑ 0 ͒2 CV The traditional expression for the volume viscosity ͓Eq. ͑2͔͒ has led to much controversy and misunderstanding. Val- where R is the universal gas constant, but they do not show ues for this constitutive property cannot be found in hand- the derivation. Even though their derivation cannot be con- books, because there are no accepted standards or even con- firmed, their expression ͑18͒ actually agrees with Tisza’s sensus for direct experimental measurement. The leading equation ͑15͒, but the authors do not mention any time-scale effort has been acoustic streaming, but as Nyborg13 has limitations. shown, the results can be easily explained as conventional Cowling and Chapman21 applied the generalized Boltz- relaxation. mann equation to derive the traditional volume viscosity, but On the theoretical side, a variety of approaches have the solution contains an unknown indefinite integral, which been undertaken. A partial list is reviewed here. The discus- they evaluated by comparison with an alternative derivation sion is confined to a single dissipative process but can based on energy and continuity equations. In fact, they did readily be extended to multiple processes. Tisza14 applied the not even need the Boltzmann equation at all. The result is Navier-Stokes equation, containing the volume viscosity term, and continuity equation and compared the result to the known expression for acoustical dispersion. He was able to 2͑d −3͒ = P, ͑19͒ make the comparison in the limit to zero frequency and ob- V d2 tained the following for a gas with internal degrees of free- dom: where d is the number of molecular degrees of freedom. For C a diatomic molecule with frozen vibration, d=5, in which = ͑␥ −1͒P i , ͑15͒ V 0 case Eq. ͑19͒ differs from ͑15͒ by a factor of 2/5. Cowling CV and Chapman recognized that Eq. ͑19͒ was “valid ␥ 0 where , P, Ci, CV, and are the gas specific-heat ratio, only¼when is small compared with the time scale of the pressure, molar specific heat of the relaxing degree of free- expansion,” and suggested a procedure for a derivation when dom, molar low-frequency specific heat at constant volume, the relaxation time is comparatively large; but they lacked and relaxation time, respectively. Later this expression was 15 16 the mathematical framework to cover the entire range of time confirmed by Herzfeld and Litovitz, Woods, and scales. Further, their expression for the relaxation time does Emanuel.17 18 not conform to that of any known physical process. DeGroot and Mazur applied the method of irreversible Pierce22 derived the volume viscosity from classical, thermodynamics and arrived at the following expression, phenomenological energy and continuity equations. For the ͑ ͒ ͑ ͒ which is a combination of their Eqs. 119 and 168 : special case of a rotational relaxation process ͑the only one ͒  RC treated , his derivation contains the quantity c / rot, where c i ͑ ͒ V = P ϱ , 16 is the mean free time and  a constant. If  is interpreted C0 C rot rot V V to be the reciprocal of the “collision number,” that is, the ϱ in which CV is the high-frequency specific heat at constant number of collisions needed to effect a successful quantum volume. Again, when applied to sound absorption, their deri- transition, then his expression for the volume viscosity can vation is valid only in the limit of zero frequency. be written as follows: 047101-5 Variational approach to the volume viscosity of fluids Phys. Fluids 18, 047101 ͑2006͒
