Finite element solution of double-diffusive boundary layer flow of viscoelastic nanofluids over a stretching sheet

This paper deals with the double-diffusive boundary layer flow of non-Newtonian nanofluid over a stretching sheet. In this model, where binary nanofluid is used, the Brownian motion and thermophoresis are classified as the main mechanisms which are responsible for the enhancement of the convection features of the nanofluid. The boundary layer equations governed by the partial differential equations are transformed into a set of ordinary differential equations with the help of group theory transformations. The variational finite element method (FEM) is used to solve these ordinary differential equations. We have examined the effects of different controlling parameters, namely, the Brownian motion parameter, the thermophoresis parameter, modified Dufour number, viscoelastic parameter, Prandtl number, regular Lewis number, Dufour Lewis number, and nanofluid Lewis number on the flow field and heat transfer characteristics. Graphical display of the numerical examine are performed to illustrate the influence of various flow parameters on the velocity, temperature, concentration, reduced Nusselt, reduced Sherwood and reduced nanofluid Sherwood number distributions. The present study has many applications in coating and suspensions, movement of biological fluids, cooling of metallic plate, melt-spinning, heat exchangers technology, and oceanography.     

Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: A Lie group analysis

In this paper, heat and mass transfer analysis for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/blowing is investigated. The governing partial differential equation and auxiliary conditions are converted to ordinary differential equations with the corresponding auxiliary conditions via Lie group analysis. The boundary layer temperature, concentration and nanoparticle volume fraction profiles are then determined numerically. The influences of various relevant parameters, namely, thermophoresis parameter Nt, Brownian motion parameter Nb, Lewis number Le, suction/injection parameter S, permeability parameter k₁, source/sink parameter λ and Prandtl parameter Pr on temperature and concentration as well as wall heat flux and wall mass flux are discussed. Comparison with published results is presented.     

Effects of heat generation/absorption on stagnation point flow of nanofluid over a surface with convective boundary conditions

Analysis has been conducted to analyze the stagnation point flow of nanofluid near a permeable stretched surface with convective boundary condition. The relevant problem formulation is presented in the presence of porous medium and internal heat generation/absorption. The effects of Brownian motion and thermophoresis occur in the transport equations. The velocity, temperature and nanoparticle concentration profiles are analyzed with respect to the involved parameters of interest namely Brownian motion parameters, thermophoresis parameter, permeability parameter, source/sink parameter, ratio of rate constants to free stream velocity and stretching velocity, Biot number and Prandtl number. A comparative study between the previous published and present results in a limiting sense is found in an excellent agreement.     

热泳和Brown运动对太阳能辐射下热分层纳米流体边界层流动的影响

在太阳辐射下的纳米流体中,数值地研究竖向延伸壁面具有可变流条件时的层流运动.使用的纳米流体模型为,在热分层中综合考虑了Brown运动和热泳的影响.应用一个特殊形式的Lie群变换,即缩放群变换,得到相应边值问题的对称群.对平移对称群得到一个精确解,对缩放对称群得到数值解.数值解依赖于Lewis数、Brown运动参数、热分层参数和热泳参数.得到结论:上述参数明显地影响着流场、温度和纳米粒子体积率的分布.显示出纳米流体提高了基流体热传导率和对流的热交换性能,基流体中的纳米粒子还具有改善液体辐射性能的作用,直接提高了太阳能集热器的吸热效率.     

Lie symmetry group analysis of magnetic field effects on free convective flow of a nanofluid over a semi-infinite stretching sheet

We investigate the steady two-dimensional flow of an incompressible water based nanofluid over a linearly semi-infinite stretching sheet in the presence of magnetic field numerically. The basic boundary layer equations for momentum and heat transfer are non-linear partial differential equations. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The dimensionless governing equations for this investigation are solved numerically using Nachtsheim–Swigert shooting iteration technique together with fourth order Runge–Kutta integration scheme. Effects of the nanoparticle volume fraction ϕ, magnetic parameter M, Prandtl number Pr on the velocity and the temperature profiles are presented graphically and examined for different metallic and non-metallic nanoparticles. The skin friction coefficient and the local Nusselt number are also discussed for different nanoparticles.     

