Weak Solvability of One Viscoelastic Fractional Dynamics Model of Continuum with Memory
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Journal of Mathematical Fluid Mechanics
Weak Solvability of One Viscoelastic Fractional Dynamics Model of Continuum with Memory V. G. Zvyagin
and V. P. Orlov
Communicated by A. V. Fursikov
Abstract. In this paper we investigate the dynamics of a multidimensional viscoelastic continuum which subjects the fractional anti-Zener constitutive law with memory along trajectories of the velocity field. A weak solvability of a corresponding initial-boundary value problem is established. Mathematics Subject Classification. Primary 76A05; Secondary 35Q35. Keywords. Viscoelastic continuum, Motion equations, Initial-boundary value problem, Weak solution, Anti-Zener model, Fractional derivative, Regular Lagrangian flow.
1. Introduction Consider the motion of a viscoelastic continuum of a constant density (it is equal to one, for simplicity) occupying a bounded domain Ω ⊂ RN , N = 2, 3, ∂Ω ∈ C 2 on time interval [0, T ], 0 < T < +∞. As it is well known, the Cauchy momentum equation for an incompressible continuum (see [13]) has the form: ∂v/∂t +
N
vi ∂v/∂xi = −∇ p + Div σ + f, div v = 0.
(1.1)
i=0
Here v = v(t, x) = (v1 (t, x), . . . , vN (t, x)) is the velocity vector, p = p(t, x) is the pressure, σ = σ(t, x) is the deviator of the stress tensor, f = f (t, x) denotes a given external force density, (t, x) ∈ QT = [0, T ]×Ω. Div σ is a vector which entries are divergences of the lines of matrix σ. The Eq. (1.1) is supplemented by the constitutive equation, which determines the type of a continuum and establishes a relationship between strain and stress. Many continua exhibit both viscous and elastic properties. There is a lot of mathematical models for stress–strain relationships which describe viscoelasticity. These models are often derived by means of the mechanical models method (structural modeling) using classical mechanics laws, i.e. Hooke’s law for elastic solids (Hook element) and Newton’s law for viscous liquids (Newton element). However these models still cannot adequately describe the behaviors of many polymers and complex biophysical fluids. The transition to models with fractional derivatives in a constitutive law is caused by the need to study a large class of materials in which creep, relaxation and elasticity effects must be taken into account. The use of fractional calculus in linear viscoelasticity leads to generalizations of the classical mechanical models: the basic Newton element is substituted by the more general Scott-Blair element of the fractional order 0 < ν < 1. Then, extending the procedures of the classical mechanical models, one get a fractional constitutive equation (that is an operator equation with fractional derivatives) (see [14]). 0123456789().: V,-vol
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V. G. Zvyagin, V. P. Orlov
JMFM
It was found that the application of concepts and methods of fractional analysis is an adequate tool for modeling the viscoelastic behavior of mechanical dynamic systems with memory, a number of physical phenomena, biological processes and hereditary elastic deformation (see [4,7,8]). Fr
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