34.2.3.3 Instationary Flows

Momentum Equations. The second scenario we will consider involves instationary flows. For instationary flows we have to solve Eq. 34.1, Eq. 34.2, and Eq. 34.3 using a suitable numerical scheme for the timesteps $\frac{\partial {v}_{x}}{\partial t},\frac{\partial {v}_{y}}{\partial t}$ and $\frac{\partial {v}_{z}}{\partial t}$ ,. Usually, we would choose a forward Euler method (see section 27.2.2) for the time step. We could then rewrite Eq. 34.1, Eq. 34.2, and Eq. 34.3 as

$\begin{array}{l}\frac{\partial {v}_{x}}{\partial t}+{v}_{x}\frac{\partial {v}_{x}}{\partial x}+{v}_{y}\frac{\partial {v}_{x}}{\partial y}+{v}_{z}\frac{\partial {v}_{x}}{\partial z}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\frac{\eta }{\rho }\left(\frac{{\partial }^{2}{v}_{x}}{\partial {x}^{2}}+\frac{{\partial }^{2}{v}_{x}}{\partial {y}^{2}}+\frac{{\partial }^{2}{v}_{x}}{\partial {z}^{2}}\right)+\frac{{k}_{x}}{\rho }\\ \frac{\partial {v}_{y}}{\partial t}+{v}_{x}\frac{\partial {v}_{y}}{\partial x}+{v}_{y}\frac{\partial {v}_{y}}{\partial y}+{v}_{z}\frac{\partial {v}_{y}}{\partial z}=-\frac{1}{\rho }\frac{\partial p}{\partial y}+\frac{\eta }{\rho }\left(\frac{{\partial }^{2}{v}_{y}}{\partial {x}^{2}}+\frac{{\partial }^{2}{v}_{y}}{\partial {y}^{2}}+\frac{{\partial }^{2}{v}_{y}}{\partial {z}^{2}}\right)+\frac{{k}_{y}}{\rho }\\ \frac{\partial {v}_{x}}{\partial }\end{array}$

Get Microfluidics: Modeling, Mechanics and Mathematics now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.