Nondimensional functions

Note that the temporal evolution of both the boundary concentration, CB/A(t), and the flux across the boundary, F(t), are controlled by the nondimensional function i /(t). This function defines a critical time scale for the switch between two regimes. We define crit as the time for which /(fcrit) = 1. From Eq. 19-39 ... 

Substituting the series (390a) and (390b) into integral (394) and introducing a nondimensional function S of the reduced complex frequency Z, one can get the result for S(Z) in an analytical form [5] ... 

Afl of the adiabatic problem [defined by equation (5-45) for the adiabatic problem, for example] with a nondimensional function o (t), and the non-dimensional heat-loss rate is... 

For brevity, it is also useful to introduce four nondimensional functions of X and t, defined as... 

The front velocity v/ is the result of the interplay among the flow characteristics (i.e., intensity U and length scale L), the diffusivity D, and the production time scale x. In this chapter we shall study the problem of front propagation in the case of cellular flows. In particular, introducing the Damkohler number Da = L/(Ux) (the ratio of advective to reactive time scales) and the Pec let number Pe = UL/D (the ratio of diffusive to advective time scales), we shall discuss how the front speed can be expressed as a nondimensional function such as Vf/vo = < >(Da,Pe). A crucial role in determining i >(Da. Pe) is played by the renormalization of the diffusion coefficient and chemical time scale [13] induced by the advection. 

We also consider an important limit case—that is, the so-called geometrical optics limit—which is realized for (D. x) - 0 maintaining D/x constant [17]. In this limit, one has a nonzero bare front speed, vo, while the front thickness c goes to zero that is, the front is sharp. Physically speaking, this limit corresponds to situations in which E, is very small compared with the other length scales of the problem. In this case we provide a simple prediction for the front speed as a nondimensional function Vf/vo = /((7/vo). 

The equilibrium statistical mechanics of the fluids can be discussed in terms of the n-particle probability density function p(l 2 n) associated with finding a particle centered at ri with orientation o>i, another particle at t2 with orientation <02, etc. we shall in addition often work with the nondimensional functions... 

Nondimensional functions used in the calibration for a butt joint with a thin adhesive... 

Nondimensional function used in the K- calibration for a rigid square inclusion embedded... 

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