By Tchen Chan-Mou

Read or Download Mean Value and Correlation Problems connected with the Motion of Small Particles suspended in a turbulent fluid PDF

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Additional resources for Mean Value and Correlation Problems connected with the Motion of Small Particles suspended in a turbulent fluid

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42 representative points in the element dydv, at the earlier instant will have had their representative points distributed over a series of elements dYldvv determined by the equation: Yl = Y - VI dt. We shall call this series of elements dYldvl the series II. When we consider another element dy dv of the y, v-diagram for the instant t, having the same value 01 y as the element dy dv considered first, and ask for the positions of the representative points of the particles at the earlier instant t - dt, we shall find that these representative points were situated in the same series II of elements dYI dv1 · It follows that the total number of particles +00 N dyIdv w(to, Yo; t, y, v), ~OO which at the instant t have their representative points in a series of elements dy dv all belonging to one definite value of y (but corresponding to all the various possible values of v), which series we shall indicate as the series I, at the earlier instant t - dt will have their representative points in elements of the series II.

In order that (I) may be valid we must have: t,+1' /dt l V(t1) V(tl + 'IlJ < for 1] 111 > f) (3) t, where m is a finite quantity such that miT' is negligible in comparison with v2 ; this inequality must apply for arbitrary values of to and T. Introducing the coefficient of correlation R('YJ) we shall write: I" d'YJ R('rJ) e= {} (£2 I" d1] R(1]). (4) () This quantity has the dimension of a time and will be called the average duration ot correlation. As R('rJ) « I, we have e < f). It is useful also to introduce the correlation integral: 00 ~ = (d'rJ v(t) v(t + 1/) = v 2 e.

K(v,-IJ,)' 7, ; A,' e---k(v-6)' 7, ' A2 = 'Y) - 1'/1; flo, Yo, CJ o = functions of 'Y)o or Yo; fll' () = o 1 Y1' CJ 1 = functions of ""12 0 - _I'·L_~_~. 2'1'/'" "/0"1 c°17'3 0 21'J'3 ... _2 _ _ 21'J'''' '/1"2 " "12 81'J'3 '/1 21'J'3 ... lL_~ It is asked to prove that the integral +00 +00 I dY1Idv l -00 WI Wn -00 for constant values of Yo, Y and v. = OJ (I ) 48 DISPERSION FUNCTION FULFILLING ALL CONDITIONS By the integration with respect to V! )' disappears from the product WIWU. In the integration with respect to YI we must take into account that terms of higher orders than T can be neglected; therefore we can write: fll = flo; YI = = Yo (Jl o1- + Yo YI - Y ' ( ) Al ,.