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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 748683, 20 pages
Asymptotic Behavior of a Viscoelastic Fluid in a Closed Loop Thermosyphon: Physical Derivation, Asymptotic Analysis, and Numerical Experiments
1Grupo de Dinámica No Lineal (DNL), Escuela Técnica Superior de Ingeniería (ICAI), Universidad Pontificia Comillas, 28015 Madrid, Spain
2ICAI, Grupo Interdisciplinar de Sistemas Complejos (GISC) and DNL, Universidad Pontificia Comillas, 28015 Madrid, Spain
Received 11 February 2013; Accepted 19 April 2013
Academic Editor: Douglas Anderson
Copyright © 2013 Justine Yasappan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fluids subject to thermal gradients produce complex behaviors that arise from the competition with gravitational effects. Although such sort of systems have been widely studied in the literature for simple (Newtonian) fluids, the behavior of viscoelastic fluids has not been explored thus far. We present a theoretical study of the dynamics of a Maxwell viscoelastic fluid in a closed-loop thermosyphon. This sort of fluid presents elastic-like behavior and memory effects. We study the asymptotic properties of the fluid inside the thermosyphon and the exact equations of motion in the inertial manifold that characterizes the asymptotic behavior. We derive, for the first time, the mathematical derivations of the motion of a viscoelastic fluid in the interior of a closed-loop thermosyphon under the effects of natural convection and a given external temperature gradient.
Instabilities and chaos in fluids subject to temperature gradients have been the subject of intense work for its applications in engineering and in atmospheric sciences. In such sort of systems, the fluid displays nontrivial behaviors (as turbulence or the formation of convective rolls) when subject to a heating that competes with buoyancy effects. A traditional approach that goes back to the pioneering work by Lorenz consists of the study of the system under some simplifications. Another approach is to study the controlled setups that capture the underlying complexity of the full system, being a thermosyphon one of those simpler cases .
In the engineering literature, a thermosyphon is a device composed of a closed-loop pipe containing a fluid where some soluble substance has been dissolved [2, 3]. The motion of the fluid is driven by the action of several forces such as gravity and natural convection. The flow inside the loop is driven by an energetic balance between thermal energy and mechanical energy. The interest on this system comes both from engineering and as a toy model of natural convection (for instance, to understand the origin of chaos in atmospheric systems).
Thus far, all the works on thermosyphons analyze the behavior of a Newtonian fluid inside the loop, consequently neglecting elastic effects in the system coming from either the fluid itself or the elastic walls of the loop. However, many interesting fluids are known to behave slightly different from the common (Newtonian) fluids in terms of their response to an applied stress and are commonly referred to as viscoelastic. Among them, it is worth emphasizing volcanic lavas, snow avalanches, flowing paint, or biological mucosas membranes.
Here, we consider a thermosyphon model in which the confined fluid is viscoelastic. This has some a priori interesting peculiarities that could affect the dynamics with respect to the case of a Newtonian fluid. On the one hand, the dynamics has memory and so its behavior depends on the whole past history, and, on the other hand, at small perturbations the fluid behaves like an elastic solid and a characteristic resonance frequency could, eventually, be relevant (e.g., consider the behavior of jelly or toothpaste).
The simplest approach to viscoelasticity comes from the so-called Maxwell constitutive equation . In this model, both Newton’s law of viscosity and Hooke’s law of elasticity are generalized and complemented through an evolution equation for the stress tensor, .
The stress tensor comes into play in the equation for the conservation of momentum:
For a Maxwellian fluid, the stress tensor takes the form where is the fluid viscosity, the Young’s modulus, and the shear strain rate (or rate at which the fluid deforms). Under stationary flow, (2) reduces to Newton’s law, and consequently, (1) reduces to the celebrated Navier-Stokes equation. On the contrary, for short times (where impulsive behavior from rest can be expected) equation (2) reduces to Hooke’s law of elasticity.
Memory effects can be well understood from (2) after performing a separation of variables and integrating. Thus, we can rewrite where it is clear that the local state of stress is calculated from the present and values of with a memory time scale of order .
