The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place this Wednesday (Oct 6, 2021) from 9:00 till 10:00 in K3. The talk will be given by Mark Dostalík. Please see the details below.
Speaker: Mark Dostalík
Title: Nonlinear stability of steady states in thermomechanics of viscoelastic fluids
Abstract: We investigate nonlinear stability of steady state solutions of partial differential equations governing the thermomechanical evolution of viscoelastic fluids. It is well-known that thermodynamical concepts can be gainfully exploited in the construction of Lyapunov functionals for nonlinear stability analysis of spatially homogeneous equilibrium rest states in thermodynamically closed systems. We demonstrate this by constructing explicit Lyapunov functionals for various types of compressible heat-conducting fluids occupying thermodynamically closed systems and we prove that the corresponding equilibrium rest states are Lyapunov stable. Further, we show that the thermodynamically oriented approach can be utilized in the nonlinear stability analysis of spatially inhomogeneous non-equilibrium steady states in thermodynamically open systems. First, we show that the spatially inhomogeneous non-equilibrium steady state of an incompressible heat-conducting viscoelastic fluid, which occupies a mechanically isolated vessel with walls kept at spatially non-uniform temperature, is globally asymptotically stable. Second, we investigate nonlinear stability of the steady internal flow of an incompressible viscoelastic fluid driven by an inflow of mechanical energy through the system’s boundary and we derive bounds on the dimensionless flow parameters which guarantee global asymptotic stability of the steady flow.