  The recent torsional oscillator results of Kim and Chan suggest a supersolid
phase transition in solid 4He. We have used a piezoelectrically driven
diaphragm to study the flow of solid helium through an array of capillaries.
Our measurements showed no indication of low temperature flow, placing
stringent restrictions on supersolid flow in response to a pressure difference.
The average flow speed at low temperatures was less than 1.2x10-14 m/s,
corresponding to a supersolid velocity at least 7 orders of magnitude smaller
than the critical velocities inferred from the torsional oscillator
measurements.
