International Journal of Non-Linear Mechanics, 1988
The equations for a fluid of third grade derived by Fosdick and Rajagopal are first studied on an... more The equations for a fluid of third grade derived by Fosdick and Rajagopal are first studied on an exterior region in three-dimensional space. A uniqueness theorem and a pointwise continuous dependence theorem (on the initial data) are proved. The conditions at infinity are weak, ...
We develop a detailed linear instability and nonlinear stability analysis for the situation of co... more We develop a detailed linear instability and nonlinear stability analysis for the situation of convection in a horizontal plane layer of fluid when there is a heat sink / source which is linear in the vertical coordinate which is in the opposite direction to gravity. This can give rise to a scenario where the layer effectively splits into three sublayers. In the lowest one the fluid has a tendency to be convectively unstable while in the intermediate layer it will be gravitationally stable. In the top layer there is again the possibility for the layer to be unstable. This results in a problem where convection may initiate in either the lowest layer, the upmost layer, or perhaps in both sublayers simultaneously. In the last case there is the possibility of resonance between the upmost and lowest layers. In all cases penetrative convection may occur where convective movement in one layer induces motion in an adjacent sublayer. In certain cases the critical Rayleigh number for thermal convection may display a very rapid increase which is much greater than normal. Such behaviour may have application in energy research such as in thermal insulation. * Research supported by a "Tipping Points, Mathematics, Metaphors and Meanings" grant of the Leverhulme Trust. I should like to thank an anonymous referee for pointing out some errors and for several helpful remarks.
Journal of Mathematical Analysis and Applications, 2015
Abstract We consider a general model for a non-Newtonian fluid of second grade which is capable o... more Abstract We consider a general model for a non-Newtonian fluid of second grade which is capable of describing shear thinning and shear thickening effects. The model incorporates parameters m and n which are varied to account for which effect, shear thinning or thickening, is desired in the tangential and normal shear rate directions, although here we restrict attention to the shear thickening case. By constructing a generalized energy-like equation and manipulating this we show that the solution may exhibit finite-time blow up if the initial data are regular enough for the solution to exist for a sufficiently long time. The nature of the blow up, or nonexistence, behavior depends critically on the normal stress coefficients α 1 and α 2 , and on the parameters m and n.
ABSTRACT We present a theory for flow through a porous material in which the saturating fluid is ... more ABSTRACT We present a theory for flow through a porous material in which the saturating fluid is such that the porosity is not too large. To achieve this we begin with the classical law of Darcy but argue that the pressure gradient will not have an instantaneous effect on the velocity. Rather, there will be a time delay. In this manner we produce a theory where the wavespeed of a non-linear wave travelling through the porous material depends explicitly on the parameters defining the fluid and the porous skeleton. We believe such a model and the resulting wave propagation analysis will be of interest to practitioners in the sound insulation industry. Exact expressions are provided for wavespeeds and wave amplitudes and these should be directly applicable to situations of immediate interest in soundproofing.
We study a model due to E.F. Keller and L.A. Segel, which describes the movement of a microbiolog... more We study a model due to E.F. Keller and L.A. Segel, which describes the movement of a microbiological population under the influence of chemical substances produced by the biological organisms themselves. There has been an enormous amount of recent activity on Keller-Segel models, especially with regard to blow-up or spike formation of the solution. In this paper, we concentrate on deriving conditions from a fully nonlinear analysis which ensure the solution will decay to a constant steady state. These thresholds are useful in understanding when organisms will not aggregate to form growths.
The problem of thermal convection is investigated when the heat flux is a nonlinear function of t... more The problem of thermal convection is investigated when the heat flux is a nonlinear function of the temperature gradient. A complete analysis of the linear instability problem is given. The nonlinear stability problem is studied in a case which is believed to be physically relevant and the stability threshold is compared directly to that found by linear instability theory.