U 2 A uniaxial deformation, as generated by a piston stroke = ͩ rotͪ P, ͑20͒ V U or an acoustical plane wave in a fluid, is composed of strongly coupled dilatational and shear deformations. The where Urot is the molecular rotational contribution to the total mathematically complex expression for the coupled dissipa- internal energy. For a diatomic molecule with frozen vibra- tion can be approximated as a conventional relaxation, at tion, Eq. ͑20͒ agrees with ͑19͒ but not ͑15͒. time scales down to the translational relaxation time, by the Recent treatments yield derivations by molecular- method of Pade’ approximants.25 The relaxation strength is dynamics simulations based on fluctuation analysis. For ex- nearly, but not exactly, equal to unity, and there are two 23 ample, Bertolini and Tari find that when “k ͓wave number͔ relaxation times—one much larger than the other. The larger goes to zero” the ratio of volume viscosity to shear viscosity relaxation time is that given by Eq. ͑23͒. When the shear is simply contribution is included, then the translational relaxation Ϸ ͑ ͒ time becomes V/ 2/3. 21 These treatments represent a microscopic approach to the 1 4 ͑␥ −1͒ M = ͩ + T ͪ, ͑24͒ relaxation of the translational degrees of freedom. Interest- tr 2 0a 3 CP ingly, the macroscopic ͑phenomenological͒ approach yields a 0 24 ratio as will be shown in Sec. V. 3 R 1 V Ϸ ͩ ͪ͑␥ −1͒ͩ ͪͩ ͪ, ͑22͒ 4 CV Pr IV. THE VARIATION where CV is the specific heat at constant volume and Pr the Prandtl number. With PrϷ2/3 for argon, the numerical value Herivel26 was able to deduce the equations of motion for found for ͑22͒ is Ϸ1/2, in fair agreement with that of ͑21͒. an inviscid, incompressible fluid by way of Hamilton’s prin- The expressions ͑15͒–͑21͒ for the traditional volume vis- ciple. Serrin27 showed subsequently, utilizing a particle iden- cosity are valid only in the limit of large time scales com- tity constraint developed by Lin,28 that the equations of mo- pared to the relaxation time, as pointed out by Cowling and tion could be evolved from Hamilton’s principle for an Chapman, and in the special case of periodic ͑acoustical͒ inviscid, compressible fluid, without invoking an irrotation- flow do not yield the correct expressions for acoustical ab- ality restriction. sorption and dispersion, even in the limit to zero frequency. Here the Hamiltonian approach is extended to include There is no prior treatment that is valid over the entire range molecular relaxation processes, using nonequilibrium ther- of time scales. The failure of these approaches to yield a modynamics formulations developed by Meixner11 and later comprehensive treatment of the volume dissipation in fluids by deGroot and Mazur18 and Woods.29 Woods has discussed illustrates the futility of attempts to describe a two-constant how the characteristic relaxation times of different types of phenomenon with a single constitutive constant. processes can be modeled in the context of reversible and irreversible thermodynamics. We are interested particularly in differentiating between thermodynamic ͑translational͒ III. TRANSLATIONAL „CLASSICAL… DISSIPATION pressure relaxation, which can be modeled as a quasirevers- ible phenomenological process, and dissipative volume vis- A dilatational disturbance imparts an excess of transla- cous behavior that evolves naturally via the Newtonian fluid tional energy to the constituent molecules of a fluid beyond constitutive transport model. The resulting constraints on the its initial equilibrium value. The return to equilibrium leads variational problem are developed to allow molecular popu- to what is