Fluid as a Continuous Medium at a Point

Continuum Hypothesis

Third, both schools of thought allow that it may be that neither CH nor ~CH is inherent in our notion of set and, hence, that there is no fact of the matter as to which is correct.

From: Philosophy of Mathematics , 2009

Basic Concepts

Nikolaos D. Katopodes , in Free-Surface Flow, 2019

1.2.2 The Fluid Continuum Hypothesis

Fluids, like all matter, consist of molecules separated by empty space. However, the mathematical determination of fluid properties, like mass density, assumes a smooth approach to the limit, as the sampling volume becomes smaller. Consider, for example, the estuary of a shallow river, in which it is reasonable to assume that salinity is uniform over the depth, but gradually increases as the ocean is approached. If the sampling volume is of the order of a few meters, a sample taken at the upstream end of the estuary will correspond to the density of fresh water. If the sample volume is of the order of hundreds of meters, however, we would expect to find a higher value for the density, as the sample now increasingly includes ocean water. Therefore, in order to capture the spatial variability of the density, we need a sampling volume that is as small as the measuring instrument allows.

Let the sampling volume be denoted by $V\text{–}{}_s$ . The mass of fluid contained in this volume is $n{M}_m$ , where n is the number of molecules contained in $V\text{–}{}_s$ , and $M_m$ the mass of an individual molecule. Therefore, the density of the sample ${\rho }_s$ is given by

(1.3) ${\rho }_s=\frac{n{M}_m}{V\text{–}{}_s}$

As the sampling size becomes smaller, the volume $V\text{–}{}_s$ is gradually reduced; however, the same is not true for the mass $n{M}_m$ , which may stay constant if the volume reduction corresponds to void space or fall abruptly if a molecule drops out of the sample, and n is reduced by one. Therefore, as the sampling volume approaches the molecular scale of the fluid, the density of the sample may experience wild fluctuations, as shown in Fig. 1.3. It is clear that when fluid properties vary so drastically, it is difficult to predict their behavior. As it will be shown in the next section, prediction implies extending known values of a fluid property in either space or time. The principle of continuum mechanics is based on the fact that many physical processes can be modeled mathematically by assuming that fluids exist as continua, thus the principles of differential calculus can be employed in solving practical problems.

Figure 1.3. Variation of density with sample size

The first hypothesis made in classical hydrodynamics concerns the concept of fluid continuum, which postulates that the substance of the fluid is distributed evenly and fills completely the space it occupies. The hypothesis abrogates the heterogeneous atomic micro-structure of matter, and allows the approximation of physical properties at the infinitesimal limit. This is accomplished by resolving fluid properties at a macroscopic level defined by a representative elementary volume (REV). Thus, the sampling volume is as small as necessary to resolve spatial variations in the properties of the fluid, but considerably larger than the scale of molecular action. Once the REV is defined, all activity below its level is essentially suppressed by a sharp cut-off filter, therefore the REV has perfectly homogeneous properties. Continuum theory postulates that the average value of any fluid property within the REV tends to a limit, as the size of the volume approaches zero, provided that the limit is reached before molecular activity prevents its attainment.

Since the REV is the smallest resolvable quantity in the fluid, it is customary to identify the sampling volume with a geometric volume of infinitesimally small size. This is not an arbitrary choice, and in fact forms the link between the fluid continuum and differential calculus. Then, as this differential volume shrinks, it degenerates to a mathematical point having unique coordinates in the flow domain, as shown in Fig. 1.4. This point contains the same amount of material at all times and is called a fluid particle. Fluid properties are constant within the particle; its linear dimensions are negligible; its moment of inertia about any axis passing through it is identically zero, and thus dynamically it behaves as a point mass. Note that the fluid particle is a fictitious entity, and should not be confused with particulate matter suspended in the fluid.

Figure 1.4. Molecular and fluid continuum scales

The foregoing set of assumptions leads to the field of continuum mechanics, in which the real fluid is replaced by a macroscopic mathematical model, consisting of infinitesimally small volumetric elements called particles. Every geometric point in three-dimensional space is occupied by a particle, thus there exists a one-to-one correspondence between particles and space points. Furthermore, the resolved medium of the continuous model contains infinitely many fluid particles having smoothly varying properties.

