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Analytical Solution of MHD Stagnation-Point Flow and Heat Transfer of Casson Fluid over a Stretching Sheet with Partial Slip  [PDF]
Samir Kumar Nandy
ISRN Thermodynamics , 2013, DOI: 10.1155/2013/108264
Abstract: This paper investigates the hydromagnetic boundary layer flow and heat transfer of a non-Newtonian Casson fluid in the neighborhood of a stagnation point over a stretching surface in the presence of velocity and thermal slips at the boundary. The governing partial differential equations are transformed into nonlinear ordinary differential equations using similarity transformations. The analytic solutions are developed by a homotopy analysis method (HAM). The results pertaining to the present study indicate that the flow and temperature fields are significantly influenced by Casson parameter ( ), the magnetic parameter , the velocity slip parameter , and the thermal slip parameter . An increase in the velocity slip parameter causes decrease in the flow velocity, while an increase in the value of the thermal slip parameter causes increase in the temperature of the fluid. It is also observed that the velocity at a point decreases with increase in . 1. Introduction The problems of flow and heat transfer in the boundary layer adjacent to a continuous moving surface have received great attention during the last decades owing to the abundance of practical applications in chemical and manufacturing processes, such as polymer extrusion, continuous casting of metals, glass fibre production, hot rolling of paper, and wire drawing. Sakiadis [1] was the first, among others, to investigate the flow behavior on continuous solid surface. Thereafter, numerous investigations were made on the flow and heat transfer over a stretching surface in different directions [2–8]. All the previous researchers restricted their analyses to flow and heat transfer for the Newtonian fluid. In recent years, it has been observed that a number of industrial fluids such as molten plastics, polymeric liquids, blood, food stuff, and slurries exhibit non-Newtonian fluid behavior. Different types of non-Newtonian fluids are viscoelastic fluid, couple stress fluid, micropolar fluid, power-law fluid, Casson fluid, and many others. Rajagopal et al. [9] and Siddappa and Abel [10] studied the flow of a viscoelastic fluid over a linear stretching sheet. Troy et al. [11], Lawrence and Rao [12], and McLeod and Rajagopal [13] discussed the problem of uniqueness/nonuniqueness of the flow of a non-Newtonian viscoelastic fluid over a stretching sheet. Rajagopal et al. [9] analyzed the solutions for the flow of viscoelastic fluid over a stretching sheet. This study was further generalized to investigate the flow of short memory fluid of type Walter's liquid B by several authors, such as Andersson [14],
Stagnation Point Flow of a Nanofluid toward an Exponentially Stretching Sheet with Nonuniform Heat Generation/Absorption  [PDF]
A. Malvandi,F. Hedayati,G. Domairry
Journal of Thermodynamics , 2013, DOI: 10.1155/2013/764827
Abstract: This paper deals with the steady two-dimensional stagnation point flow of nanofluid toward an exponentially stretching sheet with nonuniform heat generation/absorption. The employed model for nanofluid includes two-component four-equation nonhomogeneous equilibrium model that incorporates the effects of Brownian diffusion and thermophoresis simultaneously. The basic partial boundary layer equations have been reduced to a two-point boundary value problem via similarity variables and solved analytically via HAM. Effects of governing parameters such as heat generation/absorption λ, stretching parameter ε, thermophoresis , Lewis number Le, Brownian motion , and Prandtl number Pr on heat transfer and concentration rates are investigated. The obtained results indicate that in contrast with heat transfer rate, concentration rate is very sensitive to the abovementioned parameters. Also, in the case of heat generation , despite concentration rate, heat transfer rate decreases. Moreover, increasing in stretching parameter leads to a gentle rise in both heat transfer and concentration rates. 1. Introduction For years, many researchers have paid much attention to viscous fluid motion near the stagnation region of a solid body, where “body” corresponds to either fixed or moving surfaces in a fluid. This multidisciplinary flow has frequent applications in high speed flows, thrust bearings, and thermal oil recovery. Hiemenz [1] developed the first investigation in this field. He applied similarity transformation to collapse two-dimensional Navier-Stokes equations to a nonlinear ordinary differential one and then presented its exact solution. Extension of this study was carried out with a similarity solution by Homann [2] to the case of axisymmetric three-dimensional stagnation point flow. After these original studies, many researchers have put their attention on this subject [3–9]. Besides stagnation point flow, stretching surfaces have a wide range of applications in engineering and several technical purposes particularly in metallurgy and polymer industry, for instance, gradual cooling of continuous stretched metal or plastic strips which have multiple applications in mass production. Crane [10] was the first to present a similarity solution in the closed analytical form for steady two-dimensional incompressible boundary layer flow caused by the stretching plate whose velocity varies linearly with the distance from a fixed point on the sheet. The combination of stretching surface and stagnation point flow was analyzed by Yao et al. [11]. Different types fluids such
Unsteady Stagnation Point Flow and Heat Transfer over a Stretching/shrinking Sheet  [PDF]
N.M.A. Nik Long,M. Suali,A. Ishak,N. Bachok
Journal of Applied Sciences , 2011,
Abstract: This study investigates the problem of unsteady stagnation point flow and heat transfer over a stretching/shrinking sheet. The governing partial differential equations are converted into a system of nonlinear ordinary differential equations using a similarity transformation, before being solved numerically. Both stretching and shrinking cases are considered. It is found that dual solutions exist for the shrinking case while for the stretching case, the solution is unique. Moreover, it is found that the heat transfer rate at the surface increases as the stretching/shrinking parameter as well as the unsteadiness parameter increases.
