# How Does Heat Propagate in Liquids?

## Abstract

**:**

## 1. Introduction

_{q}is the heat flux, K is the thermal conductivity and T is the temperature. Equation (1) contains the information asserting that if a uniform temperature gradient is established throughout a homogeneous fluid, then the heat flow is everywhere and every time proportional to the temperature gradient. Equations such as (1) are very common in Irreversible Thermodynamics [9] because they express the Onsager’s theory property that the response of a system to an applied force is simultaneous with the application of the force [10,11]. As a general rule, such simultaneity in a macroscopic theory turns out to be an approximation to a causal behavior, where the response to a force comes after the application of the force. To account for the causality behavior, it is necessary to modify the equation for heat conduction to take into consideration a relaxation time, τ, not present in Equation (1).

_{q}/∂t is added to the Fourier law:

_{q}/∂t makes J

_{q}and ∇T no more proportional everywhere and every time).

**k**-gap” or Gapped Momentum States (GMS) [19,20,21,22]. We will show that the DML is even characterized by a

**k**-gap, whose importance is crucial in the identification of the relaxation times typical of a dual system [17]. Recently, a thorough analysis has allowed to recognize also the presence of a Gapped Energy States (GES) in dual systems [23].

## 2. Historical Background

_{C}> in connection with the collision integral of the system, doing so to demonstrate that the equation that correctly describes the diffusion is the one that will then take his name, and that he himself defines as an equation with “memory”. Solutions of the hyperbolic equation for heat transport predict a phenomenology of the diffusive type, after an initial short-lived undulatory phase consisting in a wave propagation with velocity ${\upsilon}_{C}^{wp}$ (see Equation (37)). However, no hint on the physical nature of these waves comes from the Cattaneo approach, although by digging into the depths of the literature, one can discover that already in 1946 Peshkov [35] had hypothesized that in low-temperature liquids, “a gas of thermal quanta capable of performing vibrations similar to those of sound should exist”.

_{1}, and the other by molecules, ψ

_{2}, so that q = ψ

_{1}+ ψ

_{2}. Both ψ

_{1}and ψ

_{2}will each obey a relaxation equation relating the time derivatives ψ

_{i}(i = 1,2) to ψ

_{i}and ∇T. At this point, the author claims that the inertia of molecules justifies the fact that a sudden appearance of a temperature gradient will initially affect only ${\stackrel{\u2022}{\psi}}_{i}$ but not ψ

_{i}. Then he builds a very general relaxation equation that he uses to obtain the density fluctuation and temperature-dependence in the system. What matters of this approach is that i) Nettleton assumes the liquid thermal content as made of two components, and that ii) these two components interact among themselves in a similar way to the two-field potential representing displacements and velocities, ϕ

_{1}and ϕ

_{2}, introduced in Equation (30), or, alternatively, the two sub-systems described by Equation (31).

## 3. The Dual Model of Liquids and the Heat Propagation Equation with “Memory”

#### 3.1. The Dual Model of Liquids

_{T}per unit of volume of a liquid at temperature T is

_{V}the specific heat at constant volume per unit mass, Θ is the Debye temperature of the liquid at temperature T. The fraction ${q}_{T}^{wp}$ of this energy

^{wp}is the number of wave packets per unit of volume, and ε

^{wp}is their average energy; ${m}^{*}=m\frac{{\displaystyle \underset{0}{\overset{T}{\int}}\rho {C}_{V}d\theta}}{\rho {C}_{V}T}$ is introduced only to simplify the expression. ${q}_{T}^{wp}$ is propagated through the liquid by means of inelastic interactions between the liquid particles and the lattice particles, i.e., the wave packets that transport the thermal, or generally the elastic energy. Figure 1 and Figure 2 show the way in which the interaction between a liquid particle and a lattice particle works. To make the model more intuitive, we may ideally divide this elementary interaction into two parts: one in which the lattice particle collides with the liquid particle and transfers to it momentum and energy (both kinetic and potential) and one in which the liquid particle relaxes and the energy is returned to the thermal pool through a lattice particle like in a tunnel effect. The energy ∆ε

^{wp}and momentum ∆p

^{wp}exchanged are, respectively:

^{th}is the interaction potential between the lattice particle and the liquid particle that is in turn obtained by

