I. Note on the Equations of Triple and Quadruple Correlation Functions in Isotropic Turbulence. II. On the Turbulent Mixing of Two Fluids of Different Densities. III. On the Possibility of Keeping the Electrons Inside the Dimension of the Nucleus and the Quantum Mechanical Theory of Neutron

Citation

Hu, Ning (1944) I. Note on the Equations of Triple and Quadruple Correlation Functions in Isotropic Turbulence. II. On the Turbulent Mixing of Two Fluids of Different Densities. III. On the Possibility of Keeping the Electrons Inside the Dimension of the Nucleus and the Quantum Mechanical Theory of Neutron. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9CG-4J68. https://resolver.caltech.edu/CaltechETD:etd-02152007-144149

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Part I: Recently Karman and Howarth (1938) have successfully developed a theory of isotropic turbulence by investigating the velocity correlation functions at two points in the field of flow. From the equations of motion they established a partial differential equation connecting the double and triple correlation functions, f(r,t) and h(r,t), which can be solved by using some additional physical assumptions. As suggested by the authors, the indeterminateness of two unknown functions from one equation may be due to the fact that this equation has not exhausted the information obtainable from the equations of motion; so it will be worthwhile to investigate whether this indeterminateness can be reduced by establishing the dynamical equations for the correlation functions of higher orders. This investigation has now been carried out in the present note and it is found that when we come to consider the quadruple correlation functions, the number of kinematically independent unknown functions is increased by ten, while only three dynamical equations are obtained. Consequently the consideration of quadruple correlation correlation functions gives no aid to the solution of the problem. It seems that this situation will continue when we push our consideration to correlation functions of still higher orders. This would mean that the correlation functions cannot be determined from their equations without introducing some additional assumptions. Part II: Tollmien’s calculation on the turbulent jets and half jets provides an excellent check of Prandtl’s momentum transport theory with the experiment. In the present paper we extend Tollmien’s theory to the case in which the mixing fluids are of different densities. Both the change in the distribution in mean velocity and the change of the angular divergence of the jets are investigated. Quite a number of experimental observations of turbulent jets of this kind have been published in literature on account of their importance in connection with the atomization of a spray of fuel in the internal combustion engines. It is found that the predicted density distribution across the jet agree with the experiment very well. In the following treatment we shall confine ourselves to the immiscible fluids from which an emulsion is formed by turbulent diffusion. This corresponds to the actual case of a spray of liquid fuel in the air. Part III: In the present paper an attempt is given to treat the [...]-activity along the lines of Gamow’s theory of [...]-activity without using Fermi’s conception of likening the process to the emission of light photons by the charged particle. It is found that if we consider the proton as a sphere with constant high potential inside, then the electron can be kept inside the proton for any length of time depending on the energy of the electron and the height of the potential chosen. The present theory explains satisfactorily the [...]-activity of the free neutron and gives the right order of magnitude of the nuclear force in contrast to the too small value obtained in Fermi’s theory. The latter result means that the magnitude of the nuclear force can be explained by the [...]-exchange force alone without the introduction of mesotron. At the present stage the theory cannot explain satisfactorily the different types of [...]-active nuclei, and therefore no detailed quantitative study of the mean lifes of the [...]-transmutations is given. The present model of proton can also be used to explain the binding of mesotron with the heavy particles. This provides a new way of treating the exchange interactions between the heavy particles in the mesotron theory.

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