Explicit Steady-state Solutions Research Articles

Shear Flows of Dilatant Fluids with Limited Shear Rates: Analytical Results and Linear Stability Analysis

In this paper, we study the simple shear flows of a class of dilatant fluids with a limited shear rate. This class of fluids is characterized by shear thickening behavior in which the apparent viscosity tends to infinity as the modulus of the stress approaches a finite threshold. The apparent viscosity function is a logarithmic type with two material parameters. We considered this specific form because it fits very well with the flow curves of some granular suspensions for specific values of the material parameters. Despite the nonlinearity of the constitutive law, it is possible to determine explicit steady-state solutions for a simple shear flow, namely (i) the channel flow; (ii) the flow between coaxial cylinders, and (iii) the flow down an inclined plane. We performed a two-dimensional linear stability analysis to investigate the onset of possible instabilities of the steady basic flow, putting into evidence the dependency of the critical Reynolds number on the material parameters.

Demand drives growth all the way: Goodwin, Kaldor, Pasinetti and the Steady State

AbstractA demand-driven alternative to the conventional Solow–Swan growth model is analysed. Its medium run is built around Marx–Goodwin cycles of demand and distribution. Long-run income and wealth distributions follow rules of accumulation stated by Pasinetti in combination with a technical progress function for labour productivity growth incorporating a Kaldor effect and induced innovation. An explicit steady state solution is presented along with analysis of dynamics. When wage income of capitalist households is introduced, the Samuelson–Modigliani steady state ‘dual’ to Pasinetti’s cannot be stable. Numerical simulation loosely based on US data suggests that the long-run growth rate is around 2% per year and that the capitalist share of wealth may rise from about 40 to 70% due to positive medium-term feedback of higher wealth inequality into its own growth.

BIFURCATION FROM TWO EQUILIBRIA OF STEADY STATE SOLUTIONS FOR NON-REVERSIBLE AMPLITUDE EQUATIONS

In this paper, bifurcation and stability of two kinds of constant stationary solutions for non-reversible amplitude equations on a bounded domain with Neumann boundary conditions are investigated by using the perturbation theory and weak nonlinear analysis. The asymptotic behaviors and local properties of two explicit steady state solutions, and pitch-fork bifurcations are also obtained if the bounded domain is regarded as a parameter. In addition, the stability of a new increasing or decaying local steady state solution with oscillations are analyzed.

A Study in Nucleated Polymerization Models of Protein Aggregation.

The nucleated polymerization model is a mathematical framework that has been applied to aggregation and fragmentation processes in both the discrete and continuous setting. In particular, this model has been the canonical framework for analyzing the dynamics of protein aggregates arising in prion and amyloid diseases such as as Alzheimer's and Parkinson's disease. We present an explicit steady-state solution to the aggregate size distribution governed by the discrete nucleated polymerization equations. Steady-state solutions have been previously obtained under the assumption of continuous aggregate sizes; however, the discrete solution allows for direct computation and parameter inference, as well as facilitates estimates on the accuracy of the continuous approximation.

Electron and Hole Current Characteristics of n-i-p-Type Semiconductor Quantum Dot Transistor

For the future development of semiconductor quantum dot transistors, a novel n-i-p-type semiconductor quantum dot transistor consisting of an n-type source, an i-type dot, and a p-type drain is presented in this paper. The explicit steady-state solution to a set of an infinite number of stochastic theoretical probability equations is given. The solution is applicable to both metal and semiconductor quantum dot transistors. The electron and hole current characteristics controlled by means of the gate voltage are computed numerically, and it is pointed out that the electron or hole current state can be selected by applying a positive or negative gate voltage

Effects of hydrodynamic dispersion on plume lengths for instantaneous bimolecular reactions

In this article a new two-dimensional analytical model is presented, which explores the effects of longitudinal ( α L) and transverse ( α T) dispersivity on the transient and steady state length of a plume undergoing an instantaneous bimolecular reaction. A solute B is injected into a steady, uniform flow field in a homogeneous porous medium initially filled with a solute A. Mixing of the solutes occurs only through longitudinal and transversal dispersion and reaction between the solutes is assumed to occur where both solutes are simultaneously present. Explicit steady state solutions (in the limit t → ∞ ) are presented, which describe the distributions of reactants and products in x– y space in the form of the modified Bessel function of zero order and second kind. When α T ⩽ α L ⩽ 10 α T, i.e., limits typically used in practical engineering scenarios, the steady state length of a contaminant plume is obtained as a simple algebraic expression including α T; the influence of α L upon plume length being negligible. This phenomenon has been observed experimentally and in numerical simulations. The analysis provided in this paper proves that this behavior follows directly from the governing equations. In addition, the effects of hydrodynamic dispersion on the transient plume development are investigated. The assumption that α L is only important for the transient development of the plume is verified through the use of a numerical solution.

A matrix-algebraic solution to two K m servers in a loop

An explicit steady-state solution is given for any queuing loop made up of two general servers, whose distribution functions have rational Laplace transforms. The solution is in matrix geometric form over a vector space that is itself a direct or Kronecker product of the internal state spaces of the two servers. The algebraic properties of relevant entities in this space are given in an appendix. The closed-form solution yields simple recursive relations that in turn lead to an efficient algorithm for calculating various performance measures such as queue length and throughput. A computational-complexity analysis shows that the algorithm requires at least an order of magnitude less computational effort than any previously reported algorithm.

Explicit Solutions of M/G/C/ /N-type Queueing Loops with Generalizations

Using recently developed Matrix-Algebraic techniques, we find explicit steady-state solutions of M/G/C/ /N-type loops that depend on a set of recursively defined matrices. We show that such systems are special cases of arbitrary service centers that contain (load dependent) exponential servers in which no more than C customers can be active simultaneously. We also outline a recursive algorithm that can be used to evaluate the properties of small- to moderate-sized systems. The solution to the M/G/C open system is then found by letting the overall customer population N go to infinity. The solutions for closed systems in general are not of the matrix geometric type, and only in the limit does the solution become geometric in form.

Explicit Steady-State Solutions to Some Elementary Queueing Models

Several simple queueing models, commonly used as textbook examples, have explicit steady-state solutions, which to date have escaped notice. These solutions are easily computable. In addition to their didactic interest, the results presented here are also useful in the analysis of more complex queueing models.

Explicit steady state solutions for a particular M(x)/M/1 queueing system

AbstractAn explicit steady state solution is determined for the distribution of the number of customers for a queueing system in which Poisson arrivals are bulks of random size. The number of customers per bulk varies randomly between 1 and m, m arbitrary, according to a point multinomial, and customer service is exponential. Queue characteristics are given.

Propagation of electrohydrodynamic surface waves in a conducting fluid

A comprehensive study is made of the electrohydrodynamic surface wave phenomena in an inviscid, incompressible conducting fluid in the presence of an external steady electric field due to an arbitrary periodic surface pressure distribution. The initial value wave problem has been solved by the Laplace and the generalized Fourier transforms in conjunction with asymptotic techniques — as advanced in a previous work ofDebnath andRosenblat [12]. An explicit steady state solution related to short and long wave approximations is found. The asymptotic behaviour of the transient solution for large time is demonstrated. It is shown that there are two surface wave-trains propagating in the conducting medium, one of which corresponds to the classical surface wave train and the other is originated entirely due to the interactions of the electric field with gravity. The simultaneous effects of the electric field and surface tension on the propagation of small amplitude surface waves are analyzed. The implications of the wave solutions related to very deep and shallow fluids are explored. A discussion of the electrohydrodynamic wave motions with their characteristic features is presented.