Abstract
Three of the most studied problems in fluid dynamics are revisited within implicit theories of fluids. Specifically, the onset of convection, the determination of laminar flows and the motion of a fluid down an inclined plane are studied under the assumption that the Cauchy stress tensor and the rate-of-strain tensor are related through implicit constitutive equations. Particular attention is paid to fluids whose viscosities are pressure-dependent.
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1 Introduction
The principium reddendae rationis is one of the most powerful tool that have been used in philosophical argumentations [28]. It has been used as principle by many philosophers (Spinoza, Liebniz, Descartes, Hamilton, to cite a few of them) and is very useful also in Science. Anaximander of Miletus suggestively used the principium reddendae rationis to argue that Earth was a round cylinder statically floating at the center of Universe without any support. In Anaximander’s reasoning, since Earth was equidistant from all other bodies there was no reason why it should move in any one direction.
The principium reddendae rationis has been applied more rigorously in classical continuum mechanics, and, here, we shall appeal to it to sustain the suitability of introducing implicit constitutive relations rather than explicit models. The main aim of these notes is indeed the introduction, in the framework of implicit theories of fluids, of the class of fluids whose viscosities depend on pressure and that, for such a peculiarity, are called piezo-viscous fluids [10].
It is well known that in any continuum theory the motion of a real body \(\mathcal {B}\) is governed by the balance laws of mass, linear and angular momenta, and energy as well as by the second law of Thermodynamics. In particular, if the Cauchy axiom holds, namely the internal actions in \(\mathcal {B}\) can be represented only by a vector field, in the absence of body couples the balance of angular momentum requires that the Cauchy stress tensor T is symmetric, i.e. T = TT, and the equations of mass, linear momentum and energy read
respectively. In (1)–(3) ρ denotes the mass density, v the velocity field, b the specific body force, e the specific internal energy, r the specific radiant heating, q the heat flux vector, and D the rate-of-strain tensor, i.e. the symmetric part of the velocity gradient L = ∇v. The superimposed dot denotes the material time derivative.Footnote 1 The second law of Thermodynamics is instead usually written as the Clausius-Duhem inequality
with η denoting the specific entropy and θ the temperature.
Regarding the specific body force b as known, Eqs. (1)–(3) provide seven scalar equations for 13 scalar fields—the mass density ρ, the velocity components v i (i = 1, 2, 3), and the stress tensor components T ij (i, j = 1, 2, 3). The system of PDEs (1)–(3) is then not closed. The disparity in the number of equations and unknowns is however not surprising as (1)–(3) are valid for all the non-polar materials (i.e. materials satisfying the Cauchy axiom), but do not differentiate the special material the body is made of. Therefore, to obtain a system the number of equations of which matches the number of unknowns, one has to introduce some constitutive equations characterizing the thermomechanical response of the material.
Constitutive equations are very often referred to as constitutive relations. According to the Cambridge dictionary, a relation is a connection or similarity between two things. This is exactly what modelers well educated in mechanics usually do. They connect thermodynamical quantities through an equation which is specific to a particular material or substance.
For example, in classical fluid mechanics the stress tensor is related to the rate-of-strain tensor D, the density of the fluid and the temperature through an equation of the form
Such a relationship is sufficient to close the subsystem (1)–(3) and thus there is no clear reason (no reddendae rationis) to consider a priori an explicit representation of the form
such as the Navier-Stokes constitutive equation:
where p |E is the pressure at thermodynamic equilibrium, and λ and μ are the bulk and shear viscosity, respectively. The reason of considering a representation of the form (6) stands exclusively in its mathematical ease.
The search for models of mathematical ease is common when dealing with problem of the motion of real bodies. A simple example comes from particle mechanics. It is well known that the motion of a free particle X is governed by Newton’s second law
where m is the mass of the particle, a is the acceleration vector, and F is the resultant force acting on X. In direct problems the resultant force F is usually known, and the motion is to be determined by solving (8) under prescribed initial conditions for the position x and the velocity v of X. To solve uniquely the resulting Cauchy problem, one assumes that any experimental model for the force F depends on the motion of the particle through a relation of the form \(\boldsymbol {F}=\widehat {\boldsymbol {F}}(\boldsymbol {x}, \boldsymbol {v}, t)\), where \(\widehat {\boldsymbol {F}}\) satisfies the smoothness assumptions of Cauchy’s theorem for ordinary differential equations.
Behind the introduction of a constitutive relation of the form (6) there is then the expectation that, in the framework of a field theory like continuum mechanics, any initial and boundary value problem (IBVP) governing the motion of a real body admits a unique solution as the Cauchy problem governing the motion of a free particle in the framework of Newtonian’s particle mechanics.
Jacques Hadamard [18] introduced the concept of well-posedness of IBVPs. An IBVP is well posed if it admits a unique solution which depends continuously on the data (that is on initial and boundary conditions). Well-posedness in the sense of Hadamard has always influenced strongly applied mathematicians and scholars involved in continuum mechanics research when modeling a real-world phenomenon. From a mathematical point of view it is clear that well-posedness is intriguing. From the point of view of continuum mechanics well-posedness is closely related to the concept of determinism. From this perspective, the principle of determinism for the stress stated in Truesdell and Noll’s celebrated handbook [52],
The stress in a body is determined by the history of the motion of that body,
can be regarded as a sort of ‘pathway’ to well-posedness.
John Ball and Richard James [5] commenting on the concept of well-posedness in the sense of Hadamard wrote:
Any reader of that paper will see the unmistakable influence of Truesdell. At the end of the day, perhaps it would have been realized that Hadamard’s notions of well-posedness are far too restrictive in the nonlinear setting, that non-uniqueness and even non-existence comprise acceptable behavior, and that there are probably no fundamental restrictions on the strain-energy function at all besides those arising from material symmetry and frame-indifference.
This means not only that the principle of determinism is highly questionable from a physical point of view, but also that mathematical feasibility is not a good argument to support explicit constitutive equations where stress is given in terms of motion.
Nature does not care if the relationship at the basis of its phenomena are graphs or functions. It is our idealization of natural phenomena that sometimes realizes the fact that relationships in the form of functions are more convenient. Moreover, an explicit approach rules out a priori the possibility to describe interesting phenomena like, for instance, constitutive branching. There is then no advantage to sacrifice the principium reddendae rationis on the altar of mathematical well-posedness and there are no a priori physical reason to support explicit constitutive equations.
It is well known that in an experiment it is possible to control stress or deformation. For example, when pulling a bar of steel it is possible to control the engineering stress and to record the engineering strain (the test is said to be performed in a soft device), or to control the engineering strain and to record the engineering stress (test performed in a hard device). Therefore, in the first case why can we not postulate that the motion in a body is determined by the history of the stress of that body? The fact that constitutive relationships are described by graphs and not functions is also confirmed by several experimental works (see [37] and references therein).
The aim of these lectures is to investigate classical problems in fluid mechanics (such as the onset of Rayleigh-Bénard convection, laminar flows and flows over an inclined plane) by employing implicit constitutive relations for the stress tensor. We do not claim that the use of such implicit constitutive models represents a scientific revolution. In the light of the previous discussion, we only claim that implicit constitutive equations are useful tools for investigating real-world phenomena and there is no a clear and neat reason to throw away a priori this class of relations.
These notes are not meant to be a detailed review of the literature concerning piezo-viscous fluids. The choices of the topics treated and the literature presented are based mainly on our personal tastes and pedagogical aims. The level of these lecture notes is basic and tailored for undergraduate students with an elementary knowledge of continuum mechanics.
2 Implicit Constitutive Models for the Cauchy Stress Tensor
Consider the constitutive relation for the Cauchy stress tensor T of a linearly viscous fluid given by
where
is the pressure and the positive parameter μ is the viscosity.
A quick look reveals that (9)–(10) differs from the classical constitutive model for the Cauchy stress
In fact, contrarily to (11), in (9) and (10) it is explicitly stated what it is meant by “pressure”: the negative of the mean normal stress. On using the terminology widely adopted in the literature, by “pressure” we mean the “mechanical pressure”. It is extremely important point out this from the beginning not only because we are going to examine some aspects of the flows in fluids with material parameters depending on pressure, but also because, as observed by Rajagopal [36], the term “pressure” has been used in a plethora of different contexts and, as an unavoidable consequence, it has been often misused in the literature. Referring the interested reader to [36] for a detailed discussion on the issues related with the usage of the word “pressure”, we limit to observe that, as argued by Huilgol [20], if one wishes to include pressure in the rheological material functions (as we intend to), defining the pressure through (10) is the only unambiguous way of introducing such a physical quantity. In addition, this definition can be used to interpret experimental data systematically both for compressible and incompressible fluids (the motions of which, as is well known, are subjected to the kinematic constraint divv = tr(D) = 0) [20].
