Int J Adv Eng Sci Appl Math DOI 10.1007/s12572-016-0168-z
IIT, Madras
Second-order slip flow and radiative heat and mass transfer over a vertical permeable shrinking sheet S. Das1 • A. Ali1 • R. N. Jana2 • O. D. Makinde3
Indian Institute of Technology Madras 2016
Abstract In this paper, a second-order slip flow of a viscous incompressible fluid over a shrinking sheet in the presence of thermal radiation and mass diffusion is studied numerically. The sheet is permeable and subjected to constant suction. Appropriate similarity transformation is used to transform the governing nonlinear partial differential equations to a system of coupled nonlinear ordinary differential equations. The transform equations are then solved numerically using the Runge–Kutta fourth-order method with the shooting technique. Computation is performed with the help of very robust symbolic computational software MATLAB R2012b(bvp4c). Effects of pertinent parameters on the fluid velocity, temperature, mass concentration as well as the shear stress and rate of heat and mass transfer are presented graphically and discussed. The flow and heat and mass transfer induced by shrinking sheets are important in the study of extrusion process. Keywords Second-order slip Shrinking sheet Shooting technique Heat and mass transfer
& S. Das
[email protected] 1
Department of Mathematics, University of Gour Banga, Malda 732 103, India
2
Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India
3
Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa
1 Introduction In recent times, the flow of an incompressible fluid due to a shrinking sheet acquires attention for its abnormal behaviour in the flow dynamics. The flow induced by shrinking sheet exhibits quite distinct physical phenomena from the forward stretching flow. The stretching/shrinking sheet problems have been studied extensively, much less emphasis has been given to boundary layer flow problems due to a shrinking sheet. The shrinking sheet is a surface which decreases in size to a certain area due to an imposed suction or external heat. The shrinking film is one of the common applications of shrinking problems in industries. The shrinking film is very useful in packaging of bulk products since it can be unwrapped easily with adequate heat. Shrinking problem can also be applied to study the capillary effects in smaller pores, the shrink-swell behavior and the hydraulic properties of agricultural clay soils since associated changes in hydraulic and mechanical properties of such soils will seriously hamper predictions of the flow and transport processes which are essential for agricultural development and environmental management strategies. The no-slip boundary condition is known as the central tenets of the Naviere–Stokes theory. But there are situations wherein such a condition is not appropriate. Especially no-slip condition is inadequate for most nonNewtonian liquids, as some polymer melts often exhibit microscopic wall slip and that in general is governed by a non-linear and a monotone relation between the slip velocity and the traction. The liquids exhibiting boundary slip, find applications in technological problems such as polishing of artificial heart valves and inertial cavities. The velocity slip in rarefied gases flowing over a solid surface is the Maxwell [1] slip condition and is widely implemented in current rarefied gas flow investigators. Based on
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Maxwell’s first order-slip model, Thompson developed a second-order slip model. But many researchers reported that Thompson’s model cannot predict the flow for high Knudsen number, So Beskok and Karniadakis [2] suggested and improved the second-order slip conditions. For more details see Srikanth [3] and Wu’s [4]. The boundary layer flow over a stretching/shrinking sheet plays an important role in aerodynamic, extrusion of plastic sheet, metal-spinning, manufacture of plastic and rubber sheets, paper production etc and thus, remains at the leading edge of technology development. In the industrial operation, metal or more commonly an alloy, is heated until it is molten, whereupon it is poured into a mould or dies which contains a cavity, of required shape. The hot metal issue from the die is subsequently stretched to achieve the desired product. When the super heated melt issue comes out from the die, it loses its heat and contract as it cools, this is referred as liquid state contraction. With further cooling and loss of latent heat of fusion, the atoms of the metal lose energy and become closely bound together in a regular structure. The quality of the final product greatly depends on the rate of cooling and the process of stretching. In view of such applications, Crane [5] initiated the analytical study of boundary layer flow due to a stretching sheet. He assumed the velocity of the sheet varies linearly as the distance from the slit and obtained an analytical solution. After this pioneering work, the flow over a stretching surface has drawn considerable attention and a good amount of literature in different field has been generated on this problem [6–11]. In these studies, the noslip of the fluid velocity relative to the solid boundary was considered. It is a well-known fact that a viscous fluid normally sticks to the boundary. Wang [12] reported that the partial slip between the fluid and the moving surface may occur in particulate fluid situations such as emulsions, suspensions, foams and polymer solutions. Fang et al. [13, 14] gave a closed form solution for slip MHD viscous flow over a stretching sheet. Wang [15] investigated the effect of surface slip and suction on viscous flow over a stretching sheet. Sajid et al. [16] analyzed the stretching flow with general slip condition. Sahoo [17] investigated the flow and heat transfer solution for third grade fluid with partial slip boundary condition. A wide range of applications of nano-technology and microelectro- mechanical systems have given a boost to research area where a non-continuum behavior is present. In the present context, we are interested in studying surface-fluid interaction where slip flow regime occurs. In this regard, Kundsen number (Kn ) is a deciding factor, which is a measure of molecular mean free path to characteristic length. When Kundsen number is very small, no slip is observed between the surface and the fluid and is in tune with the essence of continuum mechanics. However, when
