MEAN VALUE AND CORRELATION PROBLEMS CONNECTED WITH THE MOTION OF SMALL PARTICLES SUSPENDED IN A TURBULENT FLUID

Transcription

1 MEAN VALUE AND CORRELATION PROBLEMS CONNECTED WITH THE MOTION OF SMALL PARTICLES SUSPENDED IN A TURBULENT FLUID

2 N.V. VAN DE GARDE & CO'S DRUKKERIJ, ZALTBOMMEL

3 Mean Value and Correlation Problems connected with the Motion of Small Particles suspended in a turbulent fluid PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. J. M. TIENSTRA, HOOGLERAAR IN DE AFDELlNG DER WEG- EN WATERBOUWKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDE- DIGEN OP DINSDAG 1 JULl 1947 TE 4 UUR, DOOR TCHEN CHAN-MOU GEBOREN TE CHEKIANG CHINA Springer-Science+Business Media, B.Y. 1947

4 ISBN ISBN (ebook) DOI / Dit proefschrift en de Stellingen zijn goedgekeurd door de promotor PROF. DR. J. M. BURGERS

5 AAN MIJN OUDERS

6 CONTENTS INTRODUCTION Chapter 1. STATISTICAL CONSIDERATIONS CONCERNING THE DISPERSION OF PARTICLES IN IRREGULAR MOTION. Pages Definition of the dispersion function Further properties of the dispersion function Mean values connected with the dispersion process Series development of the dispersion function Problems with constant density Quantities which are characteristic for the mean displacement of the particles Derivation of two equations satisfied by the dispersion function Eq ua tions for isomeric dispersion functions Equation for the particle-density Partial differential equation for any property attached to the particles Isomeric diffusion; coefficient of diffusion Diffusion of particles due to irregular movements combined with a peculiar motion produced by exterior forces Introduction of a dispersion function which contains the velocity of the particles Expression for the current On the part played by the systematic velocity 29 Chapter 2. EXAMPLES OF DISPERSION FUNCTIONS ILLUSTRAT ING CERTAIN PROPERTIES DEDUCED IN CHAPTER An elementary example of isomeric dispersion Generalization Dispersion functions fu fulling the integral condition 1.11(3)

7 x CONTENTS 2.4. Dispersion function fulfilling the three conditions Relations fulfilling by the function p Integro-differential equation fulfilled by the function w(to' Yo; t, y; v) Some mean values involving the velocities Investigation of the product integral for the functon w Isomeric dispersion function constructed from Gauss' error function by introducing a discontinuous change in b 49 Chapter 3. TIME MEAN VALUES CONNECTED WITH THE MOTION OF A PARTICLE Introductory remarks Central assumption concerning the behaviour of the. function v(t) Correlations Functions with limited duration of correlation Corollary S Proofthat v = 0 for functions fulfilling (1) Correlation when v is different from zero Relative magnitudes of intervals of time Mean values in w h ithe c displacement ~ plays a role Investigation of! and y Investigations of [ Value ofvl Mean values taken over an interval of the order of the duration of correlation Mean values of the velocity and correlations..., Alternative method of calculating the new mean values Formulae containing the displacement of the particles Comparison of the new mean values with the "long range" time mean values of section , 70 Chapter 4. EQUATION OF MOTION FOR A PARTICLES SUSPENDED IN A HOMOGENEOUS FIELD Equation of Bas set, B 0 u s sin e s q and 0 see n for the motion of a particle in a fluid at rest Equation of motion of the particle in a moving fluid Equation of motion in the simplified form

8 CONTENTS XI 4.4. Integration of the equation of motion in its complete form 4.2.(4) Periodic motion I 4.6. Motion with non linear resistance. 83 Chapter 5. MEAN VALUES CONNECTED WITH THE MOTION OF A PARTICLE, CALCULATED FROM THE EQUATION OF MOTION, BOTH IN THE SIMPLIFIED AND IN THE COMPLETE FORM. 5. I. Calculation of mean values based upon the simplified equation of motion I I. Calculation of V'(O)V'(1]) and related quantities in terms of f' Calculationofv/ Expressions which depend upon the displacement y Calculation of /y Introduction of the expression for t in function of u Relation between v and u Relations between the full values of v and u and derived quantities Calculation of the coefficient of diffusion of the particles vy for the motion regulated by the complete linear equation Calculation of if = f d1]f(o)f(1]) o 5.5. Remark on Einstein's formula for the Brownian motion 104 Chapter 6. APPLICATION OF THE RESULTS TO THE PROBLEM OF THE DIFFUSION OF PARTICLES SUSPENDED IN A LIQUID IN TURBULENT MOTION Introduction I I. Influence of an inhomogeneous field of motion of the liquid surrounding a particle I IO 6. I 2. Mean value of the velocity of the particles I The relation between the mean values l2 a n d.. ~. I II 6.2. Application of the diffusion equation to the case of homogeneous turbulence , Application of the diffusion equation to particles suspended in a current of water flowing horizontally. I IS SAMENVATTING