ABSTRACT
CONTENTS 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
8.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 8.1.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 8.1.3 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
8.2 Influence of Vibration on a Porous Layer Saturated by a Pure Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.2.1 Infinite Porous Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
8.2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.2.1.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.2.1.3 Time-averaged formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8.2.1.4 Scale analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 8.2.1.5 Time-averaged system of equations . . . . . . . . . . . . . . . . . . . 335 8.2.1.6 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 8.2.1.7 Comparison of the two methods. . . . . . . . . . . . . . . . . . . . . . . 345 8.2.1.8 Effect of the direction of vibration . . . . . . . . . . . . . . . . . . . . . 348
8.2.2 Confined Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.2.2.2 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.2.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 8.2.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
8.3 Influence of Vibration on a Porous Layer Saturated by a Binary Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.3.1 Infinite Horizontal Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
8.3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.3.1.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 8.3.1.3 The time-averaged formulation . . . . . . . . . . . . . . . . . . . . . . . . 356 8.3.1.4 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
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8.3.1.5 Limiting case of the long-wave mode . . . . . . . . . . . . . . . . . 360 8.3.2 Confined Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
8.3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 8.3.2.2 Governing equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 8.3.2.3 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 8.3.2.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
8.4 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
8.1.1 Definition
Natural convection is a fluid flowmechanism inwhich the convectivemotion is produced by the density difference in a fluid subjected to a body force. This difference is usually caused by thermal and/or chemical species diffusion. Consequently, to obtain natural convection, two necessary conditions should be satisfied; the existenceof adensityvariationwithin afluidand the existence of a body force. Some common examples of body forces include gravitational, centrifugal, and electromagnetic forces, which may be constant, like gravitational force or may exhibit spatial variation as in centrifugal force. It should be noted that the existence of the body force and the density variation do not guarantee the appearance of convective motion. The relative orientation of the density gradient to the body force provides the sufficient condition for the onset of convection. The possibility of controling the hydrodynamic stability of flows by mod-
ulation has attracted the attention of researchers for many years [1]. Two types of modulations have been extensively studied; the temperature modulation and the gravity modulation. It is shown that by proper selection of the modulation parameters, dramatic modification in the stability behavior of the dynamic system can be observed [2]. In some applications, it may be desirable to operate at Rayleigh num-
bers higher than the critical one at which the convection occurs and yet have no convection. Also it is advantageous to suppress undesired chaotic motions in order to remove temperature oscillations which may exceed safe operational conditions. In the context of the temperature or heat flux modulation in porous media, we may mention the study of Caltagirone [3] and of Rees [4], Rudariah and Malashetty [5] in the Rayleigh-Bénard configuration by temperature modulation and Antohe and Lage [6] in a square cavity heated laterally by flux modulation. Thermo-vibrational convection belongs to a special class of periodic flows in which the buoyancy force is time dependent. In this class, which is different from the problems concerning spatial variations of body forces [7-9], the action of external force field
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(namely, mechanical vibration) in the presence of a nonhomogeneous scalar field (e.g., temperature or concentration) may be used to control the onset of convective motion. Under microgravity conditions, the gravitational force will be reduced drastically and consequently the buoyancy induced convection. However this situation may cause other forces, which under normal conditions are of secondary importance, to become significant. Therefore, residual vibration, which naturally exists in a spacecraft, may
be used to increase the rate of heat or mass transfer. In its simplest form, the imposed vibration can be considered as a harmonic oscillation having zero average over a vibration period. Aswith any subject concerning thermofluid science, the study of the effects of a vibration mechanism on convective motion has been motivated by practical considerations. It is a known fact that, in the presence of gravitational field, the temperature and concentration gradients may produce natural flows. This, in turn, drastically affects material processing; for example, the rate of crystal growth, etc. With the progress of the space industry, there is an opportunity to grow perfect crystals aboard a spacecraft where there exists a highly reduced gravitational environment. Further, it was thought that the unfavorable effects of natural convectionwouldbeeliminated. Therefore,manycrystal growthexperiments were conducted aboard Skylab and theMir space station. However the results were surprisingly much less interesting than expected [10]. It was confirmed experimentally that the space station did not represent an acceleration-free environment; there are transientdisturbancesdue to space stationmaneuvers, impulsive crew movement, and operation of life supporting systems. These residual accelerations are referred to as g-jitter, which can be modeled as harmonic oscillations [11-14]. The theory of thermo-vibrational convection in the fluid medium is summarized in the book written by Gershuni and Lyubimov [15] which reports the Russian studies in this field. In contrast to the thermo-vibrational problem in fluid media, work on the
vibrational problem in porous media is quite recent. We can classify these studies according to geometry, direction of vibration, range of frequency, the number of saturating fluids (mono-component or multi-component), type of boundary conditions, and transport modeling.
