Convergence of the homotopy analysis method

The homotopy analysis method is studied in the present paper. The question of convergence of the homotopy analysis method is resolved. It is proven that under a special constraint the homotopy analysis method does converge to the exact solution of the sought solution of nonlinear ordinary or partial differential equations. An optimal value of the convergence control parameter is given through the square residual error. An error estimate is also provided. Examples, including the Blasius flow, clearly demonstrate why and on what interval the corresponding homotopy series generated by the homotopy analysis method will converge to the exact solution.Comment: 12 pages, 4 combined figure

Homotopy analysis of magnetohydrodynamic convection flow in manufacture of a viscoelastic fabric for space applications

Aerospace electro-conductive polymer materials are a new family of “smart” materials being deployed in many complex applications. The precision manufacturing of such processes to manipulate properties and enhance performance can exploit magnetohydrodynamic (MHD) control and simultaneous heat transfer (thermal processing). Motivated by these applications, we develop a model for laminar free convective flow of an incompressible and electrically-conducting viscoelastic fluid (Walters’ liquid B) over a continuously moving stretching surface embedded in a porous medium in the presence of strong radiative heat flux, as a simulation of magnetic smart fabric sheet processing. A heat generation/absorption term is included in the model. Darcy’s law is used to simulate porous media bulk drag effects. The stretching is assumed to be a linear function of the coordinate along the direction of stretching. Using similarity transformations, the governing partial differential equations are converted to nonlinear ordinary differential equations. The energy equation is further rendered into confluent hypergeometric form and then solved analytically for the prescribed surface temperature (PST) case and also for the Prescribed Boundary Surface Heat Flux (PHF) case, using Kummer’s function, subject to physically realistic boundary conditions. The momentum and energy equations are also solved using the semi-numerical homotopy analysis method (HAM), which contains the auxiliary parameter , permitting relatively easy adjustment and control of the convergence region of the series solution. This method provides an efficient approximate analytical solution with high accuracy, minimal calculation, and avoidance of physically unrealistic assumptions. HAM solutions are benchmarked with robust numerical shooting quadrature and found to correlate well. The influence of magnetic field on velocity and temperature profiles is studied via the Chandrasekhar number (Q). Furthermore detailed simulations are conducted for the influence of viscoelastic parameter (k1), Eckert number (E), radiation-conduction parameter (NR), Grashof number (Gr) and heat source/sink parameter () on the flow variables. The study finds applications in electro-conductive polymeric materials processing for aerospace fabric covers and other applications with demanding safety and protection requirements in smart materials synthesis

Unsteady Boundary-Layer Flow over Jerked Plate Moving in a Free Stream of Viscoelastic Fluid

This study aims to investigate the unsteady boundary-layer flow of a viscoelastic non-Newtonian fluid over a flat surface. The plate is suddenly jerked to move with uniform velocity in a uniform stream of non-Newtonian fluid. Purely analytic solution to governing nonlinear equation is obtained. The solution is highly accurate and valid for all values of the dimensionless time 0≤τ<∞. Flow properties of the viscoelastic fluid are discussed through graphs

Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones

Abstract.: The boundary value problem for the similar stream function f = f(η;λ) of the Cheng-Minkowycz free convection flow over a vertical plate with a power law temperature distribution Tw(x) = T∞+Axλ in a porous medium is revisited. It is shown that in the λ-range −1/2 <λ < 0 , the well known exponentially decaying "first branch” solutions for the velocity and temperature fields are not some isolated solutions as one has believed until now, but limiting cases of families of algebraically decaying multiple solutions. For these multiple solutions well converging analytical series expressions are given. This result yields a bridging to the historical quarreling concerning the feasibility of exponentially and algebraically decaying boundary layers. Owing to a mathematical analogy, our results also hold for the similar boundary layer flows induced by continuous surfaces stretched in viscous fluids with power-law velocities uw(x)∼ x

Approximate Solutions for the Flow and Heat Transfer due to a Stretching Sheet Embedded in a Porous Medium with Variable Thickness, Variable Thermal Conductivity and Thermal Radiation using Laguerre Collocation Method

In this article, a numerical approach is given for studying the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with a power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by a non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing PDEs are transformed into a system of coupled non-linear ODEs which are using appropriate boundary conditions for various physical parameters. The proposed method is based on replacement of the unknown function by truncated series of well known Laguerre expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equations by means of collocation points to a system of algebraic equations with Laguerre coefficients. Thus, by solving this system of equations, the Laguerre coefficients are obtained. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. Comparison of obtained numerical results is made with previously published results in some special cases. The results attained in this paper confirm the idea that the proposed method is powerful mathematical tool and it can be applied to a large class of nonlinear problems arising in different fields of science and engineering

