Showing papers in "Journal of Engineering Mathematics in 1990"
TL;DR: In this article, a nonlinear convection-diffusion model for nonhysteretic redistribution of liquid in a finite vertical unsaturated porous column is proposed, where the nonlinear boundary value problem may be transformed to a linear problem which is exactly solvable by the method of Laplace transforms.
Abstract: We solve a versatile nonlinear convection-diffusion model for nonhysteretic redistribution of liquid in a finite vertical unsaturated porous column. With zero-flux boundary conditions, the nonlinear boundary-value problem may be transformed to a linear problem which is exactly solvable by the method of Laplace transforms. In principle, this technique applies to arbitrary initial conditions.
60 citations
TL;DR: Asymptotic solutions for large and small surface tension are developed for the profile of a symmetric sessile drop in this paper, where the respective ranges of validity are established by comparing the asymptotics with solutions obtained by numerical integration of the full equations.
Abstract: Asymptotic solutions for large and small surface tension are developed for the profile of a symmetric sessile drop. The problem for large surface tension (i.e., small Bond number) is a regular perturbation problem, where the solution may be written as a uniformly valid asymptotic expansion. The problem for small surface tension (i.e., large Bond number) is a singular perturbation problem with boundary-layer behaviour in the edge region. The solution is a matched asymptotic expansion, where some care is to be taken for the matching. The respective ranges of validity are established by comparing the asymptotic results with solutions obtained by numerical integration of the full equations.
46 citations
TL;DR: In this article, it was shown that initial conditions for the diffusion equation can be expanded in a discrete, infinite sum of mutually orthogonal similarity solutions, each having a different rate of amplitude decay.
Abstract: It is shown that expansions in similarity solutions provide a quick and economical method for assessing the large-time asymptotics of the diffusion equation on infinite and certain semi-infinite domains if Dirichlet or Neumann conditions are imposed. The similarity solutions are shown to form a basis for the Hilbert space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaCa% aaleqabaGaaGOmaaaakiaacIcaruWu5XMzHvMBaGqbbiaa-jfadaah% aaWcbeqaaiaad6gaaaGccaGGSaGaaeyzamaaCaaaleqabaWaa0baaW% qaaiaaikdaaeaacaaIXaaaaaaakmaaCaaaleqabaGaaeiFaiaa-Hha% caqG8bWaaWbaaWqabeaacaqGYaaaaaaakiaacMcaaaa!4582!\[L^2 (R^n ,{\text{e}}^{_2^1 } ^{{\text{|}}x{\text{|}}^{\text{2}} } )\]. This implies that initial conditions for the diffusion equation that are square integrable with respect to the exponentially-growing weight function % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzamaaCa% aaleqabaWaaSbaaWqaamaaDaaabaGaaeOmaaqaaiaabgdaaaGaaeiF% aerbtLhBMfwzUbacfeGaa8hEaiaabYhadaahaaqabeaacaqGYaaaaa% qabaaaaaaa!3F7E!\[{\text{e}}^{_{_{\text{2}}^{\text{1}} {\text{|}}x{\text{|}}^{\text{2}} } } \] can be expanded in a discrete, infinite sum of mutually orthogonal similarity solutions, each having a different rate of amplitude decay. This leads to a rapid, almost effortless recognition of the large-time asymptotic behaviour of the solution.
41 citations
TL;DR: In this article, a review of known exact results is given, as well as an elementary integration procedure which appears to be a general device for obtaining integrals associated with similarity solutions.
Abstract: Although the nonlinear diffusion equation has been extensively studied and there exists substantial literature in many diverse areas of science and technology, the number of exact concentration profiles is nevertheless limited. In a recent article in this journal (Hill [1]) a brief review of known exact results is given, as well as an elementary integration procedure which appears to be a general device for obtaining integrals associated with similarity solutions. This paper extends the results given in [1] and for particular power law diffusivitiescm (such asm = −/12, −1, −/32 and −2) presents a number of new exact solutions obtained by fully integrating the ordinary differential equations derived in [1]. In addition new results are found for a general nonlinear diffusion equation which includes one-dimensional diffusion with an inhomogenouus and nonlinear diffusivitycmxmas well as symmetric nonlinear diffusion in cylinders and spheres. Moreover by a separate and ad-hoc procedure a new solution is obtained of the travelling wave type but with a variable wave speed. Some of the new exact solutions obtained for one-dimensional nonlinear diffusion with power law diffusivitiescmare illustrated graphically.
