Cases of mutual compensation of the magnetic and buoyancy forces in mixed convection past a moving vertical surface

It is shown in this Note that under certain conditions, in the hydromagnetic mixed convection flow over a stretching vertical surface, between the magnetic and the buoyancy forces a mutual compensation effect can occur, such that the mixed convection problem reduces to a simple forced convection proble

Falkner-Skan flows past moving boundaries: an exactly solvable case

The Falkner-Skan flows past stretching boundaries are revisited in this paper. The usual assumption U w (x)=λ U(x), i.e. the proportionality of the stretching velocity U w (x) and the free stream velocity U(x) is adopted. For the special case of a converging channel (wedge nozzle), U(x) ~ −1/x, exact analytical solutions in terms of elementary hyperbolic functions are reported. In the range −20) flow regimes were found. In the range λ >1 unique solutions occur, while below the critical value λ c =−2 no solutions exist at all. The mechanical features of these solutions are discussed in some detai

Further Comments on "Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work”

This note is concerned with the assertion of Barletta and Nield (2009a) that "a fluid with a thermal expansion coefficient greater than that of a perfect gas (β >β perfectgas) is of marginal or no interest in the framework of convection in porous media”, and that for a remark of Magyari (Transp. Porous Media, 2009) about the forced convection eigenflow solutions, the circumstance β >β perfectgas does not represent "a sound physical basis”. Here, it is shown, however, that these assertions are in contradiction with the experimentally measured values of β for important technical fluids as e.g., air, nitrogen, carbon dioxide, and ammonia where, in the temperature range between −20 and +100°C, just the inequality β >β perfectgas hold

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 foun

Comment on "Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work” by A. Barletta and D. A. Nield, Transport in Porous Media, DOI 10.1007/s11242-008-9320-y, 2009

In a recent article by Barletta and Nield (Transport in Porous Media, DOI 10.1007/s11242-008-9320-y , 2009), the title problem for the fully developed parallel flow regime was considered assuming isoflux/isothermal wall conditions. For the limiting cases of the forced and the free convection, analytical solutions were reported; for the general case, numerical solutions were reported. The aim of the present note is (i) to give an analytical solution for the full problem in terms of the Weierstrass elliptic P-function, (ii) to illustrate this general approach by two easily manageable examples, and (iii) to rise a couple of questions of basic physical interest concerning the interplay between the viscous dissipation and the pressure work. In this context, the concept of "eigenflow” introduced by Barletta and Nield is discussed in some detai

The entrainment theorem for wall driven boundary layer flows

Boundary layer flows driven by permeable plane surfaces, stretching with power-law velocities are considered in the presence of an applied lateral mass flux. The relationship between the wall shear stress and the entrainment velocity (the transversal velocity at the outer edge of the boundary layer) as a function of the mass transfer parameter f w is examined analytically by using the Merkin transformation method. It is shown that at the value of f w where the wall shear stress vanishes, the entrainment velocity reaches a minimum or maximum value. This relationship between two characteristic quantities at the outer and inner edge of the boundary layer, respectively, is referred to as entrainment theorem. Its physical content is analyzed in the paper in some detai

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 ∞

The Free Convection Boundary-Layer Flow Induced in a Fluid Saturated Porous Medium by a Non-Isothermal Vertical Cylinder Approaches the Shape of Schlichting's Round Jet as the Cylinder Radius Tends to Zero

The title statement is proven for a circular cylinder whose surface temperature (above that of the ambient fluid) varies inversely proportional with the axial distance from the leading edg

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 reporte

Buoyancy Sustained by Viscous Dissipation

The quasi-parallel regime of a Darcy-Boussinesq boundary-layer flow over a permeable vertical flat plate adjacent to a fluid saturated porous medium is considered. ‘Quasi-parallel' means here a plane flow with a constant transversal velocity v=−v 0 directed perpendicularly towards the vertical surface, where a lateral suction with the same velocity −v 0 is applied. The plate is held at a constant temperature T w which coincides with the ambient temperature T ∞ of the fluid. The heat released by viscous dissipation induces a density gradient in the fluid. Thus, although T w=T ∞, a thermal convection occurs. The steady regime of this ‘self-sustaining buoyant flow' has been examined in detail. Wall jet-like profiles with a continuous but finite spectrum of the momentum flow have been found. These self-sustaining buoyant jets show a universal behavior, that is, there exist certain length, velocity and temperature scales such that the flow characteristics become independent of the (constant) material properties of the fluid and the porous medium as wel