Mixed convection with viscous dissipation in an inclined porous channel with isoflux impermeable walls

Combined forced and free convection flow in a fluid saturated inclined plane channel is investigated by taking into account the effect of viscous dissipation. Steady parallel flow is considered assuming that the temperature gradient in the parallel flow direction is constant, and the channel walls are subject to uniform symmetric heat fluxes. Two possible formulations of the Darcy–Boussinesq scheme are considered, based on two different choices of the reference temperature for modelling buoyancy. The first choice is a constant temperature, while the second is a streamwise changing temperature. It is shown that both approaches substantially agree in the formulation of the balance equations for the range of values of the Darcy–Rayleigh number such that viscous dissipation is important. The boundary value problem is solved analytically for any tilt angle, revealing that it admits dual solutions for assigned values of the governing parameters. The rather important effect of viscous dissipation in the special case of adiabatic channel walls is outlined. E. Magyari is on leave from Institute of Building Technology, ETH—Zürich     

Effect of Viscous Dissipation on a Quasi-Parallel Free Convection Boundary-Layer Flow over a Vertical Flat Plate in a Porous Medium

Transport in Porous Media -     

The “Butterfly Effect” in a Porous Slab

It is shown that the governing equation for the stream function of the Darcy free convection boundary layer flows past a vertical surface is invariant under arbitrary translations of the transverse coordinate y. The consequences of this basic symmetry property on the solutions corresponding to a prescribed surface temperature distribution T _w (x) are investigated. It is found that starting with a “primary solution” which describes the temperature boundary layer on an impermeable surface, infinitely many “translated solutions” can be generated which form a continuous group, the “translation group” of the given primary solution. The elements of this group describe free convection boundary layer flows from permeable counterparts of the original surface with a transformed temperature distribution [(T)\tilde]_w ( x ){\tilde {T}_w \left( x \right)}, when simultaneously a suitable lateral suction/injection of the fluid is applied. It turns out in this way that several exact solutions discovered during the latter few decades are in fact not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.     

Bloch-Ising phase transition in spin solitons

Heisenberg spin chains with ferromagnetic nearest-neighbor coupling and an easy axis of magnetisation are considered in the classical limit. The critical ratioa _c between the anisotropy coefficient and the exchange integral for which the 180°-Blochtype soliton undergoes a phase transition to an Ising-type soliton is calculated by an exact analytical algorithm. For chains including 2N=4,6,8,... spins, one finds a sequencea _c (2N) which converges rapidly to the valuea _c()=2/3 (e.g.a _c (12)=0.66664) marking the phase transition from a Bloch- to an Ising-structure of the soliton in an infinite chain.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

Evidence of quasi-intramolecular redox reactions during thermal decomposition of ammonium hydroxodisulfitoferriate(III), (NH4)2[Fe(OH)(SO3)2]·H2O

Synthesis of ammonium hydroxodisulfitoferriate(III), (diammonium catena-{bis(μ ²-sulfito-κO,κO)-μ ²-hydroxo-κ²O}ferrate(III) monohydrate) (NH₄)₂[Fe(OH)(SO₃)₂]·H₂O (compound 1) and its thermal behavior is reported. The compound is stable in air. Its thermal decomposition proceeds without the expected quasi-intramolecular oxidation of sulfite ion with ferric ions. The disproportionation reaction of the ammonium sulfite, formed from the evolved NH₃, SO₂ and H₂O in the main decomposition stage of 1, results in the formation of ammonium sulfate and ammonium sulfide. The ammonium sulfide is unstable at the decomposition temperature of 1 (150 °C) and transforms into NH₃ and H₂S which immediately forms elementary sulfur by reaction with SO₂. The formation and decomposition of other intermediate compounds like (NH₄)₂S_nO_x (n = 2, x = 3 and n = 3, x = 6) results in the same decomposition products (S, SO₂ and NH₃). Two basic iron sulfates, formed in different ratios during synthesizing experiments performed under N₂ or in the presence of air, have been detected as solid intermediates which contain ammonium ions. The final decomposition product was proved to be α-Fe₂O₃ (mineral name hematite).

     

Heat transfer characteristics of boundary-layer flows induced by continuous surfaces stretched with prescribed skin friction

A continuous surface stretched with velocity u _w=u _w (x) and having the temperature distribution T _w=T _w (x) interacts with the viscous fluid in which it is immersed both mechanically and thermally. The thermal interaction is characterized by the surface heat flux q _w=q _w (x) and the mechanical one by the skin friction τ _w=τ _w (x). In the whole previous theoretical research concerned with such processes, either (u _w and T _w) or (u _w and q _w) have been prescribed as known boundary conditions. The goal of the present paper is to initiate the investigation of the boundary layer flows induced by stretching processes for which either (τ _w and T _w ) or (τ _w and q _w) are the prescribed quantities. The case of an isothermal surface stretched with constant skin friction, (τ _w=const., T _w=const. ≠ T _∞) is worked out in detail. The corresponding flow and heat transfer characteristics are compared to those obtained for the (well known) case of a uniformly moving isothermal surface (u _w=const., T _w=const. ≠ T _∞).     

Backward Free Convection Boundary Layers in Porous Media

The well known steady free convection forward boundary layer (FBL) flows ascending over a heated upwards projecting semi-infinite flat plate embedded in a fluid saturated porous medium are compared in this paper to their less well known backward (BBL) counterparts descending over a cooled (also upwards projecting!) semi-infinite flat plate. The circumstance that the definite edge of the plate (x = 0) in the former case is a leading edge and in the latter one a trailing edge, leads to substantially different mathematical and physical features of the FBL and BBL flows, respectively. The paper considers under this aspect the case of similar flows corresponding to surface temperature distributions which are power-law functions of the distance x from the definite edge. For permeable plates the effect of an adequate lateral suction and injection of the fluid is also taken into account. The detailed investigation, however, is restricted to the particular values m = +1 and m = –1/3 of the power-law exponent m, where both FBL and BBL solutions are available in exact analytic form. For each of these values, both exponentially and algebraically decaying BBL solutions were found. In addition, the existence of an exact algebraic BBL solution valid for any value of m is reported.     

Forced Convection Flow of Power-Law Fluids Over a Flat Plate Embedded in a Darcy-Brinkman Porous Medium

The steady forced convection flow of a power-law fluid over a horizontal plate embedded in a saturated Darcy-Brinkman porous medium is considered. The flow is driven by a constant pressure gradient. In addition to the convective inertia, also the “porous Forchheimer inertia” effects are taken into account. The pertinent boundary value problem is investigated analytically, as well as numerically by a finite difference method. It is found that far away from the leading edge, the velocity boundary layer always approaches an asymptotic state with identically vanishing transverse component. This holds for pseudoplastic (0 < n < 1), Newtonian (n = 1), and dilatant (n > 1) fluids as well. The asymptotic solution is given for several particular values of the power-law index n in an exact analytical form. The main flow characteristics of physical and engineering interest are discussed in the paper in some detail.     

Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

The boundary value problem for the similar stream function f  =  f(η;λ) of the Cheng–Minkowycz free convection flow over a vertical plate with a power law temperature distribution T_w(x)  =  T_∞ + Ax^λ in a porous medium is revisited. It is shown that in the λ-range  − 1/2  < λ  <  0 , the well known exponentially decaying “first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities u_w(x)∼ x^λ.