It is widely accepted that the viscosity of fluids depends on temperature. In particular, experimental observations have shown that viscosity decreases with increasing temperature. On the other hand, there is also a vast literature on the dependence of the viscosity on pressure. For the sake of brevity, below, we report only some of the most important studies on this topic.
The first scholar to realize that the viscosity of a fluid may depend on pressure was Stokes. In fact, in his celebrated paper [49] on the constitutive response of fluids, Stokes stated
Let us now consider in what cases it is allowable to suppose μ to be independent of the pressure. It has been concluded by Du Buat from his experiments on the motion of water in pipes and canals, that the total retardation of the velocity due to friction is not increased by increasing the pressure…I shall therefore suppose that for water, and by analogy for other incompressible fluids, μ is independent of the pressure.
Stokes’s comment clearly implies that only in special circumstances the viscosity of a fluid is independent of pressure. While for flows in canals and pipes under normal conditions inclusion of the dependence of the viscosity on pressure does not affect the results of the experiments, there are several other situations where one needs to take this dependence into account. Eight decades later, Bridgman [10] gave a measure of the effect of pressure on the viscosity of water as well as of other forty-two pure liquids. In addition, Bridgman observed that, while it is true that all the physical quantities do vary with pressure, the variation in the viscosity with pressure may be far more dramatic than the variation of the other quantities with pressure. To this aim, Bridgman reported:
It may be said in general that the effects of pressure on viscosity are greater than on any other physical property hitherto measured,Footnote 2 and vary very widely with the nature of the liquid. The increase of viscosity produced by 12000 kg varies from two or three fold to millions of fold for the liquids investigated here, whereas such properties as the volume decrease under 12000 kg seldom vary by as much as a factor of two from substance to substance.
As early as 1893, based on experiments on marine glue, Barus [6] proposed an empirical relation between the viscosity μ and the pressure p of the form
where μ ref is the viscosity at the reference pressure p ref, and the piezo-viscous coefficient β is temperature dependent. Later, Andrade [2] proposed a model expressing the viscosity in terms of the pressure, the mass density and the temperature, namely
where r, s and A are constants. References to much of the literature concerning the pressure dependence of the viscosity of fluids prior 1931 can be found in the book of Bridgman [11]. More recently, Laun has modeled the viscosity of polymer melts through
where μ ref is the viscosity at the reference state (p ref, θ ref), and β and δ are positive constants. There have been numerous other experiments by Bair and coworkers that shows that the dependence of the viscosity on pressure is exponential (see the experiments of Bair and Kottke [4]). Mention must be made of the work of Martín-Alfonso and co-workers [26] wherein an intricate relationship among the temperature, viscosity and pressure is provided for bitumen. In this context, it ought to be pointed out that the pressure dependence of the properties of bitumen was recognized very early. For instance, Saal and Koens [46] not only allowed for viscosity to depend on pressure (and hence on the mean normal stress), they also allowed it to depend on the shear stresses.
In virtue of the experimental evidences reported above, it is then reasonable to assume that the viscosity of a fluid depends on pressure and temperature. Consequently, since we have defined the pressure as the negative mean normal stress, the constitutive model (9) with μ = μ(p, θ) prescribes the Cauchy stress tensor in terms of the strain-rate tensor and temperature through the implicit relation
In the following sections, we shall mainly use the implicit model (15) or its variants. However, for the sake of generality, we now determine the most general implicit model for the Cauchy stress of an isotropic fluid. We start with an implicit relation of the form
Since the fluid is isotropic, G is an isotropic tensor function of the two second-order tensors T and D, i.e. G satisfies the property
for all proper orthogonal tensors Q. Next, following Spencer [48], the most general implicit model for the Cauchy stress tensor of an isotropic fluid can be written as
where the coefficients α i, i = 0, 1, …, 8, depend on θ and the integrity basis of the two tensors T and D. The integrity basis consists of the invariants of any combination of tensor products up to second order. For the current problem, these are given by
This is a minimal set of invariants since the trace of the product of two second-order Cartesian tensors is equal to the trace of the tensor product with the factors written in reverse.
When we consider fluid models of the form (18)–(19), with
and all the remaining α i equal to zero, we recover the model (15) for a piezo-viscous fluid.
It might seem that the previous discussion on implicit constitutive models for the Cauchy stress tensor is valid only for compressible isotropic fluids. The reader might be led to such a conclusion by the usual practice in continuum mechanics to associate a Lagrange multiplier with the constraint of incompressibility, split the stress tensor into the sum of the constraint stress T C and the extra stress T E, and assume that the constraint stress is workless and independent of the state variables, and the extra stress is independent of T C [52]. This conclusion is not right and the discussion above can be easily adapted to incompressible isotropic fluids. In fact, Rajagopal [35] showed that, when dealing with incompressible fluids, appropriate choices of the material parameters α i in (18) guarantee the incompressibility without the introduction of a Lagrange multiplier and any split of the stress tensor. In these notes we shall not introduce special models for incompressible fluids because, as we shall show in the following section, incompressibility is an approximation that is valid under specific flow regimes. Therefore, when dealing with these flow regimes, there is no need at all to introduce a priori appropriate models which automatically meet the kinematic restriction of invariability of volume elements during motion.
3 Isochoric Motions of Fluids as Approximations Under Different Flow Regimes
All real bodies are compressible. In fact, if a sufficiently high pressure is employed, the body undergoes a reduction in volume. On the contrary, for most liquids in Nature, experience teaches that volume increases with increasing temperature. However, it is possible that some bodies do not undergo a significant change in volume over a sufficiently large ranges of pressures or temperatures and can hence be approximated as being incompressible in those ranges. When the ranges of pressures and temperatures are what is considered ‘normal’, in view of day to day applications, the body is considered to be incompressible. Of course, what is deemed to be a ‘significant change in volume’ is quite arbitrary and it boils down to whether neglecting the volume change and modeling the body as an incompressible body yet captures the essential features of the response of the body when subject to external stimuli. Most liquids can be approximated as incompressible liquids provided the pressures to which they are subject to are not very high and temperature changes are small enough. On the other hand, if the ranges of pressures and temperatures to which the liquid is subject are large, then volume changes do take place and, moreover, all the properties that characterize a fluid must be considered pressure- and temperature-dependent.
Müller [29] defined a body to be incompressible if the density and the internal energy depend only on temperature and do not depend on pressure. Appealing to the material frame indifference and the entropy principle Müller [29] showed that a Navier-Stokes-Fourier fluid cannot undergo changes in volume due to temperature changes. Such a behavior is clearly contradicted by experiments which show that volume changes do take place with temperature. Motivated by the fact that experimental evidence clearly contradicts Müller’s conclusion (often referred to as the Müller paradox), Gouin et al. [16] studied a class of Navier-Stokes-Fourier fluids for which the internal energy, shear and bulk viscosities, and thermal conductivity depend on pressure and temperature, while the density depends only on temperature. They referred to such materials as quasi-thermally compressible fluids and found a critical value of the pressure, denoted by p cr, below which a quasi-thermally compressible fluid behaves like a perfectly compressible fluid in Müller’s sense. Since the value of p cr is large with respect to the normal pressure conditions (for instance, for water at 20∘C, p cr ≃ 2 × 105 atm), Gouin et al. [16] concluded that a quasi thermal-incompressible fluid is experimentally similar to a perfectly incompressible fluid, removing in a such a way the Müller paradox.
The analysis of Gouin et al. [16] is based on the assumption that the density depends only on the temperature. Clearly, for a homogeneous fluid this assumption is equivalent to assuming that the deformation gradient F depends only on the temperature, i.e.
Recently, Rajagopal et al. [42] proved that assumption (21) leads to three physically unrealistic deductions:
-
the specific heat at constant volume is zero,
-
thermodynamic instability (the specific entropy fails to be a concave function of the pressure and specific volume),
-
imaginary speed of sound.
