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Variational Approach to the Volume Viscosity of Fluids Allan J Old Dominion University ODU Digital Commons Mechanical & Aerospace Engineering Faculty Mechanical & Aerospace Engineering Publications 2006 Variational Approach to the Volume Viscosity of Fluids Allan J. Zuckerwar Robert L. Ash Old Dominion University, [email protected] Follow this and additional works at: https://digitalcommons.odu.edu/mae_fac_pubs Part of the Aerodynamics and Fluid Mechanics Commons, Engineering Mechanics Commons, and the Fluid Dynamics Commons 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 ␳ D␳ץ ␳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.
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