Isothermal drag force

These are the isothermal drag force and the thermal force on a spherical particle. Various difficulties in current theory for these phenomena are cited. 

For nonequilibrium host gas, a number of particle forces, arise. These include isothermal drag force, thermal force, photophoretic force, diffusion force, stress force, and other additional cross effects in combined flows of heat, mass, and momentum. 

Of these various forces, only the isothermal drag force and the thermal force will be discussed here. The isothermal drag force presents perhaps the simplest of the nonequilibrium, noncontinuum phenomena. Yet, as will be shown, the current state of knowledge of the drag force is inadequate. The thermal force provides an example of the complexity inherent in particle motion in nonequilibrium host gas. 

Numerous experimental and theoretical investigations have been made of the isothermal drag force on a particle in the noncontinuum regime. Much of the work is summarized in several references [2.5-7]. As will be evident in this discussion, there exists even now no satisfactory comprehensive description of this phenomenon. 

Exact results are known for the isothermal drag force in the two limits Kn. 0 and Kn. -> oo for Ma. <<1. For Kn. 0, Stokes law is readily derived from the Navier-Stokes equation with stick boundary conditions... 

It is only in the two limits, Kn. are available. One may conclude that a necessary, but not sufficient condition for a satisfactory noncontinuum description of the isothermal drag force is its reduction to (2.59,61,62) or (2.63) in the appropriate limits. 

The standard of comparison for the isothermal drag force on a spherical particle for Ma - 1 has for many years been the empirical relation of MILLIKAN and coworkers [2.96] based on extensive experimental measurements ... 

Fig.2.5. Isothermal drag force on spherical particle as a function of Knudsen number for the particle. Millikan s...

With the considerable difficulties still remaining in the analysis of the isothermal drag force for spherical ultrafine particles, it is not surprising that the analysis of this problem for nonspherical particles is incomplete. This subject is reviewed by FUCHS [2.7]. Recent kinetic-theory analyses of drag on nonspherical bodies in the free-molecular regime can be found in CERCIGNANI [2.84]. The excellent series of experimental studies by STDBER and co-workers [2.80,81,116] as well as more recent work on this problem [2.82,117-119] may be consulted for further information. 

The difficulty with all the analyses cited lies in the fact that there are at present no a priori methods for determining for a given gas-particle system the phenomenological coefficients C.., Cj, or the various accommodation coefficients which arise for the thermal-force problem. Of course, C, and can be obtained from independent experimental measurements of isothermal drag force, heat transfer, and thermal transpiration, respectively [2.103], However,... 

Much additional analysis and experimental work will be required to place the theory of the thermal force on a rigorous basis. The subject is certainly much more complex than the isothermal drag force discussed earlier. Not only do the regimes Kn. > 0, Kn > 0 need further investigation, but also the regimes with Br - > 0 and Ma. > 0. Very little is known about thermophoresis of nonspherical particles, but it is undoubtedly a complex phenomenon and requires introduction of additional parameters. 

Relative Importance of the Various Terms in the Analysis of the Isothermal Fiber Spinning of a Newtonian Melt Use the data and result of Problem 14.1(b) to evaluate the importance in the isothermal fiber spinning of a Newtonian melt analysis (nylon 6-6 at 285°C) of the inertial terms and gravity relative to the viscous stress terms. Using Eq. E14.1-2 for FD, evaluate the importance of the air-drag force term. 

When (he average drag and convection coefficients are available, the drag force can be determined from Eq. 7-1 and the rate of heat transfer to or from an isothermal surface can be determined from... 

In summary, the steady-state Newtonian isothermal model is able to provide the axial velocity profile as well as the filament radius profile, and it is based on the following additional assumptions (1) slowly changing radial profile with axial distance, (2) negligible inertial and gravitational forces, (3) nonexistent radial velocity profile, (4) circular filament, (5) axial velocity profile not dependent on the radial coordinate, and (6) negligible surface tension and air drag forces. (See also Problem 9A.1 for the validity of some of the above assumptions and Schultz (1987) for a challenge of these assumptions.)... 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

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