Stability of heterogeneous flows to nonaxisymmetric disturbances

The stability of heterogeneous fluids has been extensively investigated by TAYLOl~ (1931), SYI~OE (1933), MILES (196i), HOW~a~D and GuPTA (1962) and others. These results have also been extended to include the stability Of cylindrical masses of fluid but mostly for axisymmetric disturbances. When the fluid is homogeneous and incompressible and is having a 'solid body' rotation the stability for nonaxisymmetric disturbances has been investigated by HOWARD and GO-eTA (1962), LvI)WlEO (1961) and PEDLEY (1968). In the case of a homogeneous fluid, in addition to the solid body rotation, the presence of even ~ small axial shear makes the system unstable as has been shown by LIIDWlEG (1961) for a narrow gap and by PEDLEu (1968) without this restriction. In this paper we consider the stability for non-axisymmetrical disturbances of a cylindrical mass of heterogeneous fluid, with an exponential variation of density in the radial direction and having both axial and azimuthal velocities. Assuming the dependence of the radial perturbation velocity on r ~o be of the form u = ?l-'~H(r), we discuss two cases m = t/2 and 1. The general stability criteria, for both cases, have been derived. In the second case (m-----1), as an illustration the general stability criterion is applied to the Poiseuille type flow and a bound for instability is obtained. The growth rate of the most rapidly growing disturbances is also determined

A model for manufacture of nano-sized smart materials free from impurities

ReviewThe importance of nano-sized smart materials in structural engineering, biomedical engineering, and in military applications is discussed. It is shown that the solidification of poorly conducting alloys involved in the manufacture of these materials gives rise to surface and convective instabilities. Different types of surface and convective instabilities are briefly discussed. These instabilities produce a mushy zone regarded as dendrites of nano-sized crystals. These dendrites arising from instabilities are regarded as impurities. To manufacture nano-sized smart materials free from impurities, it is essential to control both surface and convective instabilities. We discuss here briefly, different types of convective and surface instabilities in a poorly conducting fluid. We also discuss different mechanisms of control of these instabilities. Different analytical and numerical techniques used to investigate these instabilities under different boundary conditions are discussed. In this review the moment method is explained to find the condition for the onset of convection, and porous lining is used to suppress the growth rate of surface instability. This is useful in the manufacture of nano-sized smart materials free from impurities. Different methods to obtain the required basic equations and the corresponding boundary conditions are briefly discussed.published_or_final_versio

Natural convection through vertical porous stratum

The combined effect of Darcy and viscous resistances on the fully developed natural convection of a fluid between two heated vertical plates is investigated. When Darcy and viscous dissipations, in the energy equation, are negligible the energy and momentum equations become decoupled and we obtain the modified Poiseuille flow distribution through porous media. The deviation of the velocity and temperature distributions from those existing in modified Poiseuille flow are presented for various porous number σ = b k, b is the spacing between the plates and k is the permeability of the porous medium) when dissipations are not neglected. It is shown that the increase in porous number rapidly decreases the influence of both viscous and Darcy dissipations on velocity and temperature profiles. Therefore, by suitable adjustment of dissipation terms, it is possible to control the temperature distribution which will be of some use in plant physiology. © 1977

Temperature distribution in Couette flow past a permeable bed

The temperature distribution in a steady plane Couette flow having one permeable bounding wall is investigated in the presence of buoyancy force N0 when N0>0, it is shown that heat is transported both by convection and diffusion. The effect of convection is to increase the magnitude of the temperature distribution both in the free and Darcy flows. In particular, it is shown that the wall shear has no significant effect on the temperature distribution. The rate of heat transfer between the fluid and the surface is also calculated and it is shown that, it increases with the porous parameter σ. Although the viscous dissipation has very little effect on the temperature distribution yet its effect is significant on heat transfer. © 1977 Indian Academy of Sciences

Effect of buoyancy on the free surface flow past a permeable bed [Auftrieb und Wärmeübertragung an laminar parallel angeströmten Oberflächen poröser Körper]

