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Papers by Thomas Witelski
Springer Undergraduate Mathematics Series, 2015
We describe how the construction of similarity solutions of partial differential equations extend... more We describe how the construction of similarity solutions of partial differential equations extends naturally from concepts in dimensional analysis. In particular, we show how to obtain self-similar solutions through scaling invariances of linear and nonlinear PDE. We give examples illustrating how similarity solutions of PDEs can be obtained from solutions of ODE problems.
Springer Undergraduate Mathematics Series, 2015
Multi-dimensional partial differential equation problems on slender domains, having a small aspec... more Multi-dimensional partial differential equation problems on slender domains, having a small aspect ratio, can be analysed primarily in terms of solutions of simpler ODE problems for the variation of the solution in the slender direction. Using the aspect ratio as an asymptotic parameter, solutions having slow-variation (or “long-wave” dependence) in the wide direction can be constructed using matched asymptotics.
Springer Undergraduate Mathematics Series, 2015
Dimensional scaling is a process allowing for the specific physical units defining the original f... more Dimensional scaling is a process allowing for the specific physical units defining the original form of a problem to be factored out to leave a scaled mathematical problem. The solutions of the scaled problem will depend on a set of nondimensional parameters obtained from combinations of the original given quantities. Two scaling principles are introduced to guide the choices for characteristic scales that are useful for different limits under consideration. The Buckingham Pi theorem, which predicts the number of essential dimensionless parameters based on the form of the problem, is also introduced.
Springer Undergraduate Mathematics Series, 2015
Springer Undergraduate Mathematics Series, 2015
This chapter covers the modelling of two applied problems in fluid dynamics in brief case studies... more This chapter covers the modelling of two applied problems in fluid dynamics in brief case studies. The problems, on air bearing sliders and rivulet flows in wedges, build on a common core model called lubrication theory, which is derived in the beginning of the chapter. The case studies differ in many physical aspects (compressible versus incompressible flows, and free-surface versus squeeze film flows) and mathematical techniques employed (similarity solutions versus boundary layers and linear stability analysis). They serve to illustrate the power of the modelling methods to handle a diverse range of types of problems.
Methods of Mathematical Modelling, 2015
Springer Undergraduate Mathematics Series, 2015
Solutions of singularly perturbed ordinary differential equations exhibit non-uniform convergence... more Solutions of singularly perturbed ordinary differential equations exhibit non-uniform convergence as \(\varepsilon \rightarrow 0\). Regular and singular solutions (also called outer and inner solutions respectively) may be needed to satisfy all of the conditions imposed in a boundary value problem. The construction of matched asymptotic expansions gives a process for obtaining the solution of the overall problem from inner and outer solutions that hold on different parts of the domain. Determination of the scaling and placement of boundary layers will be illustrated in several examples.
Methods of Mathematical Modelling, 2015
Springer Undergraduate Mathematics Series, 2015
Springer Undergraduate Mathematics Series, 2015
Duffing equationẍ + x + x 3 = 0. Regular Perturbation Theory and Its Failure Regular perturbation... more Duffing equationẍ + x + x 3 = 0. Regular Perturbation Theory and Its Failure Regular perturbation theory: seek power series solution x(t,) = x 0 (t) + x 1 (t) + 2 x 2 (t) +. . .
Methods of Mathematical Modelling, 2015
In many problems, not all details of the structure of the solution of a partial differential equa... more In many problems, not all details of the structure of the solution of a partial differential equation are of interest. In some cases, it is possible to obtain an essential understanding of the behaviour of the system without directly solving the full equation. This chapter outlines the method of moments which can provide information about the evolution of integrals of the solution of a PDE by only solving ODEs. A related approach is used to reduce a two-dimensional problem to a modified PDE in one-dimension exhibiting Taylor-enhanced diffusion. Examination of the Turing instability for pattern formation in reaction-diffusion systems provides a cautionary note on limitations of such reduced models.
Springer Undergraduate Mathematics Series, 2015
Problems that can be written as singularly perturbed systems of first order differential equation... more Problems that can be written as singularly perturbed systems of first order differential equations can be solved using approaches that combine matched asymptotic expansions with phase plane analysis. The \(\varepsilon \rightarrow 0\) limit yields a separation of time-scales that reduces the overall system to different forms for the fast and slow dynamics over different intervals of time. Asymptotic matching is used to connect the fast and slow solutions and can be interpreted in terms of the geometric structure of nullclines and the structure of the phase plane.