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Kundsen number lies in the range 103 to 0.1, slip occurs at the surface-fluid interaction and is generally studied under the light of model Maxwell-Smoluchowski first-order slip boundary conditions. Adding, slip flow theory is an asset that enables to exploit Navier-Stokes equation even when the characteristic length approaches molecular mean free path. Slip flow theory has been validated by asymptotic solution of Boltzmann equation. In this analysis, inner kinetic solution is matched with outer (i.e., bulk) NavierStokes solution and the matching is obtained only when slip/jump coefficient are considered at boundary or at the surface (Hadjiconstantinou [18, 19]). First and second order-slip flow coefficients are therefore outcomes of above said analysis. It is important to point out that efficient slip flow model is always preferred, because of inherent simplicity over solution obtained for Boltzmann equation. The applicability of first-order slip model deteriorates as Kundsen is around or greater than 0.1. Therefore, a number of researchers have proposed second-order slip flow model. Wu [4] has proposed a second-order slip flow model for the flow of rarefied fluid along the surface based on numerical simulation of linearized Boltzmann equation. In the situation when the fluid is particulate such as emulsions, suspensions, foams and polymer solutions, the no-slip condition is inadequate. In such cases, the suitable boundary condition is the second-order slip. In spite of its importance in polymer and electrochemical industry, no proper attention has been given to the flow analysis with a second-order slip condition. It is to mention in the light of above discussion that either no slip regime or slip regime is taken as a boundary condition, the Navier- Stokes equation is still valid. The flow induced by a moving surface has its importance in industrial applications and thus has been observed considerably in the literature. The permeable stretching/ shrinking sheet is one such example, which has been studied with no slip regime or slip regime at the surface. Magyari and Keller [20] presented an exact solution of self similar boundary layer flow along a permeable stretching sheet. Andersson [21] obtained a solution of flow along stretching sheet with slip. Wang [22] analyzed the partial slip flow on stretching sheet in a quiescent medium. Miklavcic and Wang [23] studied flow on a shrinking sheet and argued the existence and non-uniqueness of a solution. Fang et al. [24] obtained an analytical solution of MHD flow along a shrinking sheet with first-order slip flow. Fang and Aziz [25] obtained exact solutions of Navier-Stokes equation for second-order slip flow along a permeable stretching sheet. Fang et al. [26] presented an analytical solution for viscous flow along a shrinking sheet considering the second-order slip model presented by Wu [4]. Qasim et al. [27] have described the effects of slip conditions on stretching flow with ohmic dissipation and thermal
Int J Adv Eng Sci Appl Math
radiation. Nanadeppanavar et al. [28] analyzed secondorder slip flow over a horizontal shrinking sheet with a non-linear Navier boundary condition. Turkyilmazoglu [29] analytically studied heat and mass transfer in MHD viscous flow with hydrodynamic and thermal first-order slip over a stretching sheet for different thermal boundary condition. Turkyilmazoglu [39] presented analytically the existence of dual and triple solution in the flow of MHD viscoelastic fluid over a shrinking surface with first-order slip. Hayat et al. [30] have studied the MHD flow and heat transfer over permeable stretching sheet with slip condition. The heat and mass transfer flow in an electrically conducting fluid over a permeable stretching sheet with ohmic dissipation have been illustrated by Hayat et al. [31]. Hayat et al. [32] have presented the MHD stagnation point flow towards heated shrinking surface subjected to heat generation/absorption. A mathematical model for studying the slip effect on peristaltic motion with heat and mass transfer has been analyzed by Hayat et al. [33]. Abbas and Hayat [34] have made an analysis on the stagnation slip flow and heat transfer over a nonlinear stretching sheet. Hayat and Javed [35] have described the wall properties and slip effects on the magnetohydrodynamic peristaltic motion of a viscous fluid with heat transfer in porous space. The effects of heat and mass transfer on peristaltic flow of Williamson fluid in a non-uniform channel with slip conditions have been discussed by Hayat and Hina [36] . Hayat et al. [37] have studied the influence of slip condition on the peristaltic transport in an asymmetric channel with heat transfer. Makinde [38] has studied the heat and mass transfer by MHD mixed convection stagnation point flow toward a vertical plate embedded in a highly porous medium with radiation and internal heat generation. Recently, Turkyilmazoglu [40] obtained analytically dual solution for MHD viscous flow with second-order hydrodynamic slip over a stretching/shrinking surface for nonisothermal/prescribed heat flux boundary condition. Singh and Chamkha [41] have obtained the dual solutions for second-order slip flow and heat transfer on a vertical permeable shrinking sheet. Bhattacharyya et al. [42] have obtained an analytic solution for magnetohydrodynamic boundary layer flow of Casson fluid over a stretching/ shrinking sheet with wall mass transfer. Sharma and Ishak [43] have studied the second order slip flow of Cu-water nanofluid over a stretching sheet with heat transfer. In the present paper, we study a second-order slip flow and heat and mass transfer on a vertical permeable shrinking sheet in the presence of thermal radiation. The Rosseland approximation for the radiative heat flux is used. The sheet surface is isothermal, reactive to the fluid and produces inert specie, which diffuses inside the boundary layer. The model of a second-order slip is the same as proposed by Wu [4]. The partial differential equations
governing the flow are equations of continuity; momentum, energy and specie diffusion, which are transformed into ordinary differential equations using the similarity transformation and solved numerically using the shooting method. The behavior of the velocity, temperature, concentration, shear stress at the sheet, temperature gradient and concentration gradient has been illustrated for variations in the physical parameters. Numerical calculations up to desired level of accuracy have been carried out for different values of the physical parameters.