To overcome these drawbacks Rajagopal et al. [42] modified the assumption (21) by postulating that
and showed that, for several classes of flow regimes of interest in the applications, the velocity field of a fluid with pressure and temperature dependent material properties is, to a first approximation, solenoidal. Therefore, in comparison to the findings by Gouin et al. [16], instead of determining pressure ranges in which a real fluid behaves like an idealized incompressible fluid, Rajagopal et al. [42] determined the flow regimes in which the motions of a fluid can be regarded, to a first approximation, as isochoric.
3.1 Equations Governing the Flows in a Piezo-Viscous Fluid
To rigorously derive the sets of approximated equations we shall employ in the following sections, we follow the same procedure as in [42]. We start by assuming that the fluid is slightly compressible due to variations in the pressure and temperature and thus assume that (22) holds.
We also assume that the motion of the fluid is sufficiently smooth so that the derivatives that are taken are meaningful. Then, differentiating the determinant of the deformation gradient with respect to time yields
where
are the isothermal compressibility and the coefficient of thermal expansion, respectively. Clearly, k T and α are related through the integrability condition
From (1) and (23) we deduce that
Hence
and, denoting v = 1∕ρ the specific volume,
Next, we introduce the enthalpy
and the Gibbs free enthalpy
and combine these two thermodynamic potentials with the balance equations (1)–(3) and the Clausius-Duhem inequality (4) to get
In our analysis we shall regard the mechanical pressure and the temperature as independent variables on which the material parameters of the fluid depend, and, since we are interested in fluids of grade 1 (see Truesdell and Noll [52]), the requirement of material frame indifference and the representation theorems for isotropic functions lead us to consider the Cauchy stress tensor to be constitutively prescribed by the implicit relation (15), the response functions of the specific internal energy and entropy to be of the form
and the heat flux vector given by the Fourier law
with k being the thermal conductivity. Finally, we introduce the specific heats at constant pressure and at constant volume through
respectively, and the specific heat ratio γ = c p∕c v [15].
Inserting (9), (23) and (33) into (31) yields the inequality
that holds true for any thermodynamical processes, i.e. for any fields ρ, v and θ satisfying the balance equations (1)–(3). Therefore, by using standard arguments in continuum thermodynamics, we deduce that
and the constitutive functions for the viscosity of the fluid μ and the thermal conductivity k are non-negative.
From (36) we deduce that the differential of the Gibbs free enthalpy is
and the constitutive functions for the specific internal energy and entropy are, respectively, of the form \(e=\hat {e}(p,\theta )\) and \(\eta =\hat {\eta }(p,\theta )\) (namely, both the specific internal energy and the specific entropy do not depend on tr(D)), with
Next, combining (29) and (30) with (37) yields
which in turn leads to
and then by virtue of (27) and (28) we obtain
As far the specific internal energy is concerned, from (36)1, (38) and (41) we have
Finally, by using (9), (23) and (42) the equations of balance of linear momentum (2) and energy (3) can be expressed as
and
respectively. Equations (26), (23), (43) and (44) constitute a system of partial differential equations for determining the thermodynamic fields ρ, v, p and θ.
3.2 Approximations
In order to introduce the most appropriate non-dimensionalization, it is necessary to record before some thermodynamic identities. Rajagopal et al. [42] observed that the specific heats ratio γ > 1, the isothermal compressibility is related to the speed of sound C in the fluid through
and the square of the coefficient of thermal expansion can be written as
Let Oxyz be a Cartesian frame of reference with orthonormal basis {i, j, k}. Let \(\varOmega _d= \mathbb {R}^2\times [0, d]\) be a horizontal fluid layer of thickness d and assume that gravity is the only body force acting on the fluid, namely b = −g k, where g is the acceleration due to gravity.Footnote 3 We assume also that no heat is supplied, i.e., r = 0. To non-dimensionalize the equations governing the fluid motion, we choose a convenient reference state (p ref, θ ref) and introduce the following scales
In (47) the subscript ‘ref’ indicates that the corresponding material parameters are evaluated at the reference state (p ref, θ ref), V is the reference velocity, θ M =maxΩ θ, θ m =minΩ θ and the isothermal compressibility has been scaled by taking into account (45). Hereinafter, we choose θ M as the reference temperature, viz θ ref = θ M.
Substituting (47) into (23), (26), (43) and (44) yields the dimensionless equations (omitting the asterisks for convenience)
where
are the Mach, second Froude, Reynolds, Brinkman, Péclet and Carnot numbers, respectively.
We now assume that the material parameters α, k T, c p and k are analytic functions and limit our analysis to the departures of the pressure and temperature from the reference state (p ref, θ M) for which we can write
and
From the integrability condition (25), the expansions (53) and (54) and the scales (47) we deduce that
for all \((j_1,j_2)\in \mathbb {N}\times \mathbb {N}_0\). In virtue of (57) we can integrate equation (49) to obtain
Evaluating the identity (46) at the reference state (p ref, θ M) yields the following relation
where
is the Eckert number. We henceforth consider thermodynamic processes for which
Therefore, as long as E∕[Fr2Ca(γ ref − 1)] is of order O(1) or smaller, from (59) we deduce that Ma2∕Fr2 is of order O(ε2) or smaller.
We are now in position to carry out a perturbation analysis with respect to the small parameter ε. Let
be the power series in ε of the thermodynamic fields v, p and θ. As far as the power series expansion of the fluid density is concerned, it may be derived from (58) and (62)2,3 by taking into account the fact that Ma2∕Fr2 is of order O(ε2). However, the expression is quite complicated and of no interest to our analysis. In our analytical scheme it suffices to know that
where o(ε) accounts for terms of order O(ε2) and higher. Therefore, by inserting (62) and (63) into (48), (50) and (51) we deduce a system of equations from which different approximations can be derived:
and
Since γ refMa2∕Fr2 is of order O(ε2) or smaller, collecting terms of order O(1) in (64) yields
whence the fluid motions can be regarded as isochoric to a first approximation.
According to the magnitude of the dimensionless numbers occurring in (64) and (65), we can derive different sets of approximate equations such as, just to mention a few of them, those which have been employed in the last few years to study the flows at low Reynolds and Froude numbers [42, 53], the effects of viscous dissipation in a piezo-viscous fluid [40], viscous stratified flows [17, 56] turbulence in forced convection [55], and heat transfer in turbulent mixed convection [54]. Here, we instead derive the flow regimes for which (63)–(66) approximate to generalizations of the celebrated Oberbeck-Boussinesq approximation and the Navier-Stokes-Fourier equations for fluids with variable material properties.
3.2.1 Generalized Oberbeck-Boussinesq Approximation
If the second Froude number is of order O(ε), the Reynolds number of order of unity, the Brinkman number of order O(ε) or smaller and the Péclet number of order of unity or greater, then at the leading order equations (65) and (66) are
and
Obviously, Eq. (68) can be integrated and, taking the atmospheric pressure as the reference pressure, we deduce that p 0 coincides with the hydrostatic pressure p h = 1 − z. We now notice that Eqs. (67) and (69) are not sufficient to determine all the thermodynamic fields at O(1). Therefore, in order to attain the closure, we collect the terms of order O(ε) in (65) and, in view of (67), get
Now equations (67), (69) and (70) form a closed system, in which p 1 can be regarded as the hydrodynamic pressure. Finally, setting P = εp 1, re-dimensionalizing (67), (69) and (70), and omitting the subscript ‘0’ yield the generalized Oberbeck-Boussinesq approximation derived by Rajagopal et al. [39]
where the dimensionalized hydrostatic pressure is given by the well-known Stevin’s law p h = ρ ref g(d − z).