traditionally known as “heat conduction losses.” lations to depart from equilibrium. The associated relaxation time hc in terms of macroscopic ͑ 24 If one restricts attention to a single degree of freedom of properties is the following: nonequilibrium͒ for simplicity, it is possible to consider a ͑␥ −1͒ M uniform mixture of N molecular species where all of the T ͑ ͒ hc = , 23 species could contribute to the nonequilibrium state by way a2C 0 0 P of the generalized chemical reaction ͑4͒. Hence, the dilata- where T, M, 0, a0, and CP are, respectively, the thermal tional fluctuations are controlled thermodynamically and we conductivity, molar mass, ambient density, low-frequency consider the limiting minimum entropy process where, to speed of sound, and molar specific heat at constant pressure. first order, DS/Dt=0. Employing the Herivel methodology, For argon at 293.15 K, Eq. ͑23͒ yields a relaxation time of and replacing Lin’s particle identity constraint with Eq. ͑8͒, 1.3ϫ10−10 s, suggesting that this loss will not be substantial one can formulate a Lagrangian except for extremely rapid excitation. It is important to note, however, that this loss occurs in all fluids, even in mon- 1 L ϵ v v − ͑U + ⍀͒, ͑25͒ atomic gases, and that it is erroneous to state that the volume 2 k k viscosity coefficient in the latter is zero—a statement often found in the literature. For slow excitation, the heat conduc- where ⍀ is the body force potential function, and a set of tion contribution to the losses will be small indeed, but not constraints. zero. Conservation of mass: 047101-6 A. J. Zuckerwar and R. L. Ash Phys. Fluids 18, 047101 ͑2006͒
Uץ vץ D + k =0. ͑26͒ ͩ ͪ = T. ץ ץ Dt xk S , Conservation of reacting species: From ͑31͒, one can write
1 Pץ ץ D − LA =0. ͑27͒ + ͩ − ͪ =− − ͑U + ⍀͒ − . vk vkvk ץ vk ץ Dt t xk 2 Material entropy constraint: ͑ ͒ Upon substituting 30 and differentiating with respect to xi, DS one obtains =0. ͑28͒ v v ץ ץ Sץ ץ ץ ץ Dt ͩ ͪ + ͩ−  − ␣ ͪ =− ͩ k k ͪ ץ vk ץ ץ ץ ץ ץ Note that the conservation constraints are all written in terms t xi xi xk xk xi 2 ͑ ͒ P ץ ץ of the material time derivative, as defined in Eq. 3 , to allow for convective flows. An explanation for the constraint ͑28͒ − ͑U + ⍀͒ − ͩ ͪ. ͑34͒ x ץ xץ is provided by Meixner11 and by Herzfeld and Litovitz,4 who i i point out that the entropy production is of “second order” Now, ͑since the entropy at equilibrium is a maximum͒ and can be vץ vץ v v ץ neglected for departures from equilibrium that are not too ͩ k k ͪ = i + ͩ n ͪ ץijkvj kmn ץ vk ץ .large xi 2 xk xm Hamilton’s principle can then be represented as and vץ t1 1 D ␦ ͫ ͑ ⍀͒ ͩ k ͪ ץ ץ ץ ץ 2ץ ץ + − + vkvk − U ͵͵͵ ͵ x vn Sץ Dt 2 t0 V k = ͫ + ͩ ͪ + ͩ␣ ͪͬ, xץ xץ xץ xץ xץ xץ x kmnץkmn D DS m m n m n m n − ␣ͩ − LAͪ −  ͬdVdt =0, ͑29͒ ͑35͒ Dt Dt where , ␣, and  are Lagrange multipliers, V the volume, where ijk is the familiar alternating tensor, obeying the iden- and the sign of has been changed to conform with the tity standard definition for the velocity potential function, as uti- ␦ ␦ ␦ ␦ ijk kmn = ijk mnk = im jn − in jm, lized by Serrin.27 The independent variations of the velocity components, density, entropy, and equilibrium departure then leaving yield ץ ץ ץ ץ ␦v : vkvk vi S k ͩ ͪ = v + ͑␦ ␦ − ␦ ␦ ͒ͫv ͩ ͪ xץ xץ x jm kn jn km jץ x 2 kץ i k m mץ Sץ ץ = +  + ␣ , ͑30͒ Sץ ץ vץ ץ ץ ץ ץ ץ vk xk xk xk + ͩ␣ ͪͬ = i + ͩ ץ ץ vj ץ vk ץ ץ vj xm xm xk xi xj ␦: ץ Sץ ץ ץ + ␣ ͪ − ͩ + ␣ ͪ, ץ ץ ץ vj ץ D 1 P = v v − ͑U + ⍀͒ − , ͑31͒ xj xj xi xi Dt 2 k k since the first term on the right-hand side of ͑35͒ vanishes ␦ S: identically. ͑ ͒ D Consequently, one can write 34 as = T, ͑32͒ ץ Sץ ץ ץ Sץ vץ ץ ץ Dt ͩ ͪ − k ͩ + ␣ ͪ − ͩ + ␣ ͪ ץ ץ ץ vk ץ ץ ץ ץ ץ ␦: t xi xi xk xk xi xk xk ץ ץ ץ ץ ץ ץ ץ ␣ vi S S D =− + ͩ + ␣ ͪ − ͩ + ␣ ͪ ץ ץ ץ vk ץ ץ ץ vk ץ A, ͑33͒ vk−= Dt xk xk xi xi xi xk xk Pץ 1ץ ⍀ Pץ Uץ where the Gibbs relation ͑5͒ has been utilized to evaluate the =− − + − . ͑36͒ ץ ץ 2 ץ ץ various partial derivatives of the internal energy: xi xi xi xi U In addition, from ͑30͒ one can rewriteץ ͩ ͪ =−A, ץ ץ Sץ ץ vץ ץ ץ ,S ͩ ͪ = i − ͩ + ␣ ͪ, xץ xץ tץ tץ xץ tץ U P i i iץ ͩ ͪ = , ͒ ͑ 2ץ ,S so that 36 becomes 047101-7 Variational approach to the volume viscosity of fluids Phys. Fluids 18, 047101 ͑2006͒
U ϱ DPץ ץ Sץ vץ ץ Sץ Dv D ͬ i − ͩ + ␣ ͪ − k ͩ + ␣ ͪ =− − ٌͫ · v + x D S Dtץ xץ xץ xץ xץ xץ Dt Dt i i i k k i = , ץ P Dtץ 1ץ ⍀ Pץ − + − . ͩ ͪ ץ ץ ץ 2 ץ xi xi xi P,S Now, where the high-frequency adiabatic compressibility
ץ ϱ 1 ץ ץ  ץ ץ ץ ץ ץ D S vk S D S S D vk S ͩ ͪ ͩ ͪ +  =  ͩ ͪ + +  S = P ץ ץ ץ ץ ץ ץ ץ ץ Dt xi xi xk Dt xi xi Dt xi xk ,S S and continuity relationship ͑26͒ץ Sץ ץ Sץ = T + ͫ ͩ + ͪ ץ vk ץ ץ ץ xi xi t xk D ٌ · v−= S Dtץ Sץ vץ Sץ vץ − k ͬ +  k = T xץ xץ xץ xץ xץ i k i k i were substituted into Eq. ͑41͒. Equation ͑40͒ then becomes DS ץ  ͩ ͪ ͑ ͒ ץ Dv ϱ DP 37 + . ͪ x Dt =− ٌ ⍀ − ٌP − ␣ ٌ ٌͫ · + ͬͲͩץ ץ i v S Dt Dt P,S and ͑42͒ ץ vץ ␣ Dץ ץ Dץ vץ ץ D ͩ␣ ͪ + k ␣ = ␣ ͩ ͪ + + k ␣ As will be shown below, compatibility with the equations of ץ ץ ץ ץ ץ ץ ץ Dt xi xi xk Dt xi xi Dt xi xk acoustical propagation leads to the following expression for : the coefficient in Eq. ͑42͒ץ ץ ץ ץ = A + ␣ͫ ͩ + ͪ ץ vk ץ ץ ץ xi xi t xk − ␣ = VS = PS = a2, ϱ 0 PS 0 0ץ S Sץ ץ vץ ץ vץ − k ͬ + ␣ k = A ͩ ͪ ץ ץ ץ ץ ץ ץ xi xk xi xk xi P,S -D where 0 and a are the low-frequency isentropic compress ץ + ␣ ͩ ͪ. ͑38͒ S 0 ץ xi Dt ibility and the low-frequency sound speed, respectively. One finds Collecting expressions, one has Dv DP ͬ ٌ ⍀ ٌ ٌͫ 2 ٌ . ⍀ =− − P + PS 0a0 · v + VSץ Uץ D ץ DS ץ ץ Sץ Dv i − T − A −  ͩ ͪ − ␣ ͩ ͪ =− − Dt Dt ץ ץ ץ ץ ץ ץ Dt xi xi xi Dt xi Dt xi xi The first term in the square brackets is the traditional volume Pץ 1ץ P + − . ͑39͒ viscosity term with xץ x ץ 2 i i 2 ͑ ͒ V = PS 0a0, 43 When Eqs. ͑5͒ and ͑28͒ are inserted into ͑39͒, one finds and the second is the additional needed term, called here the D ץ Pץ ⍀ 1ץ Dv i =− − + ␣ ͩ ͪ, “pressure relaxation” term, with x Dtץ xץ x ץ Dt i i i ͑ ͒ P = VS. 44 or in vector notation The Navier-Stokes equation, modified to include the pressure Dv 1 D relaxation term, then reads as follows: ⍀ − ٌ P + ␣ ٌ ͩ ͪ. ͑40ٌ͒−= Dt Dt Dv D 2 =−ٌͩ1− ͪP − ٌ ⍀ + ٌͫͩ − ͪ ٌ · vͬ It is more useful to express the equation of motion in terms Dt P Dt V 3 of and P rather than . One proceeds by expanding the ϫ ͑ ٌ ϫ v͒ +2ٌ͓ · ͑ ٌ ͔͒v. ͑45͒ ٌ + molar density ͑, P,S͒ about its equilibrium point: If we neglect variations in the coefficients of viscosity andץ ץ − = ͩ ͪ ͑ − ͒ + ͩ ͪ ͑P − P ͒ pressure relaxation, this equation can be written: 0 ץ 0ץ 0 P,S P ,S Dv DP 4 =−ٌP + ٌ − ٌ ⍀ + ͩ + ͪ ٌ ٌ͑ · v͒ץ + ͩ ͪ ͑S − S ͒. ͑41͒ P V Dt Dt 3 0 ץ S P, ٌ ϫ ٌ͑ ϫ v͒. ͑46͒ − ͑ ͒ In equilibrium S=S0 is an extremum, in which case first derivatives are zero and to first order the contribution can be A list of values of the volume viscosity coefficients P and dropped. It follows that V for selected fluids is given in Table I. 047101-8 A. J. Zuckerwar and R. L. Ash Phys. Fluids 18, 047101 ͑2006͒
TABLE I. Values of the volume viscosity and 4/3ϫshear viscosity coefficients for selected fluids. RH =relative humidity. S=salinity in parts per thousand.
