Application of He's methods for laminar flow in a porous‐saturated pipe

In this paper the problem of fully developed laminar steady forced convection inside a porous‐saturated pipe with uniform wall temperature is presented and the homotopy perturbation method (HPM) and the variational iteration method (VIM) are employed to solve the differential equations governing the problem. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and science. Copyright © 2009 John Wiley & Sons, Ltd.     

Couple stress fluid flow with variable properties: A second law analysis

The present work examines the combined influence of variable thermal conductivity and viscosity on the irreversibility rate in couple stress fluid flow in between asymmetrically heated parallel plates. The dimensionless fluid equations are solved by using homotopy analysis method (HAM) and validated with Runge‐Kutta shooting method (RKSM). The convergent series solution is then used for the irreversibility analysis in the flow domain. The effects of thermal conductivity and viscosity variation parameters, couple stress parameter, Reynolds number, Grashof number, Hartmann number on the velocity profile, temperature distribution, entropy production, and heat irreversibility ratio are presented through graphs, and salient features of the solutions are discussed. The computations show that the entropy production rate decreases with increased magnetic field and thermal conductivity parameters, whereas it rises with increasing values of couple stress parameter, Brinkman number, viscosity variation parameter, and Grashof number. The study is relevant to lubrication theory.     

Mining the discrete velocity method for high quality solutions for one-dimensional Poiseuille flow

A fully discretized solution for Poiseuille flow in a one-dimensional channel is presented. Unlike previous semi-analytical methods, such as the Analytical Discrete-Ordinates (ADO) or the FN methods, which have been specifically designed to avoid spatial discretization error, no analytical advantage is assumed. Instead, the solution is “mined” in a process where each discrete approximation is an element in a sequence of solutions whose convergence to the solution is accelerated. This process leads most straightforwardly to high quality benchmark results for use in algorithm verification with a minimum of theoretical and numerical complexity.     

Flow of a non‐linear (density‐gradient‐dependent) viscous fluid with heat generation,viscous dissipation and radiation

In this paper, we study the flow of a compressible (density‐gradient‐dependent) non‐linear fluid down an inclined plane, subject to radiation boundary condition. The convective heat transfer is also considered where a source term, similar to the Arrhenius type reaction, is included. The non‐dimensional forms of the equations are solved numerically and the competing effects of conduction, dissipation, heat generation and radiation are discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of semi‐analytic methods for the Fitzhugh–Nagumo equation,which models the transmission of nerve impulses

In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Fitzhugh–Nagumo equation. Numerical solutions obtained by these methods when compared with the exact solutions reveal that the obtained solutions produce high accurate results. The results show that the HPM, the VIM and the ADM are of high accuracy and are efficient for solving the Fitzhugh–Nagumo equation. Also the results demonstrate that the introduced methods are powerful tools for solving the nonlinear partial differential equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Global well‐posedness for the 2D Boussinesq system with variable viscosity and damping

In this paper, we consider the 2D Boussinesq system with variable kinematic viscosity in the velocity equation and with weak damping effect to instead of the regularity effect for the thermal conductivity. Even if without thermal diffusion in the temperature equation, we establish the global well‐posedness for the 2D Boussinesq system with general initial data.     

Effects of variable viscosity on pulsatile flow of blood in a tapered stenotic flexible artery

The objective of the present study is to investigate the effects of variable viscosity on incompressible laminar pulsatile flow of blood through an overlapping doubly constricted tapered artery. To mimic the realistic situation, wall of the artery is taken to be flexible, and physiologically relevant pulsatile flow is introduced. The governing equations of blood flow are made dimensionless. A coordinate transformation is used to make the overlapping doubly constricted wall geometry of tube to a straight tube. Taking advantage of the Stream function–Vorticity formulation, the system of partial differential equations is then solved numerically by finite difference approximations. Effects of Reynolds number, Strouhal number, degree of contraction, tapering angle, and viscosity parameters are presented graphically and analyzed. The results show that formation of stenosis and tapering disturb the flow field significantly, and degree of stenosis is more important in influencing blood flow compared with tapering.     