3.2.2 Generalized Navier-Stokes-Fourier Equations
Suppose that Fr2, Fr2∕Re and Pe are of order of unity or greater, and the Brinkman number is of order O(ε) or smaller. Then collecting the terms of order O(1) in Eqs. (65) and (66) and re-dimensionalizing lead to the Navier-Stokes-Fourier equations for a fluid with material properties depending on pressure and temperature:
Obviously, in isothermal conditions (72) reduces to the Navier-Stokes equations for piezo-viscous fluids
4 Rayleigh-Bénard Problem for Fluids with Pressure- and Temperature Dependent Viscosities
Problems involving thermal convection are amongst those that have been studied most assiduously in mechanics in virtue of their relevance to a plethora of problems in astrophysics and geophysics. Understanding thermal-convection is at the heart of explaining weather patterns, solar winds, flows in the interior of stars, thermal currents in oceans, as well as numerous important industrial applications. The prototypical theoretical model as well as experimental set up, within which one can systematically investigate the effect of thermal-convection, is the flow that occurs in a fluid layer due to a thermal gradient that is present across the layer. The earliest experiments of thermal-convection in a fluid layer, heated from below, were carried out by Bénard [7]. He found a pattern of polygonal cells, predominantly hexagonal, though a few rectangular, pentagonal and septagonal cellular structures were also present. Bénard also found that these cellular structures were also quite stable modes under certain circumstances. Lord Rayleigh [43] studied the stability of the flow in a fluid layer heated from below, when the upper layer was stress-free.
There have been numerous studies concerning the stability/instability of ‘Rayleigh-Bénard flows’. Until a critical difference in temperature is reached, the main process for the transfer of heat is conduction and upon reaching the critical temperature gradient convective rolls set in. Depending on the nature of boundary conditions (flow between solid boundaries, flow when one boundary is free of stress, etc.) one finds various types of flows are possible. A detailed discussion of the literature pertinent to various aspects of Bénard convection can be found in [8, 12, 14, 23, 25, 30]. An elegant introduction to the problem can be found in the treatise by Chandrasekhar [13].
The governing equations for the study of Bénard convection are obtained by appealing to an approximation that was independently established by Oberbeck [31, 32] and Boussinesq [9]. Such an approximation was established for fluids with constant material parameters. Thus, since here we aim at studying the problem of the onset of convection in a fluid whose viscosity varies with pressure and temperature, we cannot appeal to the classical Oberbeck-Boussinesq approximation but we have to employ its generalization derived in Sect. 3.2.1. We shall next find a necessary and sufficient condition for the linear stability of the conduction solution and compare the critical thresholds for the onset of convection in fluids with pressure and temperature dependent viscosities with the classical results for fluids whose viscosity is constant.
4.1 Conduction Solution: Evolution Equations of Perturbations
Assume that the viscosity of the horizontal fluid layer Ω d (see Sect. 3.2.1) is an analytic function of pressure and temperature, while the coefficient of thermal expansion, the specific heat at constant pressure and the thermal conductivity are constant. It is worth noting that this assumption is coherent with the experimental evidences by Bridgman [10] reported in Sect. 2 and permits to appreciate the effects of a variable viscosity on the critical threshold for the onset of convection. In this framework, the generalized Oberbeck-Boussinesq approximation (71) becomes
The appropriate boundary conditions for the temperature and hydrodynamic pressure to add to system (74) are
with θ L > θ U. Our aim is the study of stability of the steady static conduction solution m 0 to (74)–(75):
In order to study the stability of the conduction solution m 0 we introduce the perturbations u = u i + v j + w k, 𝜗 and Π to \(\bar {\boldsymbol {v}}\), \(\bar {\theta }\) and \(\bar {P}\), respectively, i.e.
Then, inserting (77) into (74) gives the evolution equations of perturbations
that are valid for all \((x,y,z,t)\in \mathbb {R}^2\times [0,d]\times [0,+\infty [\). To (78) we append the initial conditions
and the boundary conditions
and
for rigid boundaries, or
for stress-free bounding surfaces. We refer to [13] for the derivation of the boundary conditions (81) and (82). In (79) u 0 and 𝜗 0 are regular fields, with u 0 being divergence-free.
4.2 Linear Stability Analysis
Since the viscosity is an analytic function of the temperature and pressure, for sufficiently small disturbances we can approximate the two terms containing μ in (78)1 as:
and
where, henceforth, a prime denotes the derivative of a function which depends only on one variable.
Thanks to (83) and (84) we can linearize the evolution equations of perturbations (78) to obtain
It is now convenient to non-dimensionalize (85) and the boundary conditions (80)–(82) by introducing the following scales:
Inserting the dimensionless quantities (86) into (85) and (80)–(82) yields the non-dimensional equations (omitting the asterisks)
where
are the Rayleigh and Prandtl numbers, respectively, and the dimensionless boundary conditions
for rigid boundaries, and
for bounding surfaces free of stress.
As common praxis in the linear stability analysis of isochoric flows, we take the third component of the curlcurl of (87) to eliminate the disturbance Π and obtain the following coupled system in w and 𝜗:
where \(\varDelta _s=\displaystyle \frac {{\partial }^2}{{\partial } x^2}+\frac {{\partial }^2}{{\partial } y^2}\) is the so-called horizontal Laplacian. Since the coefficients in equations (91) depend only on z, the equations admit solutions which depend on x, y and t exponentially. We therefore look for solutions of the form:
in which it is understood that the real parts of these expressions must be taken into consideration to obtain physical quantities. The wave speed σ may be complex, say σ = σ r + iσ i. Thus, expressions (92) represent waves which travel in the x and y co-ordinate directions with phase speed σ i∕a, where \(a=\sqrt {a_x^2+a_y^2}\) is the two-dimensional wave number, and whose growth or decay in time is given by \(\exp (\sigma _rt)\). A wave of the form (92) is then stable if σ r ≤ 0 (marginally stable if σ r = 0), and unstable if σ r > 0.
Setting D = d∕dz and inserting (92) into (91) gives the system of ordinary differential equations
to which we add the boundary conditions
for rigid boundaries, or
for stress-free boundaries.
Denoting by the superscript ⋆ the complex conjugate, multiplying (93)1 by W⋆, (93)2 by a2 Θ⋆, summing and integrating over the interval [0, 1] taking into account the boundary conditions (94) or (95), we obtain
Considering the imaginary part of (96) yields that σ i = 0, that is the wave speed of the perturbation is real. Therefore the linearized equations of Bénard convection (85) satisfy the strong form of principle of exchange of stabilities [50] also in the case of fluids with pressure- and temperature-dependent viscosity. In addition, rewriting (96) as
with
and
we deduce that the modes (92) with two-dimensional wave number a are linearly stable if and only if
where \(\mathcal {H}\) is the space of kinematically admissible disturbances:
for rigid boundaries, or
for stress-free bounding surfaces. The existence of the maximum of the functional \(\mathcal {I}/\mathcal {D}\) in \(\mathcal {H}\) can be proved by following similar arguments as in [45].
It is easy to check that the Euler-Lagrange equations associated with the variational problem (101) coincide with (93) with σ = 0 giving the marginally stable states. Following [13], (93) with σ = 0 can be simplified further to the following sixth-order ordinary differential equation
to which we add the boundary conditions
for rigid boundaries, or
for stress-free boundaries. The square of the maximum of the functional \(\mathcal {I}/\mathcal {D}\) is then the reciprocal of the least eigenvalue of the characteristic-value problem (104) with boundary conditions (105) or (106) and thus the marginal stability curve has equation \(\mathcal {R}=\mathcal {R}_L(a)\). Finally, we introduce the so-called critical Rayleigh number
and note that if \(\mathcal {R} \leq \mathcal {R}_{cr}\) then all modes are stable, while if \(\mathcal {R} > \mathcal {R}_{cr}\) there exists at least one unstable mode. Thus, we may conclude that the conduction solution m 0 is linearly stable if and only if
To appreciate the departures from the classical results for fluids with constant viscosities, in Table 1 we display the critical thresholds for the Rayleigh number for rigid and stress-free boundaries when the viscosity depends exponentially on pressure and temperature according to (14). In this case, the nondimensional viscosity function \(\hat {\mu }\) reads
The critical thresholds \(\mathcal {R}_{cr}\) have been found for different values of the dimensionless parameter Γ by solving the eigenvalue problems (104) and (105) or (106) with the aid of the MATLAB www.bvp4c solver. Observe that, both in the rigid and stress-free case the critical threshold \(\mathcal {R}_{cr}\) increases with increasing Γ and equals the critical thresholds for fluids with constant viscosities when Γ = 0. This result is physically reasonable as the viscosity, and hence the resistance to motion from the rest state, increases as Γ increases. From the definition of Γ we can then assert that the pressure dependence of viscosity has a stabilizing effect on the onset of convection, in the sense that the critical threshold for the Rayleigh number is greater than the one that can be predicted starting from the assumption that viscosity is constant or dependent only on temperature.