͑ ͒ P V 4/3 Fluid ͑s͒ ͑Pa s͒ ͑Pa s͒ Conditions Relaxation process
a,b ϫ −6 Air, 0% RH 6640 944 24.1 10 P=1 atm O2 vibration Air, 50% RH 4.47 0.635 24.0ϫ10−6 T=293.15 K Air, 100% RH 1.97 0.280 23.9ϫ10−6 Argonb,c −8ϫ10−6 22ϫ10−6 29.6ϫ10−6 P=1 atm Translation T=293.15 K b,d ϫ −3 Sea water 2.08 4739 1.840 10 P=1 atm MgSO4 ionization T=283.16 K S=35 aSee Ref. 32. bFor values of shear viscosity see Ref. 22 ͑air and sea water͒ and Ref. 33 ͑argon͒. c See Ref. 24. This reference explains the negative value of P for argon and its physical consequences. dSee Ref. 34.
V. APPLICATION TO ACOUSTICAL WAVE − 2 k2 PROPAGATION ͫ 1 ͬ =0. ͑49͒ ΄ 1 ΅ P ͑ ͒ 1 The density and pressure for a small-signal, one- 1+i PS − 2 1+i VS dimensional ͑plane͒ propagating wave are assumed to take a0 the form The resulting solution for the complex sound speed is = + exp i͑t − kx͒ ͭ 0 1 ͮ, ͑47͒ ͑ ͒ P = P0 + P1 exp i t − kx 1 k2 1 1+i = = VS , ͑50͒ 2 2 2 where 0 and P0 are the ambient density and pressure, re- a a0 1+i PS ͑ spectively, and 1 and P1 acoustic amplitudes, k the com- plex͒ wave number, and the angular frequency, with which yields the familiar expressions for sound dispersion Ӷ , P Ӷ P . and absorption. 1 0 1 0 If the Navier-Stokes equation includes the volume vis- In the absence of convection the material time derivative cosity terms ͓Eq. ͑46͔͒, then the resulting matrix must be reduces to equivalent to the square matrix in Eq. ͑48͒ in order to yield the solution ͑50͒. The equivalent matrix is easily found by ץ D = . application of the elementary column operations to Eq. ͑49͒: tץ Dt 4,11,30 2 → For molecular dissipative processes it is assumed that in a0i PSC2 + C1 C1 Eq. ͑46͒, ͑ ͒ Ӷ and then 4/3 V,
as is justified by comparison of the third and fourth columns ͑1−i ͒C → C , ͑ ͒ VS 2 2 of Table I. However, 4/3 must be added to V for trans- lational processes, because the contributions of heat conduc- leading to tion and shear viscosity are comparable. In this case, Eq. ͑24͒ rather than Eq. ͑23͒ will be substituted into Eq. ͑43͒. The 2 2 2 2͑ ͒ traditional Navier-Stokes equation ͑without the volume vis- − + k a0i PS k 1−i VS cosity terms͒, together with the continuity equation ͑26͒ and ͫ 1 ͬ 30,31 1 =0. the acoustical equation of state, then lead to the follow- ΄ 2 2 2 ΅ P ͑ ͒ 1 1+ PS VS − 2 1+ VS ing matrix equation: a0 51͒͑ 2ץ 2ץ 2 − 2 x ץ tץ ͫ 1 ͬ =0. ͑48͒ The top row is the Navier-Stokes equation containing the -P1 volume-viscous terms, and the bottom row is the correspond ΅ ץ 1 ץ ΄ 1+ − ͩ1+ ͪ .t ing dynamic equation of state. It is readily verified that Eqץt a2 VSץPS 0 ͑51͒ recovers the solution ͑50͒. The equation of state in the Upon substituting ͑47͒ into ͑48͒, one finds time domain is then 047101-9 Variational approach to the volume viscosity of fluids Phys. Fluids 18, 047101 ͑2006͒
2 D2 1 D2 L. C. Woods, The Thermodynamics of Fluid Systems ͑Clarendon, Oxford, ͩ ͪ͑ ͒ ͩ 2 ͪ͑ ͒ 1975͒. 1− PS VS − = 1− P − P . 2 0 2 VS 2 0 3 Dt a0 Dt H. G. van Bueren, Imperfections in Crystals ͑North-Holland, Amsterdam, 1960͒. ͑52͒ 4 K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves ͑Academic, New York, 1959͒. It is noted that the equivalence operations remove the dissi- 5 ͑ ͒ J. O. Hirschfelder, C. F. Curtiss, and R. B. Byrd, Molecular Theory of pative terms from the equation of state 52 and transfers Gases and Liquids ͑Wiley, New York, 1954͒. them to the Navier-Stokes equation ͑46͒. The resulting dy- 6H. O. Kneser, “Molecular relaxation processes in gases,” in Physical namic equation of state ͑52͒ is valid for convective flows. Acoustics IIA, edited by W. P. Mason ͑Academic, New York, 1965͒,p. 193. 