First-order Quasilinear Partial Differential Equations Research Articles

Spreading of a thin film with suction or blowing including surface tension effects

In this paper we derive a fourth-order nonlinear partial differential equation modelling the effects of suction/blowing on the evolution of the free surface of a thin viscous film on a porous base. Gravity and surface tension effects are included. The suction/blowing is modelled as an arbitrary function v n ( t , r ) . The Lie group classification method is used to determine a first-order quasi-linear partial differential equation satisfied by v n ( t , r ) . This equation is coupled with the model equation and solved numerically. New results are obtained which show how the free surface and the apparent contact angle vary with suction/blowing.

Singular Control-Invariance PDEs for Nonlinear Systems

In the present study a new method is proposed that allows the derivation of control laws capable of enforcing the desirable dynamics on an invariant manifold in state space. The problem of interest naturally surfaces in broad classes of physical and chemical systems whose dynamic behavior needs to be controlled and favorably shaped by external driving forces. In particular, the formulation of the problem under consideration is mathematically realized through a system of first-order quasi-linear singular invariance partial differential equations (PDEs), and a rather general set of conditions is derived that ensures the existence and uniqueness of a solution. The solution to the above system of singular PDEs is proven to be locally analytic, thus allowing the development of a series solution method that is easily programmable with the aid of a symbolic software package. Furthermore, through the solution to the above system of singular PDEs, an analytic manifold and a nonlinear control law are computed that render the manifold invariant for the nonlinear dynamical system considered. In particular, the restriction of the system dynamics on the invariant manifold is shown to represent exactly the desirable target dynamics of the controlled system. Finally, an illustrative case study of molecular dissociation is considered, and the proposed method is evaluated through simulation studies.

Axisymmetric spreading of a thin liquid drop with suction or blowing at the horizontal base

The axisymmetric spreading under gravity of a thin liquid drop on a horizontal plane with suction or blowing of fluid at the base is considered. The thickness of the liquid drop satisfies a non-linear diffusion equation with a source term. For a group invariant solution to exist the normal component of the fluid velocity at the base, v n , must satisfy a first-order quasi-linear partial differential equation. The general form of the group invariant solution for the thickness of the liquid drop and for v n is derived. Two particular solutions are considered. Each solution depends essentially on only one parameter which can be varied to yield a range of models. In the first solution, v n is proportional to the thickness of the liquid drop. The base radius always increases even for suction. In the second solution, v n is proportional to the gradient of the thickness of the liquid drop. The thickness of the liquid drop always decreases even for blowing. A special case is the solution with no spreading or contraction at the base which may have application in ink-jet printing.

A note on exact particular solutions of the generalized shallow-water equations

This note presents a set of systems of two first-order quasi-linear partial differential equations, which can be reduced to the shallow-water equations. This set includes equations describing a two-layered fluid flow.

Singular PDEs and the problem of finding invariant manifolds for nonlinear dynamical systems

The present research work proposes a new approach to the problem of finding invariant manifolds for nonlinear real analytic dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear partial differential equations (PDEs) and a rather general set of conditions for solvability is derived using Lyapunov's auxiliary theorem. The solution of the aforementioned system of PDEs is proven to be a locally analytic invariant manifold that under certain conditions coincides with the stable or unstable manifold, and which can be easily computed with the aid of a symbolic software package.

Pressure Transients Across Constrictions

The modeling of pressure transient across constrictions is achieved by using a one-dimensional, isothermal, noncompositional, single-phase representation of the Eulerian model. A TVD scheme was used to solve the ensuing nonhomogeneous hyperbolic set of first-order quasi-linear partial differential equations. Three types of constrictions were modeled and in each case the behavior of the transient was analyzed. This analysis was used to interpret the pressure response at the inlet resulting from the reflection of the transient at the constrictions. The comparisons between the predicted and inputted data are very good, suggesting that the technique has much promise. [S0195-0738(00)00301-0]

Loss of Hyperbolicity and Ill-posedness of the Viscous–Plastic Sea Ice Rheology in Uniaxial Divergent Flow

