A remark on two-dimensional stokes flow solutions

Summary It is shown that the solution of Laplace's equation is useful to obtain a unique Stokes stream function which is valid everywhere under limited circumstances. When the Stokes stream function fails at infinity the use of the Oseen solution is required to obtain a physically acceptable solution.

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Résumé Les solutions de l'équation de Laplace permettent d'obtenir des fonctions de courant au sens de Stokes uniques et valides en tous points de l'espace, dans certains cas seulement. Quand les fonctions de courant au sens de Stokes ne sont pas valides à l'infini l'utilisation de functions de courant au sens d'Oseen est nécessaire pour obtenir une solution physiquement valable.

     

Counting the number of solutions in combustion and reactive flow problems

Summary Computational methods and comparison theory enable, when combined, an enhanced capability for counting the number of solutions in combustion equations. Very good lower bounds for the last turning point reveal a stable high temperature explosion branch for very small positive exothermicity.     

Integral representations of solutions for two-dimensional viscous flow problems

This paper concerns the steady flow of a viscous, incompressible fluid past a cylinderical obstacle. Solutions are represented in terms of simple layer potentials. Existence and uniqueness of solutions to the corresponding integral equations of the first kind are established. Emphasis is given on the asymptotic behaviors of the solutions as well as their connections to the singular perturbation results.     

The existence of approximate solutions for two-dimensional potential flow problems

A proof is given of the existence of an approximate Complex Variable Boundary Element Method solution for a Birichlet problem. This constructive proof can be used as a basis for numerical calculations. © 1996 John Wiley & Sons, Inc.     

Analytical and numerical solutions of a generalized hyperbolic non-Newtonian fluid flow

Generalized hyperbolic non-Newtonian fluid model first proposed by Al-Zahrani [1] is considered. The model was successfully applied to some drilling fluids with better performance in relating shear stress and velocity gradient compared to power-law and Hershel-Bulkley model. Special flow geometries namely pipe flow, parallel plate flow and flow between two rotating cylinders are treated. For the first two cases, analytical solutions of velocity profiles in the form of integrals are presented. For the flow between two rotating cylinders, the differential equation is solved by Runge-Kutta method combined with shooting. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Group properties and exact solutions for two-dimensional flow of a non-newtonian fluid

The Group Properties and the associated Lie Algebra are developed for the equations of motion of the unsteady two-dimensional flow of a non-Newtonian fluid in cartesian coordinates. Then by using the full one-parameter infinitesimal transformation group and its subgroups a number of exact solutions are obtained.     

Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

Asymptotic behaviour of solutions to the Navier-Stokes equations of a two-dimensional compressible flow

In this paper,we are concerned with the asymptotic behaviour of a weak solution to the Navier-Stokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p(e) = a log d(e) for large .Here d > 2,a > 0.We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity.Using properties of this function,we can prove the strong convergence of the density to its limit state.The behaviour of the velocity field and kinetic energy is also briefly discussed.     

On the solutions in the large of the two-dimensional flow of a nonviscous incompressible fluid

The Euler equations (1.1) for the motion of a nonviscous imcompressible fluid in a plane domain Ω are studied. Let E be the Banach space defined in (1.4), let the initial data v₀ belong to E, and let the external forces f(t) belong to L_loc¹(R; E). In Theorem 1.1 the strong continuity and the global boundedness of the (unique) solution v(t) are proved, and in Theorem 1.2 the strong-continuous dependence of v on the data v₀ and f is proved. In particular the vorticity rot v(t) is a continuous function in $\text{}\text{?}$gW, for every t ? R, if and only if this property holds for one value of t. In Theorem 1.3 some properties for the associated group of nonlinear operators S(t). are stated. Finally, in Theorem 1.4 a quite general sufficient condition is given on the data in order to get classical solutions.     

A local hybrid kernel meshless method for numerical solutions of two-dimensional fractional cable equation in neuronal dynamics

This study deals with obtaining numerical solutions of two-dimensional (2D) fractional cable equation in neuronal dynamics by using a recently introduced meshless method. In solution process at first stage, time derivatives that are appeared in the considered problem are discretized by using finite difference method. Then a meshless method based on hybridization of Gaussian and cubic kernels is developed in local fashion. The problem is solved both on regular and irregular domians. L_∞ and RMS error norms are calculated and compared with other numerical methods in literature as well as exact solutions. Also, obtained condition numbers are monitored. Numerical simulations show that local hybrid kernel meshless method is a thriving method for solving 2D fractional cable equation on regular and irregular domians.     