In a thermosyphon, the equations of motion can be greatly simplified because of the quasi-one-dimensional geometry of the loop. Thus, we assume that the section of the loop is constant and small compared with the dimensions of the physical device, so that the arc length coordinate along the loop () gives the position in the circuit. The velocity of the fluid is assumed to be independent of the position in the circuit; that is, it is assumed to be a scalar quantity depending only on time. This approximation comes from the fact that the fluid is assumed to be incompressible, and so besides the quasi-one-dimensional assumption. On the contrary, temperature is assumed to depend both on time and position along the loop.
The derivation of the thermosyphon equations of motion is similar to that in [2, 3]. The simplest way to incorporate (2) is by differentiating (1) with respect to time and replacing the resulting time derivative of with (2). This way to incorporate the constitutive equation allows to reduce the number of unknowns (we remove from the system of equations) at the cost of increasing the order of the time derivatives to second order.
The resulting second-order equation is then averaged along the loop section (as in ). Hence, after nondimensionalizing the variables (to reduce the number of free parameters) we arrive at our main system of equations where ; that is, we consider Newton’s linear cooling law as in [3, 5–10] also in this paper we consider the diffusion of temperature given by the term . The parameter in (5) is the nondimensional version of which has dimensions of time. Roughly speaking, it gives the (nondimensional) time scale in which the transition from elastic to fluid-like occurs in the fluid.
The model we will take into consideration forms an ODE/PDE system for the velocity , the distribution of the temperature of the fluid into the loop, (5) with , where denotes integration along the closed path of the circuit. The function describes the geometry of the loop and the distribution of gravitational forces [2, 3]. Note that .
The system of (5) is not a trivial extension of the Newtonian model in  due to the first term in the differential equation for the velocity. Specifically, the addition of a term proportional to the second derivative of is singular, in the sense that it changes qualitatively the character of the equations. The implications of this singular perturbation cannot be ascertained using a standard boundary layer analysis (see the Appendix for details). So a complementary theoretical and numerical approach is mandatory.
We assume that which specifies the friction law at the inner wall of the loop is positive and bounded away from zero. This function has been usually taken to be , a positive constant for the linear friction case  (Stokes flow), or for the quadratic law [12, 13], or even a rather general function given by , where is the Reynolds number, . Here we will consider a general function of the velocity assumed to be large [9, 14]. The functions , , and incorporate relevant physical constants of the model, such as the cross-sectional area, , the length of the loop, , and the Prandtl, Rayleigh or Reynolds number; see . Here, we consider Newton’s linear cooling law where represents the heat transfer law across the loop wall, and is a positive quantity depending on the velocity, and is the (given) ambient temperature distribution, see [3, 5, 9, 10].
Hereafter, we consider and as continuous functions, such that , and , for and positive constants.
Our contributions in this paper are the following.(i)To obtain the system of (5) governing a closed loop thermosyphon model with a viscoelastic fluid and to study the asymptotic behavior of this system which is a generalization of the previous model . Thus, although the dynamics of viscoelastic fluids has been extensively studied in the engineering literature, it has scarcely been considered in the applied mathematics community.(ii)To present an analysis beginning with the well-posedness and boundedness of solution. The existence of an attractor and an inertial manifold is shown and an explicit reduction to low-dimensional systems is obtained. It is noteworthy that we are able to obtain an exact finite-dimensional reduction (87) that may have a much lower number of degrees of freedom.(iii)To provide a detailed numerical analysis of the behavior of acceleration, velocity, and temperature which includes a thorough study of the various behaviors of the system for different values of viscoelastic fluid and ambient temperature distribution.(iv)The numerical analysis will show that viscoelasticity induces a chaotic behavior that is not captured by a boundary layer analysis (that would predict the same qualitative behaviors as in the original model in  see the Appendix for details) being the new (nontrivial) emergent behaviors induced by the viscoelasticity worth characterizing. The structure of the paper is as follows. The first section provides a general introduction to the system, explaining briefly the dynamics of a thermosyphon, viscoelastic fluids, and the objectives of this work. In Section 2, we prove the existence, uniqueness, and boundedness of the solutions. In Section 3, we provide a detailed derivation of the dynamics of the system in the inertial manifold as a reduced dimensionality version of the full system of (5). In Section 4, we present the numerical integration of the reduced system of equations valid in the manifold to understand the role of the main parameters of the physical system. Finally, in Section 5, we conclude with a proposal for future works.