The nonlinear stability of the conduction solution m 0 in a fluid whose viscosity is an analytic function of the pressure and temperature has been studied by Rajagopal et al. [38]. They proved that, under appropriate conditions on the initial disturbance of the temperature field 𝜗 0, the conduction solution is nonlinearly stable with respect to the energy of the perturbations if inequality (108) holds. In this way, Rajagopal et al. proved that (108) is a necessary and sufficient condition for the local nonlinear stability of m 0.
5 Parallel Shear Flows of Piezo-Viscous Fluids
In the last two sections we shall consider isothermal flows in fluids with variable viscosities. In particular, in this section the viscosity will be assumed to depend only on pressure, while in the next section, to include shear-thickening/thinning effects, viscosity will be expressed in terms of pressure and shear.
The main goal of this section is the determination of some classes of steady unidirectional shear flows that are possible in a piezo-viscous fluid. We shall show that, under the assumption that gravity is the only body force acting on the fluid, Couette flows are possible for any constitutive model of the viscosity, whereas Poiseuille flows are possible only in fluids with constant viscosity. To the best of our knowledge, this result is novel as results on the existence of Poiseuille flows in piezo-viscous fluids are available only in the absence of body forces. Assuming that body forces are negligible compared to viscous forces and pressure gradients, Bair et al. [3] claimed to have proven that Poiseuille flows in piezo-viscous fluids are not possible, a secondary flow being necessary. This claim is not true in general. In fact, Hron et al. [19] showed that steady unidirectional flows are possible if the dependence of the viscosity on pressure is linear, and explicit exact continuous solutions can be established even if shear-thinning effects are included. On the other hand, for other forms of the viscosity, with polynomial and exponential dependence on the pressure, Hron et al. [19] reconfirmed the results of Bair et al. [3]. Two years later, Renardy [44] gave an elegant proof on the existence/nonexistence of Poiseuille flows in piezo-viscous fluids. He proved that, in the absence of body forces, Poiseuille flows are possible only if viscosity depends linearly on pressure. In what follows we shall consider also the case in which body forces are negligible and give an alternative proof of the result by Renardy [44].
5.1 Governing Equations
When gravity is the only force acting on the fluid, the equations governing the isothermal flows in the horizontal fluid layer Ω d are the generalized Navier-Stokes equation (73).
We are here interested in two types of unidirectional flow: Couette flow, when one plate is fixed (z = 0) and the other one (z = d) moves with a prescribed velocity; and Poiseuille flow, when homogeneous Dirichlet boundary conditions at the plates z = 0, d are considered. We then look for solutions to (73) of the form
which satisfy the following boundary conditions
or
Inserting the ansatz (110) into (73) and non-dimensionalizing the resulting system of pdes by means of the scales
yield (omitting the asterisks for simplicity of notation) the dimensionless equations
where
is the shear stress. To (114) we add the dimensionless boundary conditions
or
From (114) we easily deduce that p = p(x, z) and that both the shear stress and the pressure are solutions of the wave equation ψ xx − ψ zz = 0. This leads to the representations
5.2 Couette Flows
Let us first consider the Couette flows. Assuming that the pressure at the upper boundary is constant and equal to the reference pressure, namely p(1) = 0, from (118)1 the functions Ξ and Ψ are such that Ξ(x + 1) + Ψ(x − 1) = 1 for all \(x\in \mathbb {R}\), by which one deduces that Ξ and Ψ are of the form
with a and b constants. Hence, inserting (119) into (118) we deduce that the pressure, and thus the shear stress, do not depend on x, whence a = 0 and
Finally, from (115), (116) and (120)2 the velocity component u is found to be
Couette flows are then possible in a piezo-viscous fluid with a very general response function of the viscosity. As examples, if the (dimensionless) viscosity is given by a power law of the form
then
where Lerch Φ is the Lerch Phi function; while if viscosity depends exponentially on the pressure according to the Barus law
then
Figure 1 displays the velocity profiles of the Couette flow for different models of the viscosity.
5.3 Poiseuille Flows
We now consider the Poiseuille flows and observe that, since the velocity field satisfies the boundary conditions (117), the assumption that either Ξ or Ψ is constant leads to the trivial flow u ≡ 0 (no motion). Thus, we have
Moreover, it can be easily proven that also the assumption μ′(p)u z = μ′(p)τ∕μ(p) = constant leads to the trivial motionless flow. Hence,
Next, from (115) we deduce that
whence, on using (114)1,
In the light of (126) and (127), on using the representations of p and τ (118), (129) can be rewritten as
with p as in (118)1. Therefore, because of the explicit dependence of the pressure on the vertical variable z (and not only through Ξ(x + z) and Ψ(x − z)), (130) holds if and only if the viscosity of the fluid is constant. Consequently, when gravity is the only body force acting on the fluid, Poiseuille flows are possible only if the viscosity is constant, in which case
where A 0 is the constant pressure gradient that induces the flow, and κ is an integration constant that, once A 0 is known, can be determined by measuring the pressure at a single point on the boundary.
Assume now that body forces are negligible compared to the viscous forces and pressure gradients. Then, following the same arguments as in the case in which the effects due gravity are taken into account we arrive at (130) with p given by
Since in the absence of body forces the pressure does not depend explicitly on z but only through Ξ(x + z) and Ψ(x − z), (130) with p as in (132) holds if and only if μ is a linear function of pressure. In other words, the dimensionless viscosity must be of the form
For such a dependency on the pressure, a simple manipulation of (114) and (115) gives
Integrating (134)2 yields
where ϕ(x) is an arbitrary function to be determined with the aid of (134)1. Indeed, inserting (135) into (134)1 we obtain
which holds if and only if
with A 0 being constant. Next, integrating (137) and taking into account the boundary conditions (117), we derive the Poiseuille flow
where κ is an integration constant that, as indicated before, can be determined once A 0 is known. Finally, from (138)2 it follows that the maximum speed in a Poiseuille flow decreases with increasing ϖ, and, in the limit as ϖ → 0, the velocity profile tends to the one which has been determined for a fluid with constant viscosity (see Fig. 2).
Poiseuille flow. Velocity profiles for different values of the piezo-viscous coefficient ϖ and (a) A 0 = −10 and (b) A 0 = −100
6 Flow of Fluids with Pressure and Shear Dependent Viscosity Down an Inclined Plane
In this final section, we carry out an analysis of the flow of a fluid with a pressure and shear dependent viscosity down an inclined plane within the context of the lubrication approximation. It is legitimate to ask where the pressure dependence of viscosity could become important within the context of the lubrication approximation. Thin film flows are ubiquitous in engineering, geophysics, biology and elsewhere, and low aspect ratios are often the basis for simplified fluid dynamical models. An important relevant application in geophysics is the flow of glaciers and ice sheets as well as rock glaciers. For instance, while the ice sheet covering Antarctica is several kilometers thick, it however has a horizontal extent of several thousand kilometres, yielding a length scale ratio epsilon of order 10−3 [47]. These glaciers clearly exhibit non-Newtonian characteristics in that their viscosities depend on the shear rate so that their flows are modelled using a shallow-ice approximation and Glen’s flow law [33]: in other words, as gravity currents with non-Newtonian (power law) rheology. On the other hand, there are several papers that investigate the possibility of normal stress effects in the creep of polycrystalline ice (see, for instance, [27] and [24]). In particular, Jones and Chew [22] have shown that hydrostatic pressure decreases the creep of polycristalline ice slightly and, then, above 15 MPa, a minimum creep rate is reached followed by an increase in rate with increasing hydrostatic pressure. Therefore, in view of the depths of glaciers we would expect that the pressure would also influence the viscosity. As the viscosity depends on both the shear rate as well as the pressure, it is possible that these two effects could either compete against each other thereby mitigating their effects, or join forces to enhance the qualitative and quantitative differences. As the fluid can shear-thin or shear-thicken, both possibilities may come to pass.