Interfaces for the 21st Century: New Research Directions in Fluid Mechanics and Materials Science, 2002
ABSTRACT
Modern Methods in Scientific Computing and Applications, 2002
ABSTRACT
Tribology Transactions, 1999
ABSTRACT
Tribology Transactions, 1999
ABSTRACT
Studies in Applied Mathematics, 1998
The long-time asymptotic solutions of initial value problems for the heat equation and the nonlin... more The long-time asymptotic solutions of initial value problems for the heat equation and the nonlinear porous medium equation are self-similar spreading solutions. The symmetries of the governing equations yield three-parameter families of these solutions given in terms of their mass, center of mass, and variance. Unlike the mass and center of mass, the variance, or ''timeshift,'' of a solution is not a conserved quantity for the nonlinear problem. We derive an optimal linear estimate of the long-time variance. Newman's Lyapunov functional is used to produce a maximum entropy time-shift estimate. Results are applied to nonlinear merging and time-dependent, inhomogeneously forced diffusion problems.
SIAM Journal on Applied Mathematics, 2013
The formation of singularities in models of many physical systems can be described using self-sim... more The formation of singularities in models of many physical systems can be described using self-similar solutions. One particular example is the finite-time rupture of a thin film of viscous fluid which coats a solid substrate. Previous studies have suggested the existence of a discrete, countably infinite number of distinct solutions of the nonlinear differential equation which describes the self-similar behavior. However, no analytical mechanism for determining these solutions was identified. In this paper, we use techniques in exponential asymptotics to construct the analytical selection condition for the infinite sequence of similarity solutions, confirming the conjectures of earlier numerical studies.
Physics of Fluids, 1998
ABSTRACT
Physics of Fluids, 2009
We derive a time-dependent exact solution of the free surface problem for the Navier-Stokes equat... more We derive a time-dependent exact solution of the free surface problem for the Navier-Stokes equations that describes the planar extensional motion of a viscous sheet driven by inertia. The linear stability of the exact solution to one-and two-dimensional symmetric perturbations is examined in the inviscid and viscous limits within the framework of the long-wave or slender body approximation. Both transient growth and long-time asymptotic stability are considered. For one-dimensional perturbations in the axial direction, viscous and inviscid sheets are asymptotically marginally stable, though depending on the Reynolds and Weber numbers transient growth can have an important effect. For one-dimensional perturbations in the transverse direction, inviscid sheets are asymptotically unstable to perturbations of all wavelengths. For two-dimensional perturbations, inviscid sheets are unstable to perturbations of all wavelengths with the transient dynamics controlled by axial perturbations and the long-time dynamics controlled by transverse perturbations. The asymptotic stability of viscous sheets to one-dimensional transverse perturbations and to two-dimensional perturbations depends on the capillary number ͑Ca͒; in both cases, the sheet is unstable to long-wave transverse perturbations for any finite Ca.
Springer Undergraduate Mathematics Series, 2015
We describe how the construction of similarity solutions of partial differential equations extend... more We describe how the construction of similarity solutions of partial differential equations extends naturally from concepts in dimensional analysis. In particular, we show how to obtain self-similar solutions through scaling invariances of linear and nonlinear PDE. We give examples illustrating how similarity solutions of PDEs can be obtained from solutions of ODE problems.
Springer Undergraduate Mathematics Series, 2015
Multi-dimensional partial differential equation problems on slender domains, having a small aspec... more Multi-dimensional partial differential equation problems on slender domains, having a small aspect ratio, can be analysed primarily in terms of solutions of simpler ODE problems for the variation of the solution in the slender direction. Using the aspect ratio as an asymptotic parameter, solutions having slow-variation (or “long-wave” dependence) in the wide direction can be constructed using matched asymptotics.
Springer Undergraduate Mathematics Series, 2015
Dimensional scaling is a process allowing for the specific physical units defining the original f... more Dimensional scaling is a process allowing for the specific physical units defining the original form of a problem to be factored out to leave a scaled mathematical problem. The solutions of the scaled problem will depend on a set of nondimensional parameters obtained from combinations of the original given quantities. Two scaling principles are introduced to guide the choices for characteristic scales that are useful for different limits under consideration. The Buckingham Pi theorem, which predicts the number of essential dimensionless parameters based on the form of the problem, is also introduced.
Springer Undergraduate Mathematics Series, 2015
Springer Undergraduate Mathematics Series, 2015
This chapter covers the modelling of two applied problems in fluid dynamics in brief case studies... more This chapter covers the modelling of two applied problems in fluid dynamics in brief case studies. The problems, on air bearing sliders and rivulet flows in wedges, build on a common core model called lubrication theory, which is derived in the beginning of the chapter. The case studies differ in many physical aspects (compressible versus incompressible flows, and free-surface versus squeeze film flows) and mathematical techniques employed (similarity solutions versus boundary layers and linear stability analysis). They serve to illustrate the power of the modelling methods to handle a diverse range of types of problems.
Methods of Mathematical Modelling, 2015
Springer Undergraduate Mathematics Series, 2015
Solutions of singularly perturbed ordinary differential equations exhibit non-uniform convergence... more Solutions of singularly perturbed ordinary differential equations exhibit non-uniform convergence as \(\varepsilon \rightarrow 0\). Regular and singular solutions (also called outer and inner solutions respectively) may be needed to satisfy all of the conditions imposed in a boundary value problem. The construction of matched asymptotic expansions gives a process for obtaining the solution of the overall problem from inner and outer solutions that hold on different parts of the domain. Determination of the scaling and placement of boundary layers will be illustrated in several examples.
