William Lakin - Academia.edu (original) (raw)

Papers by William Lakin

NASA STI/Recon Technical Report N, May 1, 1986

This work deals with the problem of a boundary layer on a flat plate which has a constant velocit... more This work deals with the problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream. It has previously been shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. In particular, it was proved that a solution exists only if this parameter does not exceed a certain critical value, and numerical evidence was adduced to show that this solution is nonunique. Using Crocco formulation the present work proves this nonuniqueness. Also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter.

In this work we consider a boundary-value problem arising from the transverse vibrations of a sle... more In this work we consider a boundary-value problem arising from the transverse vibrations of a slender, finite, uniform rod which rotates with constant angular velocity about an axis through the rod's fixed end. The relevant dimensionless parameter is assumed to lie in a range corresponding to rapid rotation. The differential equation in this problem is fourth-order, linear, and takes its distinctive character from the simple turning point where the coefficient of the second derivative term vanishes. A significant feature is that the turning point is also a boundary point and hence outer expansions alone are not adequate for formation of a characteristic equation. Approximations valid at and away from the turning point are obtained and related through the method of matched asymptotic expansions. Outer expansions are required to be "complete" in the sense of Olver, and approximations are found for the Stokes multipliers which describe the analytic continuations of these ...

ABSTRACT The discrete spectrum of the Orr-Sommerfeld problem of hydrodynamic stability for bounda... more ABSTRACT The discrete spectrum of the Orr-Sommerfeld problem of hydrodynamic stability for boundary layer flows in semi-infinite regions is examined. Related questions concerning the continuous spectrum are also addressed. Emphasis is placed on the stability problem for the Blasius boundary layer profile. A general theoretical result is given which proves that the discrete spectrum of the Orr-Sommerfeld problem for boundary layer profiles (U(y), 0,0) has only a finite number of discrete modes when U(y) has derivatives of all orders. Details are given of a highly accurate numerical technique based on collocation with splines for the calculation of stability characteristics. The technique includes replacement of 'outer' boundary conditions by asymptotic forms based on the proper large parameter in the stability problem. Implementation of the asymptotic boundary conditions is such that there is no need to make apriori distinctions between subcases of the discrete spectrum or between the discrete and continuous spectrums. Typical calculations for the usual Blasius problem are presented.

Integrating matrices are derived for arbitrarily spaced grid points using either interpolating or... more Integrating matrices are derived for arbitrarily spaced grid points using either interpolating or least squares fit orthogonal polynomials. Several features of the equally spaced grid case are also discussed.

By expressing partial differential equations of motion in matrix notation, utilizing the integrat... more By expressing partial differential equations of motion in matrix notation, utilizing the integrating matrix as a spatial operator, and applying the boundary conditions, the resulting ordinary differential equations can be cast into standard eigenvalue form upon assumption of the usual time dependence. As originally developed, the technique was limited to beams having continuous mass and stiffness properties along their lengths. Integrating matrix methods are extended to treat the differential equations governing the flap, lag, or axial vibrations of rotating beams having concentrated masses. Inclusion of concentrated masses is shown to lead to the same kind of standard eigenvalue problem as before, but with slightly modified matrices.

The use of integrating matrices in solving differential equations associated with rotating beam c... more The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

[

Quarterly of Applied Mathematics

We consider a group of fourth-order boundary-value problems associated with the small vibrations ... more We consider a group of fourth-order boundary-value problems associated with the small vibrations or buckling of a uniform flexible rod which is clamped at one end and rotates in a plane perpendicular to the axis of rotation. The vibrations may be in any plane relative to the plane of rotation and the rod is off-clamped, i.e. the axis of rotation does not pass through the rod’s clamped end. The governing equation for the vibrations involves a small parameter for rapid rotation and must be treated by singular perturbation methods. Further, a turning point of the equation always coincides with a boundary point. Both free and unstable vibrations are examined, and a stability boundary is obtained. Results for the unstable vibrations predict the unexpected existence of a time-independent buckled state in non-transverse planes when the rod is wholly under tension. The general buckling problem in the transverse plane is also considered.