Magneto Hydrodynamic Orthogonal Stagnation Point Flow of a Power-Law Fluid Toward a Stretching Surface  [PDF]
Manisha Patel, M.G Timol
American Journal of Computational Mathematics (AJCM) , 2011, DOI: 10.4236/ajcm.2011.12013
Abstract: Steady two dimensional MHD stagnation point flow of a power law fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation point. The fluid impinges on the surface is considered orthogonally. Numerical and analytical solutions are obtained for different cases.
MHD Stagnation-Point Flow of Casson Fluid and Heat Transfer over a Stretching Sheet with Thermal Radiation  [PDF]
Krishnendu Bhattacharyya
Journal of Thermodynamics , 2013, DOI: 10.1155/2013/169674
Abstract: The two-dimensional magnetohydrodynamic (MHD) stagnation-point flow of electrically conducting non-Newtonian Casson fluid and heat transfer towards a stretching sheet have been considered. The effect of thermal radiation is also investigated. Implementing similarity transformations, the governing momentum, and energy equations are transformed to self-similar nonlinear ODEs and numerical computations are performed to solve those. The investigation reveals many important aspects of flow and heat transfer. If velocity ratio parameter (B) and magnetic parameter (M) increase, then the velocity boundary layer thickness becomes thinner. On the other hand, for Casson fluid it is found that the velocity boundary layer thickness is larger compared to that of Newtonian fluid. The magnitude of wall skin-friction coefficient reduces with Casson parameter (β). The velocity ratio parameter, Casson parameter, and magnetic parameter also have major effects on temperature distribution. The heat transfer rate is enhanced with increasing values of velocity ratio parameter. The rate of heat transfer is enhanced with increasing magnetic parameter M for B > 1 and it decreases with M for B < 1. Moreover, the presence of thermal radiation reduces temperature and thermal boundary layer thickness. 1. Introduction In fluid dynamics the effects of external magnetic field on magnetohydrodynamic (MHD) flow over a stretching sheet are very important due to its applications in many engineering problems, such as glass manufacturing, geophysics, paper production, and purification of crude oil. The flow due to stretching of a flat surface was first investigated by Crane [1]. Pavlov [2] studied the effect of external magnetic field on the MHD flow over a stretching sheet. Andersson [3] discussed the MHD flow of viscous fluid on a stretching sheet and Mukhopadhyay et al. [4] presented the MHD flow and heat transfer over a stretching sheet with variable fluid viscosity. On the other hand, Fang and Zhang [5] reported the exact solution of MHD flow due to a shrinking sheet with wall mass suction. Bhattacharyya and Layek [6] showed the behavior of solute distribution in MHD boundary layer flow past a stretching sheet. Furthermore, many vital properties of MHD flow over stretching sheet were explored in various articles [7–12] in the literature. Several important investigations on the flow due to stretching/shrinking sheet are available in the literature [13–16]. Chiam [17] investigated the stagnation-point flow towards a stretching sheet with the stretching velocity of the plate being equal to
Unsteady Stagnation Point Flow and Heat Transfer over a Stretching/Shrinking Sheet with Suction or Injection
M. Suali,N. M. A. Nik Long,N. M. Ariffin
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/781845
Abstract: The unsteady stagnation point flow and heat transfer over a stretching/shrinking sheet with suction/injection is studied. The governing partial differential equations are converted into nonlinear ordinary differential equations using a similarity transformation and solved numerically. Both stretching and shrinking cases are considered. Results for the skin friction coefficient, local Nusselt number, velocity, and temperature profiles are presented for different values of the governing parameters. It is found that the dual solutions exist for the shrinking case, whereas the solution is unique for the stretching case. Numerical results show that the range of dual solutions increases with mass suction and decreases with mass injection.
