Arbitrary Function Of Time Research Articles

Unsteady Pressure‐Driven Viscous Flows Beneath Concrete Pavement Slabs

Water pumping flows beneath concrete pavement slabs are experimentally measured and mathematically modeled. A solution for the velocity profile and the wall shear stress is found for the condition of a constant imposed pressure gradient. The solution shows that the core flow is not significantly affected by viscosity if the time of application is less than 0.2 times the film thickness squared divided by the fluid kinematic viscosity. A finite difference method is described and used to determine the velocity profile and wall shear stress when the imposed pressure gradient is an arbitrary function of time. The velocity profile is then calculated using experimental pressure data for flow beneath a concrete pavement slab. The experimental pressure data is obtained from a 189,000‐N three‐axle truck crossing the joint between two concrete pavement slabs. The calculated mid‐plane velocity of the water between the pavement slab and the subbase compares favorably with measured velocities. The maximum water velocit...

Cost-benefit analysis of single-server n-unit imperfect switch system with adjustable repair

This paper deals with the cost-benefit analysis of a single-server n-unit system with an imperfect switch where the server becomes available immediately when summoned. The policy is to summon the server when the kth unit ( k fixed) or the switch fails. Initially, one unit is put into operation (the switch is working at t = 0) and n − 1 units are kept as cold standbys. The form of repair density function of an item (unit/switch) depends on the number of units waiting for repair. The revenue as well as the cost of repair are arbitrary functions of time. The expected net gain in (0, t) is obtained assuming that all the lifetime distributions are exponential and all the repair time densities are arbitrary.

Heat transfer in unsteady laminar flow through a channel

The temperature distribution in unsteady laminar flow of a viscous incompressible fluid in a flat channel is investigated, when the pressure gradient is an arbitrary function of time. Two techniques are presented (i) explicit finite difference scheme, and (ii) Chebyshev polynomial solution. Using both the techniques, several cases of pressure gradient are considered, with special attention paid to the linearly varying case.

Similarity solutions of the Kadomtsev-Petviashvili equation

Using the direct method introduced by Clarkson and Kruskal (CK) (1989) recently, the author obtains the similarity reductions of the Kadomtsev-Petviashvili equation (KPE). Some of them possess the same Painleve properties as that of the Boussinesq equation (BE) obtained by CK. However, the similarity solutions of KPE may include some (more than three) arbitrary functions of time. There exist some other types of similarity reductions of the KPE different from the similarity reductions of the BE obtained by CK.

Existence theorem for gauge symmetries in Hamiltonian constrained systems

It is shown that, for any constrained Hamiltonian system, one can locally construct a complete set of independent gauge transformations that depend only on the canonical coordinates and on arbitrary functions of time. The number of elements in this complete set is equal to the number of primary first-class constraints.

Gauge invariance and degree of freedom count

The precise relation between the gauge transformations in lagrangian and hamiltonian form is derived for any gauge theory. It is found that in order to define a lagrangian gauge symmetry, the coefficients of the first class constraints in the hamiltonian generator of gauge transformations must obey a set of differential equations. Those equations involve, in general, the Lagrange multipliers. Their solution contains as many arbitrary functions of time as there are primary first class constraints. If n is the number of generations of constraints (primary, secondary, tertiary…), the arbitrary functions appear in the general solution together with their successive time derivatives up to order n−1. The analysis yields as by-products: (i) a systematic way to derive all the gauge symmetries of a given lagrangian; (ii) a precise criterion for counting the physical degrees of freedom of a gauge theory directly from the form of gauge transformations in lagrangian form. This last part is illustrated by means of examples. The BRST analog of the counting of physical degrees of freedom is also discussed.

CONSOLIDATON DUE TO UNDERPUMPING AN ACCRETING LAYER OF SEDIMENT

The dissipation of pore-water pressure in a uniform layer of saturated sediment which is accumulating at a constant rate is considered. Water is free to escape from the upper (moving) boundary either to the atmosphere (the embankment-construction problem) or to a decreasing depth of water (the sedimentation problem). At the base of the sediment two conditions are examined: the water pressure there is an arbitrary function of time (case I) or water is pumped at a constant rate from wells penetrating to a highly permeable stratum beneath the sediment (case II)

Painlevé Analysis and the Complete Integrability of a Generalized Variable-Coefficient Kadomtsev-Petviashvili Equation

In this paper, we discuss the complete integrability of a generalized Kadomtsev-Petviashvili (GKP) equation with space- and time-dependent coefficients, which is known to have a physical application in the propagation of surface waves in straits or channels with varying depth and width. It is shown that the constraints which the variable-coefficient functions must satisfy for the GKP equation to pass the Painlevé tests for complete integrability are precisely the same as those in order that the equation may be transformed into either the Kadomtsev-Petviashvili equation or the Korteweg-de Vries equation, both of which are known to be completely integrable. Therefore we conclude that these are necessary and sufficient conditions for the GKP equation to be completely integrable, and when satisfied we obtain a class of completely integrable equations involving two arbitrary functions of one of the spatial variables (y) and time together with three arbitrary functions of time.

On the decoupling and solution of the Euler kinematic equations governing the motion of a Gyro

The decoupling and solution of the three strongly non-linear ordinary differential equations governing the motion of an arbitrary rigid body which is free to rotate about a fixed point (gyro) are presented. The theory developed is based on the assumption that the components of the instantaneous angular velocity are known and are smooth arbitrary functions of time. The resulting equivalent differential equation is second order with respect to the angle of the nodding motion. Furthermore, through a quantitative analysis, analytical solutions of this equation were obtained, which determine the law of motion of the gyro for some general conditions in accordance with the physical problem. Finally, two theoretical applications are investigated, the solutions of which indicate the potential value and effectiveness of the theory proposed.