2 Mathematical formulation Consider the steady two dimensional boundary layer flow of a viscous incompressible fluid past a vertical permeable shrinking sheet. Cartesian co-ordinates (x, y) are chosen such that x-axis is taken along the sheet and the y-axis is normal to it (see Fig. 1). The flow is confined in half plane y [ 0. The sheet is shrinking in its own plane with the velocity a x ða [ 0Þ. The fluid experiences a secondorder slip at the sheet surface. The surface temperature and concentration are represented by Tw and Cw respectively. All fluid properties are considered constant except the influence of density variation with temperature and concentration in the body-force term. The radiative heat flux in the x-direction is considered negligible in comparison to the y-direction. The fluid is gray, absorbing-emitting but non-scattering. Gravity acts in the opposite direction to the positive x-axis. Under the usual Boussinesq’s approximation, the free convection boundary layer flow of a radiating fluid to be governed by the following system of equations ou ov þ ¼ 0; ox oy u
ð1Þ
ou ou o2 u þ v ¼ m 2 þ gb ðT T1 Þ þ gb ðC C1 Þ; ox oy oy
ð2Þ
x Tw , Cw
g vw
o
y
Fig. 1 Geometry of the problem
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u
oT oT k o2 T 1 oqr þv ¼ ; ox oy q cp oy2 q cp oy
oC oC o2 C þv ¼D 2; u ox oy oy
ð3Þ ð4Þ
u ¼ ax þ Uslip ; v ¼ vw ; T ¼ Tw ; C ¼ Cw at y ¼ 0; u ! 0; T ! T1 ; C ! C1 as y ! 1; ð5Þ where a is stretching constant with a [ 0, Uslip the slip velocity at the sheet, vw is the mass flux velocity normal to the shrinking sheet with vw \0 for suction and vw [ 0 for injection, T1 is the free stream temperature and C1 is the free stream concentration, both assumed to be constants. Wu [4] slip velocity model (valid for arbitrary Knudsen numbers, Kn ) is given by 2 3 a l3 3 1 l2 ou 1 4 2 o2 u k l þ 2 ð1 l2 Þ k2 2 ; 3 2 Kn oy 4 Kn oy a
¼A
ou o2 u þB 2; oy oy
1
where l ¼ min Kn ; 1 , (a is the momentum accommodation coefficient with 0 a 1) and kð [ 0Þ is the mean free path. Hence, for any Kundsen number (Kn ), A is positive and B is negative. In order to simplify the physical problem, the optically thick radiation limit is considered in the present analysis. Thus, the radiative heat flux term is simplified by using the Rosseland diffusion approximation [44–46] for an optically thick fluid according to 4r oT 4 ; 3k oy
ð7Þ
where r ð¼ 5:67 108 W=m2 K4 Þ is the Stefan–Boltzman constant and k ðm1 Þ the Rosseland mean absorption coefficient. This approximation is valid at points optically far from the bounding surface and is good only for intensive absorption. We assume that the temperature difference within the flow is sufficiently small such that the term T 4 may be expressed as a linear function of temperature. This is done by expanding T 4 in a Taylor series about a free stream temperature T1 and neglecting
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u
ð8Þ
oT oT k o2 T 16r o2 T þv ¼ þ ; ox oy q cp oy2 3k q cp oy2
ð9Þ
The continuity Eq. (1) is satisfied by introducing the stream function w such that ow ow ; v¼ : ð10Þ u¼ oy ox The governing equations can be transformed in to the corresponding ordinary differential equations by the following similarity transformation rffiffiffi pffiffiffiffiffi a T T1 g¼y ; ; w ¼ am x f ðgÞ; hðgÞ ¼ m Tw T1 C C1 /ðgÞ ¼ ; ð11Þ Cw C1 where g is the similarity variable, f ðgÞ the dimensionless stream function, hðgÞ the dimensionless temperature and /ðgÞ the dimensionless species concentration. In view of Eqs. (10) and (11), the governing Eqs. (2), (3) and (9) reduce to the following dimensionless form 2
