Linear and Nonlinear Convective Instabilities in Porous Media

  • Biliana Bidin

Student thesis: Doctoral ThesisPhD


After thefirst studies on the onset of convection in porous medium by Horton &Rogers [7] and Lapwood [8], the Horton-Rogers-Lapwood problem, also well-knownas Darcy-Bénard problem, this problem hasattracted much attention from researchers to study the onset of convection inporous medium due to its industrial and environmental applications in fieldssuch as thermal insulation engineering, the growth of crystalline materials,patterned ground formation under water and application to oceanic and planetarymantle. By being derived from these early studies of the Darcy-Bénard problem, this thesis extends those studies in differentdirections and it consists of three main studies on convection in a porousmedium. Thus, it is important in this chapter (from section 1.1 to 1.5) tointroduce, explain and define all the terms and conditions used in studies ofDarcy-Bénard problems to provide betterunderstanding in the subsequent chapters.

            The first study on the onset ofconvection in porous medium is presented in Chapter 2. The given title is “the onset of convection in the unsteady thermalboundary layer above a sinusoidally heated surface embedded in a porous medium.” This study investigates the instability of theunsteady thermal boundary layer which is induced by varying the temperature ofthe horizontal boundary sinusoidally in time about the ambient temperature ofthe porous medium. This study has application in the diurnal heating andcooling from above in subsurface groundwater. The investigation of theoccurrence of the instability is undertaken by finding the criticalDarcy-Rayleigh number that marks the onset of convection. In order to do that,a linear stability analysis is applied by perturbing the basic state using adisturbance with a small amplitude. An unsteady solver is used together withNewton-Raphson iteration to find marginal instability. One finding is that the disturbance has a period which is twice that of the underlying basicstate. Cells which rotate clockwise at first tend to rise upwards from thesurface and weaken, but they induce an anticlockwise cell near the surface atthe end of one forcing period which is otherwise identical to the correspondingclockwise cell found at the start of that forcing period.

            The second study is presented inchapter 3 entitled the linear stability of the unsteady thermalboundary layer in a semi-infinite layered porous medium. The general aim of this study is to examine thestability criteria of two dimensional unsteady thermal boundary layer that isbounded from below by an impermeable surface which is induced by suddenlyraising the temperature of the lower horizontal boundary of the two layerssemi-infinite porous domain. Due to the sudden temperature increase, it issuspected that an evolving thermal boundary layer formed is potentiallyunstable. A full linear stability analysis is performed using thesmall-amplitude disturbance to perturb the basic state of the temperatureprofile and the parabolic equations are solved using the Keller box method tomark the onset of convection. The growth or decay of the disturbances ismonitored by the computation of the thermal energy of the disturbance. Thisstudy results inthe finding of the locus in parameter space where two modes with differentcritical wavenumbers have simultaneous onset, and also find cases where the twominima in the neutral curve and the intermediate maximum merge to form aquartic minimum.

            Still concerning about the layering effect, the thirdsubsequent study is presented in "The effect of layering on unsteadyconduction: an analytical solution method" which is in chapter 4. We considered a semi-infinite soliddomain which exhibits layering and the thermal conductivity and diffusivity ofeach layer is different, therefore the non-dimensional parameters areconductivity ratios and diffusivity ratios. The boundary of that domain issuddenly raised to a new temperature and detailed study is performed to the2-layer and 3-layer system over a wide range of variation of the governingnon-dimensional parameters. The analytical solution of the governing equationis obtained by employing the Laplace transform. It is concluded that thethermal diffusivity ratio and the thermal conductivity ratio are thecoefficients that play the important role in determining the manner in whichconduction occurs. In particular, the thermal diffusivity ratio affects howquickly the temperature field evolves in time.

            The first three studies in this thesis considers the caseof local thermal equilibrium (LTE), therefore we are keen to study the effectof local thermal nonequilibrium (LTNE) to the onset of Darcy-Bénard convection in porousmedium which is presented in the last chapter; chapter 5. This work is theextension of work by Banu and Rees [1] into the weakly nonlinear regime. Theaim of this chapter is achieved by employing a weakly nonlinear analysis todetermine whether the convection pattern immediately post-onset is twodimensional (rolls) or three dimensional (square cells) which will be decidedbased on the coupling coefficients value set at 1. On those occasions where thecoupling coefficient of the amplitude equation is above 1 then roll solutionsare stable i.e two dimensional. Alternatively, if the coupling coefficient isbelow 1, then the roll solutions are unstable and three-dimensional squarecells form the stable pattern. It is found in this study that the roll solutionsare stable.

Date of Award1 May 2019
Awarding Institution
  • University of Bath
SponsorsMinistry of Education Malaysia & Universiti Malaysia Perlis
SupervisorMichael Wilson (Supervisor) & Andrew Rees (Supervisor)

Cite this

Linear and Nonlinear Convective Instabilities in Porous Media
Bidin, B. (Author). 1 May 2019

Student thesis: Doctoral ThesisPhD