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The nonlinear stability analysis of double-diffusive convection with viscous dissipation effect (2023)
Journal Article
Deepika, N., Narayana, P. A. L., & Hill, A. (2023). The nonlinear stability analysis of double-diffusive convection with viscous dissipation effect. Transport in Porous Media, 150, 215–227. https://doi.org/10.1007/s11242-023-02006-3

In this article, the onset of double-diffusive convection with the effect of viscous dissipation in a horizontal fluid-saturated porous layer is examined. Two impermeable isothermal and isosolutal walls bound the porous layer, and Darcy’s law models... Read More about The nonlinear stability analysis of double-diffusive convection with viscous dissipation effect.

Thermo-convective carbon sequestration in horizontal porous layers (2019)
Journal Article
Gautam, K., Narayana, P., & Hill, A. (2019). Thermo-convective carbon sequestration in horizontal porous layers. IMA Journal of Applied Mathematics, 84(3), 650-668

This article explores the linear and nonlinear stability of double-diffusive density-driven convection in the context of carbon sequestration in deep saline aquifers. The anisotropy, due to the thermal and solutal diffusivities, is considered in a po... Read More about Thermo-convective carbon sequestration in horizontal porous layers.

Double-diffusive convection in an inclined porous layer with a concentration-based internal heat source (2017)
Journal Article
Matta, A., & Hill, A. A. (2018). Double-diffusive convection in an inclined porous layer with a concentration-based internal heat source. Continuum Mechanics and Thermodynamics, 30(1), 165-173. https://doi.org/10.1007/s00161-017-0592-8

© 2017, The Author(s). The thermosolutal instability of double-diffusive convection in an inclined fluid-saturated porous layer with a concentration-based internal heat source is investigated. The linear instability of small-amplitude perturbations t... Read More about Double-diffusive convection in an inclined porous layer with a concentration-based internal heat source.

Onset of Darcy-Brinkman convection with a uniform internal heat source and vertical throughflow (2017)
Journal Article
Deepika, N., Narayana, P., & Hill, A. A. (2017). Onset of Darcy-Brinkman convection with a uniform internal heat source and vertical throughflow. International Journal of Thermal Sciences, 117, 136-144

The problem of thermal convection in a horizontal fluid saturated porous layer is examined, where the flow is governed by the Brinkman extension of Darcy's law. A uniform internal heat source and vertical throughflow are also considered. The linear a... Read More about Onset of Darcy-Brinkman convection with a uniform internal heat source and vertical throughflow.

Double-diffusive Hadley–Prats flow in a horizontal porous layer with a concentration based internal heat source (2017)
Journal Article
Matta, A., Narayana, P. A., & Hill, A. A. (2017). Double-diffusive Hadley–Prats flow in a horizontal porous layer with a concentration based internal heat source. Journal of Mathematical Analysis and Applications, 452(2), 1005-1018. https://doi.org/10.1016/j.jmaa.2017.03.039

© 2017 Elsevier Inc. Double-diffusive Hadley–Prats flow with a concentration based heat source is investigated through linear and non-linear stability analyses. The resultant eigenvalue problems for both theories are solved numerically using Shooting... Read More about Double-diffusive Hadley–Prats flow in a horizontal porous layer with a concentration based internal heat source.

Biogeochemistry in the scales (2017)
Book Chapter
Bonnett, S., Maxfield, P., Hill, A. A., & Ellwood, M. D. F. (2017). Biogeochemistry in the scales. In J. N. Furze, K. Swing, A. K. Gupta, R. McClatchey, & D. Reynolds (Eds.), Mathematical Advances Towards Sustainable Environmental Systems. Springer International Publishing. https://doi.org/10.1007/978-3-319-43901-3

Global environmental change is challenging our understanding of how communities as a whole interact with their physical environment. Ideally, we would model the impacts of global environmental change at a global level. However, in order to mathematic... Read More about Biogeochemistry in the scales.

An MHD couple stress fluid due to a perforated sheet undergoing linear stretching with heat transfer (2016)
Journal Article
Hill, A. A., Mahabaleshwar, U. S., Sarris, I. E., Hill, A. A., Lorenzini, G., & Pop, I. (2017). An MHD couple stress fluid due to a perforated sheet undergoing linear stretching with heat transfer. International Journal of Heat and Mass Transfer, 105, 157-167. https://doi.org/10.1016/j.ijheatmasstransfer.2016.09.040

© 2016 Elsevier Ltd We investigate an MHD couple stress liquid due to a perforated sheet undergoing linear stretching with radiation. The liquid is initially at rest with its activity is restricted by pulling the two sheet ends with parallel and iden... Read More about An MHD couple stress fluid due to a perforated sheet undergoing linear stretching with heat transfer.

