Biomimetic design of microfluidic manifolds based on a generalised Murray's law

The relationship governing the optimum ratio between the diameters of the parent and daughter branches in vascular systems was first discovered by Murray using the principle of minimum work. This relationship is now known as Murray's law and states that the cube of the diameter of the parent vessel must equal the sum of the cubes of the daughter vessels. For symmetric bifurcations, an important consequence of this geometric rule is that the tangential shear stress at the wall remains constant throughout the vascular network. In the present paper, we extend this important hydrodynamic concept to arbitrary cross-sections and provide a framework for constructing a simple but elegant biomimetic design rule for hierarchical microfluidic networks. The paper focuses specifically on constant-depth rectangular and trapezoidal channels often employed in lab-on-a-chip systems. To validate our biomimetic design rule and demonstrate the application of Murray's law to microfluidic manifolds, a comprehensive series of computational fluid dynamics simulations have been performed. The numerical predictions are shown to be in very good agreement with the theoretical analysis, confirming that the generalised version of Murray's law can be successfully applied to the design of constant-depth microfluidic devices.