  Immersed nonlinear elements are prevalent in biological systems that require
a preferential flow direction, such as the venous and the lymphatic system. We
investigate here a certain class of models where the fluid is driven by
peristaltic pumping and the nonlinear elements are ideal valves that completely
suppress backflow. This highly nonlinear system produces discontinuous
solutions that are difficult to study. We show that as the density of valves
increases, the pressure and flow are well-approximated by a continuum of valves
which can be analytically treated, and we demonstrate through numeric
simulation that the approximation works well even for intermediate valve
densities. We find that the induced flow is linear in the peristaltic amplitude
for small peristaltic forces and, in the case of sinusoidal peristalsis, is
independent of pumping direction. Despite the continuum approximation used, the
physical valve density is accounted for by modifying the resistance of the
fluid appropriately. The suppression of backflow causes a net benefit in adding
valves when the valve density is low, but once the density is high enough,
valves predominately suppress forward flow, suggesting there is an optimum
number of valves per wavelength. The continuum model for peristaltic pumping
through an array of valves presented in this work can eventually provide
insights about the design and operating principles of complex flow networks
with a broad class of nonlinear elements.