2. Well-Posedness and Boundedness: Global Attractor
2.1. Existence and Uniqueness of Solutions
In this section we prove the existence and uniqueness of solutions of the thermosyphon model (5).
First, we observe that for , if we integrate the equation for the temperature along the loop taking into account the periodicity of , that is, , we have . Therefore, exponentially as time goes to infinity for every .
Moreover, if we consider , then from the second equation of system (5), we obtain that verifies the equation where .
Finally, since , we have and the equation for reads
Also, if , the operator , together with periodic boundary conditions, is an unbounded, self-adjoint operator with compact resolvent in that is positive when restricted to the space of zero average functions . Hence, the equation for the temperature in (5) is of parabolic type for .
2.1.1. The Case with Diffusion:
We consider the acceleration and write the system (5) as the following evolution system for the acceleration, velocity, and temperature: that is, with and and the initial data .
The operator is a sectorial operator in with domain and has compact resolvent, where
Theorem 1. We assume that and are locally Lipschitz, , , and . Then, given , there exists a unique solution of (5) satisfying where and for every . In particular, (5) defines a nonlinear semigroup, in , with .
Proof. We cover several steps.
Step 1. We prove the local existence and regularity. This follows easily from the variation of constants formula of . In order to prove this we write the system as (9), and we have where the operator is a sectorial operator in with domain and has compact resolvent. In this context, the operator must be understood in the variational sense; that is, for every , and coincides with the fractional space of exponent . Hereafter we denote by the norm on the space . Now, if we prove that the nonlinearity is well defined and is Lipschitz and bounded on bounded sets, we obtain the local existence for the initial data in .
Using and being locally Lipschitz together with and , we will prove the nonlinear terms, , and and satisfy , , and ; that is, is well defined, Lipschitz, and bounded on bounded sets. It is possible to prove this by considering . In order to prove these properties of the nonlinearity , let , and we note that where and adding ), and in , we have and adding in , we get and from the previous hypothesis on function , there exists such that and the rest is obvious.
Therefore, using the techniques of variations of constants formula of , we obtain the unique local solutions of (8) which are given by
with , and using again the results of , we get the regularity of solutions. In fact, from the smoothing effect of the equations, we have and , for some positive and any . Now, for we have , and since is well defined, Lipschitz, and bounded on bounded sets, we have and . Since is arbitrary, we obtain the regularity of the local solution.
Step 2. Now, we prove the solutions of (8) for every time .
To prove the global existence, we must show that the solutions are bounded in norm on finite time intervals. First, to obtain that the norm of is bounded in finite time, we note that multiplying the equations for the temperature by in and integrating by parts, we have: since .
Using Cauchy-Schwartz and the Young inequality and then the Poincaré inequality, since together with is the first nonzero eigenvalue of in , we obtain and using , we get and we conclude that the norm of in remains bounded in finite time.
Now, we note that differentiating the second equation of (5) with respect to , we obtain the same equations for , and considering now , we obtain
Thus, we show that the norm of in remains bounded in finite time. Then, using bounded for finite time, we prove that and remain bounded in finite time and we conclude.
2.1.2. The Case with No Diffusion:
We note that if is a given continuous function, then the equation for the temperature can be integrated along characteristics to obtain and plugging this into the nonlocal differential equation for the acceleration and into the equation for the velocity yields
We note that for and since in this space the translations are continuous isometries, (27) defines a continuous function of time with values in this space. Although we restrict ourselves to , many other choices of space are possible for solving problem (26). In fact any Banach space of 1-periodic functions of having zero mean and in which translations are continuous isometries can be used as an “admissible space” ; see . In particular are admissible spaces between others. Then, we can prove Lemma 2.
Lemma 2. Let , fix , and assume that where is an “admissible space”; see , in particular . Then, the function given in (27), , is an integral solution of the PDE which is satisfied only if and are differentiable. In particular, if , then is continuous with values in and satisfies the PDE as an equality in , a.e. in time. Moreover, (27) satisfies the following properties:(i)(ii) if there exist positive constants and such that satisfy for all , then satisfies positive constant independent on time;(iii) we assume that and there exist positive constants and such that satisfy for all . If we also assume that are continuous in , then is a positive constant, and .