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2001
A nonlinear stability analysis is performed for the Darcy equations of thermal convection in a fl... more A nonlinear stability analysis is performed for the Darcy equations of thermal convection in a fluid-saturated porous medium when the medium is rotating about an axis orthogonal to the layer in the direction of gravity. A best possible result is established in that we show that the global nonlinear stability Rayleigh number is exactly the same as that for linear instability. It is important to realize that the nonlinear stability boundary holds unconditionally, i.e. for all initial data, and thus for the rotating porous convection problem governed by the Darcy equations, subcritical instabilities are not possible.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010
A. E. Green, FRS and P. M. Naghdi developed a new theory of continuum mechanics based on an entro... more A. E. Green, FRS and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory, they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here develop the theory of Green and Naghdi to be applicable to thermal convection in a fluid in which is suspended a collection of minute metallic-like particles. Thus, we develop a non-Newtonian theory we believe capable of describing a nanofluid. Numerical results are presented for copper oxide or aluminium oxide particles in water or in ethylene glycol. Such combinations are used in real nanofluid suspensions.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2005
A. E. Green, F. R. S. and P. M. Naghdi developed a new theory of continuum mechanics based on an ... more A. E. Green, F. R. S. and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time t =0 and at a later time t = T . Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time. In addition, we derive similar energy bounds for a solution to the Brinkman–Forchheimer equations of viscous flow in porous media.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2006
A model for acoustic waves in a porous medium is investigated. Due to the use of lighter material... more A model for acoustic waves in a porous medium is investigated. Due to the use of lighter materials in modern buildings and noise concerns in the environment, such models for poroacoustic waves are of much interest to the building industry. The model has been investigated in some detail by P. M. Jordan. Here we present a rational continuum thermodynamic derivation of the Jordan model. We then present results for the amplitude of an acceleration wave making no approximations whatsoever.
Mathematical Models and Methods in Applied Sciences, 1991
A theoretical analysis is presented for the formation of polygonal ground which consists of stone... more A theoretical analysis is presented for the formation of polygonal ground which consists of stone borders forming regular hexagons with soil centers. The model is based on the onset of convection in a saturated soil below which is a cold permafrost layer. Darcy’s law is adopted with zero fluid inertia and the density is assumed constant except in the buoyancy term. In this term we take the equation of state proposed by Merker et al.,5 this being the one which assumes the density has a cubic dependence on temperature. Theoretical predictions of the model, particularly those for width to depth of the patterned ground cell are reported in detail.
Journal of Mathematical Analysis and Applications, 2002
We consider problems of the form d 2 u dt 2 + Au = F, αu(0) + u(T) = g, β du dt (0) + du dt (T) =... more We consider problems of the form d 2 u dt 2 + Au = F, αu(0) + u(T) = g, β du dt (0) + du dt (T) = h, for t ∈ (0, T), where A is a densely defined, linear, time independent, positive definite symmetric operator and α and β are constants. Although most of our results would hold for more general operators A, we restrict attention to the case in which A is a differential operator and determine ranges of values of α and β for which it is possible to obtain energy bounds, uniqueness results, and, in a special case, pointwise bounds. Some extensions which include a damping term or a term which arises in a generalization of the Kirchhoff string model are also discussed.
Journal of Geophysical Research: Solid Earth, 1999
When permafrost becomes submerged because of shore line erosion, the covering ocean acts as a the... more When permafrost becomes submerged because of shore line erosion, the covering ocean acts as a thermal insulator, and the submerged permafrost starts to melt. The thawed layer is bounded above by the ocean bed through which salt may intrude and by the phase boundary which for a fixed offshore position is known to progress with the square root of time. This situation gives rise to nonsteady double‐diffusion coupling Bénard convection and liquefaction which can be described by the Darcy‐Oberbeck‐Boussinesq equations. The boundary value problem is formulated, and scalings are introduced which orient themselves on the relative magnitudes of phase boundary and convective bulk velocities of the salt convective regime identified by Harrison [1982]. The multiscale perturbation analysis that is introduced not only verifies the observed thaw rates with a parabolic‐in‐time phase boundary retreat, it equally automatically generates the equations for the corresponding perturbative equations such ...
This paper numerically investigates the instability of Poiseuille flow in a fluid overlying a por... more 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, which in turn overlies a layer of Darcy-type porous material. It is shown that there are two modes of instability corresponding to the fluid and porous layers, respectively. The key parameters which affect the stability characteristics of the system are the depth ratio between the porous and fluid layers and the transition layer depth.