Die laminare Strömung entlang poröser Grenzflächen wird in Anwesenheit von Auftriebskräften theoretisch untersucht. Die Übereinstimmung zwischen Theorie und Experimenten von Rajasekhara [1] ist dann gut, wenn Strömungsgleitung an der porösen Oberfläche vorausgesetzt wird. Die Auftriebskräfte erhöhen die Geschwindigkeitsverteilung bei Wärmezufuhr (No>0) und verringern sie bei Kühlung (No0). Umgekehrte Verhältnisse liegen für No<0 vor. Insbesondere stellt sich heraus, daß der Wärmeübergang mit steigender Erwärmung der Strömung zunimmt

Frontogenesis in turbulent flow through porous media using non-linear Forchheimer equation

We present here both one- and two-dimensional models for turbulent flow through heterogeneous unbounded fluid saturated porous media using non-linear Forchheimer extended Darcy (DF) equation in the presence of gravitational field. The fluid is initially at rest and sets in motion due to a uniform horizontal density gradient. It is shown that a purely horizontal motion develops satisfying non-linear DF equation. Analytical solutions of this non-linear Initial Value Problem are obtained and limiting solutions valid for the Darcy regime in the case of laminar flow are derived. A measure of the stability of the flow is discussed briefly using Richardson number. The comparison between the nature of the solutions satisfying the non-linear and linear initial value problems are made. We found that even in the case of turbulent flow the vertical density gradient varies continuously both with space z and time t but the horizontal density gradient remains unchanged. The existence and uniqueness theorem of the Initial Value Problem is proved. The stability of these solutions are discussed and it is shown that the solutions are qualitatively and quantitatively different for View the MathML source and View the MathML source in the upper and lower half of the region. In particular, we have shown that the solution which is stable for infinitesimal perturbations is also stable for arbitrary perturbations both in time and space. In the case of two-dimensional motion, a piecewise initial density gradient with continuous distribution of density, stream function formulation is used and the solutions are obtained using time-series analysis. In this case solution shows crowding of the density profiles in the lower-half of the channel reflecting an increase in density gradient and incipient of frontogenesis there, because of the increase in circulation of the flow due to piecewise initial density gradient

Rayleigh Taylor instability in a thin film bounded by a porous layer

The Rayleigh Taylor stability in a finite thickness layer of a viscous fluid bounded below by a densely packed thick porous thick porous layer and above by a rigid surface has been studied subject to linear anal. using approxns. in effect similar to lubrication and Stokes approxns. The problem is studied anal. It is shown that the nature of the linear stability curve is controlled by the slip parameter α and the thickness of the fluid layer h, while its shape is controlled by the ratio of the surface tension to the normal stress

Natural convection past inclined porous layers

Combined Rayleigh-​Benard convection and Tollmien-​Schlichting type of instability of a fluid in an inclined layer bounded by 2 permeable bed was studied. Several types of flow, depending on the value of the Prandtl no., Pr, were studied using a fast-​convergent-​power series technique. Two different convective movements, longitudinal and transverse rolls, based on different Pr, are reported. The effect of slip at the nominal surface is to augment the instability and change the crit. Grashof no., Gr, and crit. Rayleigh no., Ra, markedly for small permeability parameter σ, being independent of Gr and Ra for large σ. The effect of inclination φ is to inhibit the onset of instability in the case of air and to augment it in the case of Hg. At max. inclination (i.e., φ = π​/2)​, the instability sets in as transverse rolls, irresp. of the value of Pr. In the case of Hg, the transverse rolls exist for all φ, whereas in the case of air, they are limited only to certain φ. The cell pattern changes dramatically in the range φ = π​/6 to π​/4

A Weak Nonlinear Stability Analysis of Double Diffusive Convection with Cross-diffusion in a Fluid-saturated Porous Medium

he effect of “Cross Diffusion” on the linear and nonlinear stability of double diffusive convection in a fluid-saturated porous medium has been studied analytically. In the case of linear theory, the normal mode technique has been used and the condition for the maintenance of “finger” and “diffusive” instabilities have been obtained. It has been found that fingers can form by taking cross diffusion terms of appropriate sign and magnitude even though both components make stabilizing contributions to the net vertical density gradient. It has also been shown that “finger” and “diffusive” instabilities can never occur simultaneously. The nonlinear theory is based on the truncated representation of Fourier series and it has been found that the finite amplitude convection may occur when both initial property gradients are stabilizing. Further, the region of finite amplitude instability always encloses the region of infinitesimal oscillatory instability. The effects of permeability and cross-diffusion terms on the heat and mass transports have also been clearly brought out