Of course, the validity of this assumption depends on the type of fluid and the scale of the physical processes to be modeled. A fluid may be represented by a continuum, if the associated molecular mean free path, λ, is small compared to a typical length scale, L, of the problem. The mean free path is the mean distance traveled by a molecule of the fluid between collisions with other molecules. For water, λ is very small, therefore it is not a relevant parameter, and the continuum hypothesis is valid for most common applications.

The validity of the continuum hypothesis is challenged in upper atmospheric dynamics where the determination of λ is based on the kinetic theory of gases. The process is complicated, but a satisfactory estimate can be obtained by assuming that air behaves as an ideal gas. Then, λ can be estimated by the average distance traveled by a molecule of air between successive collisions that modify the molecule's direction or energy. For example, at standard atmospheric pressure, $\lambda \simeq 70nm$ , but increases rapidly as the pressure drops. A quantitative measure for testing the continuum hypothesis is provided by the dimensionless ratio

(1.4) ${\mathbf{K}}_n=\frac{\lambda }{L}$

This is called the Knudsen number, named after the Danish physicist Martin Knudsen (1871–1949). The Knudsen number is used to assess the validity of the continuum hypothesis in problems such as the movement of contaminants in water or dust particles in the air. Fortunately, the conditions of the continuum hypothesis are satisfied for most environmental flows.

Example 1.2.1

Determine the applicability of the continuum hypothesis in the atmospheric boundary layer. Assume $L=50$ to $2000m$ . At what altitude does the continuum theory fail?

For air under standard conditions, $\lambda =6.1\times {10}^{-8}m$ . As a result, the Knudsen number is very small, which is a satisfactory condition for the applicability of the continuum hypothesis.

High Knudsen numbers are encountered in rarefied gas flows involving free molecular flows with ${\mathbf{K}}_n\simeq 0$ . At high altitudes, the Knudsen number increases quickly, and has a value of approximately 0.3 at around $90km$ from the surface of the earth. At $150km$ , ${\mathbf{K}}_n$ reaches a value of unity, which is considered the upper limit of the continuum hypothesis.

An additional area of concern is the flow around micro- and nano-devices, where it is possible to encounter high values of the Knudsen number due to the small length scale of the problem L. Examples include the fabrication of thin membranes that are found in liquid chemical sensors and similar devices (Martin et al., 2004).

A final remark needs to be made regarding the validity of the continuum hypothesis. The density continuity is the easiest to control, as it depends only on geometric factors. It may be more difficult to ensure continuity of the velocity field, as it also depends on the flow pattern and the boundary conditions of the specific problem under consideration. Especially near wall boundaries and in multi-phase flows, the resolution of the velocity field may require much higher detail than that needed for density continuity.

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Set Theory

Marion Scheepers , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.A.3 The Perfect Subset Property

For Ɣ a set of subsets of the real line let CH(Ɣ) denote the Continuum Hypothesis for Ɣ, which is: Each infinite set of real numbers in Ɣ has cardinality either $\aleph$ ₀ or else $2^{\mathrm{\aleph }0}$ . Thus, CH is CH( $\mathcal{P}$ ( $\mathbb{R}$ )).

Cantor proved CH(Π₀ ⁰). Later, Young extended this by proving CH(Π₁ ⁰). By 1916 Hausdorff and independently Aleksandrov proved CH(Borel). In all these results the proof was based on the notion of a perfect set: A set A of real numbers is said to be perfect if: (i) for each point in A there is a sequence in A with all terms different from the point, but it converges to the point, and (ii) for every convergent sequence with all terms from A, the limit of the sequence is also in A. The interval [0, 1] is an example of a perfect set. Cantor's famous "middle-thirds" set is a more exotic example. Cantor proved that each perfect set has cardinality $2^{\mathrm{\aleph }0}$ .

These results motivate the Perfect Subset Property: A family Ɣ of subsets of $\mathbb{R}$ has the perfect subset property if: Every infinite set in Ɣ is either countable, or else has a perfect subset. If a family of sets has the perfect subset property, then the Continuum Hypothesis for that family is witnessed by these perfect sets. Bernstein showed that not every set of real numbers of cardinality $2^{\mathrm{\aleph }0}$ has a perfect subset. Thus, having a perfect subset is a stronger property than having cardinality $2^{\mathrm{\aleph }0}$ .