On the infinite-dimensional symmetry group of the Davey–Stewartson equations

The Lie algebra of the group of point transformations, leaving the Davey–Stewartson equations (DSE’s) invariant, is obtained. The general element of this algebra depends on four arbitrary functions of time. The algebra is shown to have a loop structure, a property shared by the symmetry algebras of all known (2+1)-dimensional integrable nonlinear equations. Subalgebras of the symmetry algebra are classified and used to reduce the DSE’s to various equations involving only two independent variables.

Unsteady Porous Channel Flow of a Conducting Fluid with Suspended Particles

The flow of a viscous incompressible fluid embedded with a small spherical particle in the presence of a transverse magnetic field in a channel has been discussed. The cross-section of the channel is a porous regular hexagonal of side 4a and the walls are non-conducting. The analysis applied to the flows with pressure gradient which are arbitrary function of time. A few particular cases, flow for impulsive pressure gradient and for constant pressure gradient have been studied. The velocity of the fluid and particle decrease with increase in the intensity of the magnetic field.

Heating of a semiinfinite body by a finite heat source in the shape of a square

We determine the three-dimensional, unsteady temperature field in a semiinfinite body heated through a square region on its surface. The heat flux through the square is taken to be an arbitrary function of time.

On spherically symmetric shear-free perfect fluid configurations (neutral and charged). I

A class of solutions describing a wide variety of nonstatic, spherically symmetric, charged, shear-free perfect fluid configurations is derived. It is presented in the form of Jacobian elliptic functions characterized by seven free parameters: five constants and two arbitrary functions of time. This class of solutions is the most general charged version of the class derived by Kustaanheimo and Qvist [Comment. Phys. Math. Helsingf. 13, 12 (1948); Exact Solutions of Einstein’s Field Equations (Cambridge U. P., Cambridge, 1980), Chap. 12, Sec. 2]. It is found that many of the charged particular solutions expressible by elementary functions are new. Particular solutions, including neutral and uniform density solutions, are classified in detail. The physical interpretation of these solutions, including the study of their singularity structure, will be presented in a subsequent paper (Part II).

On the similarity solutions of the KdV equation with variable coefficients

In this paper we apply the similarity methods to a modified Korteweg-deVries equation containing an arbitrary function of time which describes the inhomogeneity effects. Some classes of functional forms for this arbitrary function are characterized and the related similarity solutions are determined.

Bäcklund transformations and the infinite-dimensional symmetry group of the Kadomtsev-Petviashvili equation

The infinite-dimensional symmetry group of the potential Kadomtsev-Petviashvili (PKP) equation is found and used to obtain a Bäcklund transformation, involving two arbitrary functions of time. This transformation is then used to generate several different types of solutions from the zero solution of the PKP equation.

Is the Stephani-Krasiński solution a viable model of the Universe?

We present a detailed study of the inhomogeneous Stephani-Krasinski solution with time-dependent curvature index. In general, the cosmological behaviour of the models depends on six arbitrary functions of time. Such models are termed ‘private universes’ and cannot be in accord with observation in the most general case. Two simple models with changing topology are considered as illustrating examples. In one of these models the pressure turns out to be negative and hence a violation of the weak energy condition in the singularity theorems is possible. A brief review of other inhomogeneous cosmologies is included for the sake of clarity. It is shown that the geodesic equation can be reduced to a complicated differential equation, which depends on the three arbitrary functions involved. Therefore, it is difficult to obtain explicit formulas for the various observational relations.

Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra

The Kadomtsev–Petviashvili (KP) equation (ut+3uux/2+ 1/4 uxxx)x +3σuyy/4=0 allows an infinite-dimensional Lie group of symmetries, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group depends on three arbitrary functions of time ‘‘t’’ and is shown to be related to a subalgebra of the loop algebra A(1)4. Low-dimensional subalgebras of the symmetry algebra are identified, specifically all those of dimension n≤3, and also a physically important six-dimensional Lie algebra containing translations, dilations, Galilei transformations, and ‘‘quasirotations.’’ New solutions of the KP equation are obtained by symmetry reduction, using the one-dimensional subalgebras of the symmetry algebra. These solutions contain up to three arbitrary functions of t.

Unsteady magnetohydrodynamic free-convection flows in a rotating fluid

The three-dimensional free-convection flow near an infinite vertical plate moving in a rotating fluid in the presence of a transverse magnetic field is studied in the case when the plate temperature undergoes a thermal transient. An exact solution has been obtained by defining a complex velocity with the help of the Laplace-transform technique, when the plate is moving with a velocity which is an arbitrary function of time. Three special cases of physical interest are also discussed.

Cost-benefit analysis of one-server two-unit hot standby system with imperfect switch subject to adjustable repair

This paper deals with the cost-benefit analysis of one-server two-unit hot standby system with imperfect switch where the repair densities depend upon the type of items (unit/standby/switch) waiting for repair. Initially one unit is put to operation (switch is working at t = 0) and the other unit is kept as a hot standby. The revenue as well as the cost of repair are arbitrary functions of time. Explicit expressions are obtained for expected net gain in [0, t] from which other characteristics like expected busy period due to repair of unit/standby/switch, expected up-time, expected number of repairs completed of unit/standby/switch etc. can be obtained.

Analysis of fluid equations by group methods

Using the machinery of Lie-group analysis several equations arising in fluid mechanics are studied. In particular, the Burgers' equation, the KdV equation, the Hopf equation, the two-dimensional KdV equation and the Lin-Tsien equation are analyzed. In all cases the particular group includes arbitrary functions of time which permit the transformation of time-dependent equations into the corresponding time-independent ones. Infinitely many time-dependent solutions are associated with each steady solution. Some solutions are constructed.