Surface Flow of Viscoelastic Membranes in Viscous Fluids

The motion of a viscoelastic membrane surrounded by viscous fluids is studied, with a view to applications to red blood cell mechanics. Slow flow with no inertial effects is assumed. The equations governing the kinematics and dynamics of the membrane motion are derived in terms of global Cartesian coordinates resulting in a form with certain practical advantages. The red blocd cell membrane is known to deform at nearly constant surface area under normal conditions and it is shown that areaanserving surface flows may be described in terms of a scalar ‘stream function.’ Using a Kelvin solid model for the viscoelastic behaviour of the membrane, minimum energy dissipation principles are derived for systems consisting of closed viscoelastic membranes containing viscous fluid, suspended in another viscous fluid of finite extent with prescribed boundary velocities. Applications to the motion of red blood cells in shear flow are discussed.