Rayleigh's transformation for thermally buoyant flows

 Rayleigh's classical transformation t=x/U connecting an unsteady parallel boundary flow to the steady plane Blasius-flow is generalized to t=C·x ^c and applied to the case of natural convection in the vicinity of a vertical flat plate. The parameters C and c of this “chronotopic transformation” (CTT) are determined self-consistently. It is shown that the analytic expressions obtained in this way for temperature and for the main component of the steady velocity field reproduce the “numerically exact” patterns to a good accuracy. Surprisingly, the transversal component of the steady flow (which in the unsteady flow is entirely missing) can also be generated by CTT with a remarkable precision. Moreover, the CTT is also able to extrapolate unsteady buoyant flows to steady ones with a good performance, even if these belong to a different set of boundary conditions (e.g. time-dependent vs. coordinate-dependent wall temperatures). Received on 6 May 1999     

Thermophoretic deposition of particles in fully developed mixed convection flow in a parallel-plate vertical channel: the full analytical solution

In a recent paper (Grosan et al. in Heat Mass Transf 45:503–509, 2009) a mostly numerical approach to the title problem has been reported. In the present paper the full analytical solution is given. Several new features emerging from this approach are discussed in detail.     

Spontaneous Breakdown of the Definiteness in Some Convective Heat Transfer Problems

In this study, it is shown that above a critical value of a governing parameter, the solutions of some convective heat transfer problems can undergo a bifurcation into a continuum of a non-denumerable infinity of solutions. Thus, the corresponding Nusselt number becomes indeterminate. The origin of this anomalous bifurcation resides in the stability change of the asymptotic state θ(∞) from that of an unstable to that of a stable equilibrium point of the system. As a consequence, the boundary condition θ(∞) = 0 becomes automatically satisfied and thus ineffective in determining the integration constants. Accordingly, the well-posed problem changes spontaneously into an ill-posed one. This remarkable phenomenon will be discussed in detail in the case of an unsteady forced and mixed convection heat transfer problem encountered in an article published recently in Transport in Porous Media. Subsequently, the mentioned loss of definiteness will be explained intuitively with the aid of a simple point-mechanical analogy.     

Forced Convection Heat Transfer from a Heated Cylinder in an Axial Background Flow in a Porous Medium: Three Exactly Solvable Cases

Assuming a background flow of velocity U = U(x) in the axial direction x of a circular cylinder with surface temperature distribution T _w = T _w (x) in a saturated porous medium, for the temperature boundary layer occurring on the cylinder three exactly solvable cases are identified. The functions {U(x), T _w (x)} associated with these cases are given explicitly, and the corresponding exact solutions are expressed in terms of the modified Bessel function K ₀ (z), the incomplete Gamma function Γ (a, z) and the confluent hypergeometric function U(a, b, z), respectively. The correlation between the Nusselt number and the Péclet number as well as the curvature effects on the heat transfer are discussed in all these cases in detail. Some “universal” features of the exponential surface temperature distribution are also pointed out.     

Vortex Instability of the Asymptotic Dissipation Profile in a Porous Medium

In this note we consider the thermoconvective stability of the recently-discovered asymptotic dissipation profile (ADP). The ADP is a uniform thickness, parallel-flow boundary layer which is induced by a cold surface in a warm saturated porous medium in the presence of viscous dissipation. We have considered destabilisation in the form of stream-wise vortex disturbances. The critical wavenumber and Rayleigh number for the onset of convection have been determined for all angles of the cooled surface between the horizontal and the vertical for which the ADP exists. The paper closes with a presentation of some strongly nonlinear computations of steady vortices.     

Wall-Driven Versus Pressure Gradient-Driven Flows in Porous Media

The heat transfer characteristics of two boundary layer flows past an isothermal plane surface adjacent to a saturated Darcy–Brinkman porous medium is compared to each other in this paper. The flows are driven either by a stretching of the adjacent plane boundary, or by an external pressure gradient. It is found that below a threshold value $\tilde{P}r_{*} $ of the modified Prandtl number $\tilde{P}r$ , the Nussselt number in case of the pressure gradient-driven flow is larger than in case of the wall- driven flow, while for $\tilde{P}r>\tilde{P}r_{*} $ the flow driven by the moving wall provides a more efficient heat transfer mechanism. The dependence of $\tilde{P}r_{*} $ on the Darcy number is also discussed in detail.     

Note on ‘A Two-Equation Analysis of Convection Heat Transfer in Porous Media’ by H. Y. Zhang and X. Y. Huang

Transport in Porous Media -     

Buoyant Poiseuille–Couette flow with viscous dissipation in a vertical channel

Steady combined forced and free convection is investigated in a vertical channel having a wall at rest and a moving wall subjected to a prescribed shear stress. The moving wall is thermally insulated, while the wall at rest is kept at a uniform temperature. The analysis deals with the fully–developed parallel flow regime. The governing equations yield a boundary value problem, that is solved analytically by employing a power series expansion of the velocity field with respect to the transverse coordinate. It is shown that the nonlinear interplay between buoyancy and viscous dissipation may determine the existence of dual solutions of the boundary value problem corresponding to fixed values of the applied shear stress on the moving wall and of the hydrodynamic pressure gradient. It is shown that a nontrivial fully separated flow may occur such that the hydrodynamic pressure gradient is zero and the shear stress vanishes on both walls. E. Magyari: On leave from Institute of Building Technology, ETH – Zürich     

The Reynolds analogy for the mixed convection over a vertical surface with prescribed heat flux

The steady mixed convection boundary layer flow over a vertical surface with prescribed heat flux is revisited in this Note. The subset of solutions which can be obtained with the aid of the Reynolds analogy is discussed in a close relationship with the dual solutions reported by Merkin and Mahmood [1] for impermeable, and more recently by Ishak et al. [2], for permeable surfaces.