f 000 þ f f 00 f 0 þ Grh þ Gc/ ¼ 0; h00 þ
ð6Þ
qr ¼
3 4 T 4 ¼ 4T1 T 3T1 :
On the use of Eqs. (7) and (8), the Eq. (3) becomes
where u and v are the velocity components along the x and y-directions, respectively, T the temperature of the fluid, C the concentration of the fluid, g the acceleration due to gravity, m the kinematic viscosity, q the fluid density, k the thermal conductivity, b the thermal expansion coefficient, b the concentration expansion coefficient, D the diffusion coefficient, cp the specific heat at constant pressure and qr the radiative heat flux. The boundary conditions are
Uslip ¼
higher order terms beyond the first degree in ðT T1 Þ to yield
ð12Þ
3RPr f h0 ¼ 0; 3R þ 4
ð13Þ
/00 þ Sc f /0 ¼ 0;
ð14Þ 3
T1 Þx where R ¼ 4rkkT 3 the radiation parameter, Gr ¼ g b ðTmw2 Re 2 1 the Grashof number that approximates the ratio of the buoyancy force to the viscous force acting on a fluid, Gc ¼ g b ðCw C1 Þx3 the solutal Grashof number which is a measure 2 2 m Re of the relative importance of transport of chemical species, qmc Pr ¼ k p the Prandtl number which measures the ratio of momentum diffusivity to the thermal diffusivity and Sc ¼ m D Schmidt number. Here primes denote differentiation with respect to g. In view of the above transformations, the boundary conditions (5) can be written as
f ð0Þ ¼ S; f 0 ð0Þ ¼ 1 þ c f 00 ð0Þ þ d f 000 ð0Þ; hð0Þ ¼ 1; /ð0Þ ¼ 1; f 0 ð1Þ ! 0; hð1Þ ! 0; /ð1Þ ! 0;
ð15Þ w pvffiffiffiffi ma
pffiffia
where S ¼ is the suction parameter, c ¼ A mð [ 0Þ the first-order slip parameter and d ¼ Bma ð\0Þ the secondorder slip parameter. It important to point out that S [ 0 for suction and S\0 for injection (blowing). It should be mentioned that in the absence of suction/blowing (S ¼ 0) and second order slip condition (c ¼ d ¼ 0), the relevant
Int J Adv Eng Sci Appl Math
results obtained are in agreement with the results reported by Makinde [38] in non-porous medium and neglecting free stream velocity. In the absence of mass transfer and thermal radiation (R ! 1), the present problem reduces to the problem studied by Singh and Chamkha [41]. In the present study, we take only suction because in the case of a shrinking surface, suction helps retaining the vorticity within the boundary layer and laminar flow is thus maintained. Suction or injection of a fluid through the bounding surface can significantly change the flow field. In general, suction tends to increase the skin friction, whereas injection acts in the opposite manner. Injection or withdrawal of fluid through a porous bounding wall is of general interest in practical problems involving boundary layer control applications such as film cooling, polymer fiber coating, and coating of wires, among others. The process of suction and blowing also has importance in many engineering activities such as in the design of thrust bearing and radial diffusers and thermal oil recovery. Suction is applied to chemical processes to remove reactants. Blowing is used to add reactants, cool the surface, prevent corrosion or scaling, and reduce drag.
