On Three-Dimensional Flow and Heat Transfer over a Non-Linearly Stretching Sheet: Analytical and Numerical Solutions

This article studies the viscous flow and heat transfer over a plane horizontal surface stretched non-linearly in two lateral directions. Appropriate wall conditions characterizing the non-linear variation in the velocity and temperature of the sheet are employed for the first time. A new set of similarity variables is introduced to reduce the boundary layer equations into self-similar forms. The velocity and temperature distributions are determined by two methods, namely (i) optimal homotopy analysis method (OHAM) and (ii) fourth-fifth-order Runge-Kutta integration based shooting technique. The analytic and numerical solutions are compared and these are found in excellent agreement. Influences of embedded parameters on momentum and thermal boundary layers are sketched and discussed.     

Formulation and Application of Optimal Homotopty Asymptotic Method to Coupled Differential - Difference Equations

In this paper we applied a new analytic approximate technique Optimal Homotopy Asymptotic Method (OHAM) for treatment of coupled differential- difference equations (DDEs). To see the efficiency and reliability of the method, we consider Relativistic Toda coupled nonlinear differential-difference equation. It provides us a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier and explicit.     

Heat Transfer Analysis of MHD Thin Film Flow of an Unsteady Second Grade Fluid Past a Vertical Oscillating Belt

This article aims to study the thin film layer flowing on a vertical oscillating belt. The flow is considered to satisfy the constitutive equation of unsteady second grade fluid. The governing equation for velocity and temperature fields with subjected initial and boundary conditions are solved by two analytical techniques namely Adomian Decomposition Method (ADM) and Optimal Homotopy Asymptotic Method (OHAM). The comparisons of ADM and OHAM solutions for velocity and temperature fields are shown numerically and graphically for both the lift and drainage problems. It is found that both these solutions are identical. In order to understand the physical behavior of the embedded parameters such as Stock number, frequency parameter, magnetic parameter, Brinkman number and Prandtl number, the analytical results are plotted graphically and discussed.     

MHD Free Convective Boundary Layer Flow of a Nanofluid past a Flat Vertical Plate with Newtonian Heating Boundary Condition

Steady two dimensional MHD laminar free convective boundary layer flows of an electrically conducting Newtonian nanofluid over a solid stationary vertical plate in a quiescent fluid taking into account the Newtonian heating boundary condition is investigated numerically. A magnetic field can be used to control the motion of an electrically conducting fluid in micro/nano scale systems used for transportation of fluid. The transport equations along with the boundary conditions are first converted into dimensionless form and then using linear group of transformations, the similarity governing equations are developed. The transformed equations are solved numerically using the Runge-Kutta-Fehlberg fourth-fifth order method with shooting technique. The effects of different controlling parameters, namely, Lewis number, Prandtl number, buoyancy ratio, thermophoresis, Brownian motion, magnetic field and Newtonian heating on the flow and heat transfer are investigated. The numerical results for the dimensionless axial velocity, temperature and nanoparticle volume fraction as well as the reduced Nusselt and Sherwood number have been presented graphically and discussed. It is found that the rate of heat and mass transfer increase as Newtonian heating parameter increases. The dimensionless velocity and temperature distributions increase with the increase of Newtonian heating parameter. The results of the reduced heat transfer rate is compared for convective heating boundary condition and found an excellent agreement.     

Mathematical Modeling of a Carrier-Mediated Transport Process in a Liquid Membrane

An analysis of the reaction diffusion in a carrier-mediated transport process through a membrane is presented. A simple approximate analytical expression of concentration profiles is derived in terms of all dimensionless parameters. Furthermore, in this work we employ the homotopy perturbation method to solve the nonlinear reaction–diffusion equations. Moreover, the analytical results have been compared to the numerical simulation using the Matlab program. The simulated results are comparable with the appropriate theories. The results obtained in this work are valid for the entire solution domain.     

A perturbation model for the oscillatory flow of a Bingham plastic in rigid and periodically displaced tubes.

