An Analytic Approach to the Liebau Problem of Valveless Pumping

Abstract

Various pumping effects have been subsumed under the name Liebau pumping. Common to these is a reciprocating injection–ejection flow, provided either by a piston source or simply by beating on a tube, as well as some form of asymmetry (e.g., diameter, elasticity, position of injection). Furthermore, there are no explicit valves in the system. Here we model such a system as a flat tube, part of which is inelastic, and part of which completes a periodic wavelike motion. The latter part is generally confined to a small region, neither at the ends nor in the center. We use Euler's equation with the appropriate boundary conditions and derive an analytic solution that yields a finite, though small pumping effect. The solution is compared to known experiments and, although the pumping effect is several orders of magnitude too small, some of the qualities of the solution correspond experiments. We find a new and perhaps useful nondimensional combination relevant to the Liebau effect, which we have named the Liebau number.