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Optimal Approximation of the Added Mass Matrix of Two Spheres of Unequal Radii by an Asymptotic Short Distance Expansion
Author(s) -
Raszillier H.,
Guiasu I.,
Durst F.
Publication year - 1990
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19900700202
Subject(s) - asymptotic expansion , pointwise , mathematics , mathematical analysis , inverse , order (exchange) , matrix (chemical analysis) , mathematical physics , combinatorics , geometry , materials science , finance , economics , composite material
An asymptotic expansion of the added mass matrix m A = {m ij }i,j=i, 2 of two spheres of radii R 1 R 2 , moving along the line of their centres in an ideal fluid (of zero vorticity) is given for small particle surface distances x (x → 0) to any order in x. The derivation starts from the classical essentially long distance expansion of m A , which is first converted by an inverse Mellin transformation into an integral representation. The expansion is deduced by the shift of the integration line in this representation and by the application of the residue theorem, and its asymptotic character is proved. The pointwise approximation of the matrix m A by a finite number of terms of the asymptotic expansion is investigated in terms of a suitably defined approximation error d ij (n) (x) for the matrix element m ij (x) and its n‐th order approximant m ij (n) (x) (n = 0, 1,…). The minimum d ij (x) of d ij (n) (x) with respect to the order n is attained for a finite value n = n ij 0 (x) because of the divergence of the expansion (lim d ij (n) (x) = ∞ for any x > 0) and defines by it the order n = n ij 0 (x) of the optimal approximant of m ij (x)., which describe the range of x where m ij (n) (x) approximates m ij with an error not larger than ϵ, are introduced, together with the setwhich gives the distance range x where an approximation of m ij (x) by the asymptotic expansion is possible within an error ϵ. By numerical computation the optimal approximation order n o ij (x) and the sets D ij (n) (ϵ) (n = 1,…, 11) are estimated. It turns out that n ij 0 (x) is large compared to the highest order (n = 1) approximant m ij (1) (x) computed previously, whereas the optimal ϵ‐approximation set D ij (ϵ) is typically by one order of magnitude larger than D ij (1) (ϵ), the ϵ‐approximation set of the approximant m ij (1) (x).