6.1 Basic Equations
We consider a fluid moving on an inclined plane, whose angle of inclination is ι. Let now Oxyz be a Cartesian frame of reference with fundamental unit vectors i, j and k, where the coordinate z is perpendicular to the plane, the x and y coordinates lie in the plane, y is horizontal and x increasing downward. We denote the components of the velocity v of the fluid in the directions x, y and z as u, v and w, respectively. The generalized Navier-Stokes equations read then
where the Cauchy stress is now given by the implicit relation
or, equivalently, since ∥D∥ = [tr(D2)]1∕2,
The viscosity is taken of the form
with χ > 0, p ref the reference pressure and, as is reasonable to expect since the fluid viscosity increases as the pressure increases, ς is a positive function whose value increases with increasing pressure. Model (141), with a viscosity of the type (142), allows for a fluid that is capable of shear thinning, when χ > 1, or shear thickening, when χ ∈ (0, 1). Here, for the sake of definiteness, we shall consider the following forms for ς:
where ς ref > 0, β ≥ 0 and n ≥ 0 are constants. In general, the material parameters that appear in (143) and (144) can be obtained by corroboration with experimental data. Here, in order to illustrate the effects due to the pressure dependence of viscosity, we merely carry out a parametric study.
We prescribe the following boundary conditions for the velocity and pressure fields
where
is the unit normal to the free surface of the current z = h(x, y, t).
Let H and L denote the characteristic thickness and characteristic length along the plane of the current free surface z = h(x, y, t), respectively. The main assumption in lubrication approximation is that the lengthscale ratio H∕L is small [51]. Here, as we are interested in fluids whose viscosity depends on the pressure, we assume that the ratio H∕L is small though H is large enough to have a significant dependence of the viscosity on the pressure.
As a consequence of the smallness of the lengthscale ratio H∕L, the component of the velocity parallel to the plane is much larger than the normal component, so that
We call U, V and W the characteristic velocities along x, y and z directions, respectively. Hence, \(U_\parallel =\sqrt {U^2+V^2}\) and W are the characteristic velocities parallel and perpendicular to the inclined plane, respectively. From Eqs. (139)1 and (147) we find that W = HU ∥∕L.
There are many ways of transforming the governing equation (139) and boundary conditions (145) into dimensionless expressions. Here we introduce a scaling which is similar to that introduced in [1]:
Substituting the dimensionless quantities (148) into equations (139), (141) and (146) and into the boundary conditions (145) leads to (omitting all asterisks)
where 𝜖 = H∕L ≪ 1, and
In this framework the dimensionless version of ς is an increasing function such that ς(0) = 1. In particular, (143) and (144) become, respectively,
and
The dimensionless quantities
are, respectively, the Reynolds and Froude numbers for a fluid film moving over an inclined plane.
Depending on the values considered for the characteristic scales, different types of flow regime occur. Here we shall focus on the following two types of flow regimes:
-
1.
The nearly steady uniform regime, where the viscous contribution is comparable to the gravitational effect. In this case, we have
$$\displaystyle \begin{aligned} U_\parallel=\left[\frac{\rho g\sin\iota H^{(\chi+1)/\chi}}{\varsigma_{\mathrm{ref}}}\right]^\chi \end{aligned} $$(155)and Fr2 = O(Re). Inertial terms and pressure gradient terms must be negligible, which means 𝜖Re ≪ 1. Therefore, from (149) and (151) the approximate equations are found to be given by
$$\displaystyle \begin{aligned} \left\{\begin{array}{ll} \displaystyle\frac{{\partial} u}{{\partial} x}+\frac{{\partial} v}{{\partial} y}+\frac{{\partial} w}{{\partial} z}=0,\\ \displaystyle\frac{\partial }{\partial z}\left\{\varsigma(p)\left[\left(\frac{{\partial} u}{{\partial} z}\right)^2+\left(\frac{{\partial} v}{{\partial} z}\right)^2\right]^{(1-\chi)/(2\chi)}\frac{{\partial} u}{{\partial} z}\right\}\\ \displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad +2^{(1-\chi)/(2\chi)}=0,\\ {} \displaystyle\frac{\partial }{\partial z}\left\{\varsigma(p)\left[\left(\frac{{\partial} u}{{\partial} z}\right)^2+\left(\frac{{\partial} v}{{\partial} z}\right)^2\right]^{(1-\chi)/(2\chi)}\frac{{\partial} v}{{\partial} z}\right\} =0,\\ \displaystyle\frac{{\partial} p}{{\partial} z}+ 1=0. \end{array}\right. \end{aligned} $$(156) -
2.
The viscous regime, where the effect of the pressure gradient is balanced by stresses induced due to the viscosity within the bulk. In this case, we have
$$\displaystyle \begin{aligned} U_\parallel=\left[\frac{\rho g\cos \iota H^{(2\chi+1)/\chi}}{\varsigma_{\mathrm{ref}}L}\right]^\chi \end{aligned} $$(157)and consequently Fr2 = 𝜖Re. Inertial terms must be small compared to the effect of the pressure gradient and the slope must be gentle (\(\tan \iota =O(\epsilon )\)). This imposes the constraint 𝜖Re ≪ 1. In such a way, from (149) and (151) we deduce the approximate equations
$$\displaystyle \begin{aligned} \left\{\begin{array}{ll} \displaystyle\frac{{\partial} u}{{\partial} x}+\frac{{\partial} v}{{\partial} y}+\frac{{\partial} w}{{\partial} z}=0,\\ \displaystyle\frac{\partial }{\partial z}\left\{\varsigma(p)\left[\left(\frac{{\partial} u}{{\partial} z}\right)^2+\left(\frac{{\partial} v}{{\partial} z}\right)^2\right]^{(1-\chi)/(2\chi)}\frac{{\partial} u}{{\partial} z}\right\}\\ \displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \quad +2^{(1-\chi)/(2\chi)}\left(\frac{\tan\iota}{\epsilon}-\frac{{\partial} p}{{\partial} x}\right)=0,\\ {} \displaystyle\frac{\partial }{\partial z}\left\{\varsigma(p)\left[\left(\frac{{\partial} u}{{\partial} z}\right)^2+\left(\frac{{\partial} v}{{\partial} z}\right)^2\right]^{(1-\chi)/(2\chi)}\frac{{\partial} v}{{\partial} z}\right\}\\ {} \displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \quad -2^{(1-\chi)/(2\chi)}\frac{{\partial} p}{{\partial} y}=0,\\ {} \displaystyle\frac{{\partial} p}{{\partial} z}+ 1=0. \end{array}\right. \end{aligned} $$(158)
Moreover, from (150) and (151), by virtue of the smallness of 𝜖, the boundary conditions (145) approximate to
Finally, we derive the evolution equation for the free surface z = h(x, y, t). We first integrate the constraint of incompressibility over the flow depth to obtain, by means of boundary condition (159)1,
But, obviously,
Therefore, combining (160) and (161) gives the required equation for h:
where we have introduced the depth-averaged variables defined as
6.2 Nearly Steady Uniform Regime
It is easy to verify that system (156) with boundary conditions (159) admits the solution
Therefore
and (162) becomes
Equation (166) is a quasilinear first order partial differential equation whose general solution can be found by the method of characteristics. If f(ξ) is an initial profile, then the corresponding solution is given by
The wave (167) breaks, i.e. its profile becomes multivalued, at time \(t_B=-\left [\,F''(f(\xi _B))\displaystyle \frac {{\mathrm {d}} f}{{\mathrm {d}}\xi }(\xi _B)\right ]^{-1}\) at the point x B = ξ B + F′(f(ξ B))t B, provided that ξ B satisfies the following two conditions
Since ς is a positive increasing function, from (168) we deduce that the pressure dependence of the viscosity has the effect of delaying the time at which the wave could break. To quantify this delaying effect we consider ς of the form (152) and assume that h(x, 0) = f(x) = 1 − x2. If the fluid is Newtonian with a constant viscosity μ 0 (i.e., χ = 1 and ς(p − p ref) = μ ref in (142)), it is easy to show that the wave breaks at time \(t_{BN}=3\sqrt {3}/8\). In order to make the differences between the non-Newtonian case that is being considered and the classical Newtonian case more evident, we have plotted the ratio between the breaking time t B in the non-Newtonian case and t BN as a function of χ (Fig. 3a) and as a function of the non-dimensional piezo-viscous coefficient ω (Fig. 3b). Furthermore, the solutions to the wave equation (166) with χ = 0.5 (Fig. 4a) and χ = 1.5 (Fig. 4b) are plotted at different times together with the profiles of the free surface z = h(x, t) in the classical Newtonian case. We find that the solutions are qualitatively similar, though quantitatively different.