^{th}, or ϕ

^{th}, are in order. First, ϕ

^{th}is supposed to be anharmonic because of the inelastic character of the interaction. The interaction is in fact not instantaneous but lasts <τ

_{p}> during which the particle is displaced by <Λ

_{p}> (here and in the rest of the paper, the two brackets < > indicate the average over a statistical ensemble of the quantity inside them). Second, f

^{th}can be positive or negative depending on whether the quantity (J

_{q}/u

_{ϕ}) increases or decreases as consequence of the interaction. In Equations (5)–(7), <ν

_{1}> and <ν

_{2}> are the wave-packet (central) frequency before and after the interaction, $\Delta {E}_{p}^{k}$ and ∆ψ

_{p}are kinetic and potential energy acquired by the liquid particle as consequence of the interaction. Finally, u

_{ϕ}is the phase velocity associated with the wave packet and σ

_{p}is the cross-section of the “obstacle”, the solid-like cluster, on the surface of which f

^{th}is applied. The reader who wants to learn more about the above is warmly addressed to the related literature [17,18].

_{2}>−<ν

_{1}>). Due to its time symmetry, we have assumed this mechanism to be the equivalent of Onsager’s reciprocity law at microscopic level [10,11,17]. In a pure isothermal liquid, energy and momentum exchanged among the icebergs are statistically equivalent, and no net effects are produced. In other words, events of type (a) are equally likely to occur as events of type (b) and will alternate to keep the balance of the two energy pools unaltered. In addition, the macroscopic equilibrium will also ensure the mesoscopic equilibrium; events (a) and (b) will be equally probable along any direction, giving a zero average over time and space. On the contrary, if a symmetry breaking is introduced, as, for instance, in the case of a temperature gradient which we are going to discuss in this paper, events of type (a) will statistically prevail over events of type (b) along the preferential direction of the externally applied temperature gradient.

_{wp}> is the extension of the wave packet and <d

_{p}> is that of the liquid particle. Once <τ

_{p}> has elapsed and the liquid particle has travelled by <Λ

_{p}>, the particle relaxes the energy stored into internal DoF; then, it travels by <Λ

_{R}> during <τ

_{R}> (not shown in the Figure) [17].

_{p}(Equation (5)) pertaining to the liquid particle internal DoF. This energy content is then released back to the heat current a <Λ> step forward, like in a tunnel effect. This is the physical mechanism in the DML constituting the relaxation effect of heat propagation during the transient phase, i.e., when the Cattaneo equation correctly describes the time evolution of the thermal energy propagation (or of the temperature distribution) in a system out of equilibrium. Similar reasoning can be applied to a flux of liquid particles when a concentration gradient is imposed to the system; it is indeed enough to follow the reverse time evolution of the elementary interaction between a liquid particle and a lattice particle, as illustrated in Figure 1b.

^{TP}be the temperature gradient that is forming across the system. To fix the ideas, let T

_{h}be the temperature of the heat source applied to the system. Because the thermal front travels through the system, it will be

_{c}(z) is the temperature of the advancing thermal front at the point z, and z is the coordinate of the advancing thermal front at time t. Due to the application of the external temperature gradient, the heat flux crossing the system will give rise to an increase in the number of wave-packet ↔ liquid particle collisions in the direction of the heat flux J

_{q}. In fact, in this case, events of type (a) in Figure 1 have a larger probability to occur than events of type (b). At equilibrium, the flux of wave packets is driven in the DML by the presence of a virtual temperature gradient [17], <δT/δz>. If <ν

_{p}> is the average number per second of wave-packet ↔ liquid particle collisions due to <δT/δz>, along the direction of J

_{q}such number is <ν

_{p}>/6 for symmetry reasons. When ∇T

^{TP}is applied, there will be an imbalance in such number, namely an increase in collisions along z; let δ<ν

_{p}> indicate the increase in the number of wave-packet ↔ liquid particle collisions per second due to ∇T

^{TP}. δ<ν

_{p}> is a quantity constant neither in time nor in space, because such is ∇T

^{TP}obtained by Equation (8). When the SS is reached, the temperature gradient and the heat flux will have reached their stationary values, δ<ν

_{p}> will even reach its stationary value (the one used in [17]).

