By A. V. Luikov

ISBN-10: 0124597564

ISBN-13: 9780124597563

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**Extra info for Analytical Heat Diffusion Theory**

**Example text**

Then, the function z = dmy/dxm will result in an effect of this operator on the function y = f(x). The effect of x and y on z is, however, of interest to us. Therefore, we assume dmy/dxm = const. It is obvious that to satisfy this condition it is necessary to assume y = Axm9 hence A = y/xm where x and y are any values of the variables obtained from the relation y = f(x). Let any values of the variables x and y be known or given according to the condition (x 0 , y0). These prescribed values of A:0 and y0 are referred to as parametric values.

12) Hence, it follows that the Fourier number has the meaning of generalized time. It may therefore be called as the homochronous number {homochronity means uniformity in time; if for two systems the relation P/a is constant, then for them homochronity obviously transforms into synchronity). Gukhman in his recent work [45] distinguishes between similarity criteria and similarity numbers. Similarity criteria are those complexes which as a whole consist of parameters given according to the condition of the problem.

7) may be rewritten as dt/At = F(Bi). 9) A transition from initial variables a, /, λ to a new variable Bi, results not only in a decrease in the number of arguments; in parallel to this, the analysis itself changes. Actually, when a, l, λ are taken as basic quantities, some particular case for each set of the given values of these parameters is obtainable. In contrast to this, any particular case does not uniquely satisfy the given value of the Biot number, because this value may be realized by the infinite number of various combinations of quantities a, /, λ.

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