FINITE-DIMENSIONAL BEHAVIOR IN A THERMOSYPHON WITH A VISCOELASTIC FLUID

We analyse the motion of a viscoelastic fluid in the interior of a closed loop thermosyphon under the effects of natural convection. We consider a viscoelastic fluid described by the Maxwell constitutive equation. This fluid presents elastic-like behavior and memory effects. We study the asymptotic properties of the fluid inside the thermosyphon and derive the exact equations of motion in the inertial manifold that characterize the asymptotic behavior. Our work is a generalization of some previous results on standard (Newtonian) fluids.

1. Introduction.Chaos in fluids subject to temperature gradients has been a subject of intense work for its applications in the field of engineering and atmospheric sciences.A thermosyphon is a device composed of a closed loop pipe containing a fluid whose motion is driven by the effect of several actions such as gravity and natural convection [7,12,11].The flow inside the loop is driven by an energetic balance between thermal energy and mechanical energy.The interest on this system comes from engineering and as a toy model of natural convection (for instance, to understand the origin of chaos in atmospheric systems).
As viscoelasticity is, in general, strongly dependent on the material composition and working regime, here we will approach this problem by studying the most essential feature of viscoelastic fluids: memory effects.To this aim we restrict ourselves to the study of the so-called Maxwell model [8].In this model, both Newton's law of viscosity and Hooke's law of elasticity are generalized and complemented through 376 A. JIM ÉNEZ-CASAS, MARIO CASTRO AND JUSTINE YASSAPAN an evolution equation for the stress tensor, σ.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 µ is the fluid viscosity, E the Young's modulus and γ the shear strain rate (or rate at which the fluid deforms).Under stationary flow, the equation ( 2) reduces to Newton's law, and consequently, the equation ( 1) reduces to the celebrated Navier-Stokes equation.On the contrary, for short times, when impulsive behavior from rest can be expected, Equation (2) reduces to Hooke's law of elasticity.
The derivation of the thermosyphon equations of motion is similar to that in [7,12,11].The simplest way to incorporate Equation (2) into Equation ( 1) is by differentiating equation ( 1) with respect to time and replacing the resulting time derivative of σ with Equation (2).This way to incorporate the constitutive equation allows to reduce the number of unknowns (we remove σ 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 Ref. [7]).Finally, after adimensionalizing the variables (to reduce the number of free parameters) we arrive at the ODE/PDE system where v(t) is the velocity, T (t, x) is the distribution of the temperature of the viscoelastic fluid in the loop, ν is the temperature diffusion coefficient, G(v) is the friction law at the inner wall of the loop, the function f is the geometry of the loop and the distribution of gravitational forces, h(x) is the general heat flux and ε is the viscoelastic parameter, which is the dimensionless version of the viscoelastic time, t V = µ/E.Roughly speaking, it gives the time scale in which the transition from elastic to fluid-like occurs in the fluid.We consider G and h are given continuous functions, such that G(v) ≥ G 0 > 0. Finally, for physical consistency, it is important to note that all functions considered must be 1-periodic with respect to the spatial variable.
Our contribution in this paper is: first, to prove the existence of an attractor and an inertial manifold and next, using this, to obtain an explicit reduction to low-dimensional systems of the behavior of viscoelastic fluids; extending the results in [9] for this kind of fluid, in order to get the similar results like [13] when we consider a given heat flux instead of Newton's linear cooling law.