Nonlinear thermal instability in a horizontal porous layer with an internal heat source and mass flow (2016)
Journal Article
Hill, A. A., Matta, A., Narayana, P. A., & Hill, A. A. (2016). Nonlinear thermal instability in a horizontal porous layer with an internal heat source and mass flow. Acta Mechanica, 227(6), 1743-1751. https://doi.org/10.1007/s00707-016-1591-8

© 2016, Springer-Verlag Wien. Linear and nonlinear stability analyses of Hadley–Prats flow in a horizontal fluid-saturated porous medium with a heat source are performed. The results indicate that, in the linear case, an increase in the horizontal th... Read More about Nonlinear thermal instability in a horizontal porous layer with an internal heat source and mass flow.

On the stability and uniqueness of the flow of a fluid through a porous medium (2016)
Journal Article
Hill, A. A., Hill, A. A., Rajagopal, K. R., & Vergori, L. (2016). On the stability and uniqueness of the flow of a fluid through a porous medium. Zeitschrift für Angewandte Mathematik und Physik, 67(3), 49. https://doi.org/10.1007/s00033-016-0645-z

© 2016, The Author(s). In this short note, we study the stability of flows of a fluid through porous media that satisfies a generalization of Brinkman’s equation to include inertial effects. Such flows could have relevance to enhanced oil recovery an... Read More about On the stability and uniqueness of the flow of a fluid through a porous medium.

Convective stability of carbon sequestration in anisotropic porous media (2014)
Journal Article
Hill, A. A., & Morad, M. R. (2014). Convective stability of carbon sequestration in anisotropic porous media. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470(2170), https://doi.org/10.1098/rspa.2014.0373

© 2014 The Author(s) Published by the Royal Society. All rights reserved. The stability of convection in an anisotropic porous medium, where the solute concentration is assumed to decay via a first-order chemical reaction, is studied. This is a simpl... Read More about Convective stability of carbon sequestration in anisotropic porous media.

Simulation of three dimensional double-diffusive throughflow in internally heated anisotropic porous media (2014)
Journal Article
Harfash, A. J., & Hill, A. A. (2014). Simulation of three dimensional double-diffusive throughflow in internally heated anisotropic porous media. International Journal of Heat and Mass Transfer, 72, 609-615. https://doi.org/10.1016/j.ijheatmasstransfer.2014.01.048

A model for double-diffusive convection in an anisotropic porous layer with a constant throughflow is explored, with penetrative convection being simulated via an internal heat source. The validity of both the linear instability and global nonlinear... Read More about Simulation of three dimensional double-diffusive throughflow in internally heated anisotropic porous media.

Stabilising solar ponds by utilising porous materials (2013)
Journal Article
Hill, A. A., Hill, A. A., & Carr, M. (2013). Stabilising solar ponds by utilising porous materials. Advances in Water Resources, 60, 1-6. https://doi.org/10.1016/j.advwatres.2013.07.005

The linear instability of a solar pond containing porous material in the lower convective zone (LCZ) is investigated. It is found that, in general, for physically realistic values, solar ponds that contain porous material are more stable than ones th... Read More about Stabilising solar ponds by utilising porous materials.

The influence of a fluid-porous interface on solar pond stability (2013)
Journal Article
Hill, A. A., Hill, A. A., & Carr, M. (2013). The influence of a fluid-porous interface on solar pond stability. Advances in Water Resources, 52, 1-6. https://doi.org/10.1016/j.advwatres.2012.08.009

The linear instability of the gradient zone of a solar pond containing a fluid-porous interface is investigated. It is found that the gradient zone can retain the same stability for lower values of the solute Rayleigh number with the introduction of... Read More about The influence of a fluid-porous interface on solar pond stability.

Double diffusive convection in a viscoelastic fluid-saturated porous layer using a thermal non-equilibrium model (2012)
Journal Article
Malashetty, M. S., Hill, A. A., & Swamy, M. (2012). Double diffusive convection in a viscoelastic fluid-saturated porous layer using a thermal non-equilibrium model. Acta Mechanica, 223(5), 967-983. https://doi.org/10.1007/s00707-012-0616-1

The stability of a binary viscoelastic fluid-saturated porous layer which is heated from below is studied, where the fluid and solid phases are not in local thermal equilibrium. The modified Darcy-Oldroyd model is employed as a momentum equation, wit... Read More about Double diffusive convection in a viscoelastic fluid-saturated porous layer using a thermal non-equilibrium model.

Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer (2012)
Journal Article
Wicks, T. J., & Hill, A. A. (2012). Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer. Continuum Mechanics and Thermodynamics, 24(3), 229-237. https://doi.org/10.1007/s00161-012-0234-0

Linear and nonlinear stability analyses were performed on a fluid layer with a concentration-based internal heat source. Clear bimodal behaviour in the neutral curve (with stationary and oscillatory modes) is observed in the region of the onset of os... Read More about Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer.

An operative method to obtain sharp nonlinear stability for systems with spatially dependent coefficients (2012)
Journal Article
Hill, A. A., & Malashetty, M. S. (2012). An operative method to obtain sharp nonlinear stability for systems with spatially dependent coefficients. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468(2138), 323-336. https://doi.org/10.1098/rspa.2011.0137

This paper explores an operative technique for deriving nonlinear stability by studying double-diffusive porous convection with a concentration-based internal heat source. Previous stability analyses on this problem have yielded regions of potential... Read More about An operative method to obtain sharp nonlinear stability for systems with spatially dependent coefficients.

Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer (2012)
Journal Article
Wicks, T. J., & Hill, A. A. (2012). Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer. Continuum Mechanics and Thermodynamics, 24(3), 229-237. https://doi.org/10.1007/s00161-012-0234-0

Linear and nonlinear stability analyses were performed on a fluid layer with a concentration-based internal heat source. Clear bimodal behaviour in the neutral curve (with stationary and oscillatory modes) is observed in the region of the onset of os... Read More about Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer.

Double-diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow (2011)
Journal Article
Capone, F., Gentile, M., & Hill, A. A. (2011). Double-diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow. International Journal of Heat and Mass Transfer, 54(7-8), 1622-1626. https://doi.org/10.1016/j.ijheatmasstransfer.2010.11.020

A model for double-diffusive convection in an anisotropic porous layer with a constant throughflow is explored, with penetrative convection being simulated via an internal heat source. Both linear instability and nonlinear stability analyses are perf... Read More about Double-diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow.

Penetrative convection in anisotropic porous media with variable permeability (2011)
Journal Article
Capone, F., Gentile, M., & Hill, A. A. (2011). Penetrative convection in anisotropic porous media with variable permeability. Acta Mechanica, 216(1-4), 49-58. https://doi.org/10.1007/s00707-010-0353-2

The onset of penetrative convection in an anisotropic porous medium, for fluids with quadratic density law, is studied. In particular, the effects of anisotropic permeability and thermal diffusivity have been taken into account. © 2010 Springer-Verla... Read More about Penetrative convection in anisotropic porous media with variable permeability.

Nonlinear stability of the one-domain approach to modelling convection in superposed fluid and porous layers (2010)
Journal Article
Hill, A. A., & Carr, M. (2010). Nonlinear stability of the one-domain approach to modelling convection in superposed fluid and porous layers. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2121), 2695-2705. https://doi.org/10.1098/rspa.2010.0014

Studies of the nonlinear stability of fluid/porous systems have been developed very recently. A two-domain modelling approach has been adopted in previous works, but was restricted to specific configurations. The extension to the more general case of... Read More about Nonlinear stability of the one-domain approach to modelling convection in superposed fluid and porous layers.

Penetrative convection via internal heating in anisotropic porous media (2010)
Journal Article
Capone, F., Gentile, M., & Hill, A. A. (2010). Penetrative convection via internal heating in anisotropic porous media. Mechanics Research Communications, 37(5), 441-444. https://doi.org/10.1016/j.mechrescom.2010.06.005

Penetrative convection in an anisotropic porous medium is simulated via internal heating. We consider both internal heat sources and sinks, where the porous medium has constant anisotropic thermal diffusivity and non-homogeneous permeability. Both li... Read More about Penetrative convection via internal heating in anisotropic porous media.

Sharp global nonlinear stability for a fluid overlying a highly porous material (2010)
Journal Article
Hill, A. A., & Carr, M. (2010). Sharp global nonlinear stability for a fluid overlying a highly porous material. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2113), 127-140. https://doi.org/10.1098/rspa.2009.0322

The stability of convection in a two-layer system in which a layer of fluid with a temperature-dependent viscosity overlies and saturates a highly porous material is studied. Owing to the difficulties associated with incorporating the nonlinear advec... Read More about Sharp global nonlinear stability for a fluid overlying a highly porous material.