Suslin generalized Hausdorff and Aleksandrov's theorem by proving that the family of analytic sets has the perfect subset property. Thus CH(Σ₁ ¹) holds. No proof was forthcoming that Π₁ ¹ has the perfect subset property. Sierpiński proved that each ${\Pi }_1^1$ set is a union of $\aleph$ ₁ Borel sets. Thus the cardinality of infinite Π₁ ¹ sets could be $\aleph$ ₀, $\aleph$ ₁ or $2^{\mathrm{\aleph }0}$ .

Gödel showed that in L there is an uncountable Π₁ ¹ set with no perfect subset. Thus for Π₁ ¹ the perfect subset property is not provable from ZFC. Is there in every model of set theory an uncountable ${\Pi }_1^1$ set with no perfect subset?

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Recent progress in pore scale reservoir simulation

Shuyu Sun , Tao Zhang , in Reservoir Simulations, 2020

Abstract

Flow and transport in geological formation are usually described within the continuum hypothesis at Darcy's scale. However, Darcy's continuum hypothesis lacks, in many cases, rigorous derivation from basic principles. This has been highlighted by the many ad hoc terms and coefficients that have been imposed into the governing equations to lump the complexities of pore-scale phenomena. Recently, there has been extensive research work trying to bridge the gap between scales and to calculate these terms based on pore-scale modeling. At a pore scale, there are a number of complexities that one needs to overcome in order to be able to carry out simulations. As an example, phenomena involving the movement of multiphase systems with either phase change and/or composition partition are among the problems that have not been very well comprehended. Currently, many oil companies, including Saudi Aramco and Schlumberger, are making great efforts in researching pore-scale flow and transport in geological formation. In this chapter, we will start from the key effort to construct the mathematical model governing the compositional multiphase flow to determine the phase compositions of the fluid mixture, and then calculate other related physical properties. Especially, capillary effect, which is often ignored in conventional reservoir phase equilibrium calculation, needs to be considered in shale and tight formations, as the nano-sized pores yield a large capillary pressure and confinement effect. Newly proposed thermodynamic consistent diffuse interface models and algorithms based on realistic equation of state will be generated to describe multicomponent multiphase flow and we will show the approach to consider partial miscibility common seen in carbon dioxide flooding.

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Viscous Fluid Flow

Nikolaos D. Katopodes , in Free-Surface Flow, 2019

5.5.2 Total Energy Equation

The energy equation represents an application of the first law of thermodynamics to a fluid satisfying the continuum hypothesis and moving with velocity V. Consider the differential element shown in Fig. 5.7. The total energy of the fluid at the centroid of the element consists of the internal energy per unit mass, e, and the kinetic energy, $\frac{1}{2}\rho V^2$ . According to the first law of thermodynamics, the time rate of change plus the net flux of the energy into the differential element is equal to the rate of work done by body and surface forces on the element, plus the rate at which heat is conducted into the element. The temporal change of energy is given by

Figure 5.7. Definition sketch for energy equation

$\frac{\partial }{\partial t}\left[\rho (e+\frac{V^2}{2})\right]\delta V\text{–}$

From Fig. 5.7, the net flux of energy through the element's faces is given by

$\{\frac{\partial }{\partial x}\left[\rho u(e+\frac{V^2}{2})\right]+\frac{\partial }{\partial y}\left[\rho v(e+\frac{V^2}{2})\right]+\frac{\partial }{\partial z}\left[\rho w(e+\frac{V^2}{2})\right]\}\delta V\text{–}$

Similarly, with reference to Fig. 5.5 and Fig. 5.6, we obtain the following expressions for the rate of work done on the element. First by body forces

(5.59) $u\rho g_x+v\rho g_y+w\rho g_z$

And then by surface forces

(5.60) $\begin{array}{c}\frac{\partial }{\partial x}(u{\sigma }_{xx}+v{\sigma }_{xy}+w{\sigma }_{xz})+\frac{\partial }{\partial y}(u{\sigma }_{yx}+v{\sigma }_{yy}+w{\sigma }_{yz})\hfill \\ \hfill +\frac{\partial }{\partial z}(u{\sigma }_{zx}+v{\sigma }_{zy}+w{\sigma }_{zz})\end{array}$

The heat fluxes are obtained in the usual manner. We define the heat flux vector q to denote the rate at which heat is conducted, per unit time and area, normal to the boundary surface of the differential element. Thus, following a Taylor series expansion, the net heat transport along the coordinate axes are given by

