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 closedloop thermosyphon. This sort of fluid presents elasticlike 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 closedloop 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 closedloop
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
The simplest approach to viscoelasticity comes from the socalled Maxwell constitutive equation [
The stress tensor comes into play in the equation for the conservation of momentum:
For a Maxwellian fluid, the stress tensor takes the form
Memory effects can be well understood from (
In a thermosyphon, the equations of motion can be greatly simplified because of the quasionedimensional 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 (
The derivation of the thermosyphon equations of motion is similar to that in [
The resulting secondorder equation is then averaged along the loop section (as in [
The model we will take into consideration forms an ODE/PDE system for the velocity
The system of (
We assume that
Hereafter, we consider
Our contributions in this paper are the following.
To obtain the system of (
To present an analysis beginning with the wellposedness and boundedness of solution. The existence of an attractor and an inertial manifold is shown and an explicit reduction to lowdimensional systems is obtained. It is noteworthy that we are able to obtain an exact finitedimensional reduction (
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.
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 [
In this section we prove the existence and uniqueness of solutions of the thermosyphon model (
First, we observe that for
Moreover, if we consider
Finally, since
Thus, with
Also, if
We consider the acceleration
The operator
Using the results and techniques of sectorial operator of [
We assume that
We cover several steps.
Using
Therefore, using the techniques of variations of constants formula of [
with
To prove the global existence, we must show that the solutions are bounded in
Using CauchySchwartz and the Young inequality and then the Poincaré inequality, since
Now, we note that differentiating the second equation of (
Thus, we show that the norm of
The system now reads
To prove the system is well posed, we use the techniques from [
We note that if
We note that for
Let
if there exist positive constants
positive constant independent on time;
we assume that
See [
Assume that
As noted earlier, we need to solve the fixed point problem
From (
To show that
First, we note that from
To prove the global existence, it is sufficient to prove that
But again from (
As noted earlier, if
In order to obtain asymptotic bounds on the solutions as
Assume that
If
If
With this, we have Lemma
If one assumes that
Integrating by parts we have
We note that the conditions (
Now, we use the asymptotic bounded for temperature to obtain the asymptotic bounded for the velocity and the acceleration functions.
Under the previous notations and hypothesis of Theorem
In particular: If
If
if
(i) From (
(ii) From (
First, we note that the hypothesis about the 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 (
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
We take a close look at the dynamics of (
Assume that
The system of (
We note that the system (
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 lowdimensional models.
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 [
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
We will prove in Proposition
Under the previous notations, for every solution of the system (
From (
We note from the previous result that we have always the upper bound for
Under the previous notations, for every solution of the system (
Using again
In particular
Then, we also have that
Finally, we note that if
We note that if
As a consequence, we have the following result on the smoothness of the attractor of (
(i) If
(ii) If
(i) From (
(ii) We note that from (i), if
Note that this result reveals in particular the asymptotic smoothing of (
Let
See, for example, [
Assume the ambient temperature given by
Then, we denote by
Note that from (
Assume that
In order to prove that
We consider the decomposition in
From (
Therefore, we have in particular that
To prove the exponential tracking property just note that the flow inside
Under the hypotheses and notations of Theorem
Thus, we will reduce the asymptotic behavior of the initial system (
Now, we will show the modes in
To make this idea more precise in terms of semigroup and attractors, we proceed as in [
We will find a reduced semigroup on the reduced space
With the previous notation, one has the following conditions.
(i) Working as aforementioned, we prove that the semigroups
Note that
If
Moreover, from the equation for the temperature in (
Now, we multiply (
Next, we pay attention to the other modes for the temperature
We will show in Proposition
Assume that
Define
If the solution of (
If
Taking real and imaginary parts of
Observe that from the previous analysis, it is possible to design the geometry of circuit and/or the external heating by properly choosing the functions
Note that the set
The physical and mathematical implications of the resulting system of ODEs which describe the dynamics at the inertial manifold need to be analyzed numerically. The role of the parameter
In this section, we describe the results of the numerical experiments obtained using the MATHEMATICA package [
Specifically, we are integrating the system of (
In order to reduce the number of parameters we make the change of variables
We have carried out two different sets of numerical experiments with regard to heat diffusion. The first numerical experiments are carried out keeping the heat diffusion to zero as it was done in [
Numerical analysis has been carried out keeping
In this section, we summarize some of the outcomes of the model equations. As we have mentioned earlier, this behavior is highly sensitive to the choice of parameters. Thus, we present those results in different subsections accounting for the most relevant signature for each set of numerical experiments.
The simulations of the numerical experiments done for large values of
In Figure
The chaotic progress of acceleration for
The inconsistent behavior of velocity for
In Figure
A chaotic global attractor of real and complex temperature for
This sort of behavior remains similar for other values of
To sum up this section, large values of the viscoelastic parameter
For values of
The stabilizing progress of acceleration for
Velocity stabilizes at 3.19981 for
Equilibrium velocity scale.
The general behavior of acceleration is that it has a chaotic outburst in the initial stages. But as time progresses it tends to stabilize, attaining equilibria. The velocity too, in the initial stages, when the time period is less than 20 units, is very inconsistent and at times unpredictable. But as the time progresses, velocity converges to a stable fixed point. Interestingly, this fixed point for the velocity is not trivial (
Equilibrium values of velocity for different values of heat flux