3 Numerical method for solution The governing non-linear ordinary differential equations (12–14) cannot be solved analytically. In order to solve numerically the transformed ordinary differential Eqs. (12– 14) along with the boundary conditions (15), we have used the fourth-order Runge–Kutta method. This method is sometimes known as RK4 and is a reasonably simple and robust scheme featuring a shooting technique. Runge–Kutta Method is a method of numerically integrating ordinary differential equations using a trial step at the midpoint of an interval to cancel out lower-order error terms. The shooting method is one among the commonly used powerful numerical methods for solving boundary value problems for ordinary differential equations. In our study, we choose the shooting method, since it has many advantages such as ease of programming in a general form, less storage is required and its suitability for automatic procedures. The shooting method is an iterative algorithm that reformulates the original boundary value problem to a related initial value problem (IVP) with its appropriate initial conditions. The solution of the IVP should converge to that of the BVP. The resulting higher order ordinary differential equations are reduced to first order differential equations by letting y1 ¼ f ; y2 ¼ f 0 ; y3 ¼ f 00 ; y4 ¼ h; y5 ¼ h0 ; y6 ¼ /; y7 ¼ /0 :
y01 ¼ y2 ; y02 ¼ y3 ; y03 ¼ y22 y1 y3 Gry4 Gcy6 ; y04 ¼ y5 ; 3R Pr y1 y5 ; ¼ 3R þ 4 y06 ¼ y7 ;
ð17Þ
y05
y07 ¼ Sc y1 y7 ; subject to the following conditions: y1 ð0Þ ¼ 0; y2 ð0Þ ¼ 1 þ c y3 ð0Þ þ d ½y22 ð0Þ y1 ð0Þy3 ð0Þ Gry4 ð0Þ Gcy6 ð0Þ; y3 ð0Þ ¼ b; y4 ð0Þ ¼ 1; y5 ð0Þ ¼ c; y6 ð0Þ ¼ 1; y7 ð0Þ ¼ d: ð18Þ where b, c and d are unknown which are to be determined as a part of the numerical solution. To solve this system of Eq. (17) we require seven initial conditions whilst we have only four initial conditions. Since y3 ð0Þ, y5 ð0Þ and y7 ð0Þ which are not prescribed, we guess three unknown initial conditions as y3 ð0Þ ¼ b, y5 ð0Þ ¼ c and y7 ð0Þ ¼ d. With this now all the seven initial conditions become known and so we solve the resultant system of seven simultaneous equations by fourth order Runge–Kutta integration scheme and get the profiles of y1 , y2 , y3 , y4 , y5 , y6 and y7 for a particular set of parameters. In the shooting method, the unspecified initial conditions b, c and d are assumed and Eq. (17) integrated numerically as an initial valued problem to a given terminal point. The accuracy of the assumed missing initial condition is checked by comparing the calculated value of the dependent variable at the terminal point with its given value there. If a difference exists, improved values of the missing initial conditions must be obtained and the process is repeated. The numerical computations are done by MATLAB bvp4c routine. For numerical calculations, a uniform step size is taken as Dg ¼ 0:01. The last value of g1 is chosen as appropriate value for that particular set of parameters. The convergence criterion largely depends on fairly good guesses of the initial conditions in the shooting technique (Chiam [47]) and is based on the relative difference between the current and the previous iterations used, when this difference reaches 106 , the solution is assumed to have converged and the iterative process is terminated.
4 Results and discussion ð16Þ
Thus, the corresponding first order differential equations are
In order to get a physical view of the problem, we examine the effects of different values of Grashof number Gr, solutal Grashof number Gc, radiation parameter R, Prandtl
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number Pr, suction parameter S, Schmidt number Sc, firstorder slip parameter c and second-order slip parameter d on the velocity, concentration, shear stress function ( velocity gradient) and surface heat transfer function (temperature gradient). We focus on the positive values of the buoyancy parameters i.e. Grashof number Gr [ 0 (which corresponds to the cooling problem) and solutal Grashof number Gc [ 0 (which indicates that the chemical species concentration in the free stream region is less than the concentration at the boundary surface). The cooling problem is often encountered in engineering applications. The values of Schmidt number Sc were chosen to be Sc ¼ 0.24, 0.62, 0.78, 2.62, representing diffusing chemical species of most common interest in air like H2 , H2 O, NH3 and Propyl Benzene respectively while the values of Prandtl number
ranges Pr ¼ 0:72 (air) to 7.1 (water). It may be noted that Pr and Sc are well-known basic transport parameters representing the contribution of thermal or mass diffusivity as compared to momentum diffusivity. Since the Prandtl number is a measure of the relative importance of the viscosity and thermal conductivity of the fluid, the case Pr ¼ 1 corresponds to those fluids whose momentum and thermal boundary layer thicknesses are of the same order of magnitude. Figures 2, 3, 4, 5, 6 present the stream function for several values of Grashof number Gr, solutal Grashof number Gc and suction parameter S. It is seen from these figures that the stream function increases with an increase in either Gr or Gc or S. Figures 5, 6 reveal that the stream
