Inner variable

It has been shown that there exists a continuous change in the physical behavior of the turbulent momentum boundary layer with the distance from the wall. The turbulent boundary layer is normally divided into several regions and sub-layers. It is noted that the most important region for heat and mass transfer is the inner region of the boundary layer, since it constitutes the major part of the resistance to the transfer rates. This inner region determines approximately 10 — 20% of the total boundary layer thickness, and the velocity distribution in this region follows simple relationships expressed in the inner variables as defined in sect 1.3.4. 

This relation may be rewritten introducing the dimensionless inner variables ... 

When written in terms of the inner variable r = p /Pe, this becomes... 

The first term matches identically with the pure conduction, leading-order approximation in the inner region, 1 /r, but the second term clearly represents an ()(Pe) mismatch between the first approximations in the two regions. The third term from the power-series approximation of the exponential is O(Pep), but we see from (9 52) that this leads to a mismatch that is 0(Pe)2 when expressed in terms of inner variables. A term of 0(Pe2) cannot appear until at least the third approximation in the inner region. 

But now we can turn to the second term, 0(Pe), in the asymptotic expansion for the inner region. We can obtain matching conditions for this 0(Pe) inner problem by expanding (9-156) for small r and expressing the result in terms of inner variables. The result is... 

Let us find the first approximation for the inner expansion. To this end, we substitute formulas (4.4.11) into (4.4.7) and pass to the inner variable r. By expanding the obtained expression into a series in Pe, we obtain i(Pe) = Pe from the matching condition (4.4.8). Hence, the first approximation for the inner expansion must be sought with regard to (4.4.9) in the form... 

In this paper time-temperature equivalence for rocks is investigated for non-linear behaviours of rocks, based upon inner-variable theory of irreversible process and creep tests. 

Biot (1954) obtained the linear motion equation for n state variables qt in a closed system subject to generalized forces Q, and absolute temperature by introducing inner variables and applying Onsager s principle, given as follows ... 

By adopting similar procedures used for linear equations by Biot (1954) and in consideration of generalized forces corresponding to the inner variables being zero, we have the non-linear constitutive equation of thermo-visco-elasticity (see Liu et al, 2001 for details) written for i =l, 2, 3... 

We remark that the inner variable r varies over the negative semiaxis, and the above condition at — oo represents the matching condition for the inner and outer solutions. 

To determine the solution in the reaction zone, we introduce the inner variables... 

One could arrive at this result by means of a more systematic analysis, namely, by nondimensionalizing the equations and introducing stretched inner variables as we did before. We will restrict ourselves to using this less formal approach. 

This means that the overlapping region lies between the boundary layer (which has a thickness of order of e) and the outer region (which has a thickness of order of unity). In this overlapping region, the intermediate variable is of order of unity. Now, we write the outer variable x and the inner variable x in terms of this intermediate variable... 

The outer variable for this problem is i, which is sometimes called slow time in the literature. The appropriate measure of the change of the dependent variable in the initial short period is an inner variable (sometimes called fast time), which is defined as... 

To carry out the matching procedure between the inner and outer expansions, the outer solutions are first written in terms of the inner variable, t, and then are expanded using a Taylor series with respect to t around the point t = 0. The final stage is to equate to the Taylor expanded outer solution to the inner expansions in the limit when t tends to infinity. The terms that are matched between the inner and outer expansions are called the common parts. 

F, z - scaled electric field and the inner variable in the ESC region... 

A survey concerning coherent structures which can play an important part in the phenomenon of drag reduction by polymer additives is given. Most of these structures have long been known. Recent measurements have shown that both the scaling of low speed streaks and the bursting frequency remain constant when made dimensionless with inner variables. Stream wise vortices have been visually verified and their intensities anemometrically determined. Various authors have independently shown that too little attention has been paid to the problem of small-scale structure in turbulence measurements. 

The inner and outer expansions for the dependent variables are related by expanding the latter in Maclaurin s series in n, expressing these in terms of the inner variable 5, and comparing the resulting series with the inner variable expansions at each order in e. This results in the following ... 

In the vicinity of the moving apparent three-phase contact line, = 0, where the profile of the outer solution (3.141) intersects the surface of the liquid film, we introduce the inner variable as before ... 

Using the new inner variable and retaining in Equation 3.140 only the leading terms, we obtain... 

Let us rewrite solution (3.233) nsing the inner variables in Eqnation 3.228 ... 

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