STABILITY OF POISEUILLE FLOW IN A FLUID OVERLYING POROUS DOMAIN

Abstract

The widespread utilization of the geological and industrial systems wherein a porous domain underlies a fluid domain urges one to study the instability mechanism of the flow through such superposed systems. For example, the underground oil reservoirs mimic a system of fluid overlying porous domain where the oil flows over the rocky, silty and sedimented bottom. To extract the oil through an outlet, pressure is to be applied through an inlet, which can be considered as an application of Poiseuille flow [1]. The stability analysis of such superposed systems plays a key role in understanding the appropriate pressure gradient required for extraction. Apart from this, there are several scenarios: e.g. contaminant flow beneath the Earth’s surface [42], solidification of alloys [22], chemical vapour deposition [41] and many more [32, 58, 70], where one encounters similar type of problems. The thesis reports the linear and weakly nonlinear stability analysis of Poiseuille flow in a horizontal channel consisting of a fluid saturated porous domain that underlies a fluid domain. The flow of the incompressible Newtonian fluid is regulated by Darcy’s law in the porous domain and by Navier-Stokes equation in the fluid domain. A permeable interface separating the fluid and porous domains allows the seepage of fluid from the fluid to porous domain. The flow in the system is induced by an external pressure gradient whereas buoyancy is introduced by the maintenance of constant temperature TU and TL (TL > TU) between top surface of the fluid domain and the bottom surface of the porous domain in case of non-isothermal flows. Both scenarios of isotropic and homogeneous as well as anisotropic and inhomogeneous porous media are considered in the thesis. The Chebyshev spectral collocation method is employed to solve the linearized perturbed system of equations of the considered problems. The wave speed (wavenumber) is considered as a complex (real) number, thus, temporal stability analysis is performed in the thesis. The weakly nonlinear stability analysis of isothermal Poiseuille flow in a fluid overlying porous domain is proposed and investigated on the basis of pioneering works of Stuart [100], Stewartson [97] and Rogers et al. [86]. The nonlinear interactions are studied by imposing finite amplitude disturbances to the Poiseuille flow in such superposed systems. The former interactions in terms of modal amplitudes dictate the fundamental mode, the distorted mean flow, the second harmonic and the distorted fundamental mode. The harmonics are solved progressively in increasing order of the least stable mode obtained from the linear theory to ascertain the cubic Landau equation, which in turn helps to determine the bifurcation phenomena. This thesis is split into six chapters, each of which is briefly described below. Chapter 1 consists of the basic geometrical framework in such superposed fluid-porous problems, some relevant preliminary definitions and concepts, a brief narrative of the hydrodynamic stability theory, the state of art of research conducted in the direction of linear and nonlinear stability analysis of buoyancy-induced as well as pressure driven flows in fluid overlying porous domain and finally the motivation and objective behind the present thesis. Chapter 2 addresses the non-isothermal Poiseuille flow in an anisotropic and inhomogeneous porous domain underlying a fluid domain. The impact of media anisotropy and inhomogeneity is inspected by virtue of linear stability analysis along with other governing parameters like depth ratio (ratio of depth of fluid domain to porous domain, dˆ), Darcy number (δ), Reynolds number (Re) and Prandtl number (Pr) concerning the stability of the fluid-porous system. The neutral curves are found to be bimodal and unimodal in nature with the anisotropy and inhomogeneity leaving its imprint on parametric variation. An increase in anisotropy or decrease in inhomogeneity parameter follows the modal change from unimodal (porous) to bimodal (both porous and fluid). Also, it has been identified that, irrespective of the considered variations in anisotropy and inhomogeneity, the least stable mode for dˆ< 0.05 is porous and for dˆ> 0.16 is fluid. Furthermore, energy budget analysis is carried out to classify the type of instability and validate the type of mode. The instability is found to be thermal-buoyant in nature with omission of low Reynolds number along with very low values of the ratio of permeability in horizontal to vertical direction, where thermal-shear instability is witnessed. Likewise, secondary flow patterns in the context of streamfunction and temperature contour are analysed to validate the least stable mode and the type of prevailing instability in the fluid-porous