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The movement of ionic solutions is an essential part of biology andtechnology. Fluidics, from nano- to micro- to microfluidics, is a burgeoningarea of technology which is all about the movement of ionic solutions, onvarious scales. Many cells, tissues, and organs of animals and plants depend onosmosis, as the movement of fluids is called in biology. Indeed, the movementof fluids through channel proteins (that have a hole down their middle) isfluidics on an atomic scale. Ionic fluids are complex fluids, with energystored in many ways. Ionic fluids flow driven by gradients of concentration,chemical and electrical potential, and hydrostatic pressure. Each flow isclassically described by its own field theory, independent of the others, butof course, in reality every gradient drives every kind of flow to a varyingextent. Combining field equations is tricky and so the theory of complex fluidsderives the equations, rather than assumes their interactions. When fieldequations are derived, rather than assumed, their variables are consistent.That is to say all variables satisfy all equations under all conditions withone set of parameters. Here we treat a classical osmotic cell in this spirit,using a sharp interface method to derive boundary conditions consistent withall flows and fields. We allow volume to change with concentration, sincechanges of volume are a property of ionic solutions known to all who make themin the laboratory. We consider flexible and inflexible membranes. We show howto combine the energetics of the membrane with the energetics of thesurrounding complex fluids. The results seem general but need application tospecific situations of technological, biological and experimental importancebefore the consequences of consistency can be understood.

Osmosis through a Semi-permeable Membrane: a Consistent Approach to Interactions