7E. Freedman, “On the use of ultrasonic absorption for the determination of very rapid reaction rates at equilibrium: application to the liquid phase VI. CONCLUSIONS associate of carboxylic acids,” J. Chem. Phys. 21, 1784 ͑1953͒. 8M. Eigen and K. Tamm, “Sound absorption in electrolyte solutions due to The variational principle of Hamilton has been applied chemical relaxation. I. Relaxation theory of multistage dissociation,” Z. to derive expressions describing the volume-viscous dissipa- Elektrochem. 66,93͑1962͒. tion in fluids. A dissipative process in a fluid that responds to 9J. Lamb, “Ultrasonic relaxation due to rotational isomers,” Z. Elektro- a dilatational excitation is either a relaxation processes, or, in chem. 64, 136 ͑1960͒. 10A. Eucken, “The association of water,” Z. Elektrochem. 52,255͑1948͒. the case of translational relaxation, can be represented as 11 J. Meixner, “Allgemeine Theorie der Schallabsorption in Gasen und Flus- such at time scales much larger than the translational relax- sigkeiten unter Berucksichtigung der Transporterscheinungen,” Acustica ation time. A relaxation process is represented by the ther- 2, 101 ͑1952͒. modynamic state variables and A, the progress variable and 12G. B. Whitham, “The Navier-Stokes equations of motion,” in Laminar ͑ affinity, respectively, in all the thermodynamic potential Boundary Layers, edited by L. Rosenhead Oxford at the Clarendon Press, Oxford, 1963͒, pp. 124–127. functions. These variables appear in the variational integral 13W. L. M. Nyborg, “Acoustic streaming,” in Physical Acoustics IIB, edited in two places: in the internal energy, and in a subsidiary by W. P. Mason ͑Academic, New York, 1965͒, pp. 265–339. constraint expressing the conservation of the participating 14L. Tisza, “Supersonic absorption and Stokes’ viscosity relation,” Phys. 2 Rev. 61, 531 ͑1942͒. The substitution P=a /␥ was introduced into his reacting species. This feature was not included in past utili- o Eq. ͑15͒ to obtain our Eq. ͑15͒. zations of the variational principle, which, although success- 15See Ref. 4, p. 159. ful in deriving the traditional Navier-Stokes equation, could 16See Ref. 2, p. 196. not account for the nonequilibrium processes occurring in a 17G. Emanuel, “Bulk viscosity of a dilute polyatomic gas,” Phys. Fluids A fluid. Consequently, the variational procedure as applied here 2, 2252 ͑1990͒. 18S. R. deGroot and P. Mazur, “Viscous flow and relaxation phenomena,” in introduces two terms into the Navier-Stokes equation: the Non-Equilibrium Thermodynamics ͑North-Holland, Amsterdam, 1962͒, traditional volume viscosity term and an independent second Chap. 7, pp. 304–333. Explicit expressions for the adiabatic and isother- term, called here the pressure relaxation term, characterized mal compressibilities were introduced into their Eq. ͑168͒ to obtain our Eq. ͑16͒. by coefficients V and P, respectively. This result is consis- 19C. S. Wang Chang, G. E. Uhlenbeck, and J. de Boer, “The heat conduc- tent with the requirement that a volume-viscous process in a tivity and viscosity of polyatomic gases, Part C,” in Studies in Statistical fluid must be represented by two independent constitutive Mechanics, edited by J. de Boer and G. E. Uhlenbeck ͑North-Holland, constants. Amsterdam, 1964͒, pp. 241–268. The expression for the volume viscosity is their Eq. ͑72͒. When applied to acoustical propagation, the “modified” 20 ͑ See Ref. 5, p. 501. Molecular properties were converted