Poiseuille flow in a fluid overlying a highly porous material (2009)
Journal Article
Hill, A. A., & Straughan, B. (2009). Poiseuille flow in a fluid overlying a highly porous material. Advances in Water Resources, 32(11), 1609-1614. https://doi.org/10.1016/j.advwatres.2009.08.007

This paper investigates the instability of Poiseuille flow in a fluid overlying a highly porous material. A two layer approach is adopted, where the Darcy-Brinkman equation is employed to describe the fluid flow in the porous medium, with a tangentia... Read More about Poiseuille flow in a fluid overlying a highly porous material.

Instability of Poiseuille flow in a fluid overlying a glass bead packed porous layer (2009)
Journal Article
Hill, A. A. (2009). Instability of Poiseuille flow in a fluid overlying a glass bead packed porous layer. Acta Mechanica, 206(1-2), 95-103. https://doi.org/10.1007/s00707-008-0099-2

The instability of Poiseuille flow in a fluid overlying a glass bead packed porous layer saturated with the same fluid is studied. A three layer configuration is adopted, where a variable effective viscosity is modelled between the outer fluid and po... Read More about Instability of Poiseuille flow in a fluid overlying a glass bead packed porous layer.

A differential constraint approach to obtain global stability for radiation-induced double-diffusive convection in a porous medium (2009)
Journal Article
Hill, A. A. (2009). A differential constraint approach to obtain global stability for radiation-induced double-diffusive convection in a porous medium. Mathematical Methods in the Applied Sciences, 32(8), 914-921. https://doi.org/10.1002/mma.1073

The stability of double-diffusive porous convection with a concentration-based internal heat source is studied. Owing to the significant sensitivity of standard energy method, a highly desirable reduction in the number of required coupling parameters... Read More about A differential constraint approach to obtain global stability for radiation-induced double-diffusive convection in a porous medium.

Anisotropy and symmetry in porous media convection (2009)
Journal Article
Capone, F., Gentile, M., & Hill, A. A. (2009). Anisotropy and symmetry in porous media convection. Acta Mechanica, 208(3-4), 205-214. https://doi.org/10.1007/s00707-008-0135-2

The linear and nonlinear stability analysis of the motionless state (conduction solution) and of a vertical throughflow in an anisotropic porous medium is performed. In particular, the effect of a nonhomogeneous porosity and a constant anisotropic th... Read More about Anisotropy and symmetry in porous media convection.

Global stability for thermal convection in a fluid overlying a highly porous material (2009)
Journal Article
Hill, A. A., & Straughan, B. (2009). Global stability for thermal convection in a fluid overlying a highly porous material. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465(2101), 207-217. https://doi.org/10.1098/rspa.2008.0303

This paper investigates the instability thresholds and global nonlinear stability bounds for thermal convection in a fluid overlying a highly porous material. A two-layer approach is adopted, where the Darcy-Brinkman equation is employed to describe... Read More about Global stability for thermal convection in a fluid overlying a highly porous material.

Penetrative convection in a fluid layer with throughflow (2008)
Journal Article
Capone, F., Gentile, M., & Hill, A. A. (2008). Penetrative convection in a fluid layer with throughflow. Ricerche di matematica, 57(2), 251-260. https://doi.org/10.1007/s11587-008-0035-8

Linear and nonlinear stability analyses of vertical throughflow in a fluid layer, where the density is quadratic in temperature, are studied. To avoid the loss of key terms a weighted functional is used in the energy analysis. Both conditional and un... Read More about Penetrative convection in a fluid layer with throughflow.

Global stability for penetrative double-diffusive convection in a porous medium (2008)
Journal Article
Hill, A. A. (2008). Global stability for penetrative double-diffusive convection in a porous medium. Acta Mechanica, 200(1-2), 1-10. https://doi.org/10.1007/s00707-007-0575-0

Linear and nonlinear stability analyses of penetrative double-diffusive convection in a porous medium are performed. Adopting a standard energy method approach yields a nonlinear threshold which is independent of the salt field. An adaptation of a ne... Read More about Global stability for penetrative double-diffusive convection in a porous medium.

Poiseuille flow in a fluid overlying a porous medium (2008)
Journal Article
Hill, A. A., & Straughan, B. (2008). Poiseuille flow in a fluid overlying a porous medium. Journal of Fluid Mechanics, 603, 137-149. https://doi.org/10.1017/S0022112008000852

This paper numerically investigates the instability of Poiseuille flow in a fluid overlying a porous medium saturated with the same fluid. A three-layer configuration is adopted. Namely, a Newtonian fluid overlying a Brinkman porous transition layer,... Read More about Poiseuille flow in a fluid overlying a porous medium.