An approximate analytical model for the pulsatile flow of an ideal Bingham plastic fluid in both a rigid and a periodically displaced tube has been developed using regular perturbation methods. Relationships are derived for the velocity field and dimensionless flow rate. The solution compares adequately with available experimentally measured oscillatory non-Newtonian fluid flow data. These solutions provide useful analytical models supporting experimental and computation studies of arterial blood flow.     

An Analytical Model of the Counter-Current Heat Exchange Phenomena

An analytical model for the counter-current heat exchange mechanism in animals has been formulated and a solution has been obtained. The nondimensional parameters that govern the mechanism have been determined in terms of the properties of the animal. The normalized temperatures are functions of normalized distance and, in general, three nondimensional heat transfer conductances. Graphical results are presented for two representative physiological systems. These results allow a delineation of those situations in which counter-current heat transfer is important, and also a quantitative prediction of the heat transfer and temperature distributions. The theory is compared to the available experimental results.     

Uncertainty analysis for temperature prediction of biological bodies subject to randomly spatial heating

An analytical solution to the Pennes bioheat transfer equation in three-dimensional geometry with practical hyperthermia boundary conditions and random heating was obtained in this paper. Uncertainties for the predicted temperatures of tissues due to approximate parameters were studied based on analyzing one-dimensional heat transfer in the biological bodies subject to a spatially decay heating. Contributions from each of the thermal parameters such as heat conductivity, blood perfusion rate, and metabolic rate of the tissues, the scattering coefficient and the surface power flux of the heating apparatus were compared and the uncertainty limit for temperature distribution in this case was estimated. The results are useful in a variety of clinical hyperthermia and biological thermal parameter measurement.     

Double Diffusive Magnetohydrodynamic (MHD) Mixed Convective Slip Flow along a Radiating Moving Vertical Flat Plate with Convective Boundary Condition

In this study combined heat and mass transfer by mixed convective flow along a moving vertical flat plate with hydrodynamic slip and thermal convective boundary condition is investigated. Using similarity variables, the governing nonlinear partial differential equations are converted into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved using a semi-numerical/analytical method called the differential transform method and results are compared with numerical results. Close agreement is found between the present method and the numerical method. Effects of the controlling parameters, including convective heat transfer, magnetic field, buoyancy ratio, hydrodynamic slip, mixed convective, Prandtl number and Schmidt number are investigated on the dimensionless velocity, temperature and concentration profiles. In addition effects of different parameters on the skin friction factor, , local Nusselt number, , and local Sherwood number are shown and explained through tables.     

Modeling and Analysis of Unsteady Axisymmetric Squeezing Fluid Flow through Porous Medium Channel with Slip Boundary

The aim of this article is to model and analyze an unsteady axisymmetric flow of non-conducting, Newtonian fluid squeezed between two circular plates passing through porous medium channel with slip boundary condition. A single fourth order nonlinear ordinary differential equation is obtained using similarity transformation. The resulting boundary value problem is solved using Homotopy Perturbation Method (HPM) and fourth order Explicit Runge Kutta Method (RK4). Convergence of HPM solution is verified by obtaining various order approximate solutions along with absolute residuals. Validity of HPM solution is confirmed by comparing analytical and numerical solutions. Furthermore, the effects of various dimensionless parameters on the longitudinal and normal velocity profiles are studied graphically.     

Analysis of diffusion delay in a layered medium. Application to heat measurements from muscle.

An analysis is presented of diffusional delays in one-dimensional heat flow through a medium consisting of several layers of different materials. The model specifically addresses the measurement of heat production by muscle, but diffusion of solute or conduction of charge through a layered medium will obey the same equations. The model consists of a semi-infinite medium, the muscle, in which heat production is spacially uniform but time varying. The heat diffuses through layers of solution and insulation to the center of the thermal element where heat flow is zero. Using Laplace transforms, transfer functions are derived for the temperature change in the center of the thermopile as a function of the temperature at any interface between differing materials or as a function of heat production in the muscle. From these transfer functions, approximate analytical expressions are derived for the time constants which scale the early and late changes in the central temperature. We find that the earliest temperature changes are limited by the diffusivities of the materials, whereas the approach to steady state depends on the total heat capacity of the system and the diffusivity of muscle. Hill (1937) analyzed a similar geometry by modeling the layered medium as a homogeneous system with an equivalent half thickness. We show that his analysis was accurate for the materials in his system. In general, however, and specifically with regard to modern thermopiles, a homogeneous approximation will lead to significant errors.(ABSTRACT TRUNCATED AT 250 WORDS)     

