Multiscale modeling

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Multiscale modeling or multiscale mathematics is the field of solving problems that have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids,^[1][2][3] solids,^[2][4] polymers,^[5][6] proteins,^{[7][8][9][10]} nucleic acids^[11] as well as various physical and chemical phenomena (like adsorption, chemical reactions, diffusion).^{[9][12][13][14]}

An example of such problems involve the Navier–Stokes equations for incompressible fluid flow.

${\displaystyle {\begin{array}{lcl}\rho _{0}(\partial _{t}\mathbf {u} +(\mathbf {u} \cdot \nabla )\mathbf {u} )=\nabla \cdot \tau ,\\\nabla \cdot \mathbf {u} =0.\end{array}}}$

In a wide variety of applications, the stress tensor ${\displaystyle \tau }$ is given as a linear function of the gradient ${\displaystyle \nabla u}$. Such a choice for ${\displaystyle \tau }$ has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, its use for more complex fluids such as polymers is dubious. In such a case, it may be necessary to use multiscale modeling to accurately model the system such that the stress tensor can be extracted without requiring the computational cost of a full microscale simulation.^[15]