Effect of suction/injection on free convection along a vertical plate in a nanofluid saturated non-Darcy porous medium with internal heat generation

The effect of internal heat generation on free convection along a vertical plate embedded in a nanofluid saturated non-Darcy porous medium in the presence of suction/injection is analyzed. The non-linear governing equations and their associated boundary conditions are initially cast into dimensionless forms by non-dimensional variables. The resulting equations are solved numerically by an accurate, implicit, iterative finite-difference methodology and the obtained results are compared favorably with previously published work. A parametric study is performed to illustrate influence of the temperature exponent, non-Darcy, suction/injection, Brownian motion and thermophoresis parameters on the profiles of the velocity components, temperature and nanoparticle volume fraction. The numerical data for the heat and nanoparticle mass transfer rates have been tabulated for various parametric conditions.     

Microorganisms swimming through radiative Sutterby nanofluid over stretchable cylinder: Hydrodynamic effect

In the present article, radiative Sutterby nanofluid flow over a stretchable cylinder is considered. The suspended swimming microorganisms have been deliberated in the fluid analysis. Different processes such as Brownian motion, thermophoresis, Joules heating, and viscous dissipation have been inspected in the presences of stratification parameters. The solutions for flow profiles have been obtained via optimal homotopy analysis method. Impacts of different physical involved variables on non-dimensional velocity, temperature, nanofluid concentration, and concentration of density of swimming microorganisms have been debated. Coefficient of skin friction, local Nusselt number, Sherwood number, and density of motile organisms have been calculated. The results reveal that Sutterby fluid parameter enhances the skin friction and has a reverse impact on the velocity, while an increase in stratification causes a declination in the flow boundary layers. The temperature of the flow is also seen to be boosted by the increment in Brownian motion parameter. Analysis of entropy generation shows that the concentration difference parameter maximizes the entropy and minimizes the dimensionless Bejan number.     

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.     

饱和纳米流体多孔介质中竖直嵌入板上边界层的非Newton流

在层流条件下,对饱和多孔介质中的竖直板,研究幂指数型非Newton流的自由对流热交换.非Newton纳米流体服从幂指数型的数学模型,模型综合考虑了Brown运动和热泳的影响.通过相似变换,将问题的偏微分控制方程组,转化为常微分方程组,得到了常微分方程组的数值解.数值解依赖于幂指数n,Lewis数Le,浮力比Nr,Brown运动参数Nb,以及热泳参数Nt.在n和Le的不同取值下,研究并讨论了对相关流体性质参数的影响和简化的Nusselt数.     

Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model

Flow and thermal field in nanofluid is analyzed using single phase thermal dispersion model proposed by Xuan and Roetzel [Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707]. The non-dimensional form of the transport equations involving the thermal dispersion effect is solved numerically using semi-explicit finite volume solver in a collocated grid. Heat transfer augmentation for copper–water nanofluid is estimated in a thermally driven two-dimensional cavity. The thermo-physical properties of nanofluid are calculated involving contributions due to the base fluid and nanoparticles. The flow and heat transfer process in the cavity is analyzed using different thermo-physical models for the nanofluid available in literature. The influence of controlling parameters on convective recirculation and heat transfer augmentation induced in buoyancy driven cavity is estimated in detail. The controlling parameters considered for this study are Grashof number (10³ < Gr < 10⁵), solid volume fraction (0 < ? < 0.2) and empirical shape factor (0.5 < n < 6). Simulations carried out with various thermo-physical models of the nanofluid show significant influence on thermal boundary layer thickness when the model incorporates the contribution of nanoparticles in the density as well as viscosity of nanofluid. Simulations incorporating the thermal dispersion model show increment in local thermal conductivity at locations with maximum velocity. The suspended particles increase the surface area and the heat transfer capacity of the fluid. As solid volume fraction increases, the effect is more pronounced. The average Nusselt number from the hot wall increases with the solid volume fraction. The boundary surface of nanoparticles and their chaotic movement greatly enhances the fluid heat conduction contribution. Considerable improvement in thermal conductivity is observed as a result of increase in the shape factor.     