Unconditional nonlinear stability for convection in a porous medium with vertical throughflow (2007)
Journal Article
Hill, A. A. (2007). Unconditional nonlinear stability for convection in a porous medium with vertical throughflow. Acta Mechanica, 193(3-4), 197-206. https://doi.org/10.1007/s00707-007-0473-5

Linear and nonlinear stability analyses of vertical throughflow in a fluid saturated porous layer, which is modelled using a cubic Forchheimer model, are studied. To ensure unconditional nonlinear results are obtainable, and to avoid the loss of key... Read More about Unconditional nonlinear stability for convection in a porous medium with vertical throughflow.

Global stability for penetrative convection with throughflow in a porous material (2007)
Journal Article
Hill, A. A., Rionero, S., & Straughan, B. (2007). Global stability for penetrative convection with throughflow in a porous material. IMA Journal of Applied Mathematics, 72(5), 635-643. https://doi.org/10.1093/imamat/hxm036

This paper investigates penetrative convection in a layer of porous material saturated with water when there is throughflow present. The density is quadratic in temperature. A linearized instability analysis is derived and compared with a weighted no... Read More about Global stability for penetrative convection with throughflow in a porous material.

Linear and non-linear stability thresholds for thermal convection in a box (2006)
Journal Article
Hill, A. A., & Straughan, B. (2006). Linear and non-linear stability thresholds for thermal convection in a box. Mathematical Methods in the Applied Sciences, 29(17), 2123-2132. https://doi.org/10.1002/mma.770

We analyse the problem of finding instability thresholds and global non-linear stability bounds for thermal convection in a linearly viscous fluid in a finite box. The vertical walls are maintained at different temperatures which gives rise to a non-... Read More about Linear and non-linear stability thresholds for thermal convection in a box.

A Legendre spectral element method for eigenvalues in hydrodynamic stability (2006)
Journal Article
Hill, A. A., & Straughan, B. (2006). A Legendre spectral element method for eigenvalues in hydrodynamic stability. Journal of Computational and Applied Mathematics, 193(1), 363-381. https://doi.org/10.1016/j.cam.2005.06.011

A Legendre polynomial-based spectral technique is developed to be applicable to solving eigenvalue problems which arise in linear and nonlinear stability questions in porous media, and other areas of Continuum Mechanics. The matrices produced in the... Read More about A Legendre spectral element method for eigenvalues in hydrodynamic stability.

Double-diffusive convection in a porous medium with a concentration based internal heat source (2005)
Journal Article
Hill, A. A. (2005). Double-diffusive convection in a porous medium with a concentration based internal heat source. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2054), 561-574. https://doi.org/10.1098/rspa.2004.1328

Linear and nonlinear stability analyses of double-diffusive convection in a fluid-saturated porous layer with a concentration based internal heat source are studied. Darcy's law and the Boussinesq approximation are employed, with the equation of stat... Read More about Double-diffusive convection in a porous medium with a concentration based internal heat source.

Penetrative convection induced by the absorption of radiation with a nonlinear internal heat source (2004)
Journal Article
Hill, A. A. (2004). Penetrative convection induced by the absorption of radiation with a nonlinear internal heat source. Dynamics of Atmospheres and Oceans, 38(1), 57-67. https://doi.org/10.1016/j.dynatmoce.2004.03.002

A model for convection induced by the selective absorption of radiation in a fluid layer is investigated. The concentration based internal heat source is modelled quadratically. A linear analysis and conditional nonlinear analysis are performed. Due... Read More about Penetrative convection induced by the absorption of radiation with a nonlinear internal heat source.

Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium (2004)
Journal Article
Hill, A. A. (2004). Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium. Continuum Mechanics and Thermodynamics, 16(4), 305-318. https://doi.org/10.1007/s00161-003-0151-3

Convection induced by the selective absorption of radiation is investigated, for the case of an internal heat source that is modelled quadratically with respect to concentration. The growth rate for the linearised system is shown to be real, and a li... Read More about Conditional and unconditional nonlinear stability for convection induced by absorption of radiation in a porous medium.

Convection induced by the selective absorption of radiation for the Brinkman model (2004)
Journal Article
Hill, A. A. (2004). Convection induced by the selective absorption of radiation for the Brinkman model. Continuum Mechanics and Thermodynamics, 16(1-2), 43-52. https://doi.org/10.1007/s00161-003-0140-6

Convection induced by the selective absorption of radiation in a porous medium is studied analytically and numerically using the Brinkman model. Both linear instability analysis and nonlinear stability analysis are employed. The thresholds show excel... Read More about Convection induced by the selective absorption of radiation for the Brinkman model.