Studies in Applied Mathematics, 1972

Canadian Journal of Physics

We consider the steady flow past a finite flat plate due to a uniform stream in the presence of a... more We consider the steady flow past a finite flat plate due to a uniform stream in the presence of a uniform magnetic field which is not aligned with the streaming motion at infinity. The Hartmann number is assumed large and positive. This leads to a singular perturbation problem involving a second-order partial differential equation. As a result, solutions of the governing equation must be obtained in three separate regions and then matched asymptotically. A related problem involving magnetohydrodynamic (MHD) flow across a finite needle is also discussed.

A model for trap-controlled charge transport in an amorphous material is considered. The model is... more A model for trap-controlled charge transport in an amorphous material is considered. The model is defined in terms of a set of coupled differential rate equations, in which each type of trap is characterized by a capture probability e*;, and a release probability r, *, This model is significant because it has recently been shown that the continuum limit of the Scher-Montroll master equation for anomalous dispersion is completely equivalent to the multiple-trapping equations. In this paper we develop the multiple-trapping model, and use asymptotic methods to treat traps whose release rates and capture rates are both different from one. We obtain an estimate of the increased transit time due to trapping, and also discuss the determination of both the number and types of different traps necessary to produce a disperse photocurrent transient.

SIAM Journal on Applied Mathematics, 1977

A generalization of the logistic equation of population biology is considered in which the specie... more A generalization of the logistic equation of population biology is considered in which the species being modeled is connected to its larger ecosystem through a lower trophic level consisting of a renewable resource. The resource adjusts rapidly to demand and is utilized by the species for population growth. The resulting initial value problem is a system of first-order differential equations with a small parameter. Singular perturbation techniques based on different time scales are used to obtain and relate approximations valid initially and for order one times. A comparison of composite approximations with numerical solutions shows that, even for moderate values of the parameter, the asymptotic results are highly accurate. Additional situations studied include a model for long term species adaptation which involves three distinct time scales.

International Journal for Numerical Methods in Engineering, 1986

Differentiating matrices allow the numerical differentiation of functions defined at points of a ... more Differentiating matrices allow the numerical differentiation of functions defined at points of a discrete grid. A type of differentiating matrix based on local approximation on a sequence of sliding subgrids is considered. Previous derivations of this type of matrix have been restricted to grids with uniformly spaced points, and the resulting derivative approximations have lacked precision, especially at endpoints. The new formulation allows grids which have arbitrarily space points. It is shown that high accuracy can be achieved through use of differentiating matrices on non-uniform grids which include near-boundary points. Use of the differentiating matrix as an operator to solve eigenvalue problems involving ordinary differential equations is also considered.

Quarterly Journal of Mechanics and Applied Mathematics, 1978

Philosophical Transactions of The Royal Society A: Mathematical, Physical and Engineering Sciences, 1978

Approximations are made to the eigenvalue relation for the Orr-Sommerfeld problem. The derived fi... more Approximations are made to the eigenvalue relation for the Orr-Sommerfeld problem. The derived first approximation is uniformly valid along the entire marginal stability curve. This approximation is found through two derivations. The first is based on the differential equation satisfied by the eigenvalue relation, the other is based on uniform approximation to solutions of the Orr-Sommerfeld equation. The theory may

Quarterly Journal of Mechanics and Applied Mathematics, 1975

Quarterly of Applied Mathematics

In this work we consider a boundary-value problem arising from the transverse vibrations of a sle... more In this work we consider a boundary-value problem arising from the transverse vibrations of a slender, finite, uniform rod which rotates with constant angular velocity about an axis through the rod’s fixed end. The relevant dimensionless parameter is assumed to lie in a range corresponding to rapid rotation. The differential equation in this problem is fourth-order, linear, and takes its distinctive character from the simple turning point where the coefficient of the second derivative term vanishes. A significant feature is that the turning point is also a boundary point and hence outer expansions alone are not adequate for formation of a characteristic equation. Approximations valid at and away from the turning point are obtained and related through the method of matched asymptotic expansions. Outer expansions are required to be “complete” in the sense of Olver, and approximations are found for the Stokes multipliers which describe the analytic continuations of these expansions acr...

Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is sh... more Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is shown that a left correction, T^o, is locally more accurate than a right correction, ^oT. For each relative error function considered, there is a left correction r and the associated differential equation. The common feature is the same integrable part whose forcing function is the difference between L-derivatives of the exact and the initial solution. Upon transformation into a Volterra integral equation, fixed point iterations generate infinite series of a lacunary type which converge globally whenever an integral equation is linear. Alternatively, when the integrable solution is used for iterative refinement, the outcomes are infinite product representations. Necessary conditions for the absolute convergence are given.

