Microfluidics: Modeling, Mechanics and Mathematics

$\begin{array}{l}\overrightarrow{\nabla }\cdot \overrightarrow{F}=\left(\frac{\partial F_{\rho }}{\partial \rho }sin\theta cos\phi +\frac{\partial F_{\rho }}{\partial \theta }\frac{1}{\rho }cos\theta cos\phi -\frac{\partial F_{\rho }}{\partial \phi }\frac{1}{\rho }\frac{sin\phi }{sin\theta }\right)sin\theta cos\phi \\ +\left(\frac{\partial F_{\theta }}{\partial \rho }sin\theta cos\phi +\frac{\partial F_{\theta }}{\partial \theta }\frac{1}{\rho }cos\theta cos\phi -\frac{\partial F_{\theta }}{\partial \phi }\frac{1}{\rho }\frac{sin\phi }{sin\theta }\right)cos\theta cos\phi \\ +\left(-\frac{\partial F_{\phi }}{\partial \rho }sin\theta cos\phi -\frac{\partial F_{\phi }}{\partial \theta }\frac{1}{\rho }cos\theta cos\phi +\frac{\partial F_{\phi }}{\partial \phi }\frac{1}{\rho }\frac{sin\phi }{sin\theta }\right)sin\phi \\ +\left(\frac{\partial F_{\rho }}{\partial \rho }sin\theta sin\phi +\frac{\partial F_{\rho }}{\partial \theta }\frac{1}{\rho }cos\theta sin\phi +\frac{\partial F_{\rho }}{\partial \phi }\frac{1}{\rho }\frac{cos\phi }{sin\theta }\right)sin\theta sin\phi \\ +\left(\frac{\partial F_{\theta }}{\partial \rho }sin\theta sin\phi +\frac{\partial F_{\theta }}{\partial \theta }\frac{1}{\rho }cos\theta sin\phi +\frac{\partial F_{\theta }}{\partial \phi }\frac{1}{\rho }\frac{cos\phi }{sin\theta }\right)cos\theta sin\phi \\ +\left(\frac{\partial F_{\phi }}{\partial \rho }sin\theta sin\phi +\frac{\partial F_{\phi }}{\partial \theta }\frac{1}{\rho }cos\theta sin\phi +\frac{\partial F_{\phi }}{\partial \phi }\frac{1}{\rho }\frac{cos\phi }{sin\theta }\right)cos\phi \\ +\left(\frac{\partial F_{\rho }}{\partial \rho }cos\theta -\frac{\partial F_{\rho }}{\partial \theta }\frac{1}{\rho }sin\theta \right)cos\theta \\ +\left(\frac{\partial F_{\theta }}{\partial \rho }cos\theta +\frac{\partial F_{\theta }}{\partial \theta }\frac{1}{\rho }sin\theta \right)sin\theta \\ +F_{\rho }\left(\frac{\partial sin\theta }{\partial x}cos\phi +\mathit{sin\theta }\frac{\partial cos\phi }{\partial x}+\frac{\partial sin\theta }{\partial y}sin\phi +\mathit{sin\theta }\frac{\partial sin\phi }{\partial y}+\frac{\partial cos\theta }{\partial z}\right)\\ +F_{\theta }\left(\frac{\partial cos\theta }{\partial x}cos\phi +cos\theta \frac{\partial cos\phi }{\partial x}+\frac{\partial cos\theta }{\partial y}sin\phi +cos\theta \frac{\partial sin\phi }{\partial y}-\frac{\partial sin\theta }{\partial z}\right)\\ +F_{\phi }\left(\frac{\partial cos\phi }{\partial y}-\frac{\partial sin\phi }{\partial x}\right)\end{array}$

$\begin{array}{l}=\frac{\partial F_{\rho }}{\partial \rho }\left({sin}^2\theta {cos}^2\phi +{sin}^2\theta {sin}^2\phi +{cos}^2\theta \right)\\ +\frac{\partial F_{\rho }}{\partial \theta }\left(\frac{1}{\rho }sin\theta cos\theta {cos}^2\phi +\frac{1}{\rho }sin\theta cos\theta {sin}^2\phi -\frac{1}{\rho }sin\theta cos\theta \right)\\ +\frac{\partial F_{\rho }}{\partial \phi }\left(-\frac{1}{\rho }sin\phi cos\phi +\frac{1}{\rho }sin\mathit{\phi cos\phi }\right)\\ +\frac{\partial F_{\theta }}{\partial \theta }\left(\frac{1}{\rho }{cos}^2\theta {cos}^2\phi +\frac{1}{\rho }{cos}^2\theta {sin}^2\right)\end{array}$