1  0.6718  0.6718 
10  1.4723  1.4723 
20  1.8601  1.8601 
30  2.1325  2.1325 
40  2.3495  2.3495 
50  2.5329  2.5329 
100  3.1998  3.1998 
1000  6.9800  6.9800 
10000  15.430  15.430 
Behavior of solutions without diffusion (









1  CS  CS  CS  CS  CS  C  C 
10  CS  CS  CS  CS  CS  C  C 
20  CS  CS  CS  CS  CS  C  C 
30  CP  CP  CP  CS  CS  C  C 
40  CP  CP  CP  CS  CS  C  C 
50  CP  CP  CP  CS  CS  C  C 
100  P  P  CP  CS  CS  C  C 
1000  CP  CP  CP  CS  CS  C  C 
10000  CP  CP  CP  CS  CS  C  C 
It is worth noting in Table
Another comment regarding Table
To sum up this section, we conclude that although the system has a chaotic initial transient, it tends to stabilize at longer times reaching a temperature gradient dependent equilibrium velocity. Notwithstanding, this asymptotic velocity depends nontrivially on temperature as a power law, being this a signature of the underlying nonlinearity of the equations.
When the viscoelastic effects are gradually less important (values of
To illustrate this, in Figure
The periodic progress of velocity for
In Figure
The chaotic but periodic plot of real and complex temperature for
Thus, in order to summarize the information covered in the last three subsections, we collect all the outcomes of the model in Table
In this case, to avoid unnecessary repetitions in the text, we focus on the main differences between this case and that in Section
Thus, in this second set of numerical experiments we introduce a nonzero value for the thermal diffusivity,
Behavior of solutions with diffusion (









1  CS  CS  CS  CS  CS  CS  CS 
10  CS  CS  CS  CS  CS  CS  CS 
20  CS  CS  CS  CS  CS  CS  CS 
30  CS  CS  CS  CS  CS  CS  CS 
40  P  P  P  CS  CS  CS  CS 
50  P  P  P  CS  CS  CS  CS 
100  P  P  P  CS  CS  CS  CS 
1000  P  P  P  CS  CS  CS  CS 
10000  P  P  P  CS  CS  CS  CS 
So, here we will only illustrate the most interesting behaviors with two examples. The first one takes the values of heat diffusion
The stabilizing process of acceleration for
Similarly, the second example (Figure
The fast stabilization of acceleration for
To sum up this section, we have found that greater values of the heat diffusion,
In this work, we have derived a novel system of equations to study the behavior of a viscoelastic material inside a thermosyphon. This model serves as a preliminary simplification of a more complex fully spatially extend system. This model illustrates the presence/absence of complex chaotic behaviors and also enables to relate them with the underlying viscoelastic (memory effects).
The main result is that we are able to prove that the original system (which involves both ordinary and partial different equations) possesses an inertial manifold in which the dynamics can be accurately described by a system of ODEs. By numerical integration of the reduced equations, we have been able to better understand the role of viscoelasticity (as opposed to a simpler Newtonian fluid) through the parameter
Our results suggest that in the absence of heat diffusion (
Physically, this induction of chaotic behaviors can be rationalized in terms of the memory effects inherent to viscoelastic models explicitly shown in (
Other interesting results are related to the effect of heat diffusion. We have found that as the heat diffusion is not zero, the system tends to stabilize either to a fixed equilibrium point or to a (
Our work can be generalized in many different ways, from changing the constitutive equation (from Maxwellian to other more complex situations) or to include shearthinning effects [
The system of (
In particular, one splits the problem into two parts: the inner problem and the outer problem. The inner problem is defined as the dynamics of the system for times up to
Besides, the outer problem is defined as the naive approximation
For instance, if the parameter
This simple analysis shows the intrinsic complexity of the physical problem when viscoelasticity is considered and, more importantly, the need to study every parameter set in detail to provide an accurate description of the type of dynamics in which the system evolves.
This work is partially supported by projects MTM200907540, GR58/08 Grupo 920894 BSCHUCM, Grupo de Investigación CADEDIF, and FIS200912964C0503, Spain, and partially supported by Grant MTM201231298 from Ministerio de Economia y Competitividad, Spain.