A Starling resistor in models of fluid flow through a collapsible tube is a resistance term that ... more A Starling resistor in models of fluid flow through a collapsible tube is a resistance term that depends on the transmural pressure across the tube wall. In a traditional Starling resistor, negative transmural pressure im- plies that the tube is fully collapsed and fluid flow is blocked. The present work develops a nonlinear expression for a Starling-like resistor that can be used to represent flow resistance in vessels, such as the intracranial transverse venous sinuses, where transmural pressure is negative in the normal resting state and the vessel is not fully collapsed. Expressing flow resistance in terms of vessel cross-sectional area, the lumped resistance between two points in the vasculature subjected to negative transmural pressure is found to depend on

Quarterly of Applied Mathematics

Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is sh... more Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is shown that a left correction, T^o, is locally more accurate than a right correction, ^oT. For each relative error function considered, there is a left correction r and the associated differential equation. The common feature is the same integrable part whose forcing function is the difference between L-derivatives of the exact and the initial solution. Upon transformation into a Volterra integral equation, fixed point iterations generate infinite series of a lacunary type which converge globally whenever an integral equation is linear. Alternatively, when the integrable solution is used for iterative refinement, the outcomes are infinite product representations. Necessary conditions for the absolute convergence are given. 1. Introduction. Two central problems in approximating the solution, <&(t,to), of a d-dimensional system of linear differential equations *' = A(t)S, *(*") = I, (1.1) are how to choose an initial approximation, 3>o(t, to), to the solution of Eq. 1.1, and how to select an error function, E(t, to). Once these decisions are made, it is usually not difficult to derive the differential equation satisfied by E(f,to). Although it is not necessarily a linear one, its integrable part is linear and its solution, Eo(t, to), represents a dominant approximation to E(t, to). It is these two functions, 4>o(t, to) and Eo(t, to), that make solving the rest of the differential equation much easier compared to the direct methods for representing or computing $(t, t0) that start from the identity matrix. The

NASA STI/Recon Technical Report N, May 1, 1986

This work deals with the problem of a boundary layer on a flat plate which has a constant velocit... more This work deals with the problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream. It has previously been shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream. In particular, it was proved that a solution exists only if this parameter does not exceed a certain critical value, and numerical evidence was adduced to show that this solution is nonunique. Using Crocco formulation the present work proves this nonuniqueness. Also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter.

In this work we consider a boundary-value problem arising from the transverse vibrations of a sle... more In this work we consider a boundary-value problem arising from the transverse vibrations of a slender, finite, uniform rod which rotates with constant angular velocity about an axis through the rod's fixed end. The relevant dimensionless parameter is assumed to lie in a range corresponding to rapid rotation. The differential equation in this problem is fourth-order, linear, and takes its distinctive character from the simple turning point where the coefficient of the second derivative term vanishes. A significant feature is that the turning point is also a boundary point and hence outer expansions alone are not adequate for formation of a characteristic equation. Approximations valid at and away from the turning point are obtained and related through the method of matched asymptotic expansions. Outer expansions are required to be "complete" in the sense of Olver, and approximations are found for the Stokes multipliers which describe the analytic continuations of these ...

ABSTRACT The discrete spectrum of the Orr-Sommerfeld problem of hydrodynamic stability for bounda... more ABSTRACT The discrete spectrum of the Orr-Sommerfeld problem of hydrodynamic stability for boundary layer flows in semi-infinite regions is examined. Related questions concerning the continuous spectrum are also addressed. Emphasis is placed on the stability problem for the Blasius boundary layer profile. A general theoretical result is given which proves that the discrete spectrum of the Orr-Sommerfeld problem for boundary layer profiles (U(y), 0,0) has only a finite number of discrete modes when U(y) has derivatives of all orders. Details are given of a highly accurate numerical technique based on collocation with splines for the calculation of stability characteristics. The technique includes replacement of &#39;outer&#39; boundary conditions by asymptotic forms based on the proper large parameter in the stability problem. Implementation of the asymptotic boundary conditions is such that there is no need to make apriori distinctions between subcases of the discrete spectrum or between the discrete and continuous spectrums. Typical calculations for the usual Blasius problem are presented.

