ABSTRACT

CONTENTS 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 9.2 Basic Thermal Energy Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

9.2.1 Darcy Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 9.2.2 Forchheimer Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 9.2.3 Brinkman Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 9.2.4 Order-of-Magnitude Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

9.3 Free Convective Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 9.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 9.3.2 Breaking the Upflow/Downflow Equivalence . . . . . . . . . . . . . . . . . 381 9.3.3 The Asymptotic Dissipation Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 9.3.4 Flow Development Toward the ADP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 9.3.5 Other Free Convective Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

9.4 Forced Convection with Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 9.4.1 Boundary-Layer Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 9.4.2 Channel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

9.5 Mixed Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 9.5.1 The Darcy-Forchheimer Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 9.5.2 Perturbation Approach for Small Gebhart Number . . . . . . . . . . . 396 9.5.3 The Aiding Up-and Downflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.5.4 Channel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

9.6 Stability Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 9.7 Research Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

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The viscous dissipation effect, which is a local production of thermal energy through the mechanism of viscous stresses, is a ubiquitous phenomenon and it is encountered in both the viscous flow of clear fluids and the fluid flow within porousmedia. When comparedwith other thermal influences on fluid motion (i.e., by means of buoyancy forces induced by heated or cooled walls, and by localized heat sources or sinks) the effect of the heat released by viscous dissipation covers a wide range of magnitudes from being negligible to being significant. Gebhart [1] discussed this range at length and stated that “a significant viscous dissipation may occur in natural convection in various devices which are subject to large decelerations or which operate at high rotational speeds. In addition, important viscous dissipation effects may also be present in stronger gravitational fields and in processes wherein the scale of the process is very large, e.g., on larger planets, in large masses of gas in space, and in geological processes in fluids internal to various bodies.” In contrast to such situations, many free convection processes are not sufficiently vigorous to result in a significant quantitative effect, although viscous dissipation sometimes serves to alter the qualitative nature of the flow. Although viscous dissipation is generally regarded as aweak effect, a prop-

erty it shares with relativistic and quantum mechanical effects in everyday life, it too has played a seminal role in history of physics. It was precisely this “weak” physical effect that allowed James Prescott Joule in 1843 to determine the mechanical equivalent of heat using his celebrated paddlewheel experiments, and thereby to set in place one of the most important milestones toward the formulation of the first principle of thermodynamics. Curiously enough, the Royal Society declined to publish Joule’s work in the famous Transactions (the Physical Review Letters of that time) and thus the paper appeared only two years later in amore liberal journal, the Philosophical Magazine. Today, papers on viscous dissipation frequently suffer a similar fate as Joule’s first paper, and it is often neglected. One of the aims of the present review is to assess the quantitative and qualitative changes brought about by the presence of viscous dissipation. From amathematical point of view the effect of viscous dissipation arises as

an additional term in the energy equation. It expresses the rate of volumetric heat generation, q′′′, by internal friction in the presence of a fluid flow. For a plane boundary-layer flow or a unidirectional flow, q′′′ takes the following forms for clear fluids and for Darcy flow through a porous medium,

q′′′clear ≡ µ ( ∂u ∂y

)2 and q′′′Darcy ≡

µ

K u2 (9.1a,b)

respectively, where µ is the dynamic viscosity and K is the permeability. It would appear that the above expression for q′′′Darcy was deduced for the first

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time by Ene and Sanchez-Palencia [2] and Bejan [3] in independent works. Other early applications of this “u2-model” for viscous dissipation in porous media are those of Nakayama and Pop [4], which discusses the external free convection from nonisothermal bodies, and of Ingham et al. [5], which deals with the mixed convection problem between two vertical walls. From a physical point of view, the difference between the two expressions

in Eqs. (9.1a) and (9.1b) originates from the fact that u denotes the actual fluid velocity for a clear fluid flow, but denotes the fluid seepage velocity (i.e., the bulk velocity divided by porosity) for a porous medium flow. At microscopic levels within a porous medium, the fluid is “extruded” through the pores of the solid matrix, and local flows are typically three dimensional even though the overall macroscopic flow is uniform and unidirectional. This microscopic process considerably enhances the rate of heat generation by viscous dissipation. Thus, as can be seen immediately for uniform forced convection flows in clear fluids (u = const. ≡ u∞), no heat is released by viscous dissipation, at least by the agency of internal frictional forces. However, in porous media the heat generation rate increases quadratically with u. In the context of boundary-layer flows it has been shown recently [6] that this fact has important consequences for far-field thermal boundary conditions for both forced and mixed convection in extended porous media. For free convection boundary-layer flows, expressions (9.1a) and (9.1b) are both compatible with the uniform asymptotic condition for the temperature, that is, T(x, y→∞) = const. = T∞. This condition isusually imposedon the temperature field since u→ 0 as y →∞. But in forced and mixed convection flows in extended porous media, this asymptotic thermal condition contradicts the corresponding energy equation because the term q′′′Darcy = (µ/K)u2∞ is nonvanishingasy→∞.Accordingly, somerecent resultspertaining tomixed convection flows in extended porous media [7,8] should be reconsidered (see Magyari et al. [9] and responses by Tashtoush [10] and Nield [11]) by taking into account suitably modified boundary conditions on T in the far field ([6] and Sections 9.4 and 9.5). Even if the quantitative effect of viscous dissipation is negligible in some

cases (see exceptions cited by Gebhart [1], Gebhart and Mollendorf [12], and Nield [13], which include situations where high accelerations exist such as in rapidly rotating systems) its qualitative effect may become significant. One interesting effect of the presence of viscous dissipation, to be discussed in more detail later, is the breaking of both the physical and mathematical equivalence that usually exists between a free convective boundary-layer flow ascending from a hot plate (Tw > T∞) and its counterpart, descending from a cold plate (Tw < T∞). For the latter case the resulting flow is strictly a parallel boundary-layer flow of constant thickness, which has been named the “asymptotic dissipation profile” orADP (seeMagyari and Keller [14] and Section 9.3). A second qualitative difference arises when viscous dissipation is included in a stability analysis of the Darcy-Benard problem — a porous layer heated from below. For a Boussinesq fluid in a Darcian medium with uniform steady temperatures on the boundaries, the basic no-flow state is

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first destabilized by two-dimensional roll patterns. The presence of viscous dissipation causes a hexagonal pattern to appear at Rayleigh numbers close to the critical value (see Rees et al. [15]). This chapter begins with a presentation of the precise mathematical formu-

lae to be used formodeling viscous dissipation, with an emphasis on the very recent debate on the correct form to use when the Brinkman terms are significant in the momentum equations. This is followed by an overview of the current state of the art in free, mixed, and forced convective boundary-layer flows, and some first tentative steps toward the application of stability theory to certain free convective flows.