$-\frac{\partial q_x}{\partial x}\delta x\left(\delta y\delta z\right),-\frac{\partial q_y}{\partial y}\delta y\left(\delta x\delta z\right),-\frac{\partial q_z}{\partial z}\delta z\left(\delta x\delta y\right)$

where the subscripts of q indicate the component of heat flux in the corresponding coordinate direction. As shown in Fig. 5.8, the total heat transfer in and out of the elementary volume is given by

Figure 5.8. Definition sketch for heat fluxes

(5.61) $-(\frac{\partial q_x}{\partial x}+\frac{\partial q_y}{\partial y}+\frac{\partial q_z}{\partial z})\delta x\delta y\delta z$

Summing up the individual terms, and using index notation, leads to the following form of the energy equation

(5.62) $\begin{array}{c}\frac{\partial }{\partial t}\left[\rho (e+\frac{u_i{u}_i}{2})\right]+\frac{\partial }{\partial x_j}\left[\rho u_j(e+\frac{u_i{u}_i}{2})\right]=\hfill \\ \hfill \rho u_i{g}_i+\frac{\partial }{\partial x_j}\left(u_i{\sigma }_{ji}\right)-\frac{\partial q_j}{\partial x_j}\end{array}$

The derivatives containing products of the velocity can be expanded, thus use of the continuity equation, i.e. Eq. (5.6), allows elimination of the associated terms, and leads to the following simplified form of the energy equation

(5.63) $\frac{D}{Dt}(e+\frac{u_i{u}_i}{2})=u_i{g}_i+\frac{1}{\rho }\frac{\partial }{\partial x_j}\left(u_i{\sigma }_{ji}\right)-\frac{1}{\rho }\frac{\partial q_j}{\partial x_j}$

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Kinematics of Motion and Deformation Measures

Martin H. Sadd , in Continuum Mechanics Modeling of Material Behavior, 2019

Abstract

Common observations indicate that material bodies undergo motion and deformation as a result of applied forces. In this chapter, we will be concerned with the development of quantitative measures of these kinematical variables based on the continuum hypothesis. Included in this effort is the description of motion and displacement, and this will lead to the construction of several strain tensor measures. Since we wish to formulate time-dependent behaviors, various dynamic variables and rate of strain tensors will also be developed. Results in this chapter by themselves will not yet be able to predict the deformation resulting from the applied loading. Other relations will need to be constructed and combined with the work here to develop a complete mechanistic theory.

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The First Law of Thermodynamics and Energy Transport Mechanisms

Robert T. Balmer , in Modern Engineering Thermodynamics, 2011

4.10 The Local Equilibrium Postulate

Surprisingly, there is no adequate definition for the thermodynamic properties of a system that is not in an equilibrium state. Some extension of classical equilibrium thermodynamics is necessary for us to be able to analyze nonequilibrium (or irreversible) processes. We do this by subdividing a nonequilibrium system into many small but finite volume elements, each of which is larger than the local molecular mean free path, so that the continuum hypothesis holds. We then assume that each of these small volume elements is in local equilibrium. Thus, a nonequilibrium system can be broken down into a very large number of very small systems, each of which is at a different equilibrium state. This technique is similar to the continuum hypothesis, wherein continuum equations are used to describe the results of the motion of discrete molecules (see Chapter 2).

The differential time quantity dt used in nonequilibrium thermodynamic analysis cannot be allowed to go to zero as in normal calculus. We require that dt > σ _s , where σ is the time it takes for one of the volume elements of the subdivided nonequilibrium system just described to "relax" from its current nonequilibrium state to an appropriate equilibrium state. This is analogous to not allowing the physical size of the element to be less than its local molecular mean free path, as required by the continuum hypothesis. The error incurred by these postulates is really quite small, because they are the result of second-order variations of the thermodynamic variables from their equilibrium values. However, just as the continuum hypothesis can be violated by systems such as rarefied gases, the local equilibrium postulate can also be violated by highly nonequilibrium systems such as explosive chemical reactions. In the case of such violations, the analysis must be carried forward with techniques of statistical thermodynamics.

Because of the similarity between the local equilibrium postulate and the continuum hypothesis, it is clear that the local equilibrium postulate could as well be called the continuum thermodynamics hypothesis.

