Flow and pressure distribution in linear discrete “ladder-type” fluidic circuits: An analytical approach

This article proposes fully analytical solutions for a certain class of networks or circuits for fluid flow, called “ladders” by analogy with the designation used in electrical engineering. Fluidic ladders comprise a discrete number of parallel channels, the ends of which are connected to a straight distributor manifold and to a straight collector manifold. The hydrodynamics are assumed to be purely linear, i.e. viscous laminar flow is assumed everywhere, inertial effects and non-linear contributions of branching singularities are neglected. The known and relatively simple case of the classical electric ladders is taken as a starting point to formulate and solve Kirchhoff's equations together with Ohm's law. The solutions for the steady-state flow-rates in each branch of the ladder are in the form of polynomials of dimensionless resistance ratios. The polynomials and their coefficients are shown to obey simple and general recurrence relations, which allow any size of ladder to be solved. A number of special cases are investigated, from a unique resistance (homogeneous ladders) to two or three different resistances, or a resistance distribution allowing a homogeneous distribution of flow among the parallel channels.