MHD free convective flow past a vertical plate

The free convective flow in incompressible viscous fluid past a vertical plate is studied under the presence of magnetic field. The flow is considered along the vertical plate at x-axis in upward direction and y-axis is taken normal to it. The governing equations are written in vector form. Afterwards, the equations are solved numerically using finite element method with automated solution techniques. Later, the effects of magnetic field strength to the velocity and temperature of the fluid are obtained. It is found that for heated plate, the velocity and the temperature of the fluid decreases when the magnetic field strength increases. Meanwhile for cooled plate, the velocity decreases but the temperature increases when the magnetic field strength increases

DETC2010-28089 HOMOTOPY ANALYSIS METHOD FOR MULTI-DEGREE-OF-FREEDOM NONLINEAR DYNAMICAL SYSTEMS

ABSTRACT In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well

Flow over Exponentially Stretching Sheet through Porous Medium with Heat Source/Sink

An attempt has been made to study the heat and mass transfer effect in a boundary layer MHD flow of an electrically conducting viscous fluid subject to transverse magnetic field on an exponentially stretching sheet through porous medium. The effect of thermal radiation and heat source/sink has also been discussed in this paper. The governing nonlinear partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations and then solved numerically using a fourth-order Runge-Kutta method with a shooting technique. Graphical results are displayed for nondimensional velocity, temperature, and concentration profiles while numerical values of the skin friction local Nusselt number and Sherwood number are presented in tabular form for various values of parameters controlling the flow system

Magneto-bioconvection flow of a Casson thin film with nanoparticles over an unsteady stretching sheet : HAM and GDQ computation

Purpose – To numerically investigate the two-dimensional unsteady laminar magnetohydrodynamic (MHD) bioconvection flow and heat transfer of an electrically-conducting non-Newtonian Casson thin film with uniform thickness over a horizontal elastic sheet emerging from a slit in the presence of viscous dissipation. The composite effects of variable heat, mass, nanoparticle volume fraction and gyrotactic micro-organism flux are considered as is hydrodynamic (wall) slip. The Buongiorno nanoscale model is deployed which features Brownian motion and thermophoretic effects. The model studies the manufacturing fluid dynamics of smart magnetic bio-nano-polymer coatings. Design/Methodology/Approach – The coupled non-linear partial differential boundary-layer equations governing the flow, heat and nano-particle and micro-organism mass transfer are reduced to a set of coupled non-dimensional equations using the appropriate transformations and then solved as an nonlinear boundary value problem with the semi-numerical Liao homotopy analysis method (HAM).Validation with a generalized differential quadrature (GDQ) numerical technique is included. Findings – An increase in velocity slip results in a significant decrement in skin friction coefficient and Sherwood number whereas it generates a substantial enhancement in Nusselt number and motile micro-organism number density. The computations reveal that the bioconvection Schmidt number decreases the micro-organism concentration and boundary-layer thickness which is attributable to a rise in viscous diffusion rate. Increasing bioconvection Péclet number substantially elevates the temperatures in the regime, thermal boundary layer thickness, nanoparticle concentration values and nano-particle species boundary layer thickness. The computations demonstrate the excellent versatility of HAM and GDQ in solving nonlinear multi-physical nanobioconvection flows in thermal sciences and furthermore are relevant to application in the synthesis of smart biopolymers, microbial fuel cell coatings etc. Originality/Value – The originality of the study is to address the simultaneous effects of unsteady and variable surface fluxes on Casson nanofluid transport of gyrotactic bio-convection thin film over a stretching sheet in the presence of a transverse magnetic field. Validation of HAM with a generalized differential quadrature (GDQ) numerical technique is included. Keywords – Magneto-hydrodynamics, Bioconvection, Nanofluid, Brownian motion, Homotopy analysis method (HAM), Generalized differential quadrature (GDQ

Explicit Solutions of a Gravity-Induced Film Flow along a Convectively Heated Vertical Wall

The gravity-driven film flow has been analyzed along a vertical wall subjected to a convective boundary condition. The Boussinesq approximation is applied to simplify the buoyancy term, and similarity transformations are used on the mathematical model of the problem under consideration, to obtain a set of coupled ordinary differential equations. Then the reduced equations are solved explicitly by using homotopy analysis method (HAM). The resulting solutions are investigated for heat transfer effects on velocity and temperature profiles