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Oberseminar Numerische Mathematik / Scientific Computing

 

Amiya Kumar Pani

Department of Mathematics, Industrial Mathematics Group, Bombay

Finite Element Approximation to the Equation of Motion Arising in the Oldroyd Model

Abstract:

Newton's model of incompressible viscous fluid is described by the wellknown Navier-Stokes equations. This has been a basic model for describing flow at moderate velocities of majority of viscous incompressible fluids encountered in practice. However, models of viscoelastic fluids have been proposed in the mid twentieth century which take into consideration the prehistory of the flow and are not subject to Newtonian flow. One such model was proposed by J.G. Oldroyd and hence, it is named after him. The equation of motion in this case gives rise to the following integro-differential equation

\begin{displaymath} \frac{\partial u}{\partial t} + u\cdot\nabla u - \mu\Delta ... ...(x,\tau)\,d \tau + \nabla p = f(x,t), \;x\in \Omega, t>0, (*) \end{displaymath}

and incompressibility condition

\begin{displaymath}\bigtriangledown.u = 0,\;\; x\in\Omega, t>0\end{displaymath}

with initial condition

\begin{displaymath}u(x,0)=u_0, u=0 , \; x\in \partial\Omega, t\geq 0.\end{displaymath}

Here, $\Omega$ is a bounded domain in $R^{d} (d=2,3)$ with boundary $\partial\Omega, \mu>0$ and the kernel $\beta(t)=\gamma exp(-\delta t)$, where both $\gamma$ and $\delta$ are positive constants. With a brief discussion on existential analysis, we, in this talk, concentrate on the finite element Galerkin method for the above system under realistically assumed regularity on the exact solutions. Since the problem (*) is an integral perturbation of the Navier-Stokes equations, we would like to discuss `how far the results on finite element analysis for the Navier-Stokes equations can be carried over to the present case.'

 

Datum: 18.05.07
Zeit:16:00 Uhr
Ort:FU Berlin, Institut für Mathematik II, Arnimallee 6, 14195 Berlin.
Raum:032 im Erdgeschoss

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