Ratio t B∕t BN as a function of (a) χ and (b) the piezo-viscous coefficient ω. The pressure-dependent parameter ς is assumed to be of the form (143) and the initial profile considered is h(x, 0) = 1 − x2
Solutions of (166) with an initial profile f(x) = 1 − x2 at t = 0, t = 0.5, t = 1. The dashed line represents the solution in the classical Newtonian case, whereas the solid line represents the solution in the case in which the dimensionless parameter ς depends on the pressure according to the law ς(p) = e0.1p and (a) χ = 0.5 and (b) χ = 1.5
Finally, in order to look for self-similar solutions of (166), we need to know whether F′ is invertible. The invertibility of F′ is linked with the equation
Indeed, if (169) admits positive roots, the least of which we denote by \(\hat {h}\), then F′ is invertible in \([0,\hat {h}[\). On the contrary, if (169) does not admit positive roots, then F′ is invertible in [0, +∞[. In any case F′ is continuous and increasing. It is interesting to show that some time after the initiation of the current, no matter what the initial shape, the solution tends to the unique self-similar solution of Eq. (166), i.e.
In order to prove (170), from (166) we deduce that h is constant along the characteristics given by
Thus, if initially h(x, 0) = f(x), the characteristics are straight lines
with x 0 being the initial value of the characteristic. The solution of (166) is then
If the viscosity does not depend on the pressure, Eq. (170) reduces to the self-similar solution found by Perazzo and Gratton [34] that in turn is the non-Newtonian counterpart of the self-similar solution derived by Huppert [21] for Newtonian fluids.
6.3 Viscous Regime
A lengthy but straightforward algebraic manipulation allows us to obtain the solution to the boundary-value problem (158)–(159):
where 
is the two-dimensional gradient:
Then
so that (162) becomes
Now let us make the further assumption that the flow depends only on the x coordinate. Then \(\displaystyle \frac {{\partial } h}{{\partial } y}=0\) (so that v = 0) and (178) reduces to
To find traveling wave solutions we assume that h depends on the single variable s ≡ x − ct, where c is a constant which represents the wave speed. Since in what follows we shall assume that the inclined plane is infinite and limit our analysis to waves propagating downwards, we take c > 0. We refer the interested reader to [41] for a more detailed study of downslope and upslope traveling wave solutions. Inserting the ansatz h = h(s) into (179) and integrating once, one obtains
c 1 being an integration constant. Let c 1 = 0 in (180). Then, (180) may be written as
In order to discuss the integrability of Eq. (181) with c > 0, we have to find the positive roots of the following equation
The roots of (182) may be found numerically. Nevertheless, we can deduce the number of positive roots of (182) by studying the function \(\mathcal {F}(h)\equiv (\tan \iota /\epsilon )^\chi F(h)/h\). \(\mathcal {F}\) is a continuous differentiable function that tends to zero as h → 0, whose derivative may be written as
From (183) it follows that \(\mathcal {F}'\) is positive in a neighbourhood of h = 0, but it might change sign away from zero if (169) admits positive roots. Here, for the sake of simplicity, we shall limit our analysis to the constitutive functions for which (169) admits at most one positive root. It is easy to recognize that models (152) and (153) meet this requirement.
We are now able to say how many positive roots (182) admits. In fact:
-
1.
if [ς(h)∕h]χ has linear growth as h → +∞, then \(\mathcal {F}\) is increasing and tends to l > 0 as h → +∞ so that (182) with c ∈ ]0, l[ admits only one positive root, whereas it does not admit a positive root for c ≥ l;
-
2.
if [ς(h)∕h]χ has sublinear growth as h → +∞, then \(\mathcal {F}\) is increasing and tends to + ∞ as h → +∞ so that, for any c > 0, (182) admits a unique positive root;
-
3.
if [ς(h)∕h]χ has superlinear growth as h → +∞, then \(\mathcal {F}\) attains its absolute maximum at h = h ∗ > 0 and tends to zero as h → +∞ so that (182) admits two positive roots if \(c\in ]0,\mathcal {F}(h_*)[\), only one positive root if \(c=\mathcal {F}(h_*)\), and no positive root for \(c >\mathcal {F}(h_*)\).
According to the number of positive roots of (182), one, two or three families of solutions to (180) may arise.
If (182) does not admit a positive root, then Eq. (180) may be numerically integrated over the range \((0,\bar {h})\) for all \(\bar {h}>0\). In this case the general solution is a decreasing function defined over the interval (−∞, c 2), c 2 being an integration constant and tends to + ∞ as s →−∞. Therefore, we do not consider these solutions as they do not satisfy the lubrication approximation.
If (182) admits only one positive root h m, then two families of solutions to (180) arise. The first is formed by bounded decreasing functions defined over the range (−∞, c 2) satisfying the inequality 0 ≤ h ≤ h m. For these solutions we have h → h m as s →−∞. Then they represent traveling waves behind a front running downslope that, far behind the front (s →−∞), tend to the steady downslope flow h = h m (see Fig. 5). The other family is formed by increasing functions bounded from below for which h ≥ h m. These solutions represents downslope traveling waves with no front for which h → h m as s →−∞ and h → +∞ as s → +∞. Therefore, they do not satisfy the lubrication approximation.
Profiles of downslope traveling waves behind a front. The solid line represents the traveling wave solution when χ = 1.5 and ς(p) = 1 + 0.2p, whereas the dashed line represents the traveling wave solution in the classical Newtonian case. We have considered \((\tan \iota )/\epsilon =1\) and c = 1
If (182) admits two positive roots, h m < h M, then, as well as the downslope traveling waves behind a front, two other families of solutions to (180) arise, representing downslope traveling waves with no front (Fig. 6). The former is constituted by bounded increasing functions satisfying the inequality h m ≤ h ≤ h M and for which we have h → h m as s →−∞ and h → h M as s → +∞. The latter is formed by decreasing functions that are bounded from below as they satisfy the inequality h ≥ h M and for which we have h → +∞ as s →−∞ and h → h M as s → +∞. We disregard the traveling wave solutions belonging to this family as the length-scale ratio fails to be small as s →−∞. On the contrary, the former class of downslope travelling waves with no front satisfies the lubrication approximation. Furthermore, as shown by Rajagopal et al. [41], the waves belonging to this family are compressive shock waves which can also be viewed as heteroclinic orbits connecting the two equilibria h = h m and h = h M of (181) (see Fig. 6).
Profiles of downslope traveling waves solutions for χ = 1.1, ς(p) = 1 + 0.05p3, \(\tan \iota /\epsilon =0.5\) and c = 1. In the case we are considering two families of downslope traveling wave solutions that satisfy the lubrication approximation arise (see the text): downslope traveling waves behind a front (solid line) and compressive shock waves (dashed line)
We finally observe that
Therefore near the wave front, where the effects of pressure can be neglected, the solution to Eq. (180) is approximated by that found by Perazzo and Gratton [34], namely
with 2 F 1(a, b, c, d) being the hypergeometric function. From (185) we deduce that h′ tends to infinity as s → c 2. Hence, near the wave front, the component of the fluid velocity normal to the incline is not small with respect to the parallel component and thus the solution does not satisfy the lubrication approximation.
Notes
- 1.
For the sake of self-consistency, if Σ is a smooth scalar, vector or tensor field defined on the trajectory of the body \(\mathcal {B}\), \(\dot {\varSigma }=\varSigma _t+(\boldsymbol {v}\cdot \nabla )\varSigma \).
- 2.
The other physical properties measured by Bridgman are the isothermal compressibility, the thermal expansion coefficient, the specific heat and the thermal conductivity.
- 3.
Assuming that Ω is a horizontal layer is convenient for deriving the set of approximations we shall adopt in this paper. However, the analysis we are going to perform can be adapted, by means of slight changes, to the case in which Ω is bounded in one direction provided that such a direction is non-horizontal.