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Boundary layer equations in fluid dynamics

Hafeez Y. Hafeez , Chifu E. Ndikilar , in Applications of Heat, Mass and Fluid Boundary Layers, 2020

Abstract

In this article we will derive the basic equations of motion for viscous flows. It is worth noting that there are two main physical principles involved: (i) conservation of mass and (ii) Newton's second law of motion, the latter leading to a system of equations expressing the balance of momentum. In addition, we will utilize Newton's law of viscosity in the guise of what will be termed a "constitute relation" and, of course, all this will be done within the confines of the continuum hypothesis. We should also note that Newton's second law of motion is formally applied to point masses, i.e., discrete particles, its application to fluid flow seems difficult at best. But we will see that because we can define fluid particles (via the continuum hypothesis), the difficulties are not actually so great. We will begin with a brief discussion of the two types of reference frame widely used in the study of fluid motion, and provide a mathematical operator that relates them. We then review some additional mathematical constructs that will be needed in subsequent derivations. Once this groundwork has been laid, we will derive the "continuity equation" which expresses the law of conservation of mass for a moving fluid, and we will consider some of its practical consequences. We then provide a similar analysis leading to the momentum equations, thus arriving at the complete set of equations known as the Navier–Stokes (NS) equations. These equations are believed to represent all the fluid motion within the confines of the continuum hypothesis. There are many ways to derive the equation of motion of a fluid; we can use the approach which requires solving a set of partial differential equations. There is another way, called the control volume approach, and in any case, Newton's second law is applied.

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Fluids

Bastian E. Rapp , in Microfluidics: Modelling, Mechanics and Mathematics, 2017

9.3.2 Continuum Hypothesis

In most cases in fluid mechanics, the smallest entity considered is a control volume of suitable size (usually in the range of several µm edge length). This control volume contains a significant number of individual atoms and thus averages all effects exerted by the individual atoms. Usually this averaging results in a significantly more predictable behavior. In this case, we treat the fluid as being a continuous piece of matter, which is why this approach is referred to as the continuum approach or the continuum hypothesis . This continuum can be treated as having average values for velocity, acceleration, entropy, or enthalpy. However, these are averaged values, so they are discontinuous when looking at the atomic scale. On the experimental scale, observing the effects in a continuum gives rise to more steady experimental data. If the resolution is increased, the measurements will become less steady as the effects of the individual atoms become more and more pronounced. The continuum hypothesis is applicable for most applications in classical macro- and microfluidic applications as the size of the control volumes can be chosen sufficiently big to contain at least some 10 000 atoms. In this case, the statistical variations are roughly in the range of 0.1 %. As we will see the length scales required for this prerequisite are in the range of about 100   nm to 1 µm for gases (see section 9.3.4) and 10   nm for liquids (see section 9.3.3). If the scale length of the fluidic system is decreased, the continuum hypothesis may not be applicable. As an example, discussing the flow of liquids in a nanopore, the control volume will contain only a few atoms that may not be treated as a homogeneous continuum.

The exact dimensions of the control volumes must be adapted to the question in mind: What level of detail is required? The bigger the control volume, the less expensive the calculation will be numerically. However, increasing the control volume size above a critical threshold may result in loss of detail on the molecular length scale (if, e.g., mass transport phenomena such as diffusion are to be considered). A suitable length scale is found once a control volume does not show significant signs of fluctuation on the molecular level, and all molecules in the control volume may be approximated as having the same field variables values (such as temperature or pressure, see section 10.1). In this state, the fluid behaves as a continuum. A fluid continuum may be approximated as being composed of individual control volumes, each of which has distinct values of the field variables.

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Deformation

Martin H. Sadd , in Elasticity (Fourth Edition), 2021

Abstract

We begin development of the basic field equations of elasticity theory by first investigating the kinematics of material deformation. As a result of applied loadings, elastic solids will change shape or deform, and these deformations can be quantified by knowing the displacements of material points in the body. Under the continuum hypothesis, this concept will establish a displacement field at all points within the elastic solid. Using appropriate geometry, particular measures of deformation can be constructed leading to the development of the strain tensor. As expected, the strain components are related to the displacement field. The purpose of this chapter is to introduce the basic definitions of displacement and strain, establish relations between these two field quantities, and finally investigate requirements to ensure single-valued, continuous displacement fields. As appropriated for linear elasticity, these kinematical results will be developed under the conditions of small deformation theory. Developments in this chapter will lead to two fundamental sets of field equations: the strain–displacement relations and the compatibility equations.

