General decay to a von Kármán system with memory

In this paper we study the von Kármán plate model with long-range memory and we show the general decay of the solution as time goes to infinity. This result generalizes and improves on earlier ones in the literature because it allows certain relaxation functions which are not necessarily of exponential or polynomial decay.     

Uniform decay rates for nonlinear viscoelastic Marguerre-von Kármán equations

In this paper, we consider a model of nonlinear viscoelastic shallow shell that is referred to as the full Marguerre-von Kármán under the presence of long-time memory. We show that the energy functional associated with the system decays exponentially to zero as time goes to infinity.     

Two-layer film flow over a rotating disk

Unsteady two-layer liquid film flow on a horizontal rotating disk is analyzed using asymptotic method for small values of Reynolds number. This analysis of non-linear evolution equation elucidates how a two-layer film of uniform thickness thins when the disk is set in uniform rotation. It is observed that the final film thickness attains an asymptotic value at large time. It is also established that viscous force dominates over centrifugal force and upper layer thins faster than lower layer at large time.     

Complementarity problem and the existence of the post-critical equilibrium state of a thin elastic plate

Using the concept of a conically bounded set, we prove a variational principle for functionals defined on a locally compact pointed convex cone. Applying this principle to the nonlinear complementarity problem, we study the existence of the post-critical equilibrium state of a thin elastic plate, subjected to unilateral conditions.     

A Series Solution of the Unsteady Von Kármán Swirling Viscous Flows

A new analytic technique is applied to solve the unsteady viscous flow due to an infinite rotating disk, governed by a set of two fully coupled nonlinear partial differential equations deduced directly from the exact Navier-Stokes equations. The system of coupled nonlinear partial differential equations is replaced by a sequence of uncoupled systems of linear ordinary differential equations. Different from all other previous analytic results, our series solution is accurate and valid for all time in the whole spatial region. Accurate expressions for skin friction coefficients are given, which are valid for all time. Such kind of series solutions have not been reported, to the best of our knowledge.      

Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas

We discuss the problem of nonlinear oscillations of a clamped plate in the presence of thermal effects in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects, in which rotational inertia is accounted for. To describe influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem.     

Solutions of the Von Kàrmàn equations via the non-variational Galerkin-B-spline approach

This paper deals with the non-variational Galerkin-B-spline method recently developed by the authors. It is applied to circular thin flexible plates within the context of Von Kàrmàn equations. Our main result shows that the low orders of approximation have been sufficient to build the solutions from the basis functions in a high quality. Comparison of solutions obtained to variational approach with known ones is performed.