# Statistical mechanical descriptions of turbulent coagulation

Lian Ping Wang, Anthony S. Wexler, Yong Zhou

Research output: Contribution to journalArticle

86 Citations (Scopus)

### Abstract

A fundamental tenet of statistical mechanics is that the rate of collision of two objects is related to the expectation value of their relative velocities. In pioneering work by Saffman and Turner [J. Fluid Mech. 1, 16 (1956)], two different formulations of this tenet are used to calculate the collision kernel Γ between two arbitrary particle size groups in a turbulent flow. The first or spherical formulation is based on the radial component wr of the relative velocity w between two particles: Γsph = 2πR2〈\wr\〉, where wr = w· R/R, R is the separation vector, and R= |R|. The second or cylindrical formulation is based on the vector velocity itself: Γcyl=2πR2(|w|), which is supported by molecular collision statistical mechanics. Saffman and Turner obtained different results from the two formulations and attributed the difference to the form of the probability function of w used in their work. A more careful examination reveals that there is a fundamental difference between the two formulations. An underlying assumption in the second formulation is that the relative velocity at any instant is locally uniform over a spatial scale on the order of the collision radius R, which is certainly not the case in turbulent flow. Therefore, the second formulation is not expected to be rigorously correct. In fact, both our analysis and numerical simulations show that the second formulation leads to a collision kernel about 25% larger than the first formulation in isotropic turbulence. For a simple uniform shear flow, the second formulation is about 20% too large. The two formulations, however, are equivalent for treating the collision rates among random molecules and the gravitational collision rates.

Original language English (US) 2647-2651 5 Physics of Fluids 10 10 Published - 1998 Yes

### Fingerprint

coagulation
Coagulation
formulations
Statistical mechanics
Turbulent flow
Shear flow
collision rates
collisions
Turbulence
Particle size
statistical mechanics
turbulent flow
Molecules
Fluids
Computer simulation
molecular collisions
isotropic turbulence
shear flow
examination

### ASJC Scopus subject areas

• Fluid Flow and Transfer Processes
• Computational Mechanics
• Mechanics of Materials
• Physics and Astronomy(all)
• Condensed Matter Physics

### Cite this

Wang, L. P., Wexler, A. S., & Zhou, Y. (1998). Statistical mechanical descriptions of turbulent coagulation. Physics of Fluids, 10(10), 2647-2651.

Statistical mechanical descriptions of turbulent coagulation. / Wang, Lian Ping; Wexler, Anthony S.; Zhou, Yong.

In: Physics of Fluids, Vol. 10, No. 10, 1998, p. 2647-2651.

Research output: Contribution to journalArticle

Wang, LP, Wexler, AS & Zhou, Y 1998, 'Statistical mechanical descriptions of turbulent coagulation', Physics of Fluids, vol. 10, no. 10, pp. 2647-2651.
Wang LP, Wexler AS, Zhou Y. Statistical mechanical descriptions of turbulent coagulation. Physics of Fluids. 1998;10(10):2647-2651.
Wang, Lian Ping ; Wexler, Anthony S. ; Zhou, Yong. / Statistical mechanical descriptions of turbulent coagulation. In: Physics of Fluids. 1998 ; Vol. 10, No. 10. pp. 2647-2651.
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