Journal of Computational and Applied Mathematics, 2006
A Legendre polynomial-based spectral technique is developed to be applicable to solving eigenvalu... more 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 corresponding generalised eigenvalue problem are sparse, reducing the computational and storage costs, where the superimposition of boundary conditions is not needed due to the structure of the method. Several eigenvalue problems are solved using both the Legendre polynomial-based and Chebyshev tau techniques. In each example, the Legendre polynomial-based spectral technique converges to the required accuracy utilising less polynomials than the Chebyshev tau method, and with much greater computational efficiency.
ABSTRACT We focus on the problem of a saline solution saturating an infinite horizontal layer of ... more ABSTRACT We focus on the problem of a saline solution saturating an infinite horizontal layer of porous material. The layer is heated from below which is an effect tending to destabilize and cause overturning motion due to thermal convection. The layer is simultaneously arranged such that the salt gradient is linear in the vertical direction and the salt concentration is greatest at the bottom of the layer. The latter effect will tend to oppose thermal convection and prevent convective overturning. These two competing effects pose a challenging nonlinear stability problem. A new way of approaching this via a change of variable in an energy method is discussed and the coincidence of the critical linear and nonlinear stability parameters is proved.
Journal de Mathématiques Pures et Appliquées, 1998
This paper deals with three fundamental modelling questions for the Darcy and Btinkman equations ... more This paper deals with three fundamental modelling questions for the Darcy and Btinkman equations for flow in a porous medium. It is first shown that in the Dirichlet initial-boundary value problem for the Brinkman equations the solution depends continuously on the viscous coefficient. Then L* convergence of the solution of this problem to the solution of an analogous problem for the Darcy equations is established. Finally, it is proved that for flow in a domain occupied by a viscous fluid in contact with a porous solid, the solution depends continuously on a coefficient in the interface boundary condition. The continuous dependence holds for Stokes flow in the fluid, and the analogous Navier-Stokes situation is discussed.
International Journal of Non-Linear Mechanics, 1988
The equations for a fluid of third grade derived by Fosdick and Rajagopal are first studied on an... more The equations for a fluid of third grade derived by Fosdick and Rajagopal are first studied on an exterior region in three-dimensional space. A uniqueness theorem and a pointwise continuous dependence theorem (on the initial data) are proved. The conditions at infinity are weak, ...
We develop a detailed linear instability and nonlinear stability analysis for the situation of co... more We develop a detailed linear instability and nonlinear stability analysis for the situation of convection in a horizontal plane layer of fluid when there is a heat sink / source which is linear in the vertical coordinate which is in the opposite direction to gravity. This can give rise to a scenario where the layer effectively splits into three sublayers. In the lowest one the fluid has a tendency to be convectively unstable while in the intermediate layer it will be gravitationally stable. In the top layer there is again the possibility for the layer to be unstable. This results in a problem where convection may initiate in either the lowest layer, the upmost layer, or perhaps in both sublayers simultaneously. In the last case there is the possibility of resonance between the upmost and lowest layers. In all cases penetrative convection may occur where convective movement in one layer induces motion in an adjacent sublayer. In certain cases the critical Rayleigh number for thermal convection may display a very rapid increase which is much greater than normal. Such behaviour may have application in energy research such as in thermal insulation. * Research supported by a "Tipping Points, Mathematics, Metaphors and Meanings" grant of the Leverhulme Trust. I should like to thank an anonymous referee for pointing out some errors and for several helpful remarks.
Journal of Mathematical Analysis and Applications, 2015
Abstract We consider a general model for a non-Newtonian fluid of second grade which is capable o... more Abstract We consider a general model for a non-Newtonian fluid of second grade which is capable of describing shear thinning and shear thickening effects. The model incorporates parameters m and n which are varied to account for which effect, shear thinning or thickening, is desired in the tangential and normal shear rate directions, although here we restrict attention to the shear thickening case. By constructing a generalized energy-like equation and manipulating this we show that the solution may exhibit finite-time blow up if the initial data are regular enough for the solution to exist for a sufficiently long time. The nature of the blow up, or nonexistence, behavior depends critically on the normal stress coefficients α 1 and α 2 , and on the parameters m and n.