Continuous and discontinuous solutions for optimum thrust nozzles of given length

In its first sections, the paper deals with optimum thrust nozzles of given length and exit radius for flows with swirl. The computation is based on a modification of methods familiar for flows without swirl. Rather extensive numerical results show that the swirl does not impair the specific impulse attainable at a given nozzle length. The analysis suggests that the assumption of isentropic continuous flows, on which this approach is based, may sometimes be too restrictive. A survey of plane nozzles shows, on the other hand, that discontinuities need to be admitted only if, besides the length, a rather large radius of the nozzle is prescribed. Discontinuous solutions have been thoroughly investigated by Shmyglevskiy. At least in principle, we use the same line of thought, but considerable simplifications are possible if one starts with the variational formulation of Rao. In its numerical discussion and also in some analytical details, the present paper goes beyond Shmyglevskiy's results. The problem is conveniently discussed in astate plane, which has the local state of the flow (flow direction and speed or Mach number) as independent variables. By taking into account second variations, one can determine the boundary of the region for which continuous solutions give the (local) maximum. This boundary coincides with the locus of points at which the solution in the physical plane would fold back into itself. Another limitation of the original approach emerges if one asks under which conditions the thrust can be increased by admitting along the control surface values of the entropy that are higher than those of the oncoming flow. The conditions for isentropic and nonisentropic jumps are formulated and evaluated next, and a survey of the discontinuities which satisfy conditions for isentropic and also for selected nonisentropic jumps is given. Up to this point, the analysis is concerned only with the state distribution along the control surface. Jumps of the state in the interior require the occurrence of centered compression waves. Sample computations show that, in most cases, flow fields of this character can be generated by the choice of the nozzle shape. In some cases, no nozzle contours exist which generate the optimizing state distribution along the control surface as determined by the present analysis. It would then be necessary to include from the very beginning conditions for the realizability of the flow field.     

Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow

We consider a strictly hyperbolic system of balance laws in one space variable, that represents a simple model for a fluid flow in the presence of phase transitions. The state variables are specific volume, velocity and mass-density fraction λ of the vapor in the fluid. A reactive source term drives the dynamics of the phase mixtures; such a term depends on a relaxation parameter and involves an equilibrium pressure, allowing for metastable states.First we prove the global existence of weak solutions to the Cauchy problem, where the initial datum for λ is close either to 0 or 1 (the pure phases) and has small total variation, while the initial variations of pressure and velocity are not necessarily small. Then we consider the relaxation limit and prove that the weak solutions of the full system converge to those of the reduced system.     

An approach to numerical modeling of the dynamics of reacting gases in nozzles

An approach is proposed to computer simulation of gas-dynamic processes in chemically nonequilibrium flows in supersonic nozzles. An algorithm for the solution of the problem is developed. Convergence of iterative processes and stability of the linearized problem are investigated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 89–95, 1986.     

Inviscid and viscous stationary waves of gas flow through contracting-expanding nozzles

In this work we study steady states of one-dimensional viscous isentropic compressible flows through a contracting-expanding nozzle. Treating the viscosity coefficient as a singular parameter, the steady-state problem can be viewed as a singularly perturbed system. For a contracting-expanding nozzle, a complete classification of steady states is given and the existence of viscous profiles is established via the geometric singular perturbation theory. Particularly interesting is the existence of a maximal sub-to-super transonic wave and its role in the formation of other complicated transonic waves consisting of a sub-to-super portion.     

A numerical solution of a two-dimensional IHCP

In this paper, we will first study the existence and uniqueness of the solution of a two-dimensional inverse heat conduction problem (IHCP) which is severely ill-posed, i.e., the solution does not depend continuously on the data. We propose a stable numerical approach based on the finite-difference method and the least-squares scheme to solve this problem in the presence of noisy data. We prove the convergence of the numerical solution, then to regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition (SVD) method.     

Verification and validation of the foredrag coefficient for supersonic and hypersonic flow of air over a cone of fineness ratio 3

The foredrag coefficient resulting from the supersonic and hypersonic flow of air over a cone was calculated numerically using a finite volume approach based on the compressible Euler and Navier-Stokes equations with constant and variable thermophysical properties. No turbulence model was considered. Simulations were carried out for a cone of fineness ratio 3 under the free-stream Mach numbers 2.73, 3.50, 4.00, 5.05 and 6.28 (the Reynolds number, based on cone length, is within 0.45 and 2.85 million). Up to six grids were employed for numerical calculations, with 60 × 60 to 1920 × 1920 volumes. The numerical error was estimated to be less than 0.01% of the numerical solution for all models. Comparisons of the numerical foredrag coefficients of the three models with the experimental data showed that the Navier–Stokes model with variable thermophysical properties agreed better with the experimental foredrag for the entire Mach number interval studied, taking into account the validation standard uncertainty.     

A numerical solution of a two-dimensional transport equation

In this paper we present a variational method for approximating solutions of the Dirichlet problem for the neutron transport equation in the stationary case. Error estimates from numerical examples are used to evaluate an approximation of the solution with respect to the steps of two grids.     

Numerical solutions of two-dimensional nonlinear integral analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.