The solution for thermally developing PoiseuiUe flow with scattering is obtained by using the method of collocation. The results show that scattering tends to decrease radiation component without affecting the connective component at low optical thicknesses. For moderate to high values of optical thicknesses, both the connective and radiation components are reduced. The relative effect of scattering both on convective and radiation Nusselt numbers appears to be unaffected by a change in the surface emissivity. A significant feature of combined radiation-convection in thermally developing flows appears to be that the total Nusselt number increases downstream of position of minimum rather than approaching an asymptotic value as is the case with corresponding pure convection. Another departure in the behavior appears to be the lack of existence of similarity in the temperature profiles, particularly at low values of conduction-radiation parameter. IntroductionA number of analyses of channel flows with combined radiation convection modes of heat transfer have been performed [1-17], A significant number among these analyses [2-9] have assumed fully developed temperature distribution which reduces the partial integro-differential equation to an ordinary integro-differential equation. This equation is identical in appearance to that for the combined radiation-conduction problem for the same geometry, except for the presence of a source term (as a function of local and bulk mean temperatures). Therefore, the equation can be solved by the same techniques as used for combined radiation-conduction problems. This assumption of fully developed temperature profile to simplify the problem has been carried over from channel flows with heat transfer by convection mode only. However, recently Lii and Ozisik [1] have shown that the thermally fully developed state for the slug flow between the parallel plates does not occur for the cases when the radiation effects are strong. Similar findings, particularly regarding the existence of a fully developed temperature profile, have been reported in analyses by Liu and Thorsen [11] and by Pearce and Emery [12]. Recently Balakrishnan and Edwards [10] compared their results for radiation and total Nusselt numbers at large distances from entry to the results of Wassel and Edwards [9] for thermally and hydrodynamically established turbulent flow in a pipe. These authors found that maximum errors in these Nusselt numbers were 2.2 percent and 7 percent, respectively, corresponding to radiation-to-conduction ratio of 10. It is clear that not all of the authors agree on the validity of the assumption of fully developed temperature profile. It is, therefore, one of the objectives of this paper to provide an appraisal of this assumption by comparing the temperature profiles at various axial locations for a variety of conditions of interest.

J k = Boltzmann constant, errr/"K = particle flux, particles/ (cm2-s) dh } , cm/s Kr = Kf/I2, s-' m M n n2 nZi n2f T = absolute temperature, OK U = approach velocity, cm/s y Greek Letters (Y B y = ld2+/dh2, erg/cm2 8 , 81 = local particle mobility, (cm/s) /dyne = mass ~f a single particle, g = ionic strength of bulk solution, ions/cm3* = number of adsorbed particles, particles/cm2 = initial value of n2, particles/cm2 = final value of n2, particles/cm2 = distance measured normal to collector, cmfor which a Boltzmann distribution of particles may be assumed, cm = largest h in the interval hmnz < h < hm, for which a Boltzmann distribution of particles may be assumed, cm L = fluid dielectric constant K = ( 8nne2/tkT)1/2, cm-l p = fluid viscosity, g/(cm-s) pf = fluid density, g/cm3 u = collision diameter, cm d = electrostatic surface charge density, statcoul* + = total potential energy of interaction, erg +s = van der Waals energy of interaction, erg +12 = Born energy of interaction, erg +DL = double-layer energy of interaction, erg ~l~~ = electrostatic potential of surface i, statvolt* o = disk rotation speed, rad/s Su brcriph max = evaluated at the maximum of 4 ( h ) m n l = evaluated at the secondary minimum of 4 ( h ) mn2 = evaluated at the primary minimum of 4 ( h ) LITERATURE CITED Brenner, H., "The Slow Motion of a Sphere through a ViscousS2 0 A consistent set of units has been specified above for each symbol, although more convenient units are sometimes employed in the text (for example, m V instead of statvolts).

A model for heat transfer from the sides of a volume heated boiling pool is proposed. Because of the density difference caused by volume boiling and by thermal expansion due to the temperature difference between the bulk fluid and the fluid near the wall, the lighter liquid and vapor bubbles cause movement of the bulk fluid in the upward direction. The rising liquid between the bubbles finds a return path along the walls or sides of the pool and forms a boundary layer which may be laminar in its initial length followed by transition to turbulent depending, of course, on the conditions prevailing at the entry to the sides and in the bulk of the pool. The analysis for the laminar case provides the definition of equivalent Grashof number for the combined two-phase and thermal expansion driven natural convection along the sides of pool. The turbulent boundary layer is analyzed by assuming a two-layer model in which the inner layer is characterized by viscous and conduction terms and the outer by mean convection terms. The similarity analysis of the governing equations yields universal profiles for temperature and velocity and the scaling laws for the inner and outer layers. An asymptotic matching of the temperature profile in the overlap region leads to a heat transfer law which correlates the available experimental data on volume heated boiling pools exceedingly well.

It is well known that the small perturbation equation governing steady or mildly unsteady potential flow in a sonic gas jet is nonlinear. However, for a sonic gas jet submerged in a liquid with a disturbance on the gas-liquid interface, it is shown that the transient motion of the gas dominates, and the nonlinear term due to accumulation of disturbances in the basic flow becomes negligible; the condition necessary for the applicability of the linearized governing equation is obtained. It is demonstrated that most gas-jet/liquid systems of physical interest satisfy this condition and that the margin with which this condition is satisfied improves as the wave velocity of the disturbance or, more particularly, as the stagnation pressure or density of the gas for a given gas-liquid system increases. The Kelvin-Helmholtz instability of the gas-liquid interface of a sonic gas jet submerged in a liquid is predominantly governed by the transfer of energy from the gas phase to the liquid layer, both through wave-drag and ‘lift’ components of the pressure perturbation; at and above the cut-off wavenumber, which only exists for very low viscosity liquids owing to the stabilizing effect of surface tension, the pressure perturbation becomes in phase with the wave amplitude. It is shown that for low viscosity liquids the phase angle between the pressure perturbation exerted by the gas phase on the liquid a t the gas-liquid interface and the wave amplitude, which is the measure of the relative effectiveness of the ‘lift’ and wave-drag components of the pressure perturbation, is a function of the density ratio (ratio of gas density a t throat conditions to liquid density). At low density ratios both of these components are operative; however, at high density ratios the wave-drag component becomes dominant. The analysis further shows that the cut-off wave- number and the wavenumber a t maximum instability decrease with increasing density ratio. For highly viscous liquids and liquids having finite viscosity the pressure perturbation is always out of phase with the wave amplitude, and no cut-off wavenumber exists, i.e. the gas-liquid interface is always unstable in spite of the stabilizing effect of viscosity and surface tension.

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