Problems Involving Multiple Length and Time Scales

1 Problems Involving Multiple Length and Time Scales 

As a more definite and concrete substantiation of our claim that a variety of problems in modeling materials are of an inherently multiscale nature, we now turn to a qualitative discussion of the emergence of multiple temporal scales in thinking about diffusion and, similarly, the emergence of multiple spatial scales in considering plasticity. 

1 Problems with Multiple Temporal Scales The Example of Diffusion 

As noted above, one of the most successful strategies for dealing with complex problems involving many different scales is to find a way to separate scales. For example, in the context of diffusion, fhe use of fhe diffusion equation (see eqn (7.18)) is an example of this approach in which the physics of the omitted temporal scales appears in the diffusion constant. This interpretation of the 

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It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

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