_{p}>, we suppose as first approximation that the application of ∇T

^{TP}increases proportionally the number of wave-packet ↔ liquid particle collisions:

_{p}> along the direction of J

_{q}, each of average length <Λ>, for a total distance travelled <Λ>δ<ν

_{p}> per second along the direction of heat flux generated by the external local temperature gradient. This quantity represents the drift velocity ${\langle {\upsilon}_{p}^{th}\rangle}^{TP}$ of the liquid particle during the TP along z due to the external temperature gradient:

^{TP}which, in the TP, as is known, can have much greater values than that of the stationary state, it being applied over much shorter distances. At steady state, ${\langle {\upsilon}_{p}^{th}\rangle}^{TP}$ will assume the value $\langle {\upsilon}_{p}^{th}\rangle $ obtained in [17].

_{h}higher than that of the system. The principle on which the reasoning is based, however, also holds true in the case in which the flow of heat inside the system is due to the presence of a heat sink at a temperature T

_{c}lower than that of the system, or even in the case in which the system is in contact with two different sources, one with a temperature higher than that of the system and the other with a lower temperature, keeping the average temperature of the system unchanged. Of course, if there is always an imbalance in the average number of collisions per second in the direction of the heat flow, the previous situations obviously differ from each other due to the different final thermal content of the system.

^{th}develops, it acts for <τ

_{p}> seconds displacing the particle by <Λ

_{p}>. Energy <∆ε

^{wp}> and momentum <∆p

^{wp}> obtained by Equations (5) and (6), respectively, are transferred to the particle during <τ

_{p}>, increasing its kinetic and potential energy and exciting internal vibrational energy levels. These DoF oscillate similarly to those pertaining to solid state, giving origin to (quasi) elastic waves with characteristic length <Λ

_{0}> and period <τ

_{0}>. <τ

_{R}> seconds later and <Λ

_{R}> meters forward, they are released back to the wave-packet pool, and the process is repeated again <τ

_{wp}> seconds later and <Λ

_{wp}> meters forward, <τ

_{wp}> being the phonon mean free flight time and <Λ

_{wp}> their mean free path, i.e., the distance travelled between two interactions with liquid particles. What matters here in order to understand the dynamics linked to the relaxation time is to recognize that <τ> is the time interval during which the energy disappears from the liquid thermal pool because it is trapped in the internal DoF; once <τ> has elapsed, it reappears in a different place. In this way, relaxation times, introduced ad hoc by Frenkel, find an immediate physical interpretation. The wave packet emerges from the collision with reduced energy and momentum, while the particle acquires the energy and momentum lost by the phonon. The effect is that of having converted part of the energy carried by the phonon into potential energy of the liquid particle.

#### 3.2. The Heat Propagation Equation “with Memory” in the DML

^{wp}= <Λ

_{wp}> / <τ

_{wp}> = λ

^{wp}·ν

^{wp}is the wave-packet velocity, and the last equality is obtained by means of the wave-packet specific heat at constant volume ${C}_{V}^{wp}$ [17,18]

^{wp}variation vs. temperature rather than the internal energy ${q}_{T}^{wp}$),

_{m}. In other words, heat propagation is inhibited for waves with wave vector below k

_{m}. This means that we are in presence of Gapped Momentum State (GMS), or simply the

**k**-gap, that is always present in all systems where the propagation of energy is described by equations such as Equation (15), or where the dispersion relation is like Equation (17) [19,20,21,22,23]. Equation (21) provides a way to calculate <ϑ>

_{M}, the maximum relaxation time, once k

_{m}is known from experiments. The presence of the

**k**-gap, or of a <ϑ>-gap, also has an influence on the relevance of the additional term $\langle \vartheta \rangle \frac{\partial {J}_{q}^{wp}}{\partial t}$ that constitutes the difference between the Cattaneo and the Fourier laws. In turn, the identification of a <ϑ>-gap recalls the presence of a GES [23]. Finally, differently from the Fourier equation, Equation (15) contains also the second time derivative of T multiplied by the relaxation time <ϑ>, thus allowing for damped propagative waves as solutions, with damping constant <ϑ> and velocity ${\upsilon}_{C}^{wp}$ obtained by Equation (18). We will deeply analyze the consequences of Equations (15), (20) and (21) in the Section 4.

## 4. Discussion

^{wp}, we obtain the momentum flux ${J}_{p}^{lp}$ transferred from the heat current to the liquid particles

_{m}introduced with Equation (20) provides, through the relation ν

_{m}=k

_{m}

^{·}u

_{ϕ}/2π, the critical frequency for the appearance of second sound (see Section 2), while in [17], it has been shown that DML is able to justify such a (second) fast sound both on a physical base and numerically.