ABSTRACT We present a theory for flow through a porous material in which the saturating fluid is ... more ABSTRACT We present a theory for flow through a porous material in which the saturating fluid is such that the porosity is not too large. To achieve this we begin with the classical law of Darcy but argue that the pressure gradient will not have an instantaneous effect on the velocity. Rather, there will be a time delay. In this manner we produce a theory where the wavespeed of a non-linear wave travelling through the porous material depends explicitly on the parameters defining the fluid and the porous skeleton. We believe such a model and the resulting wave propagation analysis will be of interest to practitioners in the sound insulation industry. Exact expressions are provided for wavespeeds and wave amplitudes and these should be directly applicable to situations of immediate interest in soundproofing.
We study a model due to E.F. Keller and L.A. Segel, which describes the movement of a microbiolog... more We study a model due to E.F. Keller and L.A. Segel, which describes the movement of a microbiological population under the influence of chemical substances produced by the biological organisms themselves. There has been an enormous amount of recent activity on Keller-Segel models, especially with regard to blow-up or spike formation of the solution. In this paper, we concentrate on deriving conditions from a fully nonlinear analysis which ensure the solution will decay to a constant steady state. These thresholds are useful in understanding when organisms will not aggregate to form growths.
The problem of thermal convection is investigated when the heat flux is a nonlinear function of t... more The problem of thermal convection is investigated when the heat flux is a nonlinear function of the temperature gradient. A complete analysis of the linear instability problem is given. The nonlinear stability problem is studied in a case which is believed to be physically relevant and the stability threshold is compared directly to that found by linear instability theory.
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2001
A nonlinear stability analysis is performed for the Darcy equations of thermal convection in a fl... more A nonlinear stability analysis is performed for the Darcy equations of thermal convection in a fluid-saturated porous medium when the medium is rotating about an axis orthogonal to the layer in the direction of gravity. A best possible result is established in that we show that the global nonlinear stability Rayleigh number is exactly the same as that for linear instability. It is important to realize that the nonlinear stability boundary holds unconditionally, i.e. for all initial data, and thus for the rotating porous convection problem governed by the Darcy equations, subcritical instabilities are not possible.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010
A. E. Green, FRS and P. M. Naghdi developed a new theory of continuum mechanics based on an entro... more A. E. Green, FRS and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory, they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here develop the theory of Green and Naghdi to be applicable to thermal convection in a fluid in which is suspended a collection of minute metallic-like particles. Thus, we develop a non-Newtonian theory we believe capable of describing a nanofluid. Numerical results are presented for copper oxide or aluminium oxide particles in water or in ethylene glycol. Such combinations are used in real nanofluid suspensions.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2005
A. E. Green, F. R. S. and P. M. Naghdi developed a new theory of continuum mechanics based on an ... more A. E. Green, F. R. S. and P. M. Naghdi developed a new theory of continuum mechanics based on an entropy identity rather than an entropy inequality. In particular, within the framework of this theory they developed a new set of equations to describe viscous flow. The new theory additionally involves vorticity and spin of vorticity. We here derive energy bounds for a class of problem in which the ‘initial data’ are given as a combination of data at time t =0 and at a later time t = T . Such problems are in vogue in the mathematical literature and may be used, for example, to give estimates of solution behaviour in an improperly posed problem where one wishes to continue a solution backward in time. In addition, we derive similar energy bounds for a solution to the Brinkman–Forchheimer equations of viscous flow in porous media.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2006
A model for acoustic waves in a porous medium is investigated. Due to the use of lighter material... more A model for acoustic waves in a porous medium is investigated. Due to the use of lighter materials in modern buildings and noise concerns in the environment, such models for poroacoustic waves are of much interest to the building industry. The model has been investigated in some detail by P. M. Jordan. Here we present a rational continuum thermodynamic derivation of the Jordan model. We then present results for the amplitude of an acceleration wave making no approximations whatsoever.
Mathematical Models and Methods in Applied Sciences, 1991
A theoretical analysis is presented for the formation of polygonal ground which consists of stone... more A theoretical analysis is presented for the formation of polygonal ground which consists of stone borders forming regular hexagons with soil centers. The model is based on the onset of convection in a saturated soil below which is a cold permafrost layer. Darcy’s law is adopted with zero fluid inertia and the density is assumed constant except in the buoyancy term. In this term we take the equation of state proposed by Merker et al.,5 this being the one which assumes the density has a cubic dependence on temperature. Theoretical predictions of the model, particularly those for width to depth of the patterned ground cell are reported in detail.
