A general class of coupled nonlinear differential equations arising in self-similar solutions of convective heat transfer problems

We establish existence and uniqueness results for a general class of coupled nonlinear third order differential equations arising in flow and heat transfer problems. We consider solutions over the semi-infinite interval. As special cases, we recover the existence and uniqueness results of solutions for the following physically meaningful scenarios (among others): (i) flow and heat transfer over a stretching sheet, (ii) flow and heat transfer over a nonlinearly stretching porous sheet, (iii) linear convective flow and heat transfer over a porous nonlinearly stretching sheet and (iv) nonlinear convective heat transfer over a porous nonlinearly stretching sheet. In all the cases the effects of viscous dissipation and the internal heat generation/absorption on the flow and heat transfer characteristics are included. Moreover, the obtained results are applicable to several problems dealing with flow and heat transfer phenomena.     

Convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation/absorption

We consider a convective flow in a porous medium of an incompressible viscous conducting fluid impinging on a permeable stretching surface with suction, and internal heat generation/absorption. Using a similarity transformation the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations. We first examine a number of special cases for which we may obtain exact solutions. We then obtain analytical solutions (by the Homotopy Analysis Method) and numerical solutions (by a boundary value problem solver), in order to further study the behavior of the nonlinear differential equations, for various values of the physical parameters. Our numerical solutions are shown to agree with the available results in the literature. We then employ the numerical results to bring out the effects of the suction parameter, heat source/sink parameter, stretching parameter, porosity parameter, the Prandtl number and the free convection parameter on the flow and heat transfer characteristics. In the absence of suction and free convection, our findings are in agreement with the corresponding numerical results of Attia [H.A. Attia, On the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation, Comput. Mater. Sci. 38 (2007) 741-745].     

Falkner-Skan equation for flow past a stretching surface with suction or blowing: Analytical solutions

The simultaneous effects of suction and injection on tangential movement of a nonlinear power-law stretching surface governed by laminar boundary layer flow of a viscous and incompressible fluid beneath a non-uniform free with stream pressure gradient is considered. The self-similar flow is governed by Falkner-Skan equation, with transpiration parameter γ, wall slip velocity λ and stretching sheet (or pressure gradient) parameter β. The exact solution for β = −1 and three closed form asymptotic solutions for β large, large suction γ, and λ → 1 have also been presented. Dual solutions are found for β = −1 for each value of the transpiration parameter, including the non-permeable surface, for each prescribed value of the wall slip velocity λ. The large β asymptotic solution also dual with respect to wall slip velocity λ, but do not depend on suction and blowing. The critical values of γ, β and λ are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solution by integral method for a trial velocity profile is presented and results are compared with the exact solutions.     

Natural convection on a vertical stretching cylinder

The flow and natural (or mixed) convection due to a vertical stretching cylinder is studied. Using similarity transforms, the Navier-Stokes and energy equations reduce to a set of nonlinear ordinary differential equations. Asymptotic analysis for large Reynolds numbers shows the relation between axisymmetric flow and two-dimensional flow. Due to the algebraic decay of the similarity functions, numerical integration is performed using a compressed coordinate. The axial velocity is composed of forced convection due to stretching and natural convection from the heated cylinder. The heat transfer increases with both the Reynolds number and the Prandtl number. The result is also a rare similarity solution of the free convection and Navier-Stokes equations.     

Approximate analytical solutions of a class of boundary layer equations over nonlinear stretching surface

Third order nonlinear ordinary differential equations, subject to appropriate boundary conditions arising in fluid dynamics, are solved using three different methods viz., the Dirichlet series, method of stretching of variables, and asymptotic function method. Similarity transformations are used to convert the governing partial differential equations into nonlinear ordinary differential equations. The numerical results obtained from the above methods for various problems are given in terms of skin friction. Our study revealed that the results obtained from these methods agree well with those of direct numerical simulation of ordinary differential equations. Also, these methods have advantages over pure numerical methods in obtaining derived quantities such as velocity profile accurately for various values of the parameters at a stretch.     

Flow,heat, and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer

In this paper, we investigate the flow, heat and mass transfer of a viscous fluid flow over a stretching sheet by including the blowing effects of mass transfer under high flux conditions. Mass transfer in this work means species transfer and is different from mass transpiration for permeable walls. The new contribution from this work is, for the first time, to consider the coupled blowing effects from massive species transfer on flow, heat, and species transfer for a stretching plate. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy and mass transfer equations are solved exactly and the effects of the blowing parameter, the Schmidt number, and the Prandtl number on the flow, heat and mass transfer are presented and discussed. The solution is given in terms of an incomplete Gamma function. It is found the coupled blowing effects due to mass transfer can have significant influences on velocity profiles, drag, heat flux, as well as temperature and concentration profiles. These solutions provide rare results with closed form analytical expressions and can be used as benchmark problem for numerical code validation.     

Strong solutions to a nonlinear fluid structure interaction system

In this paper, we prove the existence of local-in-time smooth solutions to the nonlinear fluid structure interaction model first introduced in [J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969] and considered in [V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55-82; V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J. 57 (3) (2008) 1173-1207]. In particular, the strong solutions here are obtained given initial datum for the Navier-Stokes equation in the space H¹, and initial data for the wave equation w₀ and w₁ in the spaces H²(Ωe) and H¹(Ωe) respectively.     