_{l}is its viscosity. Such a comparison shows that the ratio $\frac{{K}_{l}^{wp}}{{C}_{p}^{wp}}$ is dimensionally a viscosity, η

^{wp}, that has a similar role for the temperature as ${\eta}_{l}$ does for the fluid current. Therefore, we could speculate that if ${\eta}_{l}$ represents the capability of a fluid to transmit momentum, η

^{wp}represents the same capability for the wave-packet current. Indeed, bartering Equation (13) for ${C}_{p}^{wp}$, the ratio $\frac{{K}_{l}^{wp}}{{C}_{p}^{wp}}$ is just the momentum per unit of surface carried by wave packets.

^{11}÷ 10

^{12}Hz.

_{m}revealing the so-called

**k**-gap. This aspect has been extensively discussed elsewhere [17,19,20,21,22,23]; here, we want, however, to also focus the attention on Equation (21) and on the following expression that provides the gap in terms of wavelength instead of momentum:

_{C}> as Equation (28) is obtained by observing that it holds

**k**-gap be. Here, we shift the attention to the <Λ

_{C}>-gap or the <ϑ>-gap rather than the

**k**-gap. Recalling that a wave is well defined only if its wavelength is smaller than the propagation distance, Equations (21) and (28) tell us that there is an upper limit also for the relaxation time and for the distance travelled by wave-packets during the propagation of the thermal signal, or, which is the same, for the maximum length of the tunnel, or, alternatively, for the maximum number of liquid particles with which a lattice particle may interact before being definitively damped (remember that in Equation (11), it holds that <ϑ> = n<τ> with n ≥ 1). This offers us the possibility to illustrate the dynamics occurring in a liquid during the thermal transient. When an external temperature gradient is applied, the thermal content obtained by Equation (4) increases, determining an imbalance of the phonon flux, so that there will be an excess of interactions as well of the energy transferred from the thermal current to the liquid particles through events such as those of Figure 1a (see Equation (9)). Let us start at z = z

_{0}. Each interaction lasts <τ> = <τ

_{p}> + <τ

_{R}>, the time interval during which the energy disappears as liquid free energy to become an iceberg’s internal and kinetic energy, and the liquid particle moves by <Λ> = <Λ

_{p}> + <Λ

_{R}> forward; after that, the emerging wave packet has lost part of its initial polarization. Depending on its residual energy and momentum, it may interact with another liquid particle <τ

_{wp}> seconds later, and the above process is replicated, say, n times. The overall duration of the randomization process lasts <ϑ> = n<τ>, during which the particle, and the thermometric front, will have advanced by ${\langle \mathrm{\Lambda}\rangle}_{C}={v}_{C}^{wp}\cdot n\cdot \langle \tau \rangle $, determining a temperature increase by ∆T over <Λ

_{C}>. At the end of <ϑ>, a liquid warmer both in its molecular and gas of excitation components will be in contact with the still unperturbed medium laying beyond z = z

_{0}+ <Λ

_{C}>, setting the stage for a replica of the events. At z = z

_{0}, instead, in absence of a new advancing front, the process of heat transport after <ϑ> seconds reaches the steady state, the propagation of thermal excitations becoming at this point purely diffusive.

**k**-gap sets three distinct intervals for the propagating modes, namely (i) non-propagating shear modes, (ii) damped oscillatory shear modes and (iii) purely elastic non-dissipative shear modes. The authors recognize the presence of solid-like structures in liquids for distances where purely elastic non-dissipative modes are allowed, a picture absolutely similar to that of the DML. Comparing then <ϑ>

_{M}and <Λ

_{C}>

_{M}with <τ> and <Λ> of Equations (5) and (6), respectively, (or with their multiples, <ϑ> = n<τ> and <Λ

_{C}> = n<Λ>, with n ≥ 1), we may envisage the following correspondences: (i) the momentum carried by wave packets is too low to interact with liquid particles, and there are no propagating modes (n = 0); (ii) the momentum is large enough to allow the wave packet to interact with several liquid particles, thus obtaining damped oscillating modes and n ≥ 1; (iii) this is what happens inside a single liquid particle, a solid-like structure with the dispersionless propagation of elastic perturbations. In this last case, one experiences the propagation velocity of elastic waves similar to that of the corresponding solid [17,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61]. These limitations are therefore an indirect evidence of the presence of pseudo-crystalline structures in liquids.

