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.     

Consequences of the Translation Invariance on the Darcy Free Convection Flow Past a Vertical Surface

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 \({\tilde {T}_w \left( x \right)}\), 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.     

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.     

Effect of the source term on steady free convection boundary layer flows over a vertical plate in a porous medium. Part II

The problem of steady free convection boundary layer over a vertical isothermal impermeable flat plate which is embedded in a fluid-saturated porous medium with volumetric heat generation or absorption is studied in this paper using the Darcy equation model. The case of the externally prescribed source terms S = S(x,y) is considered in this paper. It is shown that the corresponding boundary value problem depends on the sign of the plate temperature, which implies that the source term breaks the usual upflow or downflow symmetry of the free convection problem. Looking for similarity solutions, analytical and numerical solutions of the transformed boundary value problem are obtained for several values of the problem parameters. It is also shown that, contrary to the widely spread opinion, the exponential form of the internal heat generation term is not a necessary requirement of similarity reduction.     

Normal mode analysis of the fully developed free convection flow in a vertical slot with open to capped ends

The fully developed free convection flow in a differentially heated vertical slot with open to capped ends investigated recently by Bühler (Heat Mass Transf 39:631–638, 2003) and Weidman (Heat Mass Transf Online First, February 2006) is revisited in this paper. A new method of solution of the corresponding fourth order boundary value problem, based on its reduction to “normal modes” by a complex matrix similarity transformation is presented. As a byproduct of the method, some invariant relationships involving the heat flux and the shear stress in the flow could be found.     

Buoyant MHD flows in a vertical channel: the levitation regime

Buoyant magnetohydrodynamic (MHD) flows with Joulean and viscous heating effects are considered in a vertical parallel plate channel. The applied magnetic field is uniform and perpendicular to the plates which are subject to adiabatic and isothermal boundary conditions, respectively. The main issue of the paper is the levitation regime, i.e., the fully developed flow regime for large values of the Hartmann number M, when the hydrodynamic pressure gradient evaluated at the temperature of the adiabatic wall is vanishing. The problem is solved analytically by Taylor series method and the solution is validated numerically. It is found that the fluid velocity points everywhere and for all values of M downward. For small M’s, the velocity field extends nearly symmetrically (with respect to the mid-plane) over the whole section of the channel between the adiabatic and the isothermal walls. For large values of M, by contrast, the fluid levitates over a broad transversal range of the channel, while the motion becomes concentrated in a narrow boundary layer in the neighborhood of the isothermal wall. Accordingly, the fluid temperature is nearly uniform in the levitation range and decreases rapidly within the boundary layer in front of the isothermal wall. It also turns out that not only the volumetric heat generation by the Joule effect, but also that by viscous friction increases rapidly with increasing values of M, the latter effect being even larger than the former one for all M.     

Transport in diffusion-substitution systems

The class of two-component diffusion-substitution processes including thermodynamic cross-effects is examined analytically. The diffusor approach, a physical decoupling procedure in two effective diffusion processes according to the eigenstates of the diffusivity-matrix is worked out for the linear regime of such systems explicitly. Some peculiarities of these transport processes are discussed in detail. Received on 1 September 1998     

Hamiltonian description of the heat conduction

It is shown that the linear boundary value problems of the heat conduction in a homogeneous slab can be mapped on the initial value problem for a Hamiltonian motion whose phase-space trajectories are subject to an additional restriction, the “arrival condition”. The physical consequences of this formal analogy for the macroscopic heat conduction are discussed in detail. Received on 27 April 1998     

Effect of the Source Term on Steady Free Convection Boundary Layer Flows over an Vertical Plate in a Porous Medium. Part I

The Darcy free convection boundary layer flow over a vertical flat plate is considered in the presence of volumetric heat generation/absorption. In the present first part of the paper it is assumed that the heat generation/absorption takes place in a self-consistent way, the source term q ^′′′≡ S of the energy equation being an analytical function of the local temperature difference T − T _∞. In a forthcoming second part, the case of the externally controlled source terms S = S(x,y ) will be considered. It is shown that due to the presence of S, the physical equivalence of the up- and downflows gets in general broken, in the sense that the free convection flow over the upward projecting hot plate (“upflow”) and over its downward projecting cold counterpart (“downflow”) in general become physically distinct. The consequences of this circumstance are examined for different forms of S. Several analytical solutions are given. Some of them describe algebraically decaying boundary layers which can also be recovered as limiting cases of exponentially decayingones. This asymptotic phenomenon is discussed in some detail.