Mathematical Models in Dynamics of Incompressible Viscoelastic Media

Abstract

A consistent model of incompressible viscoelastic Maxwell media is formulated. It corresponds to the choice of Jaumann rotational derivative in the constitutive relation. The governing system of equations has both real and complex characteristics. For this system, the solvability of initial‐boundary value problem in the class of analytic functions is established, and for its linearized variant the solvability is shown in the class of functions of finite smoothness. It is shown that the smallness of non‐diagonal terms of stress tensor entails absence of short‐wave instability.A wide class of exact solutions to the motion of incompressible viscoelastic Maxwell medium is found. These solutions are partially invariant with respect to some sub‐group of extended Galilei group which is admitted by equations of motion and their generalizations. The deformation of an viscoelastic strip with free boundaries is described, which moves either inertially or under the action of stretching or compressing longitudinal stresses, as well as shear stresses, applied to the free surface.The problem of filling of a spherical cavity by incompressible Maxwell medium under the action of constant pressure at infinity is considered. This is the generalization of the classic problem for viscous incompressible liquid. In both cases the cavity always shrinks to a point in a finite or infinite time. If the surface tension differs from zero, the collapse takes place in a finite time. Depending on the three dimensionless parameters (Reynolds number, capillary number and dimensionless relaxation time) both oscillatory and monotonic regimes of motion are possible. When the cavity radius is small, no oscillations can exist.A problem of filling of an spherical cavity with incompressible viscoelastic Kelvin‐Voigt medium under the action of constant pressure at infinity is also considered. Unlike the case of Maxwell medium, here both the cavity collapse and stabilization of its radius to a positive value over infinite time are possible. The finiteness of deformations was shown to lead to new qualitative effects. Three different modes of boundary behavior were found, two of which correspond to the monotonous and oscillatory regimes of achieving the limit cavity radius, while the third one corresponds to the cavity collapse in a finite time.The analogues of the classical Couette problem for the incompressible viscoelastic Kelvin‐Voigt and Maxwell media motion in the gap between two coaxial cylinders are studied. The external cylinder is fixed while the internal one is rotating either with a constant velocity (kept by an applied force) or inertially. An analogy between the propagation of acoustic cylindrical waves in a viscous gas and transversal waves in Kelvin‐Voigt medium is found. The asymptotic stabilization of the solution of the Couette motion is studied for the the first problem and for the inertial rotation of the cylinder approaching an equilibrium state. In the second problem for Maxwell motion, the asymptotic regimes of rotation retardation are also studied. Unlike the case of Kelvin‐Voigt medium where both monotonous and oscillatory regimes of rest state approaching are possible, for the case of Maxwell medium the velocity of rotation with time tends to zero in oscillatory regime.