to molar proper- Navier-Stokes equation including the volume viscosity ͑ ͒ ͑ ͒ ͒ ties related by Avogadro’s number in their Eq. 7.6-30 to obtain our Eq. terms yields the known expressions for sound dispersion ͑18͒. and absorption—another feature that has eluded past deriva- 21S. Chapman and T. G. Cowling, The Mathematical Theory of Uniform tions of the volume viscosity. Thus, acoustical measurements Gases ͑Cambridge University Press, Cambridge, U.K., 1970͒. The specific ͑ ͒ serve as a leading source of data to evaluate the volume heats in their Eq. 12.5,1 were converted to molecular degrees of freedom to obtain our Eq. ͑19͒. viscosity coefficients and their dependence upon ambient pa- 22A. D. Pierce, Acoustics: An Introduction to Its Physical Principles and rameters, although other measurement methods ͑e.g., shock Applications ͑Acoustical Society of America/American Institute of Phys- ͒  tube, impact tube, laser fluorescence, etc.͒ could serve the ics, New York, 1989 . The substitution c / rot = was introduced into his ͑ ͒ ͑ ͒ Eq. 10-7.10 to obtain our Eq. 20 , where c is the mean free time purpose as well.  between collisions and rot the reciprocal rotational collision number. Finally, it has been shown that the appearance of the 23D. Bertolini and A. Tani, “On the generalized bulk viscosity behavior,” J. volume-dissipative terms in the Navier-Stokes equation ne- Chem. Phys. 115, 6285 ͑2001͒. cessitates their removal from the dynamic equation of state. 24A. J. Zuckerwar, “Phenomenological theory of the translational relaxation A simple matrix equivalence operation, designed to transfer times in gases,” J. Acoust. Soc. Am. 105, 2210 ͑1999͒. 25Handbook of Applicable Mathematics, edited by R. F. Churchhouse the volume-dissipative terms from the dynamic equation of ͑Wiley, New York, 1981͒, pp. 216–223. state to the Navier-Stokes equation, yields a new dynamic 26J. W. Herivel, “The derivation of the equations of motion of an ideal fluid equation of state, which in the case of acoustical propagation by Hamilton’s principle,” Proc. Cambridge Philos. Soc. 51,344͑1955͒. 27 allows for dispersion but not absorption of the propagating J. Serrin, “Mathematical principles of classical fluid mechanics,” in Hand- ͑ ͒ sound wave. The resulting dynamic equation of state is valid buch der Physik, edited by S. Flugge Springer-Verlag, Berlin, 1959 ,Vol VIII/1, pp. 144–150. 28 for convective as well as acoustical flows. C. C. Lin, “Hydrodynamics of helium II,” Proc. Int. School of Physics, Course XXI ͑Academic, New York, 1963͒, pp. 93–146. 1C. Truesdall and R. A. Toupin, “The classical field theories,” in Handbuch 29L. C. Woods, Thermodynamic Inequalities in Gases and Magnetoplasmas der Physik, edited by S. Flugge ͑Springer Verlag, Berlin, 1960͒, Vol. III/1, ͑John Wiley and Sons, Chichester, 1996͒. pp. 226–793. 30H. J. Bauer, “Phenomenological theory of the relaxation phenomena in 047101-10 A. J. Zuckerwar and R. L. Ash Phys. Fluids 18, 047101 ͑2006͒
gases,” in Physical Acoustics IIA, edited by W. P. Mason ͑Academic, New 33Handbook of Chemistry and Physics, 63rd ed., edited by D. R. Lide ͑CRC York, 1965͒, pp. 47–131. Press, Boca Raton, FL, 1983͒. 31 See Ref. 4, p. 166. 34F. H. Fisher and P. F. Worcester, “Essential oceanography,” in Encyclope- 32 ANSI Standard S1.26-1995, “Method for calculation of the absorption of dia of Acoustics, edited by M. J. Crocker ͑Wiley Interscience, New York, ͑ sound by the atmosphere” Standards Secretariat of the Acoustical Society 1997͒, Vol. I, p. 381. of America, New York, 1995͒.