Proof. As noted earlier, we need to solve the fixed point problem
on a space of continuous functions. More precisely, we take , endowed with the sup norm, with and to be chosen and prove that is a contraction on .
From (29) in Lemma 2 for , we have , and this, together with the local Lipschitz property of shows that for fixed , if is sufficiently small.
To show that is a contraction, it is clear that we must prove some Lipschitz dependence on with respect to .
First, we note that from , then verify (30), that is, for all with , and given , again from (31) in Lemma 2, we have With these we find that is Lipschitz on with a Lipschitz constant depending on and that tends to zero as and then is a contraction for small enough . Therefore, local well-posedness follows.
To prove the global existence, it is sufficient to prove that is bounded on finite time intervals, since from we find that is of Cauchy type as for finite . Consequently, the limit of exists in and the solution can be prolonged.
But again from (29) together with (18) and (19), we obtain boundedness on finite time intervals and global existence follows.
As noted earlier, if , then satisfies the PDE equation as an equality in a.e. in time. In particular, we have .
2.2. Boundedness of the Solutions and Global Attractor
In order to obtain asymptotic bounds on the solutions as , we consider the friction function satisfying the hypotheses from the previous section and we also assume that there exists a constant such that We make use of L’Hopital’s lemma proved in  to prove several results in this section.
Lemma 4 (L’Hopital’s lemma). Assume that and are real differentiable functions on on and .(i)If , then .(ii)If , then .
With this, we have Lemma 5.
Remark 6. We note that the conditions (36) are satisfied for all friction functions considered in the previous works, that is, the thermosyphon models, where is constant or linear or quadratic law. Moreover, the conditions (36) are also true for , as .
Now, we use the asymptotic bounded for temperature to obtain the asymptotic bounded for the velocity and the acceleration functions.
Theorem 7. Under the previous notations and hypothesis of Theorem 1 or Theorem 3, if one assumes also that satisfies (37) for some constant , then one has the following.
Part (I). General case:(i) In particular: If , then (ii) If and , then Part (II). If and also assume that there exists a positive constant such that , then for any solution of (5) in the space , one has(i)(ii)(iii) if , In particular, (5) has a global compact and connected attractor, , in .
Part (I). General case.
(i) From (8) we have that and satisfies where . First, we rewrite (48) as with Next, for any there exists such that for any and integrating with , we obtain Using L’Hopital’s Lemma 4 proved in , we get Moreover, using again the L’Hopital’s Lemma 4 proved in , we get and from (51) together with (37), we conclude that for any .
(ii) From (47) together with Gronwall’s lemma, we get where . Consequently, for any , there exists such that for any , that is, for any , and using the previous results (i) we get (42).
Part (II). (i) From (23) together with (24), we get and by elementary integration we obtain (44). Using Part (I), the rest (ii) and (iii) are obvious. Since the sectorial operator defined in Section 2.1.1 has compact resolvent, the rest follows from [17, Theorems and ].
Remark 8. First, we note that the hypothesis about the function in Theorem 7, is satisfied when we consider Newton’s linear cooling law , where is a positive quantity; that is, , as . Moreover, this condition is also satisfied if we consider where is a positive upper bounded function.
Second, it is important to note that we prove in the next section the existence of the global compact and connected attractor and the inertial manifold for the system (8), when we consider the general Newton’s linear cooling law without the additional previous hypothesis on ; but we assume that the friction function always satisfies (36).
In order to get this, we consider the Fourier expansions and observe the dynamics of each coefficient of Fourier expansions to improve the asymptotic bounded of temperature. In particular, we will prove for every locally Lipschitz and positive function and also for every (see (72) in Proposition 9) and for every friction function satisfying (36).
3. Asymptotic Behavior: Reduction to Finite-Dimensional Systems
We take a close look at the dynamics of (5) by considering the Fourier expansions of each function and observing the dynamics of each Fourier mode. Assume that and are given by the following Fourier expansions: while the initial data is given by .