4
4
3.5 3.5
3
2.5
2.5
f(η)
f(η)
3
Gr = 0, 3, 5, 7, 10
2
S = 0, 0.5, 1, 1.5, 2 1.5 2
1 1.5
0.5
1
0
1
2
3
4
5
0
6
0
1
2
3
4
5
6
η
η
Fig. 2 Dimensionless stream function for different Gr when Gc ¼ 3, R ¼ 2, Pr ¼ 0:72, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
Fig. 4 Dimensionless stream function for different S when Gc ¼ 3, R ¼ 2, Pr ¼ 0:72, Gr ¼ 5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
5
6 5.5
4.5
5
4 4.5 4
f(η)
f(η)
3.5 3
R = 0.2, 0.3, 0.4, 0.5, 0.6
3
Gc = 2, 3, 5, 7, 10
2.5
3.5
2.5
2 2
1.5 1
1.5
0
1
2
3
η
4
5
6
Fig. 3 Dimensionless stream function for different Gc when Gr ¼ 5, R ¼ 2, Pr ¼ 0:72, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
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1
0
1
2
3
4
5
6
7
η
Fig. 5 Dimensionless stream function for different R when Gc ¼ 3, Gr ¼ 5, Pr ¼ 0:72, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
Int J Adv Eng Sci Appl Math 1.8
3.5
1.6 3
1.4 2.5
1.2
Pr = 0.72, 2, 3, 5, 7.1 2
f′(η)
f(η)
1 0.8
Gc = 2, 3, 5, 7, 10 1.5
0.6 1
0.4 0.5
0.2 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
5
η
Fig. 6 Dimensionless stream function for different Pr when Gc ¼ 3, R ¼ 2, Gr ¼ 5, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
0
1
2
3
4
η
5
6
Fig. 8 Velocity profile for different Gc when Gc ¼ 3, R ¼ 2, Pr ¼ 0:72, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
2.5
3
2.5
2
2
R = 0.2, 0.3, 0.4, 0.5, 0.6
f (η)
1.5
′
f′(η)
1.5
Gr = 0, 3, 5, 7, 10
1 1
0.5 0.5
0
0
1
2
3
η
4
5
6
7
0 0
1
2
3
4
5
6
7
8
9
η
Fig. 7 Velocity profiles for different Gr when Gc ¼ 3, R ¼ 2, Pr ¼ 0:72, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
Fig. 9 Velocity profiles for different R when Gc ¼ 3, Gr ¼ 5, Pr ¼ 0:72, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
function decreases with an increase in either radiation parameter R or Prandtl number Pr. It is meant that the boundary layer thickness increases with increasing values of Gr, Gc and S while the boundary layer thickness decreases with increasing values of R and Pr. Figure 7 shows that the fluid velocity enhances near the shrinking sheet and it decreases away from the sheet when Grashof number Gr increases. The thermal Grashof number Gr is the ratio of bouncy forces to viscous forces. Physically, an increase in the values of Gr leads to accelerate the buoyancy forces near the sheet, consequently the fluid velocity enhances near the sheet. Figure 8 reveals that the fluid velocity increases for increasing values of solutal Grashof number Gc. This is due to the fact that solutal buoyancy
force enhances the fluid velocity and increases the boundary layer thickness. Figures 9, 10, 11, 12, 13, 14 present the profiles of fluid velocity for several values of radiation parameter R, suction parameter S, Prandtl number Pr, Schmidt number Sc, first order slip parameter c and second order slip parameter d. Figure 9 shows the effect of radiation parameter R on the fluid velocity f 0 ðgÞ. It is found that the fluid velocity markedly reduces for larger values of the radiation parameter. Radiation parameter R signifies the relative contribution of thermal conduction heat transfer to thermal radiative heat transfer. It arises solely in the transformed energy Eq. (13). As R increases this corresponds to a lower radiative heat transfer contribution compared with conduction heat
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2
2
1.5
1.5
′
S = 0, 0.5, 1, 1.5, 2 1
1
0.5
0.5
0
0
1
2
3
η
4
5
0
6
Fig. 10 Velocity profiles for different S when Gc ¼ 3, R ¼ 2, Pr ¼ 0:72, Gr ¼ 5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
2.5
2
2
1.5
1
0.5
0.5
2
2
3
η
4
5
6
γ = 0.1, 1, 2, 3, 4.5
f′(η)
f′(η) 1
1
1
1.5
Pr = 0.72, 2, 3, 7.1
0
0
Fig. 12 Velocity profiles for different Sc when Gc ¼ 3, R ¼ 2, Pr ¼ 0:72, S ¼ 0:5, Gr ¼ 5, d ¼ 1 and c ¼ 0:5
2.5
0
Sc = 0.24, 0.62, 2.62
f′(η)
2.5
f (η)
2.5
3
η
4
5
6
0
0
1
2
3
η
4
5
6
Fig. 11 Velocity profiles for different Pr when when Gc ¼ 3, R ¼ 2, Gr ¼ 5, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and c ¼ 0:5
Fig. 13 Velocity profiles for different c when Gc ¼ 3, R ¼ 2, Pr ¼ 0:72, S ¼ 0:5, Sc ¼ 0:24, d ¼ 1 and Gr ¼ 5
transfer. For the case of R ¼ 1 both radiative and conductive modes offer the same contributions. As R ! 1, radiative contribution dominates conduction contribution and vice versa for R ¼ 0. Nanofluid velocity is observed to be depressed with increasing R values implying that greater radiation accelerates the flow and decreases momentum boundary layer thickness. This is associated with the boost in thermal energy imparted to the regime with lower R values owing to stronger radiative flux. In the presence of thermal radiation, the thermal boundary layer always found to thicken which implies that the thermal radiation provides an additional means to diffuse energy. Figure 10 depicts the effect of suction parameter S on the fluid velocity f 0 ðgÞ. It observed that the fluid velocity decreases significantly when suction parameter S enlarges.