References
Ancey, C.: Plasticity and geophysical flows: a review. J. Non-Newtonian Fluid Mech. 142, 4–35 (2007)
Andrade, E.N., Da, C.: The viscosity of liquids. Nature 125, 309–310 (1930)
Bair S., Khonsari M., Winer W.O.: High pressure rheology of lubricants and limitations of the Reynolds equation. Tribol. Int. 31, 573–586 (1998)
Bair S., Kottke P.: Pressure-viscosity relationship for elastohydrodynamics. Tribol. Trans. 46, 289–295 (2003)
Ball, J.M., James, R.D.: The scientific life and influence of Clifford Ambrose Truesdell III. Arch. Ration. Mech. Anal. 161, 1–26 (2002)
Barus C.: Isotherms, isopiestics and isometrics relative to viscosity. Am. J. Sci. 45, 87–96 (1893)
Bénard H.: Les tourbillons cellulaires dans une nappe liquide, Revue Gén. Sci. pures et Appl. 11, 1261–1271 and 1309–1328 (1900)
Bodenschatz, E., Pesch, W., Ahlers G.: Recent developements in Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 32, 708–778 (2000)
Boussinesq, J.: Théorie Analytique de la Chaleur. Gauthier-Villars, Paris (1903)
Bridgman, P.W.: The effect of pressure on the viscosity of forty-three pure liquids. Proc. Am. Acad. Arts Sci. 61, 57–99 (1926)
Bridgman, P.W.: The physics of high pressure. G. Bell and Sons, London (1931)
Busse, F.: Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967 (1978)
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford (1961)
Cross, M.C., Hahenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)
Fermi, E.: Thermodynamics. Dover Publications, New York (1956).
Gouin, H., Muracchini, A., Ruggeri, T.: On the Müller paradox for thermal-incompressible media. Contin. Mech. Thermodyn. 24, 505–513 (2012)
Govindarajan, R., Chandra Sahu, K.: Instabilities in viscosity-stratified flow. Annu. Rev. Fluid Mech. 46, 331–353 (2014)
Hadamard, J.: Sur les problèmes aux dérivées partielles et leur signification physique. Princeton Univ. Bull. 13, 49–52 (1902)
Hron, J., Malek, J., Rajagopal, K.R.: Simple flows of fluids with pressure-dependent viscosities. Proc. R. Soc. Lond. Ser. A 257, 1603–1622 (2001)
Huilgol, R.R.: On the definition of pressure in rheology. Rheol. Bull. 78(2), 12, 14–15 and 29 (2009)
Huppert, H.E.: Flow and instability of a viscous current down a slope. Nature 300, 427–429 (1982)
Jones, S.J., Chew H.A.M.: Creep of Ice as a Function of Hydrostatic pressure. J. Phys. Chem. 87, 4064–4066 (1983)
Koschmieder, E.L.: Bénard Cells and Taylor Vortices. Cambridge University Press, Cambridge (1993)
Man, C.S., Sun, Q.X.: On the significance of normal stresses effects in the flow of glaciers. J. Glaciol. 33, 268–273 (1987)
Manneville, P.: Rayleigh-Bénard convection: thirty years of experimental, theoretical and modelling work. In: Mutabazi, I., Wesfried, J.E., Gunyon (eds.) Dynamics of Spatio-Temporal Cellular Structures. Springer Tracts in Modern Physics, vol. 207. Springer, Berlin (2006)
Martín-Alfonso, M.J., Martínez-Boza, F.J., Navarro, F.J., Fernández, M., Gallegos, C.: Pressure-temperature-viscosity relationship for heavy petroleum fractions. Fuel 86, 227–233 (2007)
McTigue, D.F., Passman, S.L., Jones, S.J.: Normal stress effect in the creep of ice. J. Glaciol. 31, 120–126 (1985)
Melamed, Y., Lin, M.: Principle of Sufficient Reason. The Stanford Encyclopedia of Philosophy (Spring 2017 edn.) Zalta, E.N. (ed.) https://plato.stanford.edu/archives/spr2017/entries/sufficient-reason
Müller, I.: Thermodynamics. Pitman, London (1985)
Newell, A.C., Passot, T., Lega, J.: Order parameter equations for patterns. Annu. Rev. Fluid Mech. 25, 399–453 (1993)
Oberbeck, A.: Über die Wärmleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem. 7, 271–292 (1879)
Oberbeck, A.: Über die Bewegungsercheinungen der Atmosphäre. Sitz. Ber. K. Preuss. Akad. Miss. 383–395 and 1120–1138 (1888)
Paterson, W.S.B.: The Physics of Glaciers, 3rd edn. Pergamon, Trowbridge (1994)
Perazzo, C.A., Gratton, J.: Thin film of non-Newtonian fluid on an incline. Phys. Rev. E 67, 016307 (2003)
Rajagopal, K.R.: On implicit constitutive theories. J. Fluid Mech. 550, 243–249 (2006)
Rajagopal, K.R.: Remarks on the notion of pressure. Int. J. Non Linear Mech. 71, 165–172 (2015)
Rajagopal, K.R., Saccomandi, G.: A Novel approach to the description of constitutive relations. Front. Mater. 3, id.36 (2016)
Rajagopal, K.R., Saccomandi, G., Vergori, L.: Stability analysis of the Rayleigh-Bénard convection for a fluid with temperature and pressure dependent viscosity. Z. Angew. Math. Phys. 60, 739–755 (2009)
Rajagopal, K.R., Saccomandi, G., Vergori, L.: On the Oberbeck-Boussinesq approximation in fluids with pressure-dependent viscosities. Nonlinear Anal. Real World Appl. 10, 1139–1150 (2009)
Rajagopal, K.R., Saccomandi, G., Vergori, L.: Couette flow with frictional heating in a fluid with temperature and pressure dependent viscosity. Int. J. Heat Mass Transf. 54, 783–789 (2011)
Rajagopal, K.R., Saccomandi, G., Vergori, L.: Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane. J. Fluid Mech. 706, 173–189 (2012)
Rajagopal, K.R., Saccomandi, G., Vergori, L.: On the approximation of isochoric motions of fluids under different flow conditions. Proc. R. Soc. Lond. Ser. A 471(2180), 20150159 (2015)
Rayleigh, L.: On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. Philos. Mag. 32, 529–546 (1916)
Renardy, M.: Parallel shear flows of fluids with a pressure-dependent viscosity. J. Non-Newtonian Fluid Mech. 114, 229–236 (2003)
Rionero, S.: Metodi variazionali per la stabilitàasintotica in media in magnetoidrodinamica. Ann. Mate. Pura Appl. 78, 339–364 (1968)
Saal, R.N.J., Koens G.: Investigations into plastic properties of asphaltic bitumen. Physics 7, 408–412 (1933)
Schoof, C., R.C.A. Hindmarsh, R.C.A.: Thin-film flows with wall slip: an asymptotic analysis of higher order glacier flow models. Q. J. Mech. Appl. Math. 63, 73–114 (2010)
Spencer, A.J.M.: Theory of invariants. Continuum Physics Eringen, A.C. vol. 1, pt. 3. Academic, New York and London (1971)
Stokes, G.G.: On the theories of the internal friction of fluids in motion, and motion of elastic solids. Trans. Camb. Philos. Soc. 8, 287–305 (1845)
Straughan, B.: The Energy Method, Stability and Nonlinear Convection. Series in Applied Mathematical Science, vol. 91. Springer, New York (1992)
Szeri, A.Z.: Fluid Film Lubrication: Theory and Design. Cambridge University Press, Cambridge (1998)
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics. Springer, Berlin (2004)
Vergori, L.: Flows at small Reynolds and Froude numbers. Int. J. Eng. Sci. 48, 1659–1670 (2010)
Zonta, F., Soldati, A.: Effect of temperature dependent fluid properties on heat transfer in turbulent mixed convection. J. Heat Transfer-Trans. ASME 136, 022501 (2014)
Zonta, F., Marchioli, C., Soldati, A.: Modulation of turbulence in forced convection by temperature-dependent viscosity. J. Fluid Mech. 697, 150–174 (2012)
Zonta, F., Onorato, M., Soldati, A.: Turbulence and internal waves in stably-stratified channel flow with temperature-dependent fluid properties. J. Fluid Mech. 697, 175–203 (2012)
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Saccomandi, G., Vergori, L. (2018). Old Problems Revisited from New Perspectives in Implicit Theories of Fluids. In: Farina, A., Mikelić, A., Rosso, F. (eds) Non-Newtonian Fluid Mechanics and Complex Flows. Lecture Notes in Mathematics(), vol 2212. Springer, Cham. https://doi.org/10.1007/978-3-319-74796-5_2
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