Thermal and Concentration Stratifications Effects in Radiative Flow of Jeffrey Fluid over a Stretching Sheet

In this article we investigate the heat and mass transfer analysis in mixed convective radiative flow of Jeffrey fluid over a moving surface. The effects of thermal and concentration stratifications are also taken into consideration. Rosseland''s approximations are utilized for thermal radiation. The nonlinear boundary layer partial differential equations are converted into nonlinear ordinary differential equations via suitable dimensionless variables. The solutions of nonlinear ordinary differential equations are developed by homotopic procedure. Convergence of homotopic solutions is examined graphically and numerically. Graphical results of dimensionless velocity, temperature and concentration are presented and discussed in detail. Values of the skin-friction coefficient, the local Nusselt and the local Sherwood numbers are analyzed numerically. Temperature and concentration profiles are decreased when the values of thermal and concentration stratifications parameters increase. Larger values of radiation parameter lead to the higher temperature and thicker thermal boundary layer thickness.     

Optimal and Numerical Solutions for an MHD Micropolar Nanofluid between Rotating Horizontal Parallel Plates

The present analysis deals with flow and heat transfer aspects of a micropolar nanofluid between two horizontal parallel plates in a rotating system. The governing partial differential equations for momentum, energy, micro rotation and nano-particles concentration are presented. Similarity transformations are utilized to convert the system of partial differential equations into system of ordinary differential equations. The reduced equations are solved analytically with the help of optimal homotopy analysis method (OHAM). Analytical solutions for velocity, temperature, micro-rotation and concentration profiles are expressed graphically against various emerging physical parameters. Physical quantities of interest such as skin friction co-efficient, local heat and local mass fluxes are also computed both analytically and numerically through mid-point integration scheme. It is found that both the solutions are in excellent agreement. Local skin friction coefficient is found to be higher for the case of strong concentration i.e. n=0, as compared to the case of weak concentration n=0.50. Influence of strong and weak concentration on Nusselt and Sherwood number appear to be similar in a quantitative sense.     

A numerical study on dual-phase-lag model of bio-heat transfer during hyperthermia treatment

The success of hyperthermia in the treatment of cancer depends on the precise prediction and control of temperature. It was absolutely a necessity for hyperthermia treatment planning to understand the temperature distribution within living biological tissues. In this paper, dual-phase-lag model of bio-heat transfer has been studied using Gaussian distribution source term under most generalized boundary condition during hyperthermia treatment. An approximate analytical solution of the present problem has been done by Finite element wavelet Galerkin method which uses Legendre wavelet as a basis function. Multi-resolution analysis of Legendre wavelet in the present case localizes small scale variations of solution and fast switching of functional bases. The whole analysis is presented in dimensionless form. The dual-phase-lag model of bio-heat transfer has compared with Pennes and Thermal wave model of bio-heat transfer and it has been found that large differences in the temperature at the hyperthermia position and time to achieve the hyperthermia temperature exist, when we increase the value of τ_T. Particular cases when surface subjected to boundary condition of 1st, 2nd and 3rd kind are discussed in detail. The use of dual-phase-lag model of bio-heat transfer and finite element wavelet Galerkin method as a solution method helps in precise prediction of temperature. Gaussian distribution source term helps in control of temperature during hyperthermia treatment. So, it makes this study more useful for clinical applications.     