35 citations
TL;DR: In this article, the free convection boundary layer on a vertical plate with a prescribed surface heat flux proportional to (1 +x2)µ (µ a constant) is discussed.
Abstract: The free convection boundary layer on a vertical plate with a prescribed surface heat flux proportional to (1 +x2)µ (µ a constant) is discussed. For µ > −1―2 the boundary-layer solution develops from a similarity solution valid forx small to the one valid forx large. However, with µ ⩽ −1―2 the similarity equations forx large are not solvable and the behaviour for largex in this case is discussed. It is found that there are two cases to consider, namely µ < −1―2 and µ = −1―2. In both cases the leading-order problem is homogeneous involving an arbitrary constant which is determined from an integral property of the full boundary-layer problem. However, in the former case the asymptotic behaviour is algebraic, with the perturbation to the leading-order solution, arising from the heat flux boundary condition, being ofO[x1+2µ]. The latter case also involves logarithmic terms, with the perturbation to be leading-order solution now being ofO[(logx)−1].
33 citations
TL;DR: In this paper, expressions for reflection of plane incident waves from the closed end of a narrow wave tank when a number of thin vertical porous screens are introduced to damp the waves were derived.
Abstract: Expressions are derived for the reflection of plane incident waves from the closed end of a narrow wave tank when a number of thin vertical porous screens are introduced to damp the waves The results may have application to the design of wave tanks where a small amount of beach reflection is desirable
33 citations
TL;DR: In this article, the transformation group theoretic approach is applied to present an analysis of unsteady laminar free convection from a non-isothermal vertical flat plate, and the heat transfer characteristics for finite values of the Prandtl number Pr are presented, as temperature and velocity distributions.
Abstract: The transformation group theoretic approach is applied to present an analysis of the problem of unsteady laminar free convection from a non-isothermal vertical flat plate. The application of two-parameter groups reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature variations with position and time are derived. The ordinary differential equations are solved numerically using a fourth-order Runge-Kutta scheme and the gradient method. The heat-transfer characteristics for finite values of the Prandtl number Pr are presented, as temperature and velocity distributions.
21 citations
TL;DR: In this paper, a finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow, analogous to that of Riemann for gasdynamics.
Abstract: A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.
15 citations
TL;DR: In this paper, limit cycles are sought in a mathematical model of a simple hypothetical chemical reaction involving essentially only two reacting species, and strong numerical evidence is presented to assert that the limit cycle is unique and stable to infinitesimal perturbations.
Abstract: Limit cycles are sought in a mathematical model of a simple hypothetical chemical reaction involving essentially only two reacting species. Physically, these limit cycles correspond to time-periodic oscillations in the concentrations of the two chemicals. A combination of analytical and numerical methods reveals that limit-cycle behaviour is only possible in a restricted region of the parameter space. Strong numerical evidence is presented to assert that the limit cycle is unique and stable to infinitesimal perturbations. Numerical solutions are displayed and discussed.
15 citations
TL;DR: In this article, the authors study the steady state and transient motion of a system consisting of an incompressible, Newtonian fluid in an annulus between two concentric, rotating, rigid spheres.
Abstract: The research reported herein involves the study of the steady state and transient motion of a system consisting of an incompressible, Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research is to study the use of an approximate analytical method for analyzing the transient motion of the fluid in the annulus and the spheres which are started suddenly due to the action of prescribed torques. The problems include cases where: (a) one (or both) spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive torque.
14 citations
TL;DR: In this paper, it was shown that the solution of a single cosine equation and a double sine-cosine equation admits a Backlund transformation, where the latter admits a backlund transformation.
Abstract: Elementary transformations are utilized to obtain traveling wave solutions of some diffusion and wave equations, including long wave equations and wave equations the nonlinearity of which consists of a linear combination of periodic functions, either trigonometric or elliptic. In particular, a theorem is established relating the solutions of a single cosine equation and a double sine-cosine equation. It is shown that the latter admits a Backlund Transformation.