2.1.
Existence and uniqueness of solutions.First, we note if we integrate the equation for the temperature along the loop, T (t) = T 0 +t h.Therefore, T (t) is unbounded, as t → ∞, unless h = 0.However, taking θ = T − T and h * = h − h reduces to the case T (t) = T 0 = h = 0, since θ would satisfy and T f = θf, since f = 0. Therefore, hereafter we consider the system (3) where all functions have zero average.Also, the operator νA = −ν ∂ 2 ∂x 2 , together with periodic boundary conditions, is an unbounded, self-adjoint operator with compact resolvent in L 2 per (0, 1), that is positive when restricted to the space of zero average functions L2 per (0, 1).Hence, the equation for the temperature T in (3) is of parabolic type for ν > 0.
Proof.Step 1. First, we prove the local existence and regularity.This follows easily from the variation of constants formula and [4].In order to prove this we write the system as ( 5), and we have: per (0, 1) and has compact resolvent.Note that in this context the operator A = − ∂ 2 ∂x 2 must be understood in the variational sense, i.e., for every T, ϕ ∈ Ḣ1 per (0, 1), < A(T ), ϕ >= ∂T ∂x ∂ϕ ∂x and L2 per (0, 1) coincides with the fractional space of exponent 1  2 [4].Hereafter we denote by .the norm on the space L2 per (0, 1).Now, under the above notations, using that per (0, 1), we obtain that the nonlinearity ( 6) is well defined and is Lipschitz and bounded on bounded sets.
Therefore, using the techniques of variations of constants formula [4], we get the unique local solution (w, v, T per (0, 1)) of ( 4) which are given by with where (w, v, T ) ∈ C([0, τ ], Y ) and using again the results of [4](smoothing effect of the equations together with bootstrapping method), we get the above regularity of solutions.
Step 2. Now, we prove the solutions of (4) are defined for every time t ≥ 0.
To prove the global existence, we must show that solutions are bounded in the Y = R 2 × L2 per (0, 1) norm on finite time intervals.First, to obtain the norm of T is bounded in finite time, we note that multiplying the equations for the temperature by T in L2 per and integrating by parts, we have that: 1 2 Using Cauchy-Schwarz and the Young inequality and then the Poincaré inequality for functions with zero average, since T = 0, we obtain Now, by integrating we get T is bounded for finite time, and so are |v(t)| and |w(t)|, hence we have a global solution and nonlinear semigroup in Y = R 2 × L2 per (0, 1).
3. Asymptotic bounds on the solutions.In this section we use the results and techniques from [13] to prove the existence of the global attractor for the semigroup defined by (3) in the space Y = R 2 × L2 per (0, 1).In order to obtain asymptotic bounds on the solutions as t → ∞, we consider the friction function G as in [13] i.e., satisfying the hypotheses from the previous section and we also assume that there exits a constant g 0 ≥ 0 such that: Now, using the l'Hopital's lemma proved in [9] we have the following Lemma proved in [13].
Lemma 1.If we assume G(r) and H(r) = rG(r) satisfy the hypothesis from Theorem 1 together with (11), then: with Remark 1.We note that the conditions (11) are satisfied for all friction functions G considered in the previous works, i.e., the thermosyphon models where G is constant or linear or quadratic law.Moreover, its conditions (11) are also true for Finally, in order to obtain the asymptotic bounds on the solutions we obtain the asymptotic bounds for the temperature in this diffusion case and we will use the following result from [13] [Theorem 2.3 Part I] to get the asymptotic bounds for the velocity and the acceleration functions.Lemma 2. Under the above notations and hypothesis from Theorem 1, if we assume also that G satisfies (12) for some constant H 0 ≥ 0 and In particular: ii)If lim sup t →∞ T ∈ R and we denote now by Proof.First we obtain dv ds + G(s , and then from Lemma 1 together with l'Hopital's lemma we conclude (see [13] [Theorem 2.3 Part I]).

Proposition 1. If f, h ∈ L2
per (0, 1) and H(r) = rG(r) is locally Lipschitz with G(v) ≥ G 0 > 0 and satisfies (12) for some constant H 0 ≥ 0. Then for any solution of (3) in the space Y = R 2 × L2 per (0, 1) we have: i) ii) If we denote now by G * 0 = lim sup t→∞ G(v(t)) we get Therefore, (3) has a global compact and connected attractor, A, in Y = R 2 × L2 per (0, 1).Proof.i) From ( 10) we get 4. Finite-dimensional behavior.Now we take a close look at the dynamics of (3) by considering the Fourier expansions of each function and observing the dynamics of each Fourier mode.We assume that h, f, T 0 ∈ L2 per (0, 1) are given by the following Fourier expansions with Z * = Z \ {0}.Finally assume that T (t, x) ∈ L2 per (0, 1) is given by Then, the coefficients a k (t) in (21), verify the equations: Therefore, (3) is equivalent to the infinite system of ODEs consisting of (22) coupled with The two equations reflect two of the main features of (3): the coupling between modes enter only through the velocity, while diffusion acts as a linear damping term.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.
A similar explicit construction was given by Bloch and Titi in [1] for a nonlinear beam equation where the nonlinearity occurs only through the appearance of the L 2 norm of the unknown.A related construction was given by Stuart in [10] for a nonlocal reaction-diffusion equation.
We note that the system (3) is equivalent to the system (4) for the acceleration, velocity and temperature and this is equivalent now to the following infinite system of ODEs ( 23) Next, we obtain the boundedness of these coefficients that improve the boundedness of temperature of the previous section and in particular, allow as to prove the existence of the inertial manifold for the system (3).
Proposition 2. For every solution of the system (3), (w, v, T ), and for every k ∈ Z * we have and Proof.From (22), we have that and we get lim per (0, 1), then for every (w 0 , v 0 , T 0 ) ∈ A, with T 0 (x) = k∈Z * a k e 2πkix we get, In particular, if h ∈ Ḣm per with m ≥ 1, the global attractor A → R 2 × Ḣm+2 per and is compact in this space.
ii) To prove the last statement we take into account that, from i), if h In the next result we will prove that there exists an inertial manifold M for the semigroup S(t) in the phase space Y = R 2 × Ḣm per , m ≥ 1 according to [2], i.e., a submanifold of Y such that i) S(t)M ⊂ M for every t ≥ 0, ii) there exists M > 0 verifying that for every bounded set B ⊂ Y, there exists C(B) ≥ 0 such that dist(S(t), M ≤ C(B)e −M t , t ≥ 0 see, for example, [2].Assume then that h ∈ Ḣm with b k = 0 for every k ∈ K ⊂ Z * with 0 / ∈ K, since h = 0. We denote by V m the closure of the subspace of Ḣm per generated by {e 2πkix , k ∈ K}.
where |K| is the number of elements in K.
Step II.From previous assertions, T (t) • f = k∈K a k (t) • ck and the flow on M is given by ẇ Then, given (w 0 , v 0 , T 0 ) ∈ Y we decompose T 0 = T 1 0 + T 2 0 , and T (t) = T 1 (t) + T 2 (t) and we consider (w(t), v(t), T 1 (t)) ∈ M and then (w(t), v(t), T (t)) − (w(t), v(t), T 1 (t)) = (0, 0, T 2 (t)).Note that in (29) the set of equations for a k with k ∈ K ∩ J, together with the equation for v and w, are a subsystem of coupled equations.After solving this, we must solve the equations for k / ∈ K ∩ J which are linear autonomous equations.We note that 0 / ∈ K ∩ J and since K = −K and J = −J then the set K ∩ J has an even number of elements, that we denote by 2n 0 .