This is due to the fact that the effect of the suction is to take away the warm fluid on the sheet and thereby decrease the fluid velocity with an increase in S. When the wall suction ðS [ 0Þ is considered, this causes a decrease in the boundary layer thickness and the fluid velocity is reduced. The case S ¼ 0 corresponds to the impermeable shrinking sheet. It is interesting to note that a decrease in the peak value of fluid velocity within the boundary layer is observed in Figs. 11 and 12 with an increase in Schmidt number Sc and Prandtl number Pr. The physics behind this observation is that the increased Schmidt number and Prandtl number decreases chemical species molecular diffusivity and fluid thermal diffusivity, which ultimately reduces the fluid velocity. Sc is the ratio of the mass (species) and viscous diffusion time sales. It is also the
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2.5
0.9
δ = −5, −4, −3, −1
0.8
2
0.7 0.6
f′(η)
θ(η)
1.5
0.5
Gc = 2, 3, 5, 7, 10
0.4
1
0.3 0.2
0.5
0.1
0
0
0
1
2
3
4
η
5
6
Fig. 14 Velocity profiles for different d when Gc ¼ 3, R ¼ 2, Pr ¼ 0:72, S ¼ 0:5, Sc ¼ 0:24, Gr ¼ 5 and c ¼ 0:5
1
1.5
2
2.5
3
3.5
4
4.5
5
Fig. 16 Temperature profiles for different Gc when Gr ¼ 5, S ¼ 0:5, R ¼ 2 and Pr ¼ 0:72
1 0.9
0.9
0.8
0.8
0.7
0.7
0.6
θ(η)
0.6
θ(η)
0.5
η
1
0.5
Gr = 0, 3, 5, 7, 10
R = 0.2, 0.3, 0.4, 0.5, 0.6
0.5 0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0
0
0
0.5
1
1.5
2
2.5
η
3
3.5
4
4.5
5
Fig. 15 Temperature profiles for different Gr when S ¼ 0:5, Gc ¼ 3, R ¼ 2 and Pr ¼ 0:72
ratio of momentum diffusivity to species diffusivity. For Sc\1, the momentum diffusivity is lower than the species (mass) diffusivity and the species diffusion rate exceeds the momentum diffusion rate. For Sc [ 1, this scenario is reversed. Higher values of Sc correspond to higher density species diffusing in air, e.g. Sc ¼ 1 corresponds to methanol diffusing in electrically conducting air, Sc ¼ 2 implies ethylbenzene diffusing in air. Increasing Sc lowers the chemical molecular diffusivity of the species. As Sc is increased the concentration boundary layer becomes relatively thinner than the viscous (or momentum) boundary layer. The fluid velocity is therefore reduced. For the special case of Sc ¼ 1, the velocity and concentration boundary layers will be of the same order of thickness and
0
1
2
3
4
5
η
6
7
8
9
10
Fig. 17 Temperature profiles for different R when Gr ¼ 5, Gc ¼ 3, S ¼ 0:5 and Pr ¼ 0:72
both momentum and species will be diffused at the same rate. It is seen from Figs. 13 and 14 that the fluid velocity decreases with an increase in either c or d. The velocity slip parameters oppose the fluid motion. The movement of the flow in the boundary layer is because of the motion of the shrinking sheet. Therefore, the increase of velocity slip parameter decreases the effect of sheet motion on the boundary layer, and consequently the flow tends to retard. Figures 15, 16, 17, 18, 19 present the profiles of temperature for several values of thermal Grashof number Gr and solutal Grashof number Gc, radiation parameter R, suction parameter S and Prandtl number Pr. The fluid temperature is highest at the plate surface and decreases exponentially to the free stream zero value far away from the plate satisfying the boundary condition. In Figs. 15 and 16, it is observed that an
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1
0.9
0.9
0.8
0.8
0.7
0.7
Pr = 0.72, 2, 3, 7.1
0.6
φ(η)
θ(η)
0.6 0.5 0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.5
1
1.5
2
2.5
η
3
3.5
0
4
Fig. 18 Temperature profiles for different Pr when Gr ¼ 5, Gc ¼ 3, R ¼ 2 and S ¼ 0:5
0
2
3
η
4
5
6
7
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
φ(η)
0.6
0.5
S = 0, 0.5, 1, 1.5, 2
0.4
0.4 0.3
0.2
0.2
0.1
0.1 0
0.5
1
1.5
2
2.5
S = 0, 0.5, 1, 1.5, 2
0.5
0.3
0
1
Fig. 20 Concentration profiles for different Gc when Gr ¼ 5, S ¼ 0:5, R ¼ 2 and Sc ¼ 0:24
1
θ(η)
Gc = 2, 3, 5, 7, 10
0.5
3
3.5
4
4.5
5
η
0 0
1
2
3
4
η
5
6
7
8
Fig. 19 Temperature profiles for different S when Gr ¼ 5, Gc ¼ 3, R ¼ 2 and Pr ¼ 0:72
Fig. 21 Concentration profiles for different S when Gr ¼ 5, Gc ¼ 3, R ¼ 2 and Sc ¼ 0:24
