John Norbury - Academia.edu (original) (raw)

Papers by John Norbury

Journal of Marine Systems, 2015

The analysis of trophically complex mathematical ecosystem models is typically carried out using ... more The analysis of trophically complex mathematical ecosystem models is typically carried out using numerical techniques because it is considered that the number and nonlinear nature of the equations involved makes progress using analytic techniques virtually impossible. Exploiting the properties of systems that are written in Kolmogorov form, the conservative normal (CN) framework articulates a number of ecological axioms that govern ecosystems. Previous work has shown that trophically simple models developed within the CN framework are mathematically tractable, simplifying analysis. By exploiting the properties of Kolmogorov ecological systems it is possible to design particular properties, such as the property that all populations remain extant, into an ecological model. Here we demonstrate the usefulness of these results to construct a trophically complex ecosystem model. We also show that the properties of Kolmogorov ecological systems can be exploited to provide a computationally efficient method for the refinement of model parameters which can be used to precondition parameter values used in standard optimisation techniques, such as genetic algorithms, to significantly improve convergence towards a target equilibrium state.

The Quarterly Journal of Mechanics and Applied Mathematics, 1982

Steady plane flows of an ideal incompressible fluid are considered. Each flow possesses a vortici... more Steady plane flows of an ideal incompressible fluid are considered. Each flow possesses a vorticity function which is constant along the streamlines (lines on which the streamfunction is constant) of the flow. The qualitative dependence of the flow speed on variations of the bounding streamline is discussed using maximum principles for elliptic differential equations. The main theorems, involving the comparison

Mathematika, 1978

... (See Zeidler 1974 and Benjamin 1962 for further references.) With respect to the results of K... more ... (See Zeidler 1974 and Benjamin 1962 for further references.) With respect to the results of Keady and Norbury (1975) concerning Benjamin and Lighthill's conject-ures for flows without vorticity, this paper establishes that those results do indeed remain true in ideal fluid flows ...

Mathematical Proceedings of the Cambridge Philosophical Society, 1978

ABSTRACT This paper concerns steady plane periodic waves on the surface of an ideal liquid flowin... more ABSTRACT This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

Mathematical Proceedings of the Cambridge Philosophical Society, 1978

ABSTRACT This paper concerns steady plane periodic waves on the surface of an ideal liquid flowin... more ABSTRACT This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

1] Dimethylsulphide (DMS) is produced by upper ocean ecosystems and emitted to the atmosphere, wh... more 1] Dimethylsulphide (DMS) is produced by upper ocean ecosystems and emitted to the atmosphere, where it may have an important role in climate regulation. Several attempts to quantify the role of DMS in climate change have been undertaken in modeling studies. We examine a model of biogenic DMS production and describe its endogenous dynamics and sensitivities. We extend the model to develop a one-dimensional version that more accurately resolves the important processes of the mixed layer in determining the ecosystem dynamics. Comparisons of the results of the one-dimensional model with an empirical relationship that describes the global distribution of DMS, and also with vertical profiles of DMS in the upper ocean measured at the Bermuda Atlantic Time Series, suggest that the model represents the interaction between the biological and physical processes well on local and global scales. Our analysis of the model confirms its veracity and provides insights into the important processes determining DMS concentration in the oceans.

Mathematics in Industry, 2010

We show the importance of the error function in the approximation of the solution of singularly p... more We show the importance of the error function in the approximation of the solution of singularly perturbed convection-diffusion problems with discontinuous boundary conditions. It is observed that the error function (or a combination of them) provides an excellent approximation and reproduces accurately the effect of the discontinuities on the behaviour of the solution at the boundary and interior layers.

Physica D: Nonlinear Phenomena, 1999

Invasive cells variously show changes in adhesion, protease production and motility. In this pape... more Invasive cells variously show changes in adhesion, protease production and motility. In this paper the authors develop and analyse a model for malignant invasion, brought about by a combination of proteolysis and haptotaxis. A common feature of these two mechanisms is that they can be produced by contact with the extracellular matrix through the mediation of a class of surface receptors called integrins. An unusual feature of the model is the absence of cell diffusion. By seeking travelling wave solutions the model is reduced to a system of ordinary differential equations which can be studied using phase plane analysis. The authors demonstrate the presence of a singular barrier in the phase plane and a "hole" in this singular barrier which admits a phase trajectory. The model admits a family of travelling waves which depend on two parameters, i.e. the tissue concentration of connective tissue and the rate of decay of the initial spatial profile of the invading cells. The slowest member of this family corresponds to the phase trajectory which goes through the "hole" in the singular barrier. Using a power series method the authors derive an expression relating the minimum wavespeed to the tissue concentration of the extracellular matrix which is arbitrary. The model is applicable in a wide variety of biological settings which combine haptotaxis with proteolysis. By considering various functional forms the authors show that the key mathematical features of the particular model studied in the early parts of the paper are exhibited by a wider class of models which characterise the behaviour of invading cells.

