Uniqueness and Stability Results for Non-Linear Johnson-Segalman Viscoelasticity and Related Models

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2014.19.2111 DYNAMICAL SYSTEMS SERIES B Volume 19, Number 7, September 2014 pp. 2111{2132 UNIQUENESS AND STABILITY RESULTS FOR NON-LINEAR JOHNSON-SEGALMAN VISCOELASTICITY AND RELATED MODELS Franca Franchi, Barbara Lazzari and Roberta Nibbi Department of Mathematics, University of Bologna Piazza di Porta S. Donato, 5 40126 Bologna, Italy To Professor Mauro Fabrizio on his 70th birthday, in appreciation of his great ability of leading with generous friendship and an evergreen enthusiasm for research through the years. Abstract. In this paper we have proved exponential asymptotic stability for the corotational incompressible diffusive Johnson-Segalman viscolelastic model and a simple decay result for the corotational incompressible hyper- bolic Maxwell model. Moreover we have established continuous dependence and uniqueness results for the non-zero equilibrium solution. In the compressible case, we have proved a Höldercontinuous dependence theorem upon the initial data and body force for both models, whence follows a result of continuous dependence on the initial data and, therefore, uniqueness. For the Johnson-Segalman model we have also dealt with the case of negative elastic viscosities, corresponding to retardation effects. A comparison with other type of viscoelasticity, showing short memory elastic effects, is given. 1. Introduction. The flow behavior of non-Newtonian fluids has been a topic of recent extensive interest, due to their large number of physical, engineering and physiological applications; a vast amount of papers can be found in literature, which include analytical, numerical and experimental studies. This area of research has become increasingly relevant, owing to the advent and then the growth of the syn- thetic polymer industries and for the great range of non-Newtonian materials (blood, paints, food mixing, . ), playing an essential role in our everyday life. Viscoelastic response is often used as a probe in polymer rheology, as it is sensitive to the materials chemistry and microstructure. Comparative studies between various models are qualitatively carried on in the linear context, even if only a few polymers are linearly viscoelastic (see e.g. [25]). The Johnson-Segalman viscoelastic model (J-S model), firstly proposed to allow non-affine deformations [17], exhibits the ability to provide a useful description for diluted suspensions of a high-molecular-weight polymer into a largely viscous solvent, which may be generally considered a liquid. So, most recent literature is concerned only with incompressibile J-S fluids, within the single fluid approach, and often in presence of a magnetic field. 2010 Mathematics Subject Classification. Primary: 76A05, 35B30; Secondary: 76A10. Key words and phrases. Johnson-Segalman viscolestic fluids, Maxwell viscoelastic fluids, continuous dependence, stability, relaxation/retardation effects. 2111 2112 FRANCA FRANCHI, BARBARA LAZZARI AND ROBERTA NIBBI An effective method for regularizing, which is natural for viscoelastic fluids, is just to add a viscosity term in the momentum equation; in this way, the polymer is not frozen in the fluid, but it quickly relaxes to its non-stretching configuration. Mean- while, this experimental property creates an interesting link with electromagnetic theories, according to the recent Christov's revisitation [4] of a frame indifferent (continuum) formulation of electromagnetism. In view of this proposal, the diffusive J-S model allows for recovering a generalized resistive magnetohydrodynamics (MHD) theory, which circumvents the difficulty associated with the frozen-in magnetic field, and breaks the constraint of ideal MHD, as advocated by experimental observations. For a dilute polymer solution, described through the J-S model, the total (viscous) stress tensor Π is given by the sum of a Newtonian solvent part Πs and an elastic part Πe, due to the presence of the polymer solutions, which satisfy the following constitutive equations: 2µ Π = 2µ D + ζ − s (tr D) ; s s s 3 I (1) DΠe 2µe τ + Πe = 2µeD + ζe − (tr D)I ; Dt a 3 where D is the symmetric part of the velocity gradient L = rv, τ is a positive constant relaxation time and µs and µe are the constant solvent and elastic shear viscosities, while ζs and ζe are the constant solvent and elastic bulk viscosities. These parameters are supposed to meet the thermodynamic restrictions µs ≥ 0, µ = µe + µs ≥ 0, ζs ≥ 0, ζ = ζe + ζs ≥ 0 and, for a sufficiently diluted polymeric solution, it is experimentally shown that µs > µe and also µ > µe . DΠe The notation Dt a stands for the objective convected time derivative, which can be chosen from the one-parametric family of the Gordon-Schowalter derivatives (G-S