A discontinuous variational principle implying a non-equilibrium dispersion relation for damped acoustic waves
Introduction
Wave theories in general have their origins in equations stemming from underlying continuum theory; in the case of damped acoustic waves propagating in a liquid or gas these comprise the continuity and Navier–Stokes–Duhem1 equations together with an appropriate thermal conduction–convection equation. According to Olsson, Belevich, Landau and Lifschitz [1], [2], [3] they result in the following fundamental equations: for the mass density , the velocity field and the specific entropy as fundamental fields, whereas the pressure and the temperature result from the thermodynamic relations: where is the specific inner energy of the fluid. denotes the material time derivative while: is the shear rate tensor. In Eqs. (3), (6) denotes the trace of a tensor, the superscript the transpose and the symbol the dyadic product. The three coefficients in Eqs. (1) to (3), the shear viscosity , the dilatational viscosity (Lamé’s first parameter) containing the volume viscosity and the thermal conductivity , are assumed to be constant.
Sound waves can be obtained as solutions of the linearised fluid equations of motion (1)–(3). This topic has been well researched, see e.g. the review article of Jordan [4]. If transmitted through a fluid medium over a long distance, damping due to dissipation may become relevant. An alternative approach is elaborated by Marner et al. [5], starting from a variational principle for the respective continuum. Two competing mechanisms for damping exist, based on thermal conductivity and on viscosity. For some special fluids, for example pure water, the thermal conductivity can be neglected [6]. This assumption is adopted for the wave example elaborated in the present work, but the general theory presented includes thermal conductivity as a platform for a future analysis toward wave damping by thermal conductivity.
The first decisive step in our analysis is the proper formulation of the variational principle. As is well known, the use of Hamilton’s principle is ideally suited to, for example, the field of conservative Newtonian mechanics. Contrary to this, in continuum theories many open problems remain unsolved, typically in the field of viscous flow; since there are, in general, no obligatory construction rules for establishing variational principles, for certain problems a variety of suggestions have appeared from various authors based on different approaches. One has to distinguish between two major categories, namely between variational formulations based on a field description (Eulerian description), and a stochastic variational description based on a material description (Lagrangian description) and averaging particle motion. In the case of field descriptions Clebsch [7] established an approach based on the representation of the velocity by three potentials, the so-called Clebsch variables for inviscid barotropic flows [8], [9]. Subsequent modified forms of the Clebsch transformation have been applied successfully to magnetohydrodynamics [10] and plasma dynamics [11]. Seliger and Whitham [12] generalised Clebsch’s Lagrangian, supplementing two additional fields: the specific entropy and an additional field , introduced three decades previously by van Dantzig [13] as the material integral of the temperature , i.e. , and termed thermasy. Independently the notation thermal displacement is used for this quantity, for instance by Green and Naghdi, Jordan and Straughan [14], [15].
Despite including thermal degrees of freedom, Seliger and Whitham’s approach remains restricted to adiabatic and therefore reversible processes. As a continuation of their work, Zuckerwar and Ash [16], [17] suggested an extended Lagrangian considering only volume viscosity, leading to equations of motion containing qualitatively the effect of volume viscosity but differing quantitatively from the compressible Navier–Stokes equations, also known as the Navier–Stokes–Duhem equations [1], [2] without shear viscosity. They interpret their result as a generalisation of the theory of viscous flow towards thermodynamic non-equilibrium. Based on a rigorous analysis of the fundamental symmetries the Lagrangian has to fulfil, with particular regard to Galilean invariance [18], Scholle and Marner [19] suggested a Lagrangian for viscous flow considering both shear viscosity and volume viscosity reproducing Zuckerwar and Ash’s Lagrangian for the case . Again, the resulting equations of motion differ from the Navier–Stokes–Duhem equations. Considering six simple flow examples, two of them (steady shear flows) led to exact reproductions of the classical solutions, two others (transient flows) revealed the impact of viscosity on the flow at least in a qualitatively correct manner, whereas no physically reasonable solutions could be constructed for the two pressure-driven flows.
In order to resolve the above issue, Scholle and Marner [19] made use of an analogy between quantum mechanics and fluid mechanics discovered by Madelung [20] to formulate a new Lagrangian by relating the specific entropy and the thermasy to a complex field : termed the field of thermal excitation by Anthony [21], [22], [23], leading to a discontinuous Lagrangian containing an additional parameter that can be interpreted as a relaxation rate toward thermodynamic equilibrium. The reference temperature and the associated reference specific heat: enter the definition (7) as parameters necessary for dimensional reasons. 2 The field of thermal excitation is motivated in line with a rigorous analysis of symmetries and associated Noether’s balances of Schrödinger’s theory resulting in a general concept elucidated in the Appendix. By careful analysis it is proven that the dynamics resulting from Hamilton’s principle can consistently be interpreted as a generalisation of the theory of viscous flow towards thermodynamic non-equilibrium, with the parameter being the relaxation rate, giving rise to recovery of the well-known Navier–Stokes equations and the balance of inner energy when applying the limit to the resulting equations of motion.
Alternatively, scientists working on statistical physics have provided models for viscous flow based on a stochastic variational description, see e.g. [24], [25], [26], [27], [28], [29]. In their recent work Marner et al. [5] exposed conceptual similarities of their former work with the stochastic approach by interpreting the discontinuities occurring on a microscopic scale as fluctuations. Therefore, although originally motivated by previous research involving deterministic field theories, the use of the discontinuous Lagrangian seems to embrace aspects of both concepts and can therefore be considered as kind of ’in-between’ or lying betwixt deterministic and statistic approaches, since equations of motions result which are the classical ones plus ‘deterministic’ fluctuations.
This paper provides: (i) a generalisation of the Lagrangian proposed by Marner et al. [5] toward thermal conduction, (ii) an averaged wave equation for viscously damped sound waves influenced by thermodynamic non-equilibrium effects and (iii) its associated dispersion relation. It is organised as follows. In Section 2 the general approach based on the discontinuous Lagrangian method is elaborated for viscous flow with thermal conduction by generalising existing proposals for viscous flow without thermal conductivity and inviscid flow with thermal conductivity. A discontinuous Lagrangian and associated Euler–Lagrange equations are provided. Next the method is applied to damped acoustic waves due to viscosity in Section 3 by linearisation of the Euler–Lagrange equation and applying an ensemble average to the latter, ending up with a single wave equation. Its associated dispersion relation is discussed. Conclusions are drawn in Section 4 together with prospective further research topics.
Section snippets
Variational approach
In the general calculus of variations a continuous system is considered, the state of which is defined by independent fields and its evolution determined by Hamilton’s principle: based on free and independent variation of the fields and their first order spatial and temporal derivatives for fixed values at initial and final time, .
For the particular problem considered here the fields are the mass density , the velocity , the Clebsch variables3
Application to damped acoustic waves
The problem of propagating acoustic waves, the solution of which, both with and without damping, has been obtained to-date starting with the linearised Navier–Stokes–Duhem equations together with the continuity and thermal conduction equation, is considered below starting instead from the Euler–Lagrange equations derived above using a variational principle. For the sake of convenience only viscous damping is taken into account.
Conclusions and outlook
Motivated by existing Lagrangians for viscous flow without thermal conduction and inviscid flow with thermal conduction a Lagrangian has been introduced for viscous flow with thermal conduction, utilising the complex field of thermal excitation and being a generalisation of the one proposed by Marner et al. [5]. A striking feature is a discontinuity due to a complex logarithmic term leading to contributions to the resulting evolution equations that can be interpreted as small fluctuations due
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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