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Nonequilibrium thermodynamics

José M. Ortiz de Zárate , Jan V. Sengers , in Hydrodynamic Fluctuations in Fluids and Fluid Mixtures, 2006

2.1.2 Continuum hypothesis and the assumption of local equilibrium

Although a fluid consists of molecules, statistical mechanics confirms that at a macroscopic level the state of the fluid can be described by a restricted number of thermodynamic variables. To extend these concepts to systems that are not in thermodynamic equilibrium one adopts the principle of local thermodynamic states. Specifically, it is assumed that one can imagine subsystems that are infinitely small relative to the macroscopic level, but that still contain a sufficiently large number of molecules, so that one can define at any given time for these small subsystems local values of the thermodynamic properties by the methods of equilibrium statistical physics. Furthermore, one adopts a continuum hypothesis by considering the system as a continuum of local thermodynamic states with thermodynamic properties that now depend on the position r and the time t, such as ρ(r, t), T(r, t), p(r, t), etc. Thus the continuum hypothesis allows us to replace the thermodynamic quantities by corresponding thermodynamic fields that are continuous functions of space and time. In addition to the thermodynamic fields, one introduces a local center-of-mass velocity v(r, t), also called barycentric velocity, as a relevant field.

The assumption of local equilibrium implies that the local thermodynamic properties defined for the infinitesimal subsystems, as well as their derivatives, at any given time satisfy the same thermodynamic relations as those for systems in thermodynamic equilibrium. For example, the local mass density ρ(r, t) is assumed to be a functional of the local pressure p(r, t) and the local temperature T(r, t), independent of the local velocity v(r, t), with ρ[p(r, t), T(r, t)] satisfying the same equation of state as in thermodynamic equilibrium. By combining the continuum hypothesis with the local equilibrium assumption we then deduce from Eq. (2.1) a corresponding expression for the time derivative of the mass density ρ(r, t) at a given position

(2.2) $\frac{\partial }{\partial t}\rho \left(\text{r},t\right)=\rho \left(\text{r},t\right)\left[{\chi }_T\frac{\partial }{\partial t}p\left(\text{r},t\right)-{\alpha }_p\frac{\partial }{\partial t}T\left(\text{r},t\right)\right].$

and for the spatial derivative at a given time t:

(2.3) $\nabla \rho \left(\text{r},t\right)=\rho \left(\text{r},t\right)\left[{\chi }_T\nabla p\left(\text{r},t\right)-{\alpha }_p\nabla T\left(\text{r},t\right)\right].$

In the differential equations above, the thermodynamic coefficients ${\varkappa }_T$ and α_p are now, in principle, spatiotemporal fields. However, in practical applications the dependence of these thermodynamic coefficients on r and t is often neglected. The local-equilibrium assumption means that any classical thermodynamic relation can be adapted to nonequilibrium thermodynamics by replacing the thermodynamic quantities by fields, as we have done here for the equation of state.

The continuum hypothesis asserts that the local states of a nonequilibrium fluid can be described in terms of thermodynamics fields, obtained as averages over small volume elements, that depend on the position r and the time t. Since for a molecular fluid one can always define local averages of the conserved quantities, one does not need strictly speaking the assumption of local thermodynamic equilibrium in formulating equations for the conserved quantities. In practice, however, one assumes that the local values of the conserved quantities can be identified with the local equilibrium values. The local-equilibrium hypothesis implies that the thermodynamic fields satisfy at any given r and t the same relations as the equilibrium thermodynamic properties. These assumptions will be valid when the spatial and temporal dependence of the thermodynamic fields are negligible at molecular length and time scales, so that one can identify small volume elements that are homogeneous while containing so many molecules that statistical averages can be performed locally. Thus the assumption of local equilibrium will be valid when the molecular length and time scales are small compared to the hydrodynamic length and time scales. This assumption will not be valid in complex systems, such as glasses, polymer blends, colloidal systems, etc. Attempts to formulate thermodynamics "beyond" local equilibrium have been made (Villar and Rubí, 2001), but such complex systems will not be considered in this book. However, while the local-equilibrium hypothesis is valid for the thermodynamic properties in simple fluids, it is important to note that even in simple fluids the concept of local equilibrium, as formulated here, is no longer valid for the fluctuations of the thermodynamic properties. As already mentioned in Chapter 1, a major theme of this book is that the fluctuations of the (local) thermodynamic properties in nonequilibrium states are very different from those predicted by statistical mechanics for thermodynamic equilibrium.

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