Integrating matrices are derived for arbitrarily spaced grid points using either interpolating or... more Integrating matrices are derived for arbitrarily spaced grid points using either interpolating or least squares fit orthogonal polynomials. Several features of the equally spaced grid case are also discussed.

By expressing partial differential equations of motion in matrix notation, utilizing the integrat... more By expressing partial differential equations of motion in matrix notation, utilizing the integrating matrix as a spatial operator, and applying the boundary conditions, the resulting ordinary differential equations can be cast into standard eigenvalue form upon assumption of the usual time dependence. As originally developed, the technique was limited to beams having continuous mass and stiffness properties along their lengths. Integrating matrix methods are extended to treat the differential equations governing the flap, lag, or axial vibrations of rotating beams having concentrated masses. Inclusion of concentrated masses is shown to lead to the same kind of standard eigenvalue problem as before, but with slightly modified matrices.

The use of integrating matrices in solving differential equations associated with rotating beam c... more The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

[

Quarterly of Applied Mathematics

We consider a group of fourth-order boundary-value problems associated with the small vibrations ... more We consider a group of fourth-order boundary-value problems associated with the small vibrations or buckling of a uniform flexible rod which is clamped at one end and rotates in a plane perpendicular to the axis of rotation. The vibrations may be in any plane relative to the plane of rotation and the rod is off-clamped, i.e. the axis of rotation does not pass through the rod’s clamped end. The governing equation for the vibrations involves a small parameter for rapid rotation and must be treated by singular perturbation methods. Further, a turning point of the equation always coincides with a boundary point. Both free and unstable vibrations are examined, and a stability boundary is obtained. Results for the unstable vibrations predict the unexpected existence of a time-independent buckled state in non-transverse planes when the rod is wholly under tension. The general buckling problem in the transverse plane is also considered.

Studies in Applied Mathematics, 1972

Canadian Journal of Physics

We consider the steady flow past a finite flat plate due to a uniform stream in the presence of a... more We consider the steady flow past a finite flat plate due to a uniform stream in the presence of a uniform magnetic field which is not aligned with the streaming motion at infinity. The Hartmann number is assumed large and positive. This leads to a singular perturbation problem involving a second-order partial differential equation. As a result, solutions of the governing equation must be obtained in three separate regions and then matched asymptotically. A related problem involving magnetohydrodynamic (MHD) flow across a finite needle is also discussed.

A model for trap-controlled charge transport in an amorphous material is considered. The model is... more A model for trap-controlled charge transport in an amorphous material is considered. The model is defined in terms of a set of coupled differential rate equations, in which each type of trap is characterized by a capture probability e*;, and a release probability r, *, This model is significant because it has recently been shown that the continuum limit of the Scher-Montroll master equation for anomalous dispersion is completely equivalent to the multiple-trapping equations. In this paper we develop the multiple-trapping model, and use asymptotic methods to treat traps whose release rates and capture rates are both different from one. We obtain an estimate of the increased transit time due to trapping, and also discuss the determination of both the number and types of different traps necessary to produce a disperse photocurrent transient.

SIAM Journal on Applied Mathematics, 1977

A generalization of the logistic equation of population biology is considered in which the specie... more A generalization of the logistic equation of population biology is considered in which the species being modeled is connected to its larger ecosystem through a lower trophic level consisting of a renewable resource. The resource adjusts rapidly to demand and is utilized by the species for population growth. The resulting initial value problem is a system of first-order differential equations with a small parameter. Singular perturbation techniques based on different time scales are used to obtain and relate approximations valid initially and for order one times. A comparison of composite approximations with numerical solutions shows that, even for moderate values of the parameter, the asymptotic results are highly accurate. Additional situations studied include a model for long term species adaptation which involves three distinct time scales.

International Journal for Numerical Methods in Engineering, 1986

Differentiating matrices allow the numerical differentiation of functions defined at points of a ... more Differentiating matrices allow the numerical differentiation of functions defined at points of a discrete grid. A type of differentiating matrix based on local approximation on a sequence of sliding subgrids is considered. Previous derivations of this type of matrix have been restricted to grids with uniformly spaced points, and the resulting derivative approximations have lacked precision, especially at endpoints. The new formulation allows grids which have arbitrarily space points. It is shown that high accuracy can be achieved through use of differentiating matrices on non-uniform grids which include near-boundary points. Use of the differentiating matrix as an operator to solve eigenvalue problems involving ordinary differential equations is also considered.

