Microfluidics: Modeling, Mechanics and Mathematics

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

$\begin{array}{l}\frac{\partial v_x}{\partial t}+v_x\frac{\partial v_x}{\partial x}+v_y\frac{\partial v_x}{\partial y}+v_z\frac{\partial v_x}{\partial z}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\frac{\eta }{\rho }\left(\frac{{\partial }^2{v}_x}{\partial x^2}+\frac{{\partial }^2{v}_x}{\partial y^2}+\frac{{\partial }^2{v}_x}{\partial z^2}\right)+\frac{k_x}{\rho }\\ \frac{\partial v_y}{\partial t}+v_x\frac{\partial v_y}{\partial x}+v_y\frac{\partial v_y}{\partial y}+v_z\frac{\partial v_y}{\partial z}=-\frac{1}{\rho }\frac{\partial p}{\partial y}+\frac{\eta }{\rho }\left(\frac{{\partial }^2{v}_y}{\partial x^2}+\frac{{\partial }^2{v}_y}{\partial y^2}+\frac{{\partial }^2{v}_y}{\partial z^2}\right)+\frac{k_y}{\rho }\\ \frac{\partial v_x}{\partial }\end{array}$