Publications (54)62.07 Total impact
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ABSTRACT: The existence and uniqueness of solutions to the boundaryvalue problem for steady Poiseuille flow of an isothermal, incompressible, nonlinear bipolar viscous fluid in a cylinder of arbitrary crosssection is established. Continuous dependence of solutions, in an appropriate norm, is also established with respect to the constitutive parameters of the bipolar fluid model, as these parameters converge to zero, under the additional assumption that the cylinder has a circular crosssection.  [Show abstract] [Hide abstract]
ABSTRACT: The higherorder boundary conditions associated with the flow of an incompressible, nonlinear, bipolar viscous fluid in a bounded domain are derived; these boundary conditions differ from the various ad hoc sets of higherorder boundary conditions that have been used in work involving fluid dynamics models employing higherorder gradients of the velocity field. The derivation presented is based on a principle of virtual work and some deep results of Heron on higherorder traces of divergencefree vector fields. 
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ABSTRACT: A mixed hyperbolicparabolic system is derived for a lumped parameter continuum model of pulse combustion. For a regularized version of the initialboundary value problem for an associated linear system, with timedependent boundary conditions, Galerkin approximations are used to establish the existence of a suitable class of unique solutions. Standard parabolic theory is then employed to established higher regularity for the solutions of the regularized problem. Finally, a priori estimates are derived which allow for letting the artificial viscosity, in the regularized system, approach zero so as to obtain the existence of a unique solution for the original mixed hyperbolicparabolic problem.  [Show abstract] [Hide abstract]
ABSTRACT: We study a coupled nonlinear system of differential equation approximating the rotating MHD flow over a rotating sphere near the equator. In particular, using the Schauder fixed point theorem, we are able to establish existence of solutions. Other results on similar systems show that the question of existence in not obvious and, hence, that the present results are useful. Indeed, the work of McLeod in the 1970s shows some nonexistence results for similar problems. From here, we are also able to discuss some of the features of the obtained solutions. The observed behaviors of the solutions agree well with the numerical simulations present in the literature.  [Show abstract] [Hide abstract]
ABSTRACT: In this brief paper, we consider the generalized equation of steady transonic gas flow with the addition of a singular source term. While the addition of a source term often destroys selfsimilarity of such flows, we demonstrate that a selfsimilar solution can still exist in the case of a singular source. We first reduce the governing nonlinear partial differential equation into an ordinary differential equation for a class of similarity solutions. Then, we study the existence of solutions for this similarity equation. After that, several explicit solution forms are given. In constructing exact solutions analytically, we demonstrate that dual solution branches may exist for some parameter regimes. For those parameter regimes where exact or analytical solutions are not possible, we obtain numerical solutions. The results demonstrate interesting properties of the solutions which warrant further study.  [Show abstract] [Hide abstract]
ABSTRACT: A global existence theorem is established for an initialboundary value problem, with timedependent boundary data, arising in a lumped parameter model of pulse combustion; the model in question gives rise to a nonlinear mixed hyperbolicparabolic system. Using results previously established for the associated linear problem, a fixed point argument is employed to prove local existence for a regularized version of the nonlinear problem with artificial viscosity. Appropriate αpriori estimates are then derived which imply that the local existence result can be extended to a global existence theorem for the regularized problem. Finally, a different set of α priori estimates is generated which allows for taking the limit as the artificial viscosity parameter converges to zero; the corresponding solution of the regularized problem is then proven to converge to the unique solution of the initialboundary value problem for the original, nonlinear, hyperbolicparabolic system.  [Show abstract] [Hide abstract]
ABSTRACT: Solutions for a class of degenerate, nonlinear, nonlocal boundary value problems, arising in nano boundary layer fluid flows over a stretching surface, are obtained. Viscous flows over a twodimensional stretching surface and an axisymmetric stretching surface are considered. Using the Schauder fixed point theorem, existence and uniqueness results are established. The effects of the slip parameter k and the suction parameter a on the fluid velocity and on the tangential stress are investigated and discussed (through numerical results). We find that for fluid flows at nanoscales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter k.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we study the dependence of the unique positive solution of the periodic boundary value problemu″+ρ2u=w(t)us,0<s<1andρ≠0,u(0)=u(1),u′(0)=u′(1)on the parameter ρ as ρ → 0. We show that this solution u(t; ρ) is infinitely differentiable in ρ and has singularity in the order of ρ−α with α = 2/(1 − s) as ρ → 0. Furthermore, the graph of u(t; ρ) has fluctuation in t only of the order O(ρ−α+2). This will help us to determine the location of the solution in numerical computations to study the behavior of the solution when ρ is sufficiently close to 0.  [Show abstract] [Hide abstract]
ABSTRACT: An approach is described for constructing (using molecular dynamics simulations at the atomic scale) a mathematical model for the constitutive behavior of a carbon nanotube. The method is based on applying homogenization theory to a hexagonal array of carbon atoms with a specific chirality vector. The molecular dynamics simulations generate a set of periodic, rapidly varying, elastic constants for the nanotube. An example is presented to illustrate the technique for a specific array on a nanotube surface.  [Show abstract] [Hide abstract]
