Microfluidics: Modeling, Mechanics and Mathematics

$\begin{array}{lll}{\left(\left(\overrightarrow{F}\cdot \overrightarrow{\nabla }\right)\overrightarrow{G}\right)}_{\rho }\hfill & =\hfill & sin\theta cos\phi \left(F_{\rho }\left(\frac{\partial G_{\rho }}{\partial \rho }sin\theta cos\phi +\frac{\partial G_{\theta }}{\partial \rho }cos\theta cos\phi -\frac{\partial G\phi }{\partial \rho }sin\phi \right.\right)+\hfill \\ \frac{F_{\theta }}{\rho }\left(\frac{\partial G_{\rho }}{\partial \theta }sin\theta cos\phi +G_{\rho }cos\theta cos\phi +\frac{\partial G_{\theta }}{\partial \theta }cos\theta cos\phi -G_{\theta }sin\theta cos\phi \right.\hfill \\ \left.-\frac{\partial G_{\phi }}{\partial \theta }sin\phi \right)+\frac{F_{\phi }}{sin\theta \rho }\left(\frac{\partial G_{\rho }}{\partial \phi }sin\theta cos\phi -G_{\rho }sin\theta sin\phi +\right.\hfill \\ \left.\left.\frac{\partial G_{\theta }}{\partial \phi }cos\theta cos\phi -G_{\theta }cos\theta sin\phi -\frac{\partial G_{\phi }}{\partial \phi }sin\phi -G_{\phi }cos\phi \right)\right)+\hfill \\ sin\theta sin\phi \left(F_{\rho }\left(\frac{\partial G_{\rho }}{\partial \rho }sin\theta sin\phi +\frac{\partial G_{\theta }}{\partial \rho }\mathit{cos\theta }sin\phi +\frac{\partial G_{\phi }}{\partial \rho }cos\phi \right.\right)+\hfill \\ \frac{F_{\theta }}{\rho }\left(\frac{\partial G_{\rho }}{\partial \theta }sin\theta sin\phi +G_{\rho }cos\theta sin\phi +\frac{\partial G_{\theta }}{\partial \theta }cos\theta sin\phi \right.\hfill \\ -G_{\theta }sin\theta sin\phi +\left.\frac{\partial G_{\phi }}{\partial \theta }cos\phi \right)+\frac{F_{\phi }}{sin\theta \rho }\left(\frac{\partial G_{\rho }}{\partial \phi }sin\theta sin\phi +\right.\hfill \\ \left.\left.G_{\rho }sin\theta cos\phi +\frac{\partial G_{\theta }}{\partial \phi }cos\theta sin\phi +G_{\theta }cos\theta cos\phi +\frac{\partial G_{\phi }}{\partial \phi }cos\phi -G_{\phi }sin\phi \right)\right)+\hfill \\ cos\theta \left(F_{\rho }\left(\frac{\partial G_{\rho }}{\partial \rho }cos\theta -\frac{\partial G_{\theta }}{\partial \rho }sin\theta \right)+\frac{F_{\theta }}{\rho }\left(\frac{\partial G_{\rho }}{\partial \theta }cos\theta -G_{\rho }sin\theta \right.\right.\hfill \\ \left.-\frac{\partial G_{\theta }}{\partial \theta }sin\theta -G_{\theta }cos\theta \right)+\frac{F_{\phi }}{sin\theta \rho }\left.\left(\frac{\partial G_{\rho }}{\partial \phi }cos\theta -\frac{\partial G_{\theta }}{\partial \phi }sin\theta \right)\right)\hfill \\ =\hfill & F_{\rho }\left(\frac{\partial G_{\rho }}{\partial \rho }{sin}^2\theta {cos}^2\phi +\frac{\partial G_{\theta }}{\partial \rho }sin\theta cos\theta {cos}^2\phi -\frac{\partial G_{\phi }}{\partial \rho }sin\theta sin\phi cos\phi \right)+\hfill \\ \frac{F_{\theta }}{\rho }\left(\frac{\partial G_{\rho }}{\partial \theta }si{n}^2\theta {cos}^2\phi +G_{\rho }sin\theta cos\theta {cos}^2\phi +\frac{\partial G_{\theta }}{\partial \theta }sin\theta cos\theta {cos}^2\phi \right.\hfill \\ -G_{\theta }{sin}^2\theta {cos}^2\phi -\left.\frac{\partial G_{\phi }}{\partial \theta }sin\theta sin\phi cos\phi \right)+\frac{F_{\phi }}{sin\theta \rho }\left(\frac{\partial G_{\rho }}{\partial \phi }{sin}^2\theta {cos}^2\phi \right.\hfill \\ -G_{\rho }{sin}^2\theta sin\phi cos\phi +\frac{\partial G_{\theta }}{\partial \phi }sin\theta cos\theta {cos}^2\phi -G_{\theta }sin\theta cos\theta sin\phi cos\phi \hfill \\ \left.-\frac{\partial G_{\phi }}{\partial \phi }sin\theta sin\phi cos\phi -G_{\phi }sin\theta {cos}^2\phi \right)+\hfill \\ F_{\rho }\left(\frac{\partial G_{\rho }}{\partial \rho }{sin}^2\theta {cos}^2\phi +\frac{\partial G_{\theta }}{\partial \rho }sin\theta cos\theta {sin\; ...}^{}\right)\hfill \end{array}$