Numerical study of homotopy‐perturbation method applied to Burgers equation in fluid

In this article, the problem of Burgers equation is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. Comparison is made between the HPM and Exact solutions. The obtained solutions, in comparison with the exact solutions, admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

The influence of variable viscosity and viscous dissipation on the non-Newtonian flow: An analytical solution

An analytical solution for the flow of a third grade fluid in a pipe is obtained using homotopy analysis method (HAM). The fluid considered is with variable space dependence viscosity. The temperature of the pipe is taken to be higher than the temperature of the fluid. Expressions for velocity and temperature profiles are constructed analytically and explained with the help of graphs.     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect

In this paper, we obtain approximate‐analytical solutions of a cancer chemotherapy effect model involving fractional derivatives with exponential kernel and with general Mittag‐Leffler function. Laplace homotopy perturbation method and the modified homotopy analysis transform method were applied. The first method is based on a combination of the Laplace transform and homotopy methods, while the second method is an analytical technique based on homotopy polynomial. The cancer chemotherapy effect equations are solved numerically and analytically using the aforesaid methods. Illustrative examples are included to demonstrate the validity and applicability of the presented technique with new fractional‐order derivatives with exponential decay law and with general Mittag‐Leffler law.     

Homotopy perturbation method for numerical solutions of KdV‐Burgers' and Lax's seventh‐order KdV equations

In this article, we applied homotopy perturbation method to obtain the solution of the Korteweg‐de Vries Burgers (for short, KdVB) and Lax's seventh‐order KdV (for short, LsKdV) equations. The numerical results show that homotopy perturbation method can be readily implemented to this type of nonlinear equations and excellent accuracy. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Heat transfer and Couette flow of a chemically reacting non‐linear fluid

In this paper we study the flow and heat transfer in a chemically reacting non‐linear fluid between two long horizontal parallel flat plates that are at different temperatures. The top plate is sheared, whereas the bottom plate is fixed. The fluid is modeled as a generalized power‐law fluid whose viscosity is also assumed to be a function of the concentration. The effects of radiation are neglected. The equations are made dimensionless and the boundary value problem is solved numerically; the velocity and temperature profiles are obtained for various dimensionless numbers. Published in 2009 by John Wiley & Sons, Ltd.     

On the effective viscosity of a periodic suspension – analysis of primal and dual formulations for Newtonian and non‐Newtonian solvents

Computer simulations of the injection molding process of fiber‐reinforced plastics critically depend on the accuracy of the constitutive models. Of prime importance for the process simulation is the precise knowledge of the viscosity. Industrial applications generally feature both high shear rates and high fiber volume fractions. Thus, both the shear‐thinning behavior of the melt and the strong anisotropic effects induced by the fibers play a dominant role. Unfortunately, the viscosity cannot be determined experimentally in its full anisotropy, and analytical models cease to be accurate for the high fiber volume fractions in question. Computing the effective viscosity by a simplified homogenization approach serves as a possible remedy. This paper is devoted to the analysis of a cell problem determining the effective viscosity. We provide primal as well as dual formulations and prove corresponding existence and uniqueness theorems for Newtonian and Carreau fluids in suitable Sobolev spaces. In the Newtonian regime, the primal formulation leads to a saddle point problem, whereas a dual formulation can be obtained in terms of a coercive and symmetric bilinear form. This observation has deep implications for numerical formulations. As a by‐product, we obtain the invertibility of the effective viscosity, considered as a function, mapping the macroscopic shear rate to the macroscopic shear stress. Copyright © 2016 John Wiley & Sons, Ltd.     

The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

In this paper, the steady‐state Oseen viscous flow equations past a known or unknown obstacle are solved numerically using the method of fundamental solutions (MFS), which is free of meshes, singularities, and numerical integrations. The direct problem is linear and well‐posed, whereas the inverse problem is nonlinear and ill‐posed. For the direct problem, the MFS computations of the fluid flow characteristics (velocity, pressure, drag, and lift coefficients) are in very good agreement with the previously published results obtained using other methods for the Oseen flow past circular and elliptic cylinders, as well as past two circular cylinders. In the inverse obstacle problem the boundary data and the internal measurement of the fluid velocity are minimized using the MATLAB^© optimization toolbox lsqnonlin routine. Regularization was found necessary in the case the measured data are contaminated with noise. Numerical results show accurate and stable reconstructions of various star‐shaped obstacles of circular, bean, or peanut cross‐section.     