Flow of charge within an ion channel is considered. Continuum models which de-scribe the distribu... more Flow of charge within an ion channel is considered. Continuum models which de-scribe the distribution of mobile ionic charge are presented with dimensional analysis used to derive simplified solutions. The question of accounting for the transition from regions where continuum models are adequate to regions where individual molecules must be considered is also discussed.

Abstract-Convection-induced instability in reactiondiffusion systems produces complicated pattern... more Abstract-Convection-induced instability in reactiondiffusion systems produces complicated patterns of oscillations behind propagating wavefronts. We transform the system twice: into lambda-omega form, then into polar variables. We find analytical estimates for the wavefront speed which we confirm numerically. Our previous work examined a simpler system [E. H. Flach, S. Schnell, and J. Norbury, Phys. Rev. E 76, 036216 ]; the onset of instability is qualitatively different in numerical solutions of this system. We modify our estimates and connect the two different behaviours. Our estimate explains how the Turing instability fits with pattern found in reactiondiffusion-convection systems. Our results can have important applications to the pattern formation analysis of biological systems.

SIAM Journal on Applied Mathematics, 2000

ABSTRACT Members of a family of traveling wave solutions for a simple two-variable model of malig... more ABSTRACT Members of a family of traveling wave solutions for a simple two-variable model of malignant invasion driven by haptotaxis are shown to have shocks (which satisfy an entropy condition) and to be computationally stable. By seeking traveling wave solutions, a PDE model, equivalent to three first-order PDEs defined on the real space and time axes, may be reduced to an ordinary differential equation system, which can be studied in a two-dimensional phase plane. This phase plane is shown to contain a singular barrier and a ”hole” in this barrier which admits two singular phase trajectories. Thus orbits are able to smoothly cross this barrier along either of the two singular trajectories. Once this has occurred, the barrier seems to prevent any heteroclinic steady-state connections. However, the authors show that shocks admitted by the PDEs allow the orbit to jump back over the singular barrier. This process leads to a previously unnoticed family of solutions for malignant invasion which have slower wavespeeds than the family of smooth traveling wave solutions (extending over the whole real axis) admitted by the model. The existence of a minimum-speed wave with semi-infinite support is demonstrated. The practical stability of these new shock solutions is confirmed via numerical solutions of the model. The minimum speed solution is found to evolve from specific nonsmooth initial data, whereas smooth initial data extending over the whole real axis evolves to faster waves.

SIAM Journal on Applied Mathematics, 2009

We wish to predict ionic currents that flow through narrow protein channels of biological membran... more We wish to predict ionic currents that flow through narrow protein channels of biological membranes in response to applied potential and concentration differences across the channel, when some features of channel structure are known. We propose to apply singular perturbation analysis to the coupled Poisson-Nernst-Planck equations, which are the basic continuum model of ionic permeation and semiconductor physics. In semiconductor physics the problem is a singular perturbation, because the ratio of the Debye length to the width of the channel is a very small parameter that multiplies the Laplacian term in the Poisson equation. In contrast to semiconductors, the atomic scale geometry of narrow ion channels sometimes makes this ratio a large parameter, which, surprisingly, renders the problem a singular perturbation in a different sense. We construct boundary layers and match them asymptotically across the different regions of the channel to derive good approximations for Fick's and Ohm's laws. Our aim is to extend the asymptotic analysis to a class of nonlinear problems hitherto intractable. Analytical and numerical results for the mass flux and the electric current serve as a tool for molecular biophysicists and physiologists to understand, study and control protein channels, thereby aiding clinical and technological applications.

SIAM Journal on Applied Mathematics, 1994

A coupled system of partial differential equations that describes the evolution of solid and gas ... more A coupled system of partial differential equations that describes the evolution of solid and gas temperatures and gas concentration inside a catalytic converter is derived and examined. The use of a reaction term, which is a discontinuous function of the solid ...