Methods of Mathematical Modelling, 2015
Springer Undergraduate Mathematics Series, 2015
Springer Undergraduate Mathematics Series, 2015
Duffing equationẍ + x + x 3 = 0. Regular Perturbation Theory and Its Failure Regular perturbation... more Duffing equationẍ + x + x 3 = 0. Regular Perturbation Theory and Its Failure Regular perturbation theory: seek power series solution x(t,) = x 0 (t) + x 1 (t) + 2 x 2 (t) +. . .
Methods of Mathematical Modelling, 2015
In many problems, not all details of the structure of the solution of a partial differential equa... more In many problems, not all details of the structure of the solution of a partial differential equation are of interest. In some cases, it is possible to obtain an essential understanding of the behaviour of the system without directly solving the full equation. This chapter outlines the method of moments which can provide information about the evolution of integrals of the solution of a PDE by only solving ODEs. A related approach is used to reduce a two-dimensional problem to a modified PDE in one-dimension exhibiting Taylor-enhanced diffusion. Examination of the Turing instability for pattern formation in reaction-diffusion systems provides a cautionary note on limitations of such reduced models.
Springer Undergraduate Mathematics Series, 2015
Problems that can be written as singularly perturbed systems of first order differential equation... more Problems that can be written as singularly perturbed systems of first order differential equations can be solved using approaches that combine matched asymptotic expansions with phase plane analysis. The \(\varepsilon \rightarrow 0\) limit yields a separation of time-scales that reduces the overall system to different forms for the fast and slow dynamics over different intervals of time. Asymptotic matching is used to connect the fast and slow solutions and can be interpreted in terms of the geometric structure of nullclines and the structure of the phase plane.
Interfaces for the 21st Century: New Research Directions in Fluid Mechanics and Materials Science, 2002
ABSTRACT
Modern Methods in Scientific Computing and Applications, 2002
ABSTRACT
Tribology Transactions, 1999
ABSTRACT
Tribology Transactions, 1999
ABSTRACT
Studies in Applied Mathematics, 1998
The long-time asymptotic solutions of initial value problems for the heat equation and the nonlin... more The long-time asymptotic solutions of initial value problems for the heat equation and the nonlinear porous medium equation are self-similar spreading solutions. The symmetries of the governing equations yield three-parameter families of these solutions given in terms of their mass, center of mass, and variance. Unlike the mass and center of mass, the variance, or ''timeshift,'' of a solution is not a conserved quantity for the nonlinear problem. We derive an optimal linear estimate of the long-time variance. Newman's Lyapunov functional is used to produce a maximum entropy time-shift estimate. Results are applied to nonlinear merging and time-dependent, inhomogeneously forced diffusion problems.
SIAM Journal on Applied Mathematics, 2013
The formation of singularities in models of many physical systems can be described using self-sim... more The formation of singularities in models of many physical systems can be described using self-similar solutions. One particular example is the finite-time rupture of a thin film of viscous fluid which coats a solid substrate. Previous studies have suggested the existence of a discrete, countably infinite number of distinct solutions of the nonlinear differential equation which describes the self-similar behavior. However, no analytical mechanism for determining these solutions was identified. In this paper, we use techniques in exponential asymptotics to construct the analytical selection condition for the infinite sequence of similarity solutions, confirming the conjectures of earlier numerical studies.
Physics of Fluids, 1998
ABSTRACT
Physics of Fluids, 2009
We derive a time-dependent exact solution of the free surface problem for the Navier-Stokes equat... more We derive a time-dependent exact solution of the free surface problem for the Navier-Stokes equations that describes the planar extensional motion of a viscous sheet driven by inertia. The linear stability of the exact solution to one-and two-dimensional symmetric perturbations is examined in the inviscid and viscous limits within the framework of the long-wave or slender body approximation. Both transient growth and long-time asymptotic stability are considered. For one-dimensional perturbations in the axial direction, viscous and inviscid sheets are asymptotically marginally stable, though depending on the Reynolds and Weber numbers transient growth can have an important effect. For one-dimensional perturbations in the transverse direction, inviscid sheets are asymptotically unstable to perturbations of all wavelengths. For two-dimensional perturbations, inviscid sheets are unstable to perturbations of all wavelengths with the transient dynamics controlled by axial perturbations and the long-time dynamics controlled by transverse perturbations. The asymptotic stability of viscous sheets to one-dimensional transverse perturbations and to two-dimensional perturbations depends on the capillary number ͑Ca͒; in both cases, the sheet is unstable to long-wave transverse perturbations for any finite Ca.