From (27) taking into account that b
Corollary 2. Under the notations and hypotheses of the Theorem 2, we suppose that the set K ∩ J is finite and then |K ∩ J| = 2n 0 .Then the asymptotic behavior of the system (3), is described by a system of N = 2n 0 + 2 coupled equations in R N , which determine (w, v, a k ), k ∈ K ∩ J, and a family of |K \ (K ∩ J)| linear non-autonomous equations.
Proof.Note that on the inertial manifold T Thus, the dynamics of the system depends only on the coefficients in K∩J.Moreover the equations for a −k are conjugated of the equations for a k and therefore we have From this, and taking real and If K ∩ J = ∅, and G(v) = G 0 then on the inertial manifold we get a homogeneous linear equation for the velocity with positive coefficients, and by this lim sup t→∞ |v(t)| = 0, and therefore the equation for w on the inertial manifold is   3) is given by a reduced explicit system in R N , where N = 2n 0 + 2 (w(t), v(t), a k 1 (t), a k 2 (t), k ∈ (K ∩ J) + ) and n 0 = |(K ∩ J) + |.Observe that from the analysis above, it is possible to design the geometry of circuit and/or the external heating, by properly choosing the functions f and/or the heat flux, h, so that the resulting system has an arbitrary number of equations of the form N = 2n 0 + 2.
Note that it may be the case that K and J are infinite sets, but their intersection is finite.Also, for a circular circuit we have f (x) ∼ asin(x)+bcos(x), i.e.; J = {±1} and then K ∩ J is either {±1} or the empty set.Also, if in the original variables for (3), h is constant we get K ∩ J = ∅ for any choice of f .This conclusion about the finite-dimensional asymptotic behavior is similar to the previous models like [5,6,9] between others.Next, using these results the physical and mathematical implications of the resulting system of ODEs which describes the dynamics at the inertial manifold has been analyzed numerically in [14].
we consider the following decomposition in Ḣm per , T = T 1 + T 2 , where T 1 is the projection of T on V m and T 2 is the projection of T on the subspace generated by {e 2πkix , k ∈ Z * \ K} i.e.T 1 = k∈K a k e 2πkix and T 2 = k∈Z * \K a k e 2πkix = T − T 1 .

4. 2 .
The reduced explicit system.Under the hypotheses and notations of Theorem 2, we suppose moreover thatf (x) = k∈J c k e 2πkix ,with c k = 0 for every k ∈ J ⊂ Z.Note that since all functions involved are real, one has āk = a −k , bk = b −k and ck = c −k .Then, on the inertial manifoldT (t) • f = k∈K a k (t) ck = k∈K∩J a k (t).c −k .So, the evolution of the velocity v, and the acceleration w depends only on the coefficients of T which belong to the set K ∩ J.

2 5 .
and using v(t) → 0 we prove that θ(t) → 0 in L2 per .Conclusion.Taking real and imaginary parts of coefficients a k (t) (temperature), b k (heat flux at the wall of the loop) and c k (geometry of circuit)a k (t) = a k 1 (t) + ia k 2 (t), b k = b k 1 + ib k 2 and c k = c k 1 + ic k2, the asymptotic behavior of the system ( we have that |a k (t)| ≤ |a k0 |e −νπ 2 k 2 t and this together with νπ 2 k 2 t ≥ νπ 2 t for every k ∈ Z * implies that T 2 (t) Ḣm −νπ 2 t i.e., T 2 (t) → 0 in Ḣm per if t → ∞.Therefore, we have that T 2 (t) Ḣm per → 0 as t → ∞ with exponential decay rate e −νπ 2 t .Thus M attracts (w(t), v(t), T (t)) with exponential rate e −νπ 2 t .
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