SIAM Journal on Applied Mathematics, 1991

ABSTRACT Atmospheric or oceanic flows strongly constrained by rotation and stratification can be ... more ABSTRACT Atmospheric or oceanic flows strongly constrained by rotation and stratification can be described by a set of Lagrangian partial differential equations called the semigeostrophic equations. In these equations the trajectories must be determined implicitly. Generalized solutions of these equations are defined as a sequence of rearrangements of the fluid, which need not be smooth. These solutions are closely related to generalized solutions of the Monge-Ampere equation. Existence and uniqueness of such solutions is proved. The evolution is shown to be a sequence of minimum-energy states of the fluid, giving strong physical plausibility to the solutions.

Quarterly Journal of the Royal Meteorological Society, 2007

ABSTRACT Discussion of the proper boundary conditions to use in limited area forecasting models r... more ABSTRACT Discussion of the proper boundary conditions to use in limited area forecasting models requires knowledge of the properties of the governing equations. the theory of Oliger and Sundstrom states that the primitive hydrostatic equations commonly used are not hyperbolic and no local boundary conditions can be chosen. This paper shows their result to be incorrect, the equations are hyperbolic but not all the characteristics involve the time variable. Implications for the boundary conditions are discussed.

The Quarterly Journal of Mechanics and Applied Mathematics, 2005

A novel hybrid finite-element/finite-volume numerical method is developed to determine the capill... more A novel hybrid finite-element/finite-volume numerical method is developed to determine the capillary rise of a liquid with a free surface (under surface tension and gravitational forces). The few known exact analytical solutions are used to verify the numerical computations and establish their accuracy for a range of liquid contact angles. The numerical method is then used to ascertain the limitations of a number of theoretical approximations to solutions for the capillary rise in the linearized limit, for special geometries such as plane walls, concentric cylinders and in a wedge of arbitrary included angle. The existence of a critical wedge angle for a given contact angle is verified. However, the effect of slight practical rounding of wedge corners dramatically reduces the theoretical corner height.

The Quarterly Journal of Mechanics and Applied Mathematics, 2005

The upper free surface z = u(x, y) of a static fluid with gravity acting in the z direction, occu... more The upper free surface z = u(x, y) of a static fluid with gravity acting in the z direction, occupying a volume V , satisfies the Laplace-Young equation. The fluid wets the vertical boundaries of V so that the usual capillary contact conditions hold. This paper considers wedge shaped volumes V with corner angle 2α, that belong to the intermediate corner angle case of π/2 − γ < α < π/2 where γ is the contact angle, and determines explicitly, a regular power series expansion for the height u(r, θ) of the fluid near the corner, r = 0, to all orders in r. shows that it is possible to have logarithmic terms for a general corner expansion of the Laplace-Young equation, with appropriate boundary conditions. However, we suggest that the usual practical cases do not possess any singular terms near the corner, and we analytically and explicitly produce a non-singular series to any order in r, and propose that near the corner the far field effects are lost through any "interior or inner flat" region in exponentially small terms. We give computational solutions for these regular (energy minimising) cases based on a numerical finite volume method on an unstructured mesh, which fully support our assertions and our analytical series results, including the (minor) influence of the far field on local corner behaviour.

Proceedings of the Royal Society B: Biological Sciences, 1998

Cells use a combination of changes in adhesion, proteolysis and motility (directed and random) du... more Cells use a combination of changes in adhesion, proteolysis and motility (directed and random) during the process of migration. Proteolysis of the extracellular matrix (ECM) results in the creation of haptotactic gradients, which cells use to move in a directed fashion. The proteolytic creation of these gradients also results in the production of digested fragments of ECM. In this study

Nonlinearity, 2001

A haptotaxis-dominated model of cell invasion is considered for small cell diffusion and fast pro... more A haptotaxis-dominated model of cell invasion is considered for small cell diffusion and fast protease adjustment to the cell-collagen matrix interaction. A simplified limit model has travelling wave cell invasion profiles that are blunt, that is end with a 'shock-like' step, and that evolve stably from initial data that lie to one side of some initial plane. In common with diffusiondominated systems, the travelling wave which evolves from such initial data has the minimum wavespeed permissible in the model. This minimum wavespeed is not, however, determined by the local stability of the steady states in the travelling wave phase plane, but by a novel combination of singular behaviour within the phase plane and hyperbolic shock conditions. It is shown that more accurate models including the detailed fast dynamics of the protease require small amounts of diffusion (of the same order as the fast dynamics timescale) in order to remain stable. However, small diffusion and fast protease adjustment then give physically relevant and interesting solutions that evolve from semicompact initial data and stably invade at speeds well predicted by the simple model.

Journal of Marine Systems, 2015

The analysis of trophically complex mathematical ecosystem models is typically carried out using ... more The analysis of trophically complex mathematical ecosystem models is typically carried out using numerical techniques because it is considered that the number and nonlinear nature of the equations involved makes progress using analytic techniques virtually impossible. Exploiting the properties of systems that are written in Kolmogorov form, the conservative normal (CN) framework articulates a number of ecological axioms that govern ecosystems. Previous work has shown that trophically simple models developed within the CN framework are mathematically tractable, simplifying analysis. By exploiting the properties of Kolmogorov ecological systems it is possible to design particular properties, such as the property that all populations remain extant, into an ecological model. Here we demonstrate the usefulness of these results to construct a trophically complex ecosystem model. We also show that the properties of Kolmogorov ecological systems can be exploited to provide a computationally efficient method for the refinement of model parameters which can be used to precondition parameter values used in standard optimisation techniques, such as genetic algorithms, to significantly improve convergence towards a target equilibrium state.

The Quarterly Journal of Mechanics and Applied Mathematics, 1982

Steady plane flows of an ideal incompressible fluid are considered. Each flow possesses a vortici... more Steady plane flows of an ideal incompressible fluid are considered. Each flow possesses a vorticity function which is constant along the streamlines (lines on which the streamfunction is constant) of the flow. The qualitative dependence of the flow speed on variations of the bounding streamline is discussed using maximum principles for elliptic differential equations. The main theorems, involving the comparison

Mathematika, 1978

... (See Zeidler 1974 and Benjamin 1962 for further references.) With respect to the results of K... more ... (See Zeidler 1974 and Benjamin 1962 for further references.) With respect to the results of Keady and Norbury (1975) concerning Benjamin and Lighthill&amp;amp;#x27;s conject-ures for flows without vorticity, this paper establishes that those results do indeed remain true in ideal fluid flows ...

Mathematical Proceedings of the Cambridge Philosophical Society, 1978

ABSTRACT This paper concerns steady plane periodic waves on the surface of an ideal liquid flowin... more ABSTRACT This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

Mathematical Proceedings of the Cambridge Philosophical Society, 1978

ABSTRACT This paper concerns steady plane periodic waves on the surface of an ideal liquid flowin... more ABSTRACT This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

1] Dimethylsulphide (DMS) is produced by upper ocean ecosystems and emitted to the atmosphere, wh... more 1] Dimethylsulphide (DMS) is produced by upper ocean ecosystems and emitted to the atmosphere, where it may have an important role in climate regulation. Several attempts to quantify the role of DMS in climate change have been undertaken in modeling studies. We examine a model of biogenic DMS production and describe its endogenous dynamics and sensitivities. We extend the model to develop a one-dimensional version that more accurately resolves the important processes of the mixed layer in determining the ecosystem dynamics. Comparisons of the results of the one-dimensional model with an empirical relationship that describes the global distribution of DMS, and also with vertical profiles of DMS in the upper ocean measured at the Bermuda Atlantic Time Series, suggest that the model represents the interaction between the biological and physical processes well on local and global scales. Our analysis of the model confirms its veracity and provides insights into the important processes determining DMS concentration in the oceans.

Mathematics in Industry, 2010

We show the importance of the error function in the approximation of the solution of singularly p... more We show the importance of the error function in the approximation of the solution of singularly perturbed convection-diffusion problems with discontinuous boundary conditions. It is observed that the error function (or a combination of them) provides an excellent approximation and reproduces accurately the effect of the discontinuities on the behaviour of the solution at the boundary and interior layers.

Physica D: Nonlinear Phenomena, 1999

Invasive cells variously show changes in adhesion, protease production and motility. In this pape... more Invasive cells variously show changes in adhesion, protease production and motility. In this paper the authors develop and analyse a model for malignant invasion, brought about by a combination of proteolysis and haptotaxis. A common feature of these two mechanisms is that they can be produced by contact with the extracellular matrix through the mediation of a class of surface receptors called integrins. An unusual feature of the model is the absence of cell diffusion. By seeking travelling wave solutions the model is reduced to a system of ordinary differential equations which can be studied using phase plane analysis. The authors demonstrate the presence of a singular barrier in the phase plane and a "hole" in this singular barrier which admits a phase trajectory. The model admits a family of travelling waves which depend on two parameters, i.e. the tissue concentration of connective tissue and the rate of decay of the initial spatial profile of the invading cells. The slowest member of this family corresponds to the phase trajectory which goes through the "hole" in the singular barrier. Using a power series method the authors derive an expression relating the minimum wavespeed to the tissue concentration of the extracellular matrix which is arbitrary. The model is applicable in a wide variety of biological settings which combine haptotaxis with proteolysis. By considering various functional forms the authors show that the key mathematical features of the particular model studied in the early parts of the paper are exhibited by a wider class of models which characterise the behaviour of invading cells.

