Kinematics, Conservation Equations, and Boundary Conditions for Incompressible Flow

Abstract

This text describes liquid flow in microsystems, primarily flow of water and aqueous solutions. To this end, this chapter describes basic relations suitable for describing the flow of water. For flows in microfluidic devices, liquids are well approximated as incompressible , i.e., having approximately uniform density, so this text describes incompressible flow exclusively. This chapter describes the kinematics of flow fields, which describes the motion and deformation of fluids. As part of this process, key concepts are introduced, such as streamlines, pathlines, streaklines, the stream function, vorticity, circulation, and strain rate and rotation rate tensors. These concepts provide the language used throughout the text to communicate the modes of fluid motion and deformation. We discuss conservation of mass and momentum for incompressible flows of Newtonian fluids. Finally, we discuss boundary conditions for the governing equations, including solid and free interfaces with surface tension, and in particular we give attention to the no-slip condition and its applicability in micro- and nanoscale devices. This chapter assumes familiarity with vector calculus, which is reviewed in Appendix C. Importantly, Appendix C also covers the notation and coordinate systems used throughout. We define a fluid as a material that deforms continuously when experiencing a nonuniform stress of any magnitude. We are primarily interested in a continuum description of the fluid flow, meaning that we are interested in the macroscopic manifestation of the motions of the individual molecules that make up the fluid, i.e., the velocity and the pressure of the fluid as a function of time and space.