ABSTRACT: Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists To > 0, ν* > 0 and a unique family of strong solutions uν of the Euler or NavierStokes initialboundary value problem on the time interval (0, T0), depending continuously on the viscosity coefficient v for 0 < ν < ν*. The solutions of the NavierStokes problem satisfy a Naviertype boundary condition. We give the information on the rate of convergence of the solutions of the NavierStokes problem to the solution of the Euler problem for ν → 0+.  [Show abstract] [Hide abstract]
ABSTRACT: This study numerically investigates the fully developed flow of a Newtonian fluid in a poroussaturated corrugated pipe, on the basis of a Brinkman model. The variable coefficient Helmholtz equation, which is obtained by means of an epitrochoid transformation, is solved using a spectral–Galerkin method. The effects of both the Darcy number and corrugation on the velocity field are discussed and presented graphically. The nature of these effects is documented for the first time.  [Show abstract] [Hide abstract]
ABSTRACT: Provided the initial velocity and the external body force are sufficiently smooth, there exist T(0) > 0, nu* > 0 and a unique continuous family of strong solutions u(v) (0 <= nu <= nu*) of the Euler or NavierStokes initialboundary value problem on the time interval (0, T(0)). The solutions of the NavierStokes problem satisfy a Naviertype boundary condition. To cite this article: H. Bellout et al., C R. Acad. Sci. Paris, Ser. 1347 (2009). (C) 2009 Published by Elsevier Masson SAS on behalf of Academie des sciences.  [Show abstract] [Hide abstract]
ABSTRACT: Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T 0 > 0, ν 0 > 0 and a unique continuous family of strong solutions u ν (0 ≤ ν < ν 0) of the Euler or Navier–Stokes initialboundary value problem on the time interval (0, T 0). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ν and curl 2 u ν .  [Show abstract] [Hide abstract]
ABSTRACT: Solutions for a class of nonlinear secondorder differential equations, arising in diffusion of chemically reactive species of a nonNewtonian fluid immersed in a porous medium over a stretching sheet, are obtained. Furthermore, using the Brouwer fixed point theorem, existence results are established. Moreover, the exact analytical solutions (for some special cases) are obtained. The results obtained for the diffusion characteristics reveal many interesting behaviors that warrant further study of the effects of reaction rate on the transfer of chemically reactive species.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper a twodimensional quasivariational inequality arising in elastohydrodynamic lubrication is studied for nonconstant viscosity. So far, existence results for such piezo–viscous problems require an L∞L∞ property for an auxiliary problem. For the usual pressure–viscosity relations, this property needs small data assumptions which are not observed in experimental conditions. In the present work, such small data assumptions are proved unnecessary for existence results. Besides wellestablished monotonicity behavior for the viscosity–pressure relation, the only condition used here is on the asymptotic behavior for this law as the pressure tends to infinity. If the procedure used here, namely the introduction of a reduced pressure by Grubin transform followed by a regularization procedure, appears somewhat classical, the way in which an upper bound is obtained is completely new. 
Article: Chaos in the thermal convection of a Newtonian fluid with a temperature dependent viscosity
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ABSTRACT: The onset of chaotic motion in Newtonian fluid with a temperature dependent viscosity is explored in the context of the Rayleigh–Bénard thermal convection setup. Galerkin truncation is used to derive a loworder dynamical system (generalized Lorenz system) from the governing equations which reduces to the classical Lorenz system for a Newtonian fluid with a constant viscosity. The effect of the temperature dependent viscosity on the nonlinear solutions is analyzed by considering projections in the phasespace. Also, time signature of the solutions is investigated. The onset of chaotic motion is also discussed in detail for different values of temperature dependent viscosity in this paper.  [Show abstract] [Hide abstract]
ABSTRACT: Exact solutions for a class of nonlinear secondorder differential equations arising in a third grade fluid flow, at a rotating cylinder (unbounded domain case) and between rotating cylinders (bounded domain case), are obtained. Furthermore, the exact solutions are compared with the numerical ones. It is observed that the difference between the exact and the numerical solutions is about 1% for small R (the nondimensional distance between the cylinders) and is about 3% when R=100. This difference increases with an increasing R. Moreover, for large R it is not easy to obtain meaningful results numerically. Hence, these exact solutions for various values of the parameters R and ω (rotation parameter) are useful for experimental and numerical studies, and warrant further study.  [Show abstract] [Hide abstract]
ABSTRACT: We treat the Stokes and the NavierStokes equation with the conditions curlku · n = 0 (k = 0, 1, 2) on the boundary of the flow field. The approach is based on a spectral analysis and properties of operator curl. (© 2004 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim)  [Show abstract] [Hide abstract]
ABSTRACT: Thermally developing laminar flow of a dipolar fluid in a duct (pipe or channel) including axial conduction (Graetz problem extended) is investigated. The solutions are based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation for laminar flow into a pair of firstorder partial differential equations. This approach, which is based on the solution method of Paputsakis et al. for a laminar pipe flow of a Newtonian fluid, is not plagued by any uncertainties arising from expansions in terms of eigenfunctions belonging to a nonselfadjoint operator. Then the eigenvalue problem is solved by means of the method of the weighted residual. Following this, the effect of the dipolar constant on the Nusselt number and temperature field are discussed in detail. Finally, it is shown that the Newtonian solution is a special case of the present result.
Publication Stats
659  Citations  
62.07  Total Impact Points  
Top Journals
Institutions

20132015

Illinois State University
Worcester, Massachusetts, United States


19882013

Northern Illinois University
 Department of Mathematical Sciences
DeKalb, Illinois, United States


19932002

Charles University in Prague
Praha, Praha, Czech Republic


1992

Wichita State University
Wichita, Kansas, United States


1987

Purdue University
 Department of Mathematics
West Lafayette, Indiana, United States