Flow of charge within an ion channel is considered. Continuum models which de-scribe the distribu... more Flow of charge within an ion channel is considered. Continuum models which de-scribe the distribution of mobile ionic charge are presented with dimensional analysis used to derive simplified solutions. The question of accounting for the transition from regions where continuum models are adequate to regions where individual molecules must be considered is also discussed.

Abstract-Convection-induced instability in reactiondiffusion systems produces complicated pattern... more Abstract-Convection-induced instability in reactiondiffusion systems produces complicated patterns of oscillations behind propagating wavefronts. We transform the system twice: into lambda-omega form, then into polar variables. We find analytical estimates for the wavefront speed which we confirm numerically. Our previous work examined a simpler system [E. H. Flach, S. Schnell, and J. Norbury, Phys. Rev. E 76, 036216 ]; the onset of instability is qualitatively different in numerical solutions of this system. We modify our estimates and connect the two different behaviours. Our estimate explains how the Turing instability fits with pattern found in reactiondiffusion-convection systems. Our results can have important applications to the pattern formation analysis of biological systems.

SIAM Journal on Applied Mathematics, 2000

ABSTRACT Members of a family of traveling wave solutions for a simple two-variable model of malig... more ABSTRACT Members of a family of traveling wave solutions for a simple two-variable model of malignant invasion driven by haptotaxis are shown to have shocks (which satisfy an entropy condition) and to be computationally stable. By seeking traveling wave solutions, a PDE model, equivalent to three first-order PDEs defined on the real space and time axes, may be reduced to an ordinary differential equation system, which can be studied in a two-dimensional phase plane. This phase plane is shown to contain a singular barrier and a ”hole” in this barrier which admits two singular phase trajectories. Thus orbits are able to smoothly cross this barrier along either of the two singular trajectories. Once this has occurred, the barrier seems to prevent any heteroclinic steady-state connections. However, the authors show that shocks admitted by the PDEs allow the orbit to jump back over the singular barrier. This process leads to a previously unnoticed family of solutions for malignant invasion which have slower wavespeeds than the family of smooth traveling wave solutions (extending over the whole real axis) admitted by the model. The existence of a minimum-speed wave with semi-infinite support is demonstrated. The practical stability of these new shock solutions is confirmed via numerical solutions of the model. The minimum speed solution is found to evolve from specific nonsmooth initial data, whereas smooth initial data extending over the whole real axis evolves to faster waves.

SIAM Journal on Applied Mathematics, 2009

We wish to predict ionic currents that flow through narrow protein channels of biological membran... more We wish to predict ionic currents that flow through narrow protein channels of biological membranes in response to applied potential and concentration differences across the channel, when some features of channel structure are known. We propose to apply singular perturbation analysis to the coupled Poisson-Nernst-Planck equations, which are the basic continuum model of ionic permeation and semiconductor physics. In semiconductor physics the problem is a singular perturbation, because the ratio of the Debye length to the width of the channel is a very small parameter that multiplies the Laplacian term in the Poisson equation. In contrast to semiconductors, the atomic scale geometry of narrow ion channels sometimes makes this ratio a large parameter, which, surprisingly, renders the problem a singular perturbation in a different sense. We construct boundary layers and match them asymptotically across the different regions of the channel to derive good approximations for Fick's and Ohm's laws. Our aim is to extend the asymptotic analysis to a class of nonlinear problems hitherto intractable. Analytical and numerical results for the mass flux and the electric current serve as a tool for molecular biophysicists and physiologists to understand, study and control protein channels, thereby aiding clinical and technological applications.

SIAM Journal on Applied Mathematics, 1994

A coupled system of partial differential equations that describes the evolution of solid and gas ... more A coupled system of partial differential equations that describes the evolution of solid and gas temperatures and gas concentration inside a catalytic converter is derived and examined. The use of a reaction term, which is a discontinuous function of the solid ...

