Event Detail

Event Type: 
Department Colloquium
Tuesday, January 25, 2005 - 07:00
Kidd 364

Speaker Info

Department of Mechanical Engineering, U. of Pittsburgh

Over the last 40 years the study of the motion of small particles in a viscous liquid has become one of the main focuses of engineering research. The presence of the particles affects the flow of the liquid, and this, in turn, affects the motion of the particles, so that the problem of determining the flow characteristics is highly coupled. It is just this latter feature that makes any fundamental mathematical problem related to liquid-particle interaction a particularly challenging one.
Interestingly enough, even though the mathematical theory of the motion of rigid particles in a liquid is one of the oldest and most classical problems in fluid mechanics, owed to the seminal contributions of Stokes, Kirchhoff, and Lord Kelvin, only very recently have mathematicians become interested in a systematic study of the basic problems related to liquid-particle interaction.
In this talk we focus on the mathematical analysis of two problems related to the motion of a sphere in a channel. Specifically, in the first problem a sphere is moving under the gradient of shear generated by a unidirectional two-dimensional Poiseuille flow of a viscoelastic liquid in a horizontal channel. In the second problem a sphere is sedimenting in a vertical channel under the action of gravity. In both problems the setting is two-dimensional and the viscoelastic liquid is taken to be a second-order liquid model. Our main goal is to evaluate, at first order in a suitable Reynolds number and in the Weissenberg number, the equilibrium position of the sphere with respect to one of the walls, and its translational and angular velocities. Moreover, we investigate the attainability and the stability of the equilibria and their dependence on the (effective) mass of the sphere and on the physical properties of the liquid.