increase in thermal Grashof number and solutal Grashof number cause a decrease in the thermal boundary layer thickness and consequently the fluid temperature decreases due to buoyancy effect. An increase in radiation parameter R causes a decrease in the fluid temperature within the boundary layer and consequently the thermal boundary layer thickness decreases as shown in Fig. 17. It is seen from Fig. 18 that increase in Pr brings a decrease in the thermal boundary layer thickness. At high Prandtl number, the fluid velocity decreases, which in turn implies lower thermal diffusivity leading to a decrease in the fluid temperature. It is seen from Fig. 19 that the fluid temperature decreases with an increase in suction parameter S. The temperature profiles decrease with an increase in the suction effect and as the suction rate is increased, more warm fluid is taken away, and
thus the thermal boundary layer thickness decreases. Figures 20, 21, 22 present the profiles of concentration for several values of Grashof number Gc, suction parameter S and Schmidt number Sc. The chemical species concentration is highest at the plate surface and decreases exponentially to the free stream zero value far away from the sheet. A decrease in the concentration boundary layer is observed with an increase in solutal Grashof number due to buoyancy effect as depicted in Fig. 20. It is seen from Fig. 21 that the concentration decreases with an increase in suction parameter S. Fluid suction has tendency to reduce the solute boundary layer thickness. It is observed in Fig. 22 that an increase in Schmidt number Sc leads to decrease in the species concentration. Actually, the Schmidt number is
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4
0.9
3 0.8
2
0.7
Sc = 0.24, 0.62, 0.78, 0.95, 2.62
1
f (η)
0.5
′′
φ(η)
0.6
0
0.4 0.3
−1
0.2
−2 0.1 0
S = 0, 0.5, 1, 1.5, 2 0
1
2
3
η
4
5
6
−3 −1
7
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
η
Fig. 22 Concentration profiles for different Sc when Gr ¼ 5, Gc ¼ 3, R ¼ 2 and S ¼ 0:5
Fig. 24 Shear stress profiles for different S when Gr ¼ 5, Gc ¼ 3,Pr ¼ 0:72, R ¼ 0:5 and c ¼ 0:5 0
0.2
−0.2
0
−0.4
−0.2
−0.6
f (η)
−0.4
Pr = 0.72, 2, 3, 7.1
′′
f′′(η)
R = 0.2, 0.3, 0.4, 0.5, 0.6 −0.6
−0.8
−0.8
−1
−1
−1.2
−1.2
−0.8
−1.4
0
1
2
3
4
5
η
6
7
8
9
10
0
1
2
3
4
5
6
7
η
Fig. 23 Shear stress profiles for different R when Gr ¼ 5, Gc ¼ 3, Pr ¼ 0:72, S ¼ 0:5 and c ¼ 0:5
Fig. 25 Shear stress profiles for different Pr when Gr ¼ 5, Gc ¼ 3, c ¼ 0:5, R ¼ 0:5 and S ¼ 0:5
inversely proportional to the diffusion coefficient D. A fluid with a higher Schmidt number has relatively low mass diffusivity and thereby reduces the solute boundary layer thickness. With thinner concentration boundary layers, the concentration gradients will be enhanced causing a decrease in concentration of species in the boundary layer. For Sc\1, the species diffusivity exceeds momentum diffusivity and this accounts for the greater concentration values for Sc ¼ 0.24, 0.62, 0.78, 0.95 compared with the minimized concentration profile for Sc ¼ 2:62, since in this latter case Sc [ 1, i.e. momentum diffusivity exceeds mass (species) diffusivity. The implication for chemical engineering designers is that in such a regime, a lower Schmidt number
diffusing species must be employed to enhance concentration distributions in the medium. Figures 23, 24, 25, 26 show that the shear stress decreases near the sheet surface and it increases away from the sheet with an in increase in either R or S or Pr while it increases with an increase in c. It is interesting that there are interception points for the shear stress profiles. It is also observed from Figs. 23, 24, 25 that the wall shear stress (the drag force) f 00 ð0Þ reduces due to the increasing values of R, S and Pr. The reason is that smaller values of Pr are equivalent to increase in the thermal conductivity of the fluid and therefore, heat is able to diffuse away from the heated surface more rapidly for higher values of Pr. Hence,
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2
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1
5
0
4
− θ′(η)
f′′(η)
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−1
γ = 0, 0.3, 0.5, 1, 2
−2
3
2
−3
−4 −1
Pr = 0.72, 2, