Thermoviscous Fluid Research Articles

Thermodynamically Consistent Evolution Equations in Continuum Mechanics

This paper addresses the modelling of material behaviour in terms of differential (or rate) equations. To comply with the objectivity principle, recourse is made to invariant fields in the Lagrangian description or to objective time derivatives in the Eulerian description. The thermodynamic consistency is investigated in terms of the Clausius–Duhem inequality with two unusual features. Firstly, the (non-negative) entropy production is viewed as a constitutive function per se. Secondly, the inequality is viewed as a constraint on the pertinent fields and it is solved by using a representation formula, which allows for the the admissibility of a class of models. For definiteness, models of heat conduction are established, within Lagrangian descriptions, while models of the Navier–Stokes–Voigt fluid are investigated within Eulerian descriptions. In connection with thermo-viscous fluids, evolution equations are investigated within the Eulerian description. It is shown that the thermodynamic consistency is compatible with both objective and non-objective evolution equations.

Analytical method for studying acoustic fields radiated by general spherical sound sources in thermoviscous fluids.

The study of acoustic radiation from spherical sound sources plays a crucial role in understanding the thermoviscous effects in practical acoustic problems. However, finding a general solution of acoustic radiation from spherical sound sources in thermoviscous fluids remains a formidable challenge. To advance this issue, an analytical method is developed in this paper to calculate the acoustic field radiated by spherical sound sources with the isothermal boundary condition and arbitrary velocity boundary condition. The developed method is utilized to present the solutions of the acoustic field generated by an oscillating sphere and a general spherical sound source, and the accuracy and validity of these solutions are verified through analytical and numerical methods.

Time-weighted estimates for the Blackstock equation in nonlinear ultrasonics

High frequencies at which ultrasonic waves travel give rise to nonlinear phenomena. In thermoviscous fluids, these are captured by Blackstock’s acoustic wave equation with strong damping. We revisit in this work its well-posedness analysis. By exploiting the parabolic-like character of this equation due to strong dissipation, we construct a time-weighted energy framework for investigating its local solvability. In this manner, we obtain the small-data well-posedness on bounded domains under less restrictive regularity assumptions on the initial conditions compared to the known results. Furthermore, we prove that such initial boundary-value problems for the Blackstock equation are globally solvable and that their solution decays exponentially fast to the steady state.

Toward a unified formulation of the acoustic transfer admittance of cylindrical cavities for reciprocity calibration of microphones

The pressure reciprocity technique for calibration of Laboratory Standard microphones provides the basis for primary measurement standards for sound pressure. This calibration method is described in the International Electrotechnical Commission (IEC) standard 61094-2:2009 where the key aspect relating to the calculation of the acoustic transfer admittance of couplers was recently completed and clarified in the amendment IEC 61094-2:2009/AMD1:2022. Three models are currently provided for calculating this quantity: the low-frequency solution suitable to low frequencies, the extended low-frequency solution suitable to low and medium frequencies, and the broadband solution suitable to medium and high frequencies. It is established that neither of these models is correct for all frequencies routinely considered in many National Metrology Institutes, namely from 2 Hz to 25 kHz, so that a transition must be made. Based on the fundamental equations of acoustics in thermoviscous fluid, this paper provides a unified formulation for the acoustic transfer admittance of cylindrical cavities. Under the common quasi-plane wave approximation, this paper provides a new form of solution for the Fourier equation, not restricted by any assumption on the pressure variation and leading to a new form of propagation equation in the coupler. Two techniques are used to solve the problem, the Laplace Adomian Decomposition Method and a stepped duct approximation technique. Numerical results are presented, providing substantial evidence to support the validity of these formulations. A high level of agreement is observed between the two models, approximately 10−6 dB for the amplitude and 10−4 degrees for the phase, excluding the resonance frequencies of couplers. Finally, this paper evaluates the compatibility of the extended low-frequency solution provided in the IEC amendment with the fundamental equations established in this study. A new simplified solution is provided, highlighting discrepancies in the application of the heat conduction corrective factor as compared to the guidelines outlined in the IEC amendment.