system. Additionally, the stability analysis in case of highly porous media, i.e., when the porous domain is modelled by Brinkman equations is also analysed in Appendix E of Chapter 2. A comparison between the results obtained by the two models is also made. Chapter 3 investigates the onset of thermosolutal convection for a pressure driven flow in superposed fluid-porous systems. This Chapter is an extension of Chapter 2 by introducing the salinity gradient to the thermal Poiseuille flow. A higher temperature was preserved at the bottom plate of the porous domain and consequently, the salinity followed the other way round. The analysis is once again consummated by means of linear stability theory and the influence of major controlling parameters, dˆ, Re, Lewis number (Le), thermal Rayleigh number in porous domain (Ram) and δ on the system’s stability is extensively discussed along the lines of neutral stability curves as well as instability boundaries. It is observed that the neutral curves exhibit unimodal, bimodal and trimodal characteristics with the dominant mode as porous ubiquitously. Along the instability boundary, it is observed that, for Ram > 1, with increasing dˆ, the critical value of solutal Rayleigh number (Rac sm) first decreases and then increases before finally saturating to a constant value, whereas with increasing values of Le, Rasm increases continuously till Le < 60. On the other hand, a linear-type characteristic is seen between Rasm and Ram for varying Re. An insight into the energy budget spectrum to understand the underlying instability mechanism revealed the prevailing type of instability as solutal-buoyant. Further, in general, it is noticed that, whenever the type of mode is porous (fluid), the solutal-buoyant forces are balanced by the dissipation due to surface drag (combination of viscous dissipation, inter-facial stresses and thermal-buoyant forces). The onset of instability in the present study is achieved very much earlier (Rasm ∼ 28) in contrast to existing results on natural convection (Rasm ∼ 83). Chapter 4 proposes and examines the weakly nonlinear stability analysis of isothermal Poiseuille flow in a fluid overlying porous domain. The nonlinear interactions are studied by imposing finite amplitude disturbances to the classical model deliberated in Chang et al. (J. Fluid Mech., vol. 564, 2006, 287-303). The order parameter theory is used to ascertain the cubic Landau equation and the regimes of instability for the bifurcations are determined henceforth. The well-established controlling parameters viz. the depth ratio (dˆ= depth of fluid domain/depth of porous domain), Beavers-Joseph constant (α) and δ are inquired upon for the bifurcation phenomena. The imposed finite amplitude disturbances are viewed for bifurcations along the neutral stability curves and away from the critical point as a function of the wavenumber (a) and Reynolds number (Re). The even-fluid-layer (porous) mode along the neutral stability curves correlates to the subcritical (supercritical) bifurcation phenomena. On perceiving the bifurcations as a function of a and Re by moving away from the bifurcation/critical point, subcritical bifurcation is observed for increasing dˆ,α and decreasing δ. In contrast to only fluid flow through a channel, it is found that the inclusion of porous domain aids in the early appearance of subcritical bifurcation when α = 0.2,dˆ= 0.13,δ = 0.003. A considerable difference between the computed skin friction coefficient for the base and the distorted state is observed for small (large) values of dˆ (α). An extension of Chapter 4 to anisotropic and inhomogeneous porous domain gives way to Chapter 5. In this Chapter, the presented weakly nonlinear theory once again predicts the existence of subcritical transition to turbulence of Poiseuille flow an anisotropic and inhomogeneous porous domain that underlies a fluid domain. In general, on moving away from the bifurcation point, it is found that a decrease in the value of inhomogeneity (in terms of Ai) favours subcritical bifurcation. For the considered variation of parameters, the bifurcation either subcritical or supercritical remains same irrespective of the value of media anisotropy (ξ ) in the vicinity of the bifurcation point except for dˆ= 0.2,Ai = 1. In such a situation, subcritical (supercritical) bifurcation is witnessed for ξ = 0.001,0.01,0.1 (1, 3). A correspondence between type of mode via linear theory and the type of bifurcation via nonlinear theory is witnessed, which is further affirmed by the secondary flow patterns. Eventually, the weakly nonlinear stability analysis helps to bridge the gap between the onset of instability and transition to turbulence. Finally, Chapter 6 presents summary and closing remarks of the thesis and some potential directions for further research.