Journal of Mathematical Analysis and Applications, 2002
We consider problems of the form d 2 u dt 2 + Au = F, αu(0) + u(T) = g, β du dt (0) + du dt (T) =... more We consider problems of the form d 2 u dt 2 + Au = F, αu(0) + u(T) = g, β du dt (0) + du dt (T) = h, for t ∈ (0, T), where A is a densely defined, linear, time independent, positive definite symmetric operator and α and β are constants. Although most of our results would hold for more general operators A, we restrict attention to the case in which A is a differential operator and determine ranges of values of α and β for which it is possible to obtain energy bounds, uniqueness results, and, in a special case, pointwise bounds. Some extensions which include a damping term or a term which arises in a generalization of the Kirchhoff string model are also discussed.
Journal of Geophysical Research: Solid Earth, 1999
When permafrost becomes submerged because of shore line erosion, the covering ocean acts as a the... more When permafrost becomes submerged because of shore line erosion, the covering ocean acts as a thermal insulator, and the submerged permafrost starts to melt. The thawed layer is bounded above by the ocean bed through which salt may intrude and by the phase boundary which for a fixed offshore position is known to progress with the square root of time. This situation gives rise to nonsteady double‐diffusion coupling Bénard convection and liquefaction which can be described by the Darcy‐Oberbeck‐Boussinesq equations. The boundary value problem is formulated, and scalings are introduced which orient themselves on the relative magnitudes of phase boundary and convective bulk velocities of the salt convective regime identified by Harrison [1982]. The multiscale perturbation analysis that is introduced not only verifies the observed thaw rates with a parabolic‐in‐time phase boundary retreat, it equally automatically generates the equations for the corresponding perturbative equations such ...
This paper numerically investigates the instability of Poiseuille flow in a fluid overlying a por... more 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, which in turn overlies a layer of Darcy-type porous material. It is shown that there are two modes of instability corresponding to the fluid and porous layers, respectively. The key parameters which affect the stability characteristics of the system are the depth ratio between the porous and fluid layers and the transition layer depth.
Journal of Computational and Applied Mathematics, 2006
A Legendre polynomial-based spectral technique is developed to be applicable to solving eigenvalu... more 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 corresponding generalised eigenvalue problem are sparse, reducing the computational and storage costs, where the superimposition of boundary conditions is not needed due to the structure of the method. Several eigenvalue problems are solved using both the Legendre polynomial-based and Chebyshev tau techniques. In each example, the Legendre polynomial-based spectral technique converges to the required accuracy utilising less polynomials than the Chebyshev tau method, and with much greater computational efficiency.
ABSTRACT We focus on the problem of a saline solution saturating an infinite horizontal layer of ... more ABSTRACT We focus on the problem of a saline solution saturating an infinite horizontal layer of porous material. The layer is heated from below which is an effect tending to destabilize and cause overturning motion due to thermal convection. The layer is simultaneously arranged such that the salt gradient is linear in the vertical direction and the salt concentration is greatest at the bottom of the layer. The latter effect will tend to oppose thermal convection and prevent convective overturning. These two competing effects pose a challenging nonlinear stability problem. A new way of approaching this via a change of variable in an energy method is discussed and the coincidence of the critical linear and nonlinear stability parameters is proved.
Journal de Mathématiques Pures et Appliquées, 1998
This paper deals with three fundamental modelling questions for the Darcy and Btinkman equations ... more This paper deals with three fundamental modelling questions for the Darcy and Btinkman equations for flow in a porous medium. It is first shown that in the Dirichlet initial-boundary value problem for the Brinkman equations the solution depends continuously on the viscous coefficient. Then L* convergence of the solution of this problem to the solution of an analogous problem for the Darcy equations is established. Finally, it is proved that for flow in a domain occupied by a viscous fluid in contact with a porous solid, the solution depends continuously on a coefficient in the interface boundary condition. The continuous dependence holds for Stokes flow in the fluid, and the analogous Navier-Stokes situation is discussed.
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