Acoustic radiation force on a spherical thermoviscous particle in a thermoviscous fluid including scattering and microstreaming.

We derive general analytical expressions for the time-averaged acoustic radiation force on a small spherical particle suspended in a fluid and located in an axisymmetric incident acoustic wave. We treat the cases of the particle being either an elastic solid or a fluid particle. The effects of particle vibrations, acoustic scattering, acoustic microstreaming, heat conduction, and temperature-dependent fluid viscosity are all included in the theory. Acoustic streaming inside the particle is also taken into account for the case of a fluid particle. No restrictions are placed on the widths of the viscous and thermal boundary layers relative to the particle radius. We compare the resulting acoustic radiation force with that obtained from previous theories in the literature, and we identify limits, where the theories agree, and specific cases of particle and fluid materials, where qualitative or significant quantitative deviations between the theories arise.

Acoustics of thermoviscous fluids: The Kirchhoff-Helmholtz representation in generalized form.

The Kirchhoff-Helmholtz representation of linear acoustics is generalized to thermoviscous fluids, by deriving separate bounded-region equations for the acoustic, entropy, and vorticity modes in a uniform fluid at rest. For the acoustic and entropy modes we introduce modal variables in terms of pressure and entropy perturbations, and develop asymptotic approximations to the mode equations that are valid to specified orders in two thermoviscous parameters. The introduction of spatial windowing for the mode variables leads to surface source and dipole distributions as a way of representing boundary conditions for each mode. For the acoustic mode the boundary source distribution is expressible in terms of the fluid normal velocity, the normal heat flux, and the vector ω×n̂, where ω is the vorticity on the boundary and n̂ is the unit normal; only the first of these is present in the usual lossless-fluid version of the Kirchhoff-Helmholtz representation. Use of the generalized thermoviscous representation to project exterior sound fields from surface data, where the data may contain contributions from all three linear modes, is shown to be robust to cross-modal contamination. The asymptotic limitations of the thermoviscous modal equations are discussed in an appendix.

Investigation of eigenfunctions of perturbation of the radial component of flow velocity

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of a generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The influence of taking into account the exponential dependence of fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of incompressible fluid flow in an annular channel is investigated. The eigenvalue spectra were obtained numerically. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity disturbances, their possible growth or decay over time. Graphs of the eigenfunctions of the generalized flow stability equation in an annular channel have been constructed. It is shown that the structure of the spectra largely depends on both the properties of the liquid, determined by the functional dependence of viscosity, and on the geometry of the channel. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It has been established that at small values of the thermoviscosity parameter the spectrum is comparable to the spectrum for isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are a greater number of points at which the problem has non-zero amplitudes of transverse velocity disturbances. It is worth noting that the diversity of eigenvalue spectra corresponds to the diversity of eigenfunctions, which have a nontrivial distribution of the oscillation amplitude over the cross section in each case. Smooth curves are obtained for a narrow channel, as for the case of a flat channel. However, as the ratio of the channel radii increases, “jumps” appear. Also, note that the eigenfunctions do not have the property of symmetry; this follows from the fact that the velocity profile in the unperturbed state also does not have symmetry. The maximum values of the eigenfunctions are shifted to the right from the center of the channel, which corresponds to the fact that disturbances arise and grow intensively near the hot wall.

Aerodynamic sound generation in thermoviscous fluids: A canonical problem revisited

Although the Lighthill–Curle acoustic analogy theory is formally exact, the presence of linear source terms related to viscous stresses and non-isentropic density changes makes it unsuitable for studying aerodynamic sound generation in low Reynolds number thermoviscous flows. Here we use an extension of the Ffowcs Williams and Hawkings formulation, with thermoviscous effects explicitly included, to find an analytical solution to the canonical problem of sound radiation from a circular cylinder immersed in a viscous heat-conducting fluid and rotating sinusoidally about its axis. Existing published solutions are compared and an earlier null result is explained. The new analysis reveals the dominant source of sound at low Mach numbers to be unsteady viscous dissipation rather than Reynolds-stress quadrupoles, unless the fluid parameter B=αc2/cp is zero.