_{0}> and lifetime <τ

_{0}> [17] within a liquid particle (see Figure 2) that are connected with <Λ

_{C}> and <ϑ>. They can be deduced from the experimental values obtained in light scattering experiments for the wave-vector k. Indeed, <Λ

_{0}> will be a multiple of the phonon wavelength λ

^{0}, <Λ

_{0}> =aλ

^{0}, and <τ

_{0}> that of τ = 1/ν

^{0}, <τ

_{0}> = a/ν

^{0}, with a > 1. Using the data for water of [58,60], typical values for the parameters characterizing a phonon (variation range is function of temperature, pressure and k orientation) are (central) frequency n/<τ

_{0}> = <ν

^{0}> ≈ 0.95 ÷ 2.5 THz, wave-length <Λ

_{0}>/n = <λ

^{0}> ≈ 1 ÷ 3 nm and velocity <Λ

_{0}>/<τ

_{0}> = <λ

^{0}>·<ν

^{0}> = <u

^{0}> ≈ 3.1 ÷ 3.4·10

^{3}m/s. Interestingly, this last value is in very good agreement with the experimental data obtained for the propagation velocity of thermal waves in water [46] of 3.2·10

^{3}m/s (see also Figure 5 in [17]).

_{C}>-gap is a direct consequence of the presence of dissipation that takes place during <ϑ>. The absence of dissipation would imply an infinite propagation range, as it happens in perfect ideal crystals and as is described by the classical Fourier law. Because of the relevance of these aspects, they will be carefully dealt with in a separate paper [62], being outside the main topic of the present one. However, it is worth anticipating some additional comments regarding the presence of

**k**-gap [19,20,21,22,23]. Because the

**k**-gap in liquids is present only in the transverse spectrum, while the longitudinal one remains gapless, it is reasonable to assume the

**k**-gap in liquids to be related to a finite propagation length of shear waves. For the

**k**-gap to emerge in the wave spectrum, two essential ingredients are mandatory. The first is to obtain a wave-like component enabling wave propagation; this is represented in the DML by the wave packets that in turn allow for propagation of the thermal (and generally of the elastic) signal through progressive waves. The second consists in producing a dissipative effect that disrupts the wave continuity and dissipates it over a given distance, thus destroying waves and giving origin to the

**k**-gap. The latter is represented by the wave-packet ↔ liquid particle interaction, which works by moving the wave packets from where they are absorbed by the liquid particle to where they return to the system energy pool, like in a tunnel effect. If <ϑ> is the time during which the shear stress relaxes, then ${\langle \mathrm{\Lambda}\rangle}_{C}={\upsilon}_{C}^{wp}\cdot \langle \vartheta \rangle $ represents the shear wave propagation length (or liquid elasticity length).

_{M}, also has an influence on the additional (negative) term $\langle \vartheta \rangle \frac{\partial {J}_{q}^{wp}}{\partial t}$ characteristic of the Cattaneo equation, this being limited by the upper limit for <ϑ>

_{M}.

**k**-gap) constituted by two mutually interacting sub-systems that we identify in the DML with the lattice particles and the liquid particles. They introduce a two-field potential representing displacements and velocities, ϕ

_{1}and ϕ

_{2}, respectively. Neglecting the details of the mathematical formalism, what matters is that the equations of motion for the two scalar fields decouple, leading to two separate Cattaneo-like equations for ϕ

_{1}and ϕ

_{2}, whose solutions are the following:

_{1}and ϕ

_{2}reduce and grow over time <τ>, respectively (as electrical current and voltage do for an inductance); this implies that the two interacting sub-systems, represented by the two scalar fields, exchange energy and momentum, like wave-packets and liquid particles do. Therefore, we may hypothesize that the interaction between the population of wave packets and that of liquid particles is described by such couple of mutually interacting potentials. In addition, because the total scalar field is the product of ϕ

_{1}and ϕ

_{2}, the total energy of the whole system does not vary with time, i.e., it is a constant of motion, as expected in the DML for systems constituted by the two populations of mutually interacting sub-systems (as in an electrical circuitry without sources or sinks). Finally, the motion is dissipative and hydrodynamic, and could represent a possible microscopic origin for the viscosity. Theories in which the system interacts with its environment have been used to explain important effects involving dissipation [19,23].