Local contact interactions between sea ice floes can be modeled on the large scale by treating the pack as a two-dimensional continuum with granular properties. One such model, which has gained prominence, is the viscous plastic constitutive rheology, using an elliptical yield curve and normal flow law. It has been used extensively in ice and coupled ice‐ocean studies over the past two decades. It is shown that in uniaxial flow this model reduces to a system of three quasi-linear first-order partial differential equations, which are hyperbolic in convergent flow and have mixed elliptic/hyperbolic behavior in divergence with two imaginary wave speeds. A linear stability analysis shows that the change in type causes the equations to be unstable and ill posed in uniaxial divergence. The root cause is a positive feedback mechanism that becomes stronger and stronger with smaller wavelengths. Numerical computations are used to demonstrate that fingersform and break the ice into discrete blocks. The frequency and growth rate of the fingers increase as the numerical resolution is increased, which implies that the model does not converge to a solution as the grid is refined. Two new models are proposed that are well posed. The first retains the positive feedback mechanism and introduces higher-order derivatives to suppress the unbounded growth rate of the instability. The second eliminates the positive feedback mechanism, and the instability, by repositioning the elliptical yield curve in principal stress space. Numerical simulations show that this model diverges without becoming unstable.

Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [ J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.

Semianalytical solution for multicomponent pressure swing adsorption

A theoretical study of multicomponent pressure swing adsorption (PSA) in which the system is considered to be one-dimensional, isothermal, locally at equilibrium with linear adsorption isotherms, and is considered to have negligible diffusion effects and axial total pressure variation is presented in this paper. The resulting mathematical model for multicomponent PSA is a set of first-order quasilinear hyperbolic partial differential equations. The similarities between this model and the model for multicomponent elution chromatography are used to define regions of constant state in the physical plane of time and distance and to obtain the solution in these regions semianalyticaly. Simple relations in terms of algebraic and ordinary differential equations which provide quick and simple estimation of the maximum product purity, maximum productivity, and maximum feed and purge times for PSA processes with an arbitrary number of components are derived. The application of the theory is illustrated by the construction of solutions for a four-step PSA process.

Singularities of solution surfaces for quasilinear 1st order partial differential equations

We study singularities of solution surfaces of the characteristic Cauchy problem for quasilinear first-order partial differential equations as an application of the previous result on vector fields near a generic submanifold.

First-order partial differential equations and Henstock-Kurzweil integrals

The existence and uniqueness of solutions to the Cauchy problem for a first-order quasi-linear partial differential equation is studied in this paper using the Henstock-Kurzweil integral. The classical theory requires certain differentiability and continuity conditions on the coefficients of the derivatives in the equation. It is shown here that in the Henstock-Kurzweil integral setting these conditions can be relaxed and that the resulting solution is differentiable though the derivatives need not be continuous. This sharpens the classical result and provides a bridge between classical and weak solutions in the linear case.

Simulation of Transients in Natural Gas Pipelines

Summary Transient flow of natural gas in pipelines is simulated without neglecting the inertia term. The governing equations constitute a non-homogeneous hyperbolic set of first-order quasilinear partial differential equations. The first-order, three-point, explicit Godunov scheme and the second-order, five-point, total variation diminishing (TVD) scheme are used to solve this set of equations. Two examples are simulated. The first is the propagation of a slow transient, with 24-hour cycle, in a 45-mile long, 8-in. inside diameter (ID) transmission pipeline, while the second is propagation of a fast transient in a 24-in., 300-ft long pipe. Comparisons between the predicted results and the measured data are fairly good and appear to be better than the predictions reported in the literature. This suggests that the mathematical model presented in the present study, which includes the inertia term in the momentum equation, is more realistic and the numerical schemes used are more robust.

Modeling and computation of the onset of failure in polymeric liquid filaments

In this paper we study a one-dimensional (1-D) quasilinear hyperbolic model for the elongation and onset of failure of polymeric liquid filaments. Our model is derived from the full 3-D free surface boundary value problem, assuming a Maxwell constitutive law for the polymeric liquid, by (a) positing a dominant balance of inertia, surface tension, gravity, viscosity and elastic relaxation, and (b) applying the slender axisymmetric perturbation methods of Bechtel et al., J. Stability Appl. Anal. Continua, 2 (1992) 1. The transient 1-D approximate equations in Lagrangian form consist of a coupled system of five, first-order quasilinear partial differential equations which admit explicit characteristic analysis and classification. The system may be genuinely hyperbolic or mixed elliptic-hyperbolic type. Importantly, a simulation which begins hyperbolic may evolve into the ill-posed mixed type regime at a particular time and location. Such a transition corresponds to a departure from slender behavior and breakdown of the 1-D theory, which we equate with onset of filament failure. The detailed nature of this transition from slender behavior is given particular scrutiny here. Our 1-D model is applied with appropriate initial and boundary data to simulate the experimental conditions in which one end of a filament is attached to a ceiling and the other to a falling drop which pulls a slender filament behind it. We show a representative sample of extensive parameter studies in which the model for the filament is initially a well-posed hyperbolic system, and numerical predictions are made on the filament behavior as it stretches. A remarkable variety of filament behavior is predicted within this 1-D Maxwell model. As we vary the competing physical effects, we predict qualitatively different locations, timescales and modes of onset of failure.