TL;DR: In this paper, the shape of the fluid surface is calculated by means of a conformal-mapping technique, which leads to a Hilbert problem, and the results are compared with known finite-element simulations.
Abstract: The injection of a viscous fluid into a mould formed by two parallel plates is considered. The flow front is supposed to move at constant speed. It is assumed that there is complete adherence between the fluid and the mould walls, and that the environmental pressure is constant. For a Newtonian fluid the problem is described in terms of two analytic complex functions. The shape of the fluid surface is calculated by means of a conformal-mapping technique, which leads to a Hilbert problem. The results are compared with known finite-element simulations.
TL;DR: In this article, normal oscillation modes of an incompressible fluid in an open container, part of the wall of which may be flexible, are modeled by a membrane and the dependence of the two modes on the Bond number and the Reynolds number is investigated.
Abstract: In this paper we study normal oscillation modes of an incompressible fluid in an open container, part of the wall of which may be flexible. The flexible part of the container wall is modelled by a membrane. We first investigate the eigenfrequencies of an inviscid fluid in a flexible container. We are able to show, by analytical means, that the eigenfrequencies of an inviscid fluid decrease when part of the rigid container wall is replaced by a membrane. The problem of viscous fluid oscillations in a flexible container is then studied numerically using the finite-element technique. Two different types of eigenmodes are observed: free-surface oscillation modes and structural vibration modes. The dependence of the two modes on the Bond number (measure of the ratio of gravitational and tension forces) and the Reynolds number is investigated.
TL;DR: In this article, it is shown that it is unnecessary to take special precautions for the sudden change in boundary conditions at the edge of the disk except if one is interested in the flow at distances which are smaller than about 10−3a from the edge.
Abstract: The boundary-layer equations outside a rotating disk of radius a have been solved. It is shown that it is unnecessary to take special precautions for the sudden change in boundary conditions at the edge of the disk except if one is interested in the flow at distances which are smaller than about 10−3
a from the edge. The behaviour of the flow at large distances from the disk is investigated analytically with results which are confirmed by the numerical computations.
TL;DR: For large values of the Reynolds number Re two terms of the asymptotic series for the torque have been calculated in this paper, of order Re−1/2 and Re−13/14, respectively.
Abstract: For large values of the Reynolds number Re two terms of the asymptotic series for the torque have been calculated. They are of order Re−1/2 and Re−13/14, respectively. The second term has been obtained after investigation of the double-deck structure which is present near the edge of the disk over a length of order Re−3/7.
TL;DR: In this article, the behavior of steady jet-like flows is examined in a low-Rossby-number rotating fluid, and the asymptotic properties of the flow leading up to this singular point are calculated for jets of various inflow widths and a structure which resolves the singularity that occurs in each case is described.
Abstract: The behaviour of steady jet-like flows is examined in a low-Rossby-number rotating fluid. Unlike the corresponding non-rotating flow, the momentum flux of a jet in a rotating fluid is not conserved with distance downstream and, as a consequence, the jet loses all of its momentum at a finite distance from the source, apparently developing a singularity as this occurs. The asymptotic properties of the flow leading up to this singular point are calculated for jets of various inflow widths and a structure which resolves the singularity that occurs in each of these cases is described. The properties on the approach to the singularity are shown to be similar to those of the exact solution described by Gadgil [12]. Both the asymptotic structure and the resolution of the singularity are, however, applicable to the expected breakdown of any form of jet in rotating fluid under similar conditions. The consequences of this are discussed, particularly in relation to the separated-flow structure proposed for motion past a cylindrical obstacle in Page [3].
TL;DR: In this paper, the authors considered the natural convective flow in a fluid-saturated porous medium with the bottom wall being partially heated or cooled, and analyzed the flow and heat transfer for a range of values of the two non-dimensional parameters which define the problem.
