Band 65, Heft 11

In the present investigation we have studied a tangent hyperbolic fluid in a uniform inclined tube. The governing equations are simplified using long wavelength and low Reynold number approximations. The solutions of the problem in simplified form are calculated with two methods namely (i) the perturbation method and (ii) the homotopy analysis method. The comparison of the solutions show a very good agreement between the two results. At the end of the article the expressions of the pressure rise and the frictional force are calculated with the help of numerical integration. The graphical results are presented to show the physical behaviour of Weissenberg number We, amplitude ratio φ , and tangent hyperbolic power law index n.

This paper outlines the conversion of partial differential equations (PDEs) into the corresponding ordinary differential equations (ODEs) by a complex transformation which is widely used in the exp-function method. The proposed homotopy perturbation method (HPM) is employed to solve the travelling wave solutions. Several examples are given to reveal the reliability and efficiency of the algorithm.

This article deals with the variable viscosity effects on the channel flow of a third-order fluid. The walls of the asymmetric channel are of different temperatures. Continuity, momentum, and energy equations are utilized in the mathematical analysis. Three types of solutions, namely, the perturbation, homotopy analysis, and numerical are derived. These solutions are compared. The perturbation, homtopy analysis, and numerical solutions are identical up to three digits. The expressions for pressure rise and frictional forces have been calculated using numerical integration. The expressions for pressure rise, frictional forces, velocity profile, temperature profile, pressure gradient, and streamlines have been discussed graphically at the end of the article.

In this paper, we define the new generalized difference sequence spaces c 0 (Δ m v ,M,u, p,q ), c (Δ m v , M ,u, p,q ), and l ∞(Δ m v ,M,u, p,q ).We also study some inclusion relations between these spaces

In this paper, the exp-function method is generalized to construct N -soliton solutions of nonlinear differential-difference equations. With the aid of symbolic computation, we choose the Toda lattice to illustrate the validity and advantages of the generalized work. As a result, 1-soliton, 2-soliton, and 3-soliton solutions are obtained, from which the uniform formula of N -soliton solutions is derived. It is shown that the exp-function method may provide us with a straightforward and effective mathematical tool for generating N -soliton solutions of nonlinear differential-difference equations in mathematical physics.

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

An analysis has been carried out to study the combined effects of heat and mass transfer on the unsteady flow of a micropolar fluid over a stretching sheet. The thermal radiation effects are presented. The arising nonlinear partial differential equations are first reduced to a set of nonlinear ordinary differential equations and then solved by the homotopy analysis method (HAM). Plots for various interesting parameters are presented and discussed. Numerical data for surface shear stress, Nusselt number, and Sherwood number in steady case are also tabulated. Comparison between the present and previous limiting results is given.

This investigation describes the effects of heat transfer on magnetohydrodynamic (MHD) axisymmetric flow of a viscous fluid between two radially stretching sheets. Navier-Stokes equations are transformed into the ordinary differential equations by utilizing similarity variables. Solution computations are presented by using the homotopy analysis method. The convergence of obtained solutions is checked. Skin friction coefficient and Nusselt number are given in tabular form. The dimensionless velocities and temperature are also analyzed for the pertinent parameters entering into the problem.

This paper is devoted to the study of a stagnation point flow of an incompressible third-order fluid towards a shrinking sheet (with heat transfer). The governing nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by means of a similarity transformation and then solved by the homotopy analysis method. Two types of flow problems, namely, (i) two dimensional stagnation flow toward a shrinking sheet and (ii) axisymmetric stagnation flow towards an axisymmetric shrinking surface have been discussed. Also, two types of boundary conditions are taken into account: (i) prescribed surface temperature (PST) and (ii) prescribed heat flux (PHF) case. The effects of various emerging parameters of non-Newtonian fluid have been investigated in detail and shown pictorically. The convergence of the solutions have been discussed through ¯ h -curves and residual error. For further validity, the homotopy Pad´e approximation is also applied.

The paper presents the analytical description of diffraction phenomena of sound at the opening of a two dimensional semi-infinite acoustically soft duct. This soft duct is symmetrically located inside an infinite duct with normal impedance boundary conditions in the case where the surface acoustic impedances of the upper and lower infinite plates are different from each other. A matrix Wiener- Hopf equation associated with a new canonical scattering problem is solved explicitly. A new kernel function arose for the problem and has been factorized. The graphical results are also presented which show how effectively the unwanted noise can be reduced by proper selection of different parameters.

The intercalation of dimethyl sulphoxide (DMSO), pyridine (Py), ethanolamine (Ea), and Nmethyl formamide (NMF) molecules into the kaolinite interlayers led to an appreciable decrease of 3697 cm − 1 of the hydroxyl band. The appearance of the peaks at 3662, 3541, and 3504 cm −1 proved that the DMSO species are intercalated between the kaolinite layers through forming H-bonds with internal-surface hydroxyl groups. The intensities of the 942 and 796 cm -1 bending peaks arising from inner-surface hydroxyls decreased and new vibrational features appeared due to the intercalation of the guest species. The d 001 value of pure kaolinite was found at 7.18 A° , and the d 001 values were seen at 11.26, 11.62, 10.77, and 10.67 °A for kaolinite-dimethyl sulphoxide (K-DMSO), kaolinite-pyridine (K-Py), kaolinite-ethanolamine (K-EA), and kaolinite-N-methyl formamide (K-NMF) composites, respectively. The endothermic differential thermal analysis (DTA) peaks at a temperature of 108 - 334 ◦ C reflected the changes in the physicochemical properties of the intercalated species. The thermal stability increase followed the order of K-Py < K-NMF < K-Ea < K-DMSO. Based on the thermal analysis data, the intercalation ratios of the composites above were determined as 80.0, 40.0, 81.6, and 82.0%, respectively. The specific surface areas are affected by the intercalation geometry of the composites within the gallery spacing. The surface areas of the K-DMSO, K-Py, and K-EA complexes increased whereas the surface area of K-NMF decreased with respect to that of untreated kaolinite.

Quantum-chemical calculations of the parameters of the n M + ・ NbClF 2 − 6 type particles have been performed, where M stands for Na, K, Cs and n = 0 - 6. Under certain conditions such particles may exist in melts of alkali metal chlorides.Within the framework of this approximation, compositions for themost stable particles in molten salts were obtained. Relative stability of the particles containing the NbF − 6 , NbClF 2 − 6 , and NbF 2 − 7 complexes has been calculated. Energies and some other characteristics of the electron structure and the particle geometry structure were determined depending on the n and M values. For estimation of the third sphere’s influence, the systems (M 2 NbF 7 +8MCl) and (M 2 NbF 7 +15MCl) are surveyed. The chlorine anion enters the first coordination sphere in the Nasystem only, i. e. in this system the true complex NbClF 3 − 7 is formed.