Heat Transfer in a Micropolar Fluid over a Stretching Sheet with Newtonian Heating

This article looks at the steady flow of Micropolar fluid over a stretching surface with heat transfer in the presence of Newtonian heating. The relevant partial differential equations have been reduced to ordinary differential equations. The reduced ordinary differential equation system has been numerically solved by Runge-Kutta-Fehlberg fourth-fifth order method. Influence of different involved parameters on dimensionless velocity, microrotation and temperature is examined. An excellent agreement is found between the present and previous limiting results.     

Numerical Solution to Generalized Burgers'-Fisher Equation Using Exp-Function Method Hybridized with Heuristic Computation

In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers''-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers''-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems.     

Boundary Layer Flow Past a Stretching Surface in a Porous Medium Saturated by a Nanofluid: Brinkman-Forchheimer Model

In this study, the steady forced convection flow and heat transfer due to an impermeable stretching surface in a porous medium saturated with a nanofluid are investigated numerically. The Brinkman-Forchheimer model is used for the momentum equations (porous medium), whereas, Bongiorno’s model is used for the nanofluid. Uniform temperature and nanofluid volume fraction are assumed at the surface. The boundary layer equations are transformed to ordinary differential equations in terms of the governing parameters including Prandtl and Lewis numbers, viscosity ratio, porous medium, Brownian motion and thermophoresis parameters. Numerical results for the velocity, temperature and concentration profiles, as well as for the reduced Nusselt and Sherwood numbers are obtained and presented graphically.     

On the steady laminar flow of a viscous incompressible fluid in an elastic tube

The conditions are examined under which an approximate relation between the radius of the tube and the distance along the axis, as obtained by N. Rashevsky (1945), is consistent with the assumptions made in the solution. These conditions are reduced to relations between two dimensionless parameters of the system. Second approximations are found for three different ranges of values of these parameters.     

An evaluation of the wind chill factor: its development and applicability.

The wind chill factor has become a standard meteorologic term in cold climates. Meteorologic charts provide wind chill temperatures meant to represent the hypothetical air temperature that would, under conditions of no wind, effect the same heat loss from unclothed human skin as does the actual combination of air temperature and wind velocity. As this wind chill factor has social and economic significance, an investigation was conducted on the development of this factor and its applicability based on modern heat transfer principles. The currently used wind chill factor was found to be based on a primitive study conducted by the U.S. Antarctic Service over 50 years ago. The resultant equation for the wind chill temperature assumes an unrealistic constant skin temperature and utilizes heat transfer coefficients that differ markedly from those obtained from equations of modern convective heat transfer methods. The combined effect of these two factors is to overestimate the effect of a given wind velocity and to predict a wind chill temperature that is too low.     

A parametric study of wind chill equivalent temperatures by a dimensionless steady-state analysis

A first order analytical approximation of steady-state heat conduction in a hollow cylinder exchanging heat at its external surface by convection with a cold and windy environment is presented. The model depicts the thermal behavior of certain body elements, e.g., head/face, when exposed to such environments. The results are presented by dimensionless parameters and facilitate the estimation of wind chill equivalent temperatures (WCETs). The effects of several variables on determining WCETs were studied using specific examples, leading to the following generalizations: (1) the conditions assumed for "calm" wind speed appear to be a dominant factor in determining WCET; (2) the effects, on both (skin) surface temperature and on WCET, of a 1°C change in environmental temperature appear to be more pronounced than those of a 1 m/s change in wind speed; (3) similarly, predicted WCETs are more sensitive to the geometrical dimensions assumed for the modeled entity than they are to wind speeds; and (4) tissue thermal conductivity, the angle at which the convective heat transfer coefficient is measured relative to wind direction, and the factor used to establish "effective" wind speeds in the domain occupied by humans relative to reported values, all seem to have relatively small effects on the determination of WCET. These conclusions strongly suggest, among other things, that for any given combination of environmental conditions, wind chill indices may best be presented as ranges rather than as single values. This seems to apply even when worst-case scenarios are considered. Also emphasized is the need for careful and realistic selection of all the parameter values used in the determination of WCETs.