Abstract: The natural convective flow in a fluid-saturated porous medium is considered for an infinite horizontal channel with the bottom wall being partially heated or cooled. The flow and heat transfer are analysed for a range of values of the two non-dimensional parameters which define the problem, namely the Rayleigh number Ra and aspect ratio ɛ. Numerical solutions are obtained for ɛ = 1 and ɛ = 0.1 for both heated and cooled cases and for a range of values of Ra. In the heated case, the nature of the flow is seen to change from unicellular for smaller values of Ra to multicellular for large Ra, with the value of Ra at this changeover being decreased as ɛ is decreased. Also, for this case, a range of values of Ra is found over which both unicellular and multicellular flows are possible. For the cooled case, a boundary layer is seen to develop on the bottom wall as Ra is increased for both the values of e taken. Finally, a solution for ɛ≪1 is obtained and is compared with the numerical solutions for ɛ = 0.1.
TL;DR: In this paper, the numerical solution of the Navier-Stokes equations for the steady planar laminar flow of an incompressible viscous fluid past a finite flat plane, at various angles of incidence to an oncoming uniform stream at infinity and at various Reynolds numbers Re.
Abstract: This paper deals with the numerical solution of the Navier-Stokes equations for the steady planar laminar flow of an incompressible viscous fluid past a finite flat plane, at a various angles of incidence to an oncoming uniform stream at infinity and at various Reynolds numbers Re.
TL;DR: In this article, it was shown that the structure of free shear layers in a rotating fluid change only slightly when an asymmetry is introduced into the geometry; in fact, once a Poisson equation which describes the flow in the interior Taylor column is solved, the behaviour in both % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% 4rNCHbGeaGqi
Abstract: There are two basic conclusions reached in this paper. It is shown first that the structure of free shear layers in a rotating fluid change only slightly when an asymmetry is introduced into the geometry; in fact, once a Poisson equation which describes the flow in the interior Taylor column is solved, the behaviour in both % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaG4maaaaaaa!3775!\[\frac{1}{3}\] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3776!\[\frac{1}{4}\]-layers is understood for all situations. The second result derived is that when the Stewartson layers are along solid sidewalls in an asymmetric configuration, then the velocity within the interior has additional components induced by the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3776!\[\frac{1}{4}\]-layer; the first perturbation is O(E% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGinaaaaaaa!3776!\[\frac{1}{4}\]).
TL;DR: The appearance of chaotic behavior in numerical treatment of a two-dimensional system is examined from an analytical point of view in this article, where the original model exhibiting chaotic-like solutions is unfolded to a three-parameter model.
Abstract: The appearance of apparently chaotic behaviour in numerical treatment of this two-dimensional system is examined from an analytical point of view. The original two-parameter model exhibiting chaotic-like solutions is unfolded to a three-parameter model. This enlarged model is shown to have a condimension-two degenerate Hopf bifurcation, the unfolding of which contains phase portraits with three concentric limit cycles. Some segments of these limit cycles are so close to each other that numerical integration causes transitions across the unstable limit cycle, thus giving the appearance of chaotic behaviour. The region in parameter space where this occurs is quite significant and it includes part of the plane of the original two-parameter model.
TL;DR: In this paper, a die made up of two parallel plates is modeled as a free streamline and its shape is calculated with the use of complex-function theory and conformal-mapping techniques.
Abstract: When a viscous fluid is extruded from a capillary or an annular die, the thickness of the fluid jet is in general unequal to the width of the die. This phenomenon is called “die-swell” and is studied in this paper for a die made up of two parallel plates. It is assumed that no slip will occur between the fluid and the plates, and that the pressure in the space into which the fluid is emitted is constant and uniform. The fluid surface is a free streamline. Its shape is calculated with the use of complex-function theory and conformal-mapping techniques. The predicted ratio of swell is found to be in full agreement with known finite-element results.
TL;DR: In this paper, an analytic solution for a three-dimensional contact problem, in linear elasticity, is constructed through the separation of Laplace's equation in paraboloidal coordinates, where a rigid punch under normal loading is applied to an isotropic elastic medium occupying an infinite half-space where the contact region is parabolic and the punch profile is prescribed.
Abstract: In this paper an analytic solution for a three-dimensional contact problem, in linear elasticity, is constructed through the separation of Laplace's equation in paraboloidal coordinates. A rigid punch under normal loading is applied to an isotropic elastic medium occupying an infinite half-space where the contact region is parabolic and the punch profile is prescribed. This treatment allows for a general punch profile provided it is physically reasonable so as to ensure the convergence of the solution.