  Recent experiments have shown that when a near-hemispherical lipid vesicle
attached to a solid surface is subjected to a simple shear flow it exhibits a
pattern of membrane circulation much like a dipole vortex. This is in marked
contrast to the toroidal circulation that would occur in the related problem of
a drop of immiscible fluid attached to a surface and subjected to shear. This
profound difference in flow patterns arises from the lateral incompressibility
of the membrane, which restricts the observable flows to those in which the
velocity field in the membrane is two-dimensionally divergence free. Here we
study these circulation patterns within the simplest model of membrane fluid
dynamics. A systematic expansion of the flow field based on Papkovich--Neuber
potentials is developed for general viscosity ratios between the membrane and
the surrounding fluids. Comparison with experimental results [C. V\'ezy, G.
Massiera, and A. Viallat, Soft Matter 3, 844 (2007)] is made, and it is shown
how such studies could allow measurements of the membrane viscosity. Issues of
symmetry-breaking and pattern selection are discussed.