derivative) [8], defined as follows DΠe = Π_ e + ΠeW − WΠe + a(DΠe + ΠeD) ; −1 ≤ a ≤ 1; (2) Dt a where the spin tensor W is the antisymmetric part of L, a is the dimensionless slip parameter, accounting for the slip between the molecular network and the continuum medium, and the superposed dot denotes the material time derivative. For a = ±1,2 leads to the upper/lower convected derivatives, whereas for a = 0 gives the corototational or Jaumann derivative. In the non-objective case the corotational (or upper/lower convected) derivative reduces to the material one and becomes a partial derivative in the linear limit. The non-linear invariant J-S model provides a non instaneous response charac- ter, which is quite embodied in the constitutive equation of rate type for the elastic stress tensor, with a single relaxation time and elastic viscosities. For constant rheological parameters, the J-S viscoelastic model is of Oldroyd B type; further- more, as already noted, the introduction of the regularizing Newtonian component is empirically motivated and allows for viscoelastic hysteresis, in analogy with other electromagnetic and thermal theories with lagging (see e.g. [5, 29]). The J-S model ranges from the classical Newtonian and Maxwell models, according as Πe or Πs be null; in turn, the simplified Maxwell model furnishes a fairly good description of the rheological behavior of purely polymer solutions or melts. Moreover, also for deformations occurring at time scales either longer or shorter than the relaxation UNIQUENESS AND STABILITY FOR NON-LINEAR J-S VISCOELASTICITY 2113 time, the viscoelastic fluid behaves either like a simple Newtonian fluid or diplays an elastic behavior. Indeed, a polymer liquid/fluid has a spectrum of relaxation times, so the relaxation time present in1 2 must be interpreted as the longest relaxation time exhibited by the real polymeric situation. Indeed, the J-S model, for constant parameters, has the additional advantage of being described through a constitutive equation of Jeffreys type, with a relaxation time τΠ = τ, a retardation time τD and a dimensionless retardation parameter r, given by the ratio between τD and τΠ. In linear/linearized (about the rest null state) viscoelasticity, for low relaxation/retardation times, this creates a special link with other dual-phase-lagging (DPL) thermal and electromagnetic theories, accounting for short elastic memory effects. So, besides the relaxation effects, another important mathematical and physical aspect, which has not yet received particular attention, is the retardation effect of the deformation rate tensor on the elastic stress tensor (corresponding to negative elastic viscosities or, equivalently, to suppose the dimensionless retardation parameter r > 1) on the stability analysis. It is empirically proven that, strongly negative viscosities would lead to instabilities and, consequently, to ill- posed problems. Meanwhile, one needs to realize that viscoelastic instabilities are of fundamental experimental importance in understanding the physics of complex fluids and also of practical relevance to material processing and fluid characteriza- tions (see e.g. [23] for a comparison between experimental data and scission model prediction). The incompressible corotational J-S model is very versatile and strong, and provides an almost complete mathematical and also numerical formulation, which is easily incorporated into boundary layer theories for engineering applications; further, also in presence of a magnetic field, it has been used to describe the non instantaneous response of blood, under physiological conditions, and the peristaltic motions in a flux tube, like an endoscope (see e.g.[13, 14]). In this connection, it is noteworhy to emphasize that the peristaltic pumps occur in many practical applications involving biomechanical systems, such as in dialysis and heart-lung-machines, but also in nuclear industry, and have been the object of many recent papers (see e.g. [15]). An interesting result about structural stability is given in [24]; the wave propa- gation properties have been recently investigated in [20,21] and, from a Lagrangian point of view, in [11]. In the framework of non-Newtonian fluids another interesting tool is the ability to control viscoelastic stresses in ascertaining characterizing properties of the materials; the controllability of the J-S model is investigated in [31] and in [26] for the Maxwell fluid. The interaction between (microscopic) elastic properties and the (macroscopic) fluid motion, in analogy with other theories with micro/macro scales effects accounting for lagging phenomena, gives not only a variety of interesting rheological phenomena, but also a powerful challenge in analysis and numerical simulation. Moreover, as in recent years there

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