Investigation of the spectral characteristics of a thermoviscous fluid flow in an annular channel

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of the generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The effect of taking into account the exponential dependence of the fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of an incompressible fluid flow in a flat channel for various wall temperatures is studied. Spectral patterns of eigenvalues of the generalized equation are constructed. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity perturbations, their possible growth or decay with time. It is shown that the structure of the spectra largely depends both on the properties of the liquid, determined by the index of the functional dependence of viscosity, and on the geometry of the channel. It has been established that for small values of the thermoviscosity parameter, the spectrum is comparable to the spectrum for an isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has nonzero amplitudes of transverse velocity perturbations. The stability of a thermoviscous fluid flow depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It is shown that, at fixed values of the Reynolds number and wave number, the flow can become unstable with an increase in the thermoviscosity parameter.

Thermoviscous liquid conjugate heat exchange simulation in cylindrical channel

The article is concerned with the flow of a thermoviscous fluid in a cylindrical channel for various types of viscosity functions in the joint solution of the problem of conjugate heat exchange between the channel and the surrounding material. A model of a cylindrical channel with a thermoviscous liquid flowing inside at a given velocity and viscosity is considered. The methodology of the process consists in applying a system of model equations that describes the dynamics of thermoviscous fluid flow processes based on the laws of phase energy conservation, the continuity equation, the Navier-Stokes momentum conservation equation taking into account conjugate heat transfer and the heat transfer equation for a solid. Numerical implementation of model representations was carried out using the Comsol Multiphysics digital package. Velocity and temperature profiles were obtained in the cross section of the channel. The influence of the viscosity function on the flow profile is shown, as well as the quantitative characteristics of the flow and heat transfer. An analysis was made of the dependence of the nature of the velocity and temperature profiles on the viscosity of the liquid and the type of dependence of viscosity on temperature. The result of the work done is the found features and regularities that arise in the process of conjugate heat transfer of a thermoviscous liquid in a cylindrical channel when comparing the linear and exponential dependences of viscosity on temperature. The analysis of the obtained dependences of viscosity and temperature in the flow profile, the value of the maximum flow rate was carried out. It is shown that viscous friction causes a local increase in temperature near the pipe walls, followed by an increase in fluid velocity due to a decrease in viscosity. The results obtained are basic for the further development of this topic and will be used to solve new problems in the field of modeling the thermoviscous fluid flow and conjugate heat transfer.

Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube

Abstract

Gravity-driven thermoviscous liquid film down a heated or cooled vertical cylinder

Gravity-driven flow of a liquid over an isothermal cylinder is unconditionally unstable. The flow of a thermoviscous fluid over a heated or cooled substrate shows interesting stability behavior. The relevant parameters in the model affect the spatiotemporal nature of the instability.

Turbulent kinetic energy transfer and dissipation in thermoviscous fluid flow

Turbulent flow of thermoviscous liquid is studied in a three-dimensional region with periodicity in two directions. Flow characteristics are described in the terms of equation for turbulent kinetic energy: this allows to differentiate contributions from different components related to generation, transport, and dissipation of turbulent kinetic energy. Those terms can be calculated from averaging the moments of different order. The previous studies demonstrated that thermoviscous liquid flow occurs through several stages of evolution, including the unsteady turbulence. This allows discussing the problem of mathematical rigorous statement and applicability of different methods for averaging. Existence of spatial periodicity allow using a combined spatial-time averaging for different values on the interval of steady turbulence. Results are presented as a set of Z—t-diagrams. Besides, the paper presents analysis of flow development on the basis of direct visualization of velocity and temperature.

Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality.

This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders . While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works.

Thermoviscous fluid flow in nonisothermal layer: structures, scales, and correlations

The paper studies turbulent mixing in thermoviscous fluid flow in a 3D cubic domain which is extended periodically in two directions (X and Y). The flow turbulization develops under the impact of two-dimensional chaotic disturbances at mass average Reynolds number Re1 = 4704. The vortex field structure is discussed in terms of an isosurface of Q-criterion and local enstrophy ζ1. For the advanced stages of flow evolution, the study considers Eulerian correlation coefficients for velocity fluctuations (auto-correlation functions) and the cross-correlations of pressure and temperature. The Eulerian correlation coefficient is split for analysis of correlation characteristics in periodicity and wall-normal directions. The integral scale is evaluated depending on the distance to the walls. The flow analysis is performed in the terms of viscous scale. The mesh resolution is evaluated for the flow regions corresponding to the logarithmic boundary layer and the near-wall thermal layers.