Exact multiplicity results for a class of two-point boundary value problems

This paper is concerned with the exact number of positive solutions for boundary value problems ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which the nonlinearity f is positive on (0,∞) and (p−1)f(u)−uf^′(u) changes sign from negative to positive.     

Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in presence of external magnetic field

This paper presents a mathematical analysis of MHD flow and heat transfer to a laminar liquid film from a horizontal stretching surface. The flow of a thin fluid film and subsequent heat transfer from the stretching surface is investigated with the aid of similarity transformation. The transformation enables to reduce the unsteady boundary layer equations to a system of non-linear ordinary differential equations. Numerical solution of resulting non-linear differential equations is found by using efficient shooting technique. Boundary layer thickness is explored numerically for some typical values of the unsteadiness parameter S and Prandtl number Pr, Eckert number Ec and Magnetic parameter Mn. Present analysis shows that the combined effect of magnetic field and viscous dissipation is to enhance the thermal boundary layer thickness.     

Exact number of positive solutions for semipositone problems

This paper is concerned with the exact number of positive solutions for the boundary value problem ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf^′(u) change sign exactly once from negative to positive on (0,∞).     

Entire functions that share a small function with their derivatives

Let f be an entire function of finite order and a an entire function of order less than f's. If f−a and f^′−a have the same zeros with the same multiplicities, then f^′−a≡c(f−a) for some non-zero constant c.     

Free convection heat and mass transfer in a power law fluid past an inclined surface with thermophoresis

The problem of heat and mass transfer in a power law, two-dimensional, laminar, boundary layer flow of a viscous incompressible fluid over an inclined plate with heat generation and thermophoresis is investigated by the characteristic function method. The governing non-linear partial differential equations describing the flow and heat transfer problem are transformed into a set of coupled non-linear ordinary differential equation which was solved using Runge–Kutta shooting method. Exact solutions for the dimensionless temperature and concentration profiles, are presented graphically for different physical parameters and for the different power law exponents 0 < n < 0.5 and for n > 0.5.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Similarity solutions for liquid metal systems near a sharply cornered conductive region

We study the boundary value problem (Pm,a): on (0,+∞), subject to the boundary conditions f(0)=a∈R, f^′(0)=−1 and f^′(+∞)=0. The problem arises in the study of similarity solutions for high frequency excitation of liquid metal systems in an antisymmetric magnetic field. We give a complete picture of solutions of (Pm,a) for the physical interesting case: m<−1 and a?0.     

On a ramification bound of torsion semi-stable representations over a local field

Let p be a rational prime, k be a perfect field of characteristic p, W=W(k) be the ring of Witt vectors, K be a finite totally ramified extension of Frac(W) of degree e and r be a non-negative integer satisfying r<p−1. In this paper, we prove the upper numbering ramification group for j>u(K,r,n) acts trivially on the pn-torsion semi-stable GK-representations with Hodge-Tate weights in {0,…,r}, where u(K,0,n)=0, u(K,1,n)=1+e(n+1/(p−1)) and u(K,r,n)=1−p−n+e(n+r/(p−1)) for 1<r<p−1.     

On the structure of similarity solutions of a differential equation arising from boundary layer problem

We consider an ordinary differential equation with f(0)=a, f^′(0)=1, f^′(∞):=limt→∞f^′(t)=0, where β is a real constant. The given problem may arise from the study of steady free convection flow over a vertical semi-infinite flat plate in a porous medium, or the study of a boundary layer flow over a vertical stretching wall. In this paper, the structure of solutions for the cases of β?−2 is studied. Combining the results of [B. Brighi, T. Sari, Blowing-up coordinates for a similarity boundary layer equation, Discrete Contin. Dyn. Syst. 5 (2005) 929-948; J.-S. Guo, J.-C. Tsai, The structure of solution for a third order differential equation in boundary layer theory, Japan J. Indust. Appl. Math. 22 (2005) 311-351; J.-C. Tsai, Similarity solutions for boundary layer flows with prescribed surface temperature, Appl. Math. Lett. 21 (1) (2008) 67-73], we conclude that the given problem may possess at most two types solutions for β∈R. Moreover, multiple solutions are also verified for various pairs of (a,β).