_{k}along the decay process

^{wp}(we limit here to consider an idealized dispersionless elastic medium). In an isothermal system, they move in random directions, so that we may consider the following relation holding for u

^{wp}[40,64,65]:

^{wp}multiplied by the velocity of advancement of the wave-front, ${\upsilon}_{C}^{wp}$, provides the power per unit cross-section, $\Delta {J}_{q}^{wp}$, dissipated by the wave-packets:

_{C}>, one obtains the thermal power per unit of volume dissipated by phonons along <Λ

_{C}>:

_{V}, one obtains the temperature variation vs. time of the liquid contained in a volume of unitary cross-section and height <Λ

_{C}>:

## 5. Conclusions

^{th}that allows the propagation not only of energy but also of momentum. The interaction is characterized by a tunnel effect, by means of which the energy subtracted to the phonon pool is sequestered for a time lapse <τ> in a non-propagating form, the internal DoF of the liquid particle. Once <τ> has elapsed, the liquid molecule has travelled through the liquid by <Λ> and in that point the energy emerges from the tunnel and returns to the phonon pool. This interaction has allowed to apply a hyperbolic equation for the first time to describe the heat (or mass) propagation in a liquid, thus solving the dilemma of infinite diffusion velocity which is typical of the Fourier law. The additional term distinguishing the Cattaneo-type equation from the Fourier law is indeed justified from the physical point of view just by the anharmonic interaction liquid particle ↔ lattice particle. Tunneling in DML has various consequences. First, it represents the missing link to justify the use of a hyperbolic equation to describe the propagation of a thermal (and elastic) signal in a liquid. The hyperbolic equation is of course representative of the transient phase, when the effect of the propagation delay is also present at the macroscopic level. At stationary state, the consequences of the tunnel effect are no longer appreciable at the macroscopic level, although still present at the microscopic level but equally distributed throughout the system. The hyperbolic equation reduces to the parabolic Fourier equation once the steady state is reached. Furthermore, the tunnel effect justifies the presence of shear waves on distances of the order of the dimensions of the molecular clusters. The above picture also allows to justify the additional initial condition needed to solve the hyperbolic equation that results to be dependent only upon constitutive properties of the medium. Therefore, another capability of DML is demonstrated.

_{V}deduced in the Phonon Theory of Liquid Thermodynamics [72,73], a thermodynamic model of liquids in which these are assumed to be Dual Systems as in the DML. They develop a method to calculate the thermal conductivity of liquids, with the highest agreement with experimental data ever obtained, even if compared with previous models of liquids.

^{−5}–10

^{−6}times that of the earth is present.

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic representation of inelastic collisions between wave packets and liquid particles. The event represented in (

**a**), in which an energetic wave packet transfers energy and momentum to a liquid particle, is commuted upon time reversal into the one represented in (

**b**), where a liquid particle transfers energy and momentum to a wave packet. The particle changes velocity and the frequency of wave packet is shifted by the amount (ν

_{2}−ν

_{1}). Due to its time symmetry, this mechanism has been assumed the equivalent of Onsager’ reciprocity law at microscopic level [10,11,17]. In a pure isothermal liquid, energy and momentum exchanged among the icebergs are statistically equivalent, and no net effects are produced. Events of type (

**a**) will alternate with events of type (

**b**) so that the balance of the two energy pools is unaltered. In addition, the macroscopic equilibrium will also ensure the mesoscopic equilibrium; events (

**a**) and (

**b**) will be equally probable along any direction to have a zero average over time and space. On the contrary, if a symmetry breaking is introduced, as, for instance, a temperature or a concentration gradient, one type of event will prevail over the other along a preferential direction (Re-drawn after Peluso, F., [18]).

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**MDPI and ACS Style**

Peluso, F.
How Does Heat Propagate in Liquids? *Liquids* **2023**, *3*, 92-117.
https://doi.org/10.3390/liquids3010009

**AMA Style**

Peluso F.
How Does Heat Propagate in Liquids? *Liquids*. 2023; 3(1):92-117.
https://doi.org/10.3390/liquids3010009

**Chicago/Turabian Style**

Peluso, Fabio.
2023. "How Does Heat Propagate in Liquids?" *Liquids* 3, no. 1: 92-117.
https://doi.org/10.3390/liquids3010009