Microfluidics: Modeling, Mechanics and Mathematics

Chapter 13

Continuity and Navier-Stokes Equations in Different Coordinate Systems

13.1 Cartesian Coordinates

13.1.1 Continuity Equation: Compressible

We derived the continuity equation in section 10.4 for Cartesian coordinates. For compressible fluids, it is given by Eq. 10.16 as

$\begin{array}{ll}\frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{\upsilon }\right)\hfill & =0\hfill \\ \frac{\partial \rho }{\partial t}+{\upsilon }_x\frac{\partial \rho }{\partial x}+{\upsilon }_y\frac{\partial \rho }{\partial y}+{\upsilon }_z\frac{\partial \rho }{\partial z}+\rho \left(\frac{\partial {\upsilon }_x}{\partial x}+\frac{\partial {\upsilon }_y}{\partial y}+\frac{\partial {\upsilon }_z}{\partial z}\right)\hfill & =0\hfill \end{array}$

13.1.2 Continuity Equation: Incompressible

For incompressible fluids, the equation simplifies to Eq. 10.17 given by

$\begin{array}{ll}\overrightarrow{\nabla }\cdot \overrightarrow{\upsilon }\hfill & =0\hfill \\ \frac{\partial {\upsilon }_x}{\partial x}+\frac{\partial {\upsilon }_y}{\partial y}+\frac{\partial {\upsilon }_z}{\partial z}\hfill & =0\hfill \end{array}$

13.1.3 Navier-Stokes Equation: Incompressible Newtonian Fluids ...