A new family of unsteady boundary layers over a stretching surface

In this paper, a new family of unsteady boundary layers over a stretching flat surface was proposed and studied. This new class of unsteady boundary layers involves the flows over a constant speed stretching surface from a slot, and the slot is moving at a certain speed. Depending on the slot moving parameter, the flow can be treated as a stretching sheet problem or a shrinking sheet problem. Both the momentum and thermal boundary layers were studied. Under special conditions, the solutions reduce to the unsteady Rayleigh problem and the steady Sakiadis stretching sheet problem. Solutions only exist for a certain range of the slot moving parameter, α. Two solutions are found for −53.55° < α < −45°. There are also two solution branches for the thermal boundary layers at any given Prandtl number in this range. Compared with the upper solution branch, the lower solution branch leads to simultaneous reduction in wall drag and heat transfer rate. The results also show that the motion of the slot greatly affects the wall drag and heat transfer characteristics near the wall and the temperature and velocity distributions in the fluids.     

Analytical solution of magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface

A new kind of analytic technique, namely the homotopy analysis method (HAM), is employed to give an explicit analytical solution of the steady two-dimensional stagnation-point flow of an electrically conducting power-law fluid over a stretching surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. A uniform transverse magnetic field is applied normal to the surface. An explicit analytical solution is given by recursive formulae for the first-order power-law (Newtonian) fluid when the ratio of free stream velocity and stretching velocity is not equal to unity. For second and real order power-law fluids, an analytical approach is proposed for magnetic field parameter in a quite large range. All of our analytical results agree well with numerical results. The results obtained by HAM suggest that the solution of the problem under consideration converges.     

Exact solution of thermal radiation on MHD flow over a stretching porous sheet

The effect of radiation on MHD steady asymmetric flow of an electrically conducting fluid past a stretching porous sheet in the presence of radiation has been analyzed. Exact solutions for the velocity and temperature fields have been derived and the effects of radiation, magnetic, Prandtl number, wall temperature and suction (or injection) parameters have been studied with the help of graphs.     

Three-dimensional flow over a stretching surface in a viscoelastic fluid

This article looks at the hydrodynamic elastico-viscous fluid over a stretching surface. The equations governing the flow are reduced to ordinary differential equations, which are analytically solved by applying an efficient technique namely the homotopy analysis method (HAM). The solutions for the velocity components are computed. The numerical values of wall skin friction coefficients are also tabulated. The present HAM solution is compared with the known exact solution for the two-dimensional flow and an excellent agreement is found.     

Note on unsteady viscous flow on the outside of an expanding or contracting cylinder

In this paper, the viscous flow on the outside of an expanding or contracting cylinder is studied. The governing Navier-Stokes equations are transformed into a similarity equation, which is solved by a shooting method. The solution is an exact solution to the unsteady Navier-Stokes equations. Results show both trivial and non-trivial solutions. For trivial solutions, there is no axial flow induced during the cylinder expansion or contraction. However, for the non-trivial solutions which only exist for cylinder expansion, an axial flow is generated and its strength increases with the increase in expansion speed.     

Renormalized solutions to a system of type Navier-Stokes

The aim of this paper is to show the existence of weak-renormalized solutions to a system of type Navier-Stokes in the two-dimensional case. This system may be regarded as a modified version of the non-isothermal solidification problem with melt convection.     

Existence of solutions and Gevrey class regularity for Leray-alpha equations

In this paper we consider some equations similar to Navier-Stokes equations, the three-dimensional Leray-alpha equations with space periodic boundary conditions. We establish the regularity of the equations by using the classical Faedo-Galerkin method. Our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the Leray-alpha equations, furthermore, the strong solutions are analytic in time with values in the Gevrey class of functions (for the space variable). The relations between the Leray-alpha equations and the Navier-Stokes equations are also considered.     

Numerical methods for a coupled system of differential equations arising from a thermal ignition problem

This article is concerned with monotone iterative methods for numerical solutions of a coupled system of a first‐order partial differential equation and an ordinary differential equation which arises from fast‐igniting catalytic converters in automobile engineering. The monotone iterative scheme yields a straightforward marching process for the corresponding discrete system by the finite‐difference method, and it gives not only a computational algorithm for numerical solutions of the problem but also the existence and uniqueness of a finite‐difference solution. Particular attention is given to the “finite‐time” blow‐up property of the solution. In terms of minimal sequence of the monotone iterations, some necessary and sufficient conditions for the blow‐up solution are obtained. Also given is the convergence of the finite‐difference solution to the continuous solution as the mesh size tends to zero. Numerical results of the problem, including a case where the continuous solution is explicitly known, are presented and are compared with the known solution. Special attention is devoted to the computation of the blow‐up time and the critical value of a physical parameter which determines the global existence and the blow‐up property of the solution. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Controllability of classical solutions implies controllability of weak solutions for a coupled system of wave equations and its application

In this paper, for a coupled system of one‐dimensional wave equations with Dirichlet boundary controls, we show that the controllability of classical solutions implies the controllability of weak solutions. This conclusion can be applied in proving some results that are hardly obtained by a direct way in the framework of classical solutions. For instance, we strictly derive the necessary conditions for the exact boundary synchronization by two groups in the framework of classical solutions for the coupled system of wave equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Positive solutions for a coupled system of nonlinear fractional differential equations

Using the Krasnoselskii–Zabreiko fixed point theorem, we establish two existence theorems for positive solutions of a coupled system of nonlinear fractional differential equations. Power functions and nonnegative matrices are used to characterize coupling behavior of our nonlinearities, so nonlinearities may grow differently; in fact, one may grow superlinearly, and the other may grow sublinearly. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives

In this paper, by introducing the fractional derivative in the sense of Caputo, the Adomian decomposition method is directly extended to study the coupled Burgers equations with time- and space-fractional derivatives. As a result, the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good accuracy of the proposed method.