Quarterly Journal of Mechanics and Applied Mathematics, 1978

Philosophical Transactions of The Royal Society A: Mathematical, Physical and Engineering Sciences, 1978

Approximations are made to the eigenvalue relation for the Orr-Sommerfeld problem. The derived fi... more Approximations are made to the eigenvalue relation for the Orr-Sommerfeld problem. The derived first approximation is uniformly valid along the entire marginal stability curve. This approximation is found through two derivations. The first is based on the differential equation satisfied by the eigenvalue relation, the other is based on uniform approximation to solutions of the Orr-Sommerfeld equation. The theory may

Quarterly Journal of Mechanics and Applied Mathematics, 1975

Quarterly of Applied Mathematics

In this work we consider a boundary-value problem arising from the transverse vibrations of a sle... more In this work we consider a boundary-value problem arising from the transverse vibrations of a slender, finite, uniform rod which rotates with constant angular velocity about an axis through the rod’s fixed end. The relevant dimensionless parameter is assumed to lie in a range corresponding to rapid rotation. The differential equation in this problem is fourth-order, linear, and takes its distinctive character from the simple turning point where the coefficient of the second derivative term vanishes. A significant feature is that the turning point is also a boundary point and hence outer expansions alone are not adequate for formation of a characteristic equation. Approximations valid at and away from the turning point are obtained and related through the method of matched asymptotic expansions. Outer expansions are required to be “complete” in the sense of Olver, and approximations are found for the Stokes multipliers which describe the analytic continuations of these expansions acr...

Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is sh... more Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is shown that a left correction, T^o, is locally more accurate than a right correction, ^oT. For each relative error function considered, there is a left correction r and the associated differential equation. The common feature is the same integrable part whose forcing function is the difference between L-derivatives of the exact and the initial solution. Upon transformation into a Volterra integral equation, fixed point iterations generate infinite series of a lacunary type which converge globally whenever an integral equation is linear. Alternatively, when the integrable solution is used for iterative refinement, the outcomes are infinite product representations. Necessary conditions for the absolute convergence are given.

A Starling resistor in models of fluid flow through a collapsible tube is a resistance term that ... more A Starling resistor in models of fluid flow through a collapsible tube is a resistance term that depends on the transmural pressure across the tube wall. In a traditional Starling resistor, negative transmural pressure im- plies that the tube is fully collapsed and fluid flow is blocked. The present work develops a nonlinear expression for a Starling-like resistor that can be used to represent flow resistance in vessels, such as the intracranial transverse venous sinuses, where transmural pressure is negative in the normal resting state and the vessel is not fully collapsed. Expressing flow resistance in terms of vessel cross-sectional area, the lumped resistance between two points in the vasculature subjected to negative transmural pressure is found to depend on

Quarterly of Applied Mathematics

Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is sh... more Given an initial approximation <&0 to the fundamental matrix of solutions for = A(t), it is shown that a left correction, T^o, is locally more accurate than a right correction, ^oT. For each relative error function considered, there is a left correction r and the associated differential equation. The common feature is the same integrable part whose forcing function is the difference between L-derivatives of the exact and the initial solution. Upon transformation into a Volterra integral equation, fixed point iterations generate infinite series of a lacunary type which converge globally whenever an integral equation is linear. Alternatively, when the integrable solution is used for iterative refinement, the outcomes are infinite product representations. Necessary conditions for the absolute convergence are given. 1. Introduction. Two central problems in approximating the solution, <&(t,to), of a d-dimensional system of linear differential equations *' = A(t)S, *(*") = I, (1.1) are how to choose an initial approximation, 3>o(t, to), to the solution of Eq. 1.1, and how to select an error function, E(t, to). Once these decisions are made, it is usually not difficult to derive the differential equation satisfied by E(f,to). Although it is not necessarily a linear one, its integrable part is linear and its solution, Eo(t, to), represents a dominant approximation to E(t, to). It is these two functions, 4>o(t, to) and Eo(t, to), that make solving the rest of the differential equation much easier compared to the direct methods for representing or computing $(t, t0) that start from the identity matrix. The