The Unsteady Flow of a Fluid of Finite Depth with an Oscillating Bottom

In this paper, the unsteady flow of a fluid of finite depth with an oscillating bottom is examined. The flow is assumed in the absence of viscous dissipation. The governing equations of the flow are decoupled in the velocity and temperature fields. The velocity and temperature fields have been obtained analytically. The effects of various material parameters on these fields have been discussed with the help of graphical illustrations. It is noticed that the upward thrust (ρfy) vanishes when Reiner Rivlin coefficient of viscosity (μc) is zero and the transverse force (ρfz) perpendicular to the flow direction vanishes for thermo-viscosity coefficient (α8) is zero. The external forces generated perpendicular to the flow direction is a special feature of thermo-viscous fluid when compared to the other type of fluids.

Some Features of Hydrodynamic Instability of a Plane Channel Flow of a Thermoviscous Fluid

In this work we study the effect of the exponential temperature dependence of the incompressible fluid viscosity on the critical parameters of the flow hydrodynamic stability in a flat channel at various specified values of the wall temperature. The temperature field perturbations are thought to be absent. The eigenvalue spectra for the generalized Orr-Sommerfeld equation are constructed at sufficiently large Peclet numbers. The spectra structure, neutral stability curves, and the critical Reynolds number are shown to largely depend on the fluid properties that are determined by the exponential viscosity function.

Isolated Convection Modes for the Anomalous Thermoviscous Liquid in a Plane Cell

This work is devoted to the numerical research of free convection by a Newtonian anomalous thermoviscous fluid in a flat cell. The cell is heated from below, cooled from above; and the lateral boundaries are heat insulated. The viscosity anomaly of the fluid is modeled by the Gaussian function of temperature and is characterized by two parameters. A flow regime with isolated convective cells separated by a region of high viscosity, the so-called viscous barrier, was detected at a certain set of control parameters. For these flow regimes, current lines, heat fluxes, temperature fields, and contours of the components of the velocity vector of the fluid are given.

Numerical Modelling of Boundary Layers and Far Field Acoustic Propagation in Thermoviscous Fluid

To model linear acoustics in a thermoviscous fluid in open domain and time-harmonic regime, a Finite Element formulation in a bounded meshed domain is combined with the integral representation of the field for the propagative solution. The integrals are non-singular and involve the only Finite Element node values for temperature variation and particle velocity variables. To overcome the non-uniqueness of solutions at fictitious resonant frequencies, a Burton-Miller combination of integral representation is used. This formulation is suitable to compute acoustic radiation, scattering and diffraction by objects or mutual interaction between transducers. Two-dimensional computational experiments are presented in an infinite, open domain (exterior), showing that the model can be achieved in meshing only a thin domain surrounding the physical boundaries of a device.

Dynamic equations of a transversely isotropic, highly porous, fibrous material including oscillatory heat transfer effects.

The dynamic equations of a transversely isotropic fibrous, highly porous material are presented in terms of microstructure-derived analytical expressions for viscous dissipation, and analytical expressions for the oscillatory heat transfer between the thermal fields of the solid cylindrical glassfibres and the surrounding viscous fluid. This represents the non-equilibrium thermal expansion of the fluid, occurring when waves propagate in the porous material, and results in a frequency-dependent scaling of the fluid dilatation term. A state-space transfer matrix solution of the governing equations has been introduced, allowing the numerical acoustical performance of the fibrous material to be investigated, including the acoustical effects of heat transfer. In order to understand the dissipation mechanisms of the viscous and thermal boundary layers on the surface of the fibres and the validity of the assumptions made in the current model, a thermoviscous acoustic fluid finite element procedure has also been introduced. The results from these simulations illustrate the frequency-dependent interaction of the boundary layers between neighbouring fibres in the porous material.