Problems in Particle-Fluid Mechanics

Citation

Liu, Joseph Tsu Chieh (1964) Problems in Particle-Fluid Mechanics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/11Y4-2H39. https://resolver.caltech.edu/CaltechETD:etd-10172002-122957

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. The continuum equations describing the motion of a fluid containing small solid particles are discussed and stated. The examples considered fall into two categories: (1) when the fluid is incompressible and viscous, with simultaneous occurrence of particle-fluid momentum relaxation and fluid viscous diffusion; and (2) when the fluid can be considered as "inviscid" but compressible, with simultaneous occurrence of coupled particle-fluid momentum and thermal relaxations and fluid compressibility. Under (1), the low Mach-number Rayleigh problem is studied. Many of the physical features of the non-linear steady (constant pressure) laminar boundary-layer problem are recovered from appropriate expansions from this exact solution. One obtains answers to questions about the modifications on the boundary layer growth and skin friction; particularly their transition from the "frozen" value near the leading edge, where the viscous layer is "thin" and the fluid viscous diffusion behaves as if in the absence of particles with the ordinary fluid kinematic viscosity,[.....], to the ultimate "equilibrium" value far downstream where the mixture then behaves as a single heavier fluid and viscous diffusion takes place with the "equilibrium" kinematic viscosity augmented by the particle density [.....].The uncoupled thermal Rayleigh problem (small relative temperature differences) is directly inferred, and this answers questions about the modifications on the surface heat-transfer rates and particularly about the possibility of similarity with the velocity boundary layer. Similarity of the two boundary layers is possible when, in addition to lateral diffusion effects being similar as indicated by Prandtl number unity, the streamwise relaxation processes must also be similar. The infinite flat plate oscillating in its own plane is studied, and appropriate expansions from the exact solutions point out how approximate treatment of periodic boundary layers in the absence of a mean flow may be made. Under (2), the first-order small perturbation theory is discussed, leading from the equation for acoustic propagation to that for linearized supersonic flow. The two-dimensional steady case, or the Ackeret problem, is considered in detail. The Mach wave structure induced by a thin obstacle is deduced and shows a rapid damping of the disturbance along the "frozen" Mach wave (based on the sound speed of a gas in the absence of particles), both damping and diffusiveness along an intermediary Mach wave, and diffusiveness along the "equilibrium" Mach wave (based on the sound speed of an equilibrium mixture of gas and particles) and along which the bulk of the disturbance is carried to regions far from the obstacle. An exact form of the pressure coefficient is obtained for any surface shape (consistent with the linear theory), and involves a convolution integral of two Bessel functions with imaginary argument which is analytically evaluated. When the particle-fluid density ratio is small, the "frozen" and "equilibrium" Mach waves are very closely clustered together. A "boundary layer technique", based on the fact that changes across the Mach waves are rapid compared to changes along Mach waves, is then applied to obtain a simplified version of the linearized equation that describes Mach waves inclined toward the downstream direction only. While the Mach wave structure is consistent with the exact treatment, the pressure coefficient takes on the much simpler form of decreasing exponentials. The transition is, again, from the "frozen" value at the leading edge towards the "equilibrium" value in the downstream direction insofar as the surface shape permits.

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