Numerical solution of the free‐surface viscous flow on a horizontal rotating elliptical cylinder

The numerical solution of the free‐surface fluid flow on a rotating elliptical cylinder is presented. Up to the present, research has concentrated on the circular cylinder for which steady solutions are the main interest. However, for noncircular cylinders, such as the ellipse, steady solutions are no longer possible, but there will be periodic solutions in which the solution is repeated after one full revolution of the cylinder. It is this new aspect that makes the investigation of noncircular cylinders novel. Here we consider both the time‐dependent and periodic solutions for zero Reynolds number fluid flow. The numerical solution is expedited by first mapping the fluid film domain onto a rectangle such that the position of the free‐surface is determined as part of the solution. For the time‐dependent case a simple time‐marching method of lines approach is adopted. For the periodic solution the discretised nonlinear equations have to be solved simultaneously over a time period. The resulting large system of equations is solved using Newton's method in which the form of the Jacobian enables a straightforward decomposition to be implemented, which makes matrix inversion manageable. In the periodic case all derivatives have been approximated pseudospectrally with the time derivative approximated by a differentiation matrix which has been specially derived so that the weight of fluid is algebraically conserved. Of interest is the solution for which the weight of fluid is at its maximum possible value, and this has been obtained by increasing the weight until a consistency break‐down occurs. Time‐dependent solutions do not produce the periodic solution after a long time‐scale but have protuberances which are constantly appearing and disappearing. Periodic solutions exhibit spectral accuracy solutions and maximum supportable weight solutions have been obtained for ranges of eccentricity and angular velocity. The maximum weights are less than and approximately proportional to those obtained for the circular case. The shapes of maximum weight solutions is distinctly different from sub‐maximum weight solutions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Further results on the impulsive synchronization of uncertain complex‐variable chaotic delayed systems

Synchronization and control of nonlinear dynamical systems with complex variables has attracted much more attention in various fields of science and engineering. In this article, we investigate the problem of impulsive synchronization for the complex‐variable delayed chaotic systems with parameters perturbation and unknown parameters in which the time delay is also included in the impulsive moment. Based on the theories of adaptive control and impulsive control, synchronization schemes are designed to make a class of complex‐variable chaotic delayed systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Sufficient conditions are derived to synchronize the complex‐variable chaotic systems include delayed impulses. To illustrate the effectiveness of the proposed schemes, several numerical examples are given. © 2014 Wiley Periodicals, Inc. Complexity 21: 131–142, 2016     

Indirect boundary integral formulation for the solution of shear‐thinning flow inside single rotor mixers

An efficient indirect boundary integral formulation for the evaluation of inelastic non‐Newtonian shear‐thinning flows at low Reynolds number is presented in this article. The formulation is based on the solution of a homogeneous Stokes flow field and the use of a particular solution for the nonlinear non‐Newtonian terms that yields the complete solution to the problem. Matrix multiplications are reduced in comparison to other means of handling nonlinear terms in boundary integral formulations such as the dual reciprocity method. The iterative solution of the nonlinear system of equations has been performed with a modified Newton‐Raphson method obtaining accurate results for values of the power law index as low as 0.4 without domain partitioning. Geometries such as Couette flow and a typical industrial polymer mixer have been analyzed with the proposed method obtaining good results with a reduction in computational cost compared with other equivalent formulations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:1610–1627, 2011     

A study of the fractional‐order mathematical model of diabetes and its resulting complications

Diabetes is a worldwide problem that affects one of every 11 persons nowadays. The IDF Diabetes Atlas (Eighth edition, 2017) states that approximately 415 million people in the world are living with the disease and that this number will rise to 629 million by the year 2045. It is a very serious problem of the world. A major part of the world population is affected by this disease and its resulting complications. In this paper, we propose to investigate a fractional‐order model of diabetes and its resulting complications. The mathematical model's parameters define the population of diabetic patients and those who are diabetic with complications at a given time t. We have also discussed the existence, uniqueness, and stability of the fractional‐order model, which we consider here. We make use of the homotopy decomposition method (HDM) in order to solve the problem.     

Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.