SIAM Journal on Applied Mathematics, 1991

ABSTRACT Atmospheric or oceanic flows strongly constrained by rotation and stratification can be ... more ABSTRACT Atmospheric or oceanic flows strongly constrained by rotation and stratification can be described by a set of Lagrangian partial differential equations called the semigeostrophic equations. In these equations the trajectories must be determined implicitly. Generalized solutions of these equations are defined as a sequence of rearrangements of the fluid, which need not be smooth. These solutions are closely related to generalized solutions of the Monge-Ampere equation. Existence and uniqueness of such solutions is proved. The evolution is shown to be a sequence of minimum-energy states of the fluid, giving strong physical plausibility to the solutions.

Quarterly Journal of the Royal Meteorological Society, 2007

ABSTRACT Discussion of the proper boundary conditions to use in limited area forecasting models r... more ABSTRACT Discussion of the proper boundary conditions to use in limited area forecasting models requires knowledge of the properties of the governing equations. the theory of Oliger and Sundstrom states that the primitive hydrostatic equations commonly used are not hyperbolic and no local boundary conditions can be chosen. This paper shows their result to be incorrect, the equations are hyperbolic but not all the characteristics involve the time variable. Implications for the boundary conditions are discussed.

The Quarterly Journal of Mechanics and Applied Mathematics, 2005

A novel hybrid finite-element/finite-volume numerical method is developed to determine the capill... more A novel hybrid finite-element/finite-volume numerical method is developed to determine the capillary rise of a liquid with a free surface (under surface tension and gravitational forces). The few known exact analytical solutions are used to verify the numerical computations and establish their accuracy for a range of liquid contact angles. The numerical method is then used to ascertain the limitations of a number of theoretical approximations to solutions for the capillary rise in the linearized limit, for special geometries such as plane walls, concentric cylinders and in a wedge of arbitrary included angle. The existence of a critical wedge angle for a given contact angle is verified. However, the effect of slight practical rounding of wedge corners dramatically reduces the theoretical corner height.

The Quarterly Journal of Mechanics and Applied Mathematics, 2005

The upper free surface z = u(x, y) of a static fluid with gravity acting in the z direction, occu... more The upper free surface z = u(x, y) of a static fluid with gravity acting in the z direction, occupying a volume V , satisfies the Laplace-Young equation. The fluid wets the vertical boundaries of V so that the usual capillary contact conditions hold. This paper considers wedge shaped volumes V with corner angle 2α, that belong to the intermediate corner angle case of π/2 − γ < α < π/2 where γ is the contact angle, and determines explicitly, a regular power series expansion for the height u(r, θ) of the fluid near the corner, r = 0, to all orders in r. shows that it is possible to have logarithmic terms for a general corner expansion of the Laplace-Young equation, with appropriate boundary conditions. However, we suggest that the usual practical cases do not possess any singular terms near the corner, and we analytically and explicitly produce a non-singular series to any order in r, and propose that near the corner the far field effects are lost through any "interior or inner flat" region in exponentially small terms. We give computational solutions for these regular (energy minimising) cases based on a numerical finite volume method on an unstructured mesh, which fully support our assertions and our analytical series results, including the (minor) influence of the far field on local corner behaviour.

Proceedings of the Royal Society B: Biological Sciences, 1998

Cells use a combination of changes in adhesion, proteolysis and motility (directed and random) du... more Cells use a combination of changes in adhesion, proteolysis and motility (directed and random) during the process of migration. Proteolysis of the extracellular matrix (ECM) results in the creation of haptotactic gradients, which cells use to move in a directed fashion. The proteolytic creation of these gradients also results in the production of digested fragments of ECM. In this study

Nonlinearity, 2001

A haptotaxis-dominated model of cell invasion is considered for small cell diffusion and fast pro... more A haptotaxis-dominated model of cell invasion is considered for small cell diffusion and fast protease adjustment to the cell-collagen matrix interaction. A simplified limit model has travelling wave cell invasion profiles that are blunt, that is end with a 'shock-like' step, and that evolve stably from initial data that lie to one side of some initial plane. In common with diffusiondominated systems, the travelling wave which evolves from such initial data has the minimum wavespeed permissible in the model. This minimum wavespeed is not, however, determined by the local stability of the steady states in the travelling wave phase plane, but by a novel combination of singular behaviour within the phase plane and hyperbolic shock conditions. It is shown that more accurate models including the detailed fast dynamics of the protease require small amounts of diffusion (of the same order as the fast dynamics timescale) in order to remain stable. However, small diffusion and fast protease adjustment then give physically relevant and interesting solutions that evolve from semicompact initial data and stably invade at speeds well predicted by the simple model.