Assume that is given by Then, we find that the coefficient in (60) is a solution of Since all the functions involved are real, we have , and . Therefore, (5) is equivalent to the infinite system of ODEs consisting of (61) coupled with
The system of (5) reflects two of the main features: (i) the coupling between the modes enter only through the velocity, while diffusion acts as a linear damping term, and (ii) it is important to note in this model that we have also the nonlinear term given by Newton’s linear cooling law.
In what follows, we will exploit this explicit equation for the temperature modes to analyze the asymptotic behavior of the system and to obtain the explicit low-dimensional models.
3.1. Attractors and Inertial Manifolds
The existence of an inertial manifold does not rely, in this case, on the existence of large gaps in the spectrum of the elliptic operator but on the invariance of certain sets of Fourier modes.
A similar explicit construction was given by Bloch and Titi in  for a nonlinear beam equation where the nonlinearity occurs only through the appearance of the norm of the unknown. A related construction was given by Stuart in  for a nonlocal reaction-diffusion equation.
In order to get the inertial manifold for this system, we first improve the bounds on acceleration, velocity, and temperature of the previous section for all situations with and general locally Lipschitz function under the hypotheses of Lemma 5 for some , in particular satisfying (36).
We will prove in Proposition 9 that we have always an upper bounded for the temperature in independent of the velocity and the function , considered in Newton’s linear cooling law and also independent of the diffusion coefficient. That is,
Proposition 9. Under the previous notations, for every solution of the system (5), , and for every , one has(i)(ii) and a positive constant such that ;(iii) In particular, one has a global compact and connected attractor where are the upper bounds for acceleration and velocity as given in (67) and (66) and .
Proof. From (61), we have with Thus, we obtain and we get . Using Theorem 7 together with , we get From this upper bounded for the velocity, we also have the upper bound for the continuous positive function , and using Part (II) from Theorem 7 we conclude.
We note from the previous result that we have always the upper bound for and from Theorem 7 for the velocity. Therefore, we can consider the upper bound for the continuous positive function ; we note that and we prove in Proposition 10 the bound of solutions to show the influence of diffusion coefficient .
Proposition 10. Under the previous notations, for every solution of the system (5), , and for every , one has Moreover, one has that and positive constant such that ;
Proof. Using again
if we assume that , then we obtain
and we get
that is, if , then .
In particular for every and also ; that is, .
Then, we also have that
Finally, we note that if , then given , there exists such that for every , and integrating in , we obtain , for any , and Using again Theorem 7, we conclude.
We note that if , then . If we observe in the numerical experiments, as is bigger, that the solution becomes stable or periodic.
As a consequence, we have the following result on the smoothness of the attractor of (5).
(i) If , then for every .
(ii) If is the global attractor in the space , then for every , with , one gets In particular, if with , the global attractor and is compact in this space.
Proof. (i) From (77) we have , therefore, if , then for every and .
(ii) We note that from (i), if , then , and therefore ; that is, where are the upper bounds for acceleration and velocity as given in (74) and (73). But, the set is compact in since for any sequence in we can extract a subsequence that we still denote such that it converges weakly to a function and such that for any , the Fourier coefficients verify as , where is the th Fourier coefficient of . Therefore, and for every integer , where denotes the norm in . Hence, the first term goes to zero as and the second term can be made arbitrarily small as . Consequently, and in , and the result follows.
Note that this result reveals in particular the asymptotic smoothing of (5). In the next result we will prove that the dynamical system induced by (8) in the phase space , , has an inertial manifold. According to , we have the following definition.
Definition 12. Let , be a nonlinear semigroup in a Banach space in that has a global attractor . Then, a smooth manifold is called an inertial manifold if(i) is positively invariant, that is, for every ;(ii) contains the attractor, that is, ;(iii) is exponentially attracting in the sense that there exists a constant such that for every bounded set there exists such that for every .
Assume the ambient temperature given by where , that is, with for every with , since .
Then, we denote by the closed linear subspace of spanned by and consider the following spectral decomposition in , where denotes the projection of onto and the projection onto the space generated by . Note that (8) is equivalent to