Stagnation-point flow over a permeable stretching/shrinking sheet in a copper-water nanofluid
Norfifah Bachok, Anuar Ishak, Roslinda Nazar and Norazak Senu
Boundary Value Problems , 2013, DOI: 10.1186/1687-2770-2013-39
Abstract: An analysis is carried out to study the heat transfer characteristics of steady two-dimensional stagnation-point flow of a copper (Cu)-water nanofluid over a permeable stretching/shrinking sheet. The stretching/shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation-point. Results for the skin friction coefficient, local Nusselt number, velocity as well as the temperature profiles are presented for different values of the governing parameters. It is found that dual solutions exist for the shrinking case, while for the stretching case the solution is unique. The results indicate that the inclusion of nanoparticles into the base fluid produces an increase in the skin friction coefficient and the heat transfer rate at the surface. Moreover, suction increases the surface shear stress and in consequence increases the heat transfer rate at the fluid-solid interface.
Unsteady stagnation point flow and heat transfer over a stretching/shrinking sheet with prescribed surface heat flux  [cached]
M. Suali,N. M. A. Nik Long,A. Ishak
Applied Mathematics and Computational Intelligence , 2012,
Abstract: An analysis is carried out to study the unsteady two dimensional stagnation point flow and heat transfer over a stretching/shrinking sheet with prescribed surface heat flux. The governing partial differential equations are converted into nonlinear ordinary differential equations using similarity variables, and solved numerically. The effects of the unsteadiness parameter A, stretching/shrinking parameter ε and Prandtl number Pr on the flow and heat transfer characteristics are studied. It is found that the skin friction f′′(0) and the local Nusselt number 1θ(0) increase as the the unsteadiness parameter A increases. Moreover, the velocity and temperature increase as ε and Pr increase.
Some Aspects of Non-Orthogonal Stagnation-Point Flow towards a Stretching Surface  [PDF]
Motahar Reza, Anadi Sankar Gupta
Engineering (ENG) , 2010, DOI: 10.4236/eng.2010.29091
Abstract: The problem of steady two-dimensional oblique stagnation-point flow of an incompressible viscous fluid towards a stretching surface is reexamined. Here the surface is stretched with a velocity proportional to the distance from a fixed point. Previous studies on this problem are reviewed and the errors in the boundary conditions at infinity are rectified. It is found that for a very small value of shear in the free stream, the flow has a boundary layer structure when , where and are the free stream stagnation-point velocity and the stretching velocity of the sheet, respectively, being the distance along the surface from the stagnation-point. On the other hand, the flow has an inverted boundary layer structure when . It is also observed that for given values of and free stream shear, the horizontal velocity at a point decreases with increase in the pressure gradient parameter.
Numerical and Series Solutions for Stagnation-Point Flow of Nanofluid over an Exponentially Stretching Sheet  [PDF]
Meraj Mustafa, Muhammad A. Farooq, Tasawar Hayat, Ahmed Alsaedi
PLOS ONE , 2013, DOI: 10.1371/journal.pone.0061859
Abstract: This investigation is concerned with the stagnation-point flow of nanofluid past an exponentially stretching sheet. The presence of Brownian motion and thermophoretic effects yields a coupled nonlinear boundary-value problem (BVP). Similarity transformations are invoked to reduce the partial differential equations into ordinary ones. Local similarity solutions are obtained by homotopy analysis method (HAM), which enables us to investigate the effects of parameters at a fixed location above the sheet. The numerical solutions are also derived using the built-in solver bvp4c of the software MATLAB. The results indicate that temperature and the thermal boundary layer thickness appreciably increase when the Brownian motion and thermophoresis effects are strengthened. Moreover the nanoparticles volume fraction is found to increase when the thermophoretic effect intensifies.
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