Orthogonally transitive G2 cosmologies

The authors provide a new framework for analysing orthogonally transitive G2 cosmologies, with a view to describing their asymptotic behaviour near the big bang and at late times. They assume a perfect fluid source with a linear equation of state and zero cosmological constant. The Einstein field equations are written as an autonomous system of first-order quasi-linear partial differential equations without constraints, in terms of dimensionless variables. The equilibrium points of this system are referred to as dynamical equilibrium states, and they show that the corresponding cosmological models are self-similar, but not necessarily spatially homogeneous.

Sensitivity Analysis of Two-Phase Flow Problems

The mathematical derivation and application of two deterministic sensitivity analysis methods, the direct approach of sensitivity (DAS) and the adjoint sensitivity method (ASM), are presented for two-phase flow problems. The physical problems investigated are formulated by the transient one-dimensional two-phase flow diffusion model, which consists of a system of four coupled quasi-linear first-order partial differential equations. The DAS method provides the sensitivity coefficients of all primary dependent variables at each time and space location with respect to a single input parameter. On the other hand, the ASM provides the sensitivity coefficients of a single response function at a specified time and space location with respect to all input parameters.

On the structure of balance equations and extended field theories of mechanics

A new approach to thermodynamics has been proposed recently, in which, in addition to the densities of mass, momentum and energy, the densities of momentum flux and energy flux are also taken as independent state variables. Extra equations of balance motivated by the moment equations of the kinetic theory are postulated. In this paper, we explore the general structure of the moment equations and formulate extended field theories based on these equations together with very simple types of constitutive equations. The resulting field equations form a system of first-order quasi-linear partial differential equations. For the extended field theories, the diversity of material models is no longer characterized by the constitutive equations alone, rather it is also characterized by the choice of the system of balance equations.

Frontogenesis in an Advective Mixed-Layer Model

Through an analysis of a multi-dimensional extension of the bulk mixed layer model of Kraus and Turner (1967) it is shown how, even under spatially uniform atmospheric conditions, an initially smooth horizontal temperature gradient in the surface mixed layer can still develop into a front. For this to occur, it is essential that the net downward heat flux at the air-water interface be related to the difference between the mixed layer temperature T and an apparent atmospheric temperature TA (Haney, 1971), so that the initial horizontal gradient in T corresponds to a horizontal variation in the surface buoyancy flux. As a result the mixed layer depth also differs from place to place. Depending on the direction of the wind-driven transport, this produces either a steepening or a flattening of the initial temperature profile. A lower bound condition for the initial horizontal advective heat flux is derived in terms of the initial surface heat flux, the stirring by the wind, and the time rate of chance of the apparent atmospheric temperature. If that condition is satisfied, a front develops. If not, then frontogenesis is prevented by the damping effect of the local atmospheric heating. It is shown that the condition can be satisfied on the scales of lakes (or small seas) and in near-equatorial oceanic regions. Mathematically, the physical process can be described by a first-order quasilinear hyperbolic partial differential equation that is solved exactly by the method of characteristics.

Exact renormalization group equations for the two-dimensional Ising model

By explicit construction we show the existence of an exact renormalization group for triangular Ising models with only nearest-neighbor interactions. The recursion relations take the form of a set of three quasilinear first-order partial differential equations for the interactions. We determine a nontrivial fixed point and study the linearized flow around it. This yields the specific-heat exponent $\ensuremath{\alpha}=0$, in agreement with the Onsager and Houtappel solutions, and demonstrates universality. The free energy is expressed as the trajectory integral of an explicitly given function of the interactions.

Reflector design as an initial-value problem

The problem of synthesizing a reflector surface to produce a two-variable generalized far-field pattern under the geometrical optics approximation can, under certain conditions, be formulated as an initial-value problem. The problem involves solving a set of four simultaneous quasi-linear first-order partial differential equations which can be handled using standard numerical techniques.

Reflector design for two-variable beam shaping in the hyperbolic case

The design of a reflector capable of producing a generalized two-variable beam shape when illuminated by a point source is investigated under the geometric-optics approximation. The problem is formulated as a mapping problem between points on a unit sphere in which areas are related by energy considerations. The solution of a resulting set of nonlinear partial differential equations is required. An investigation of the hyperbolic form of these equations shows that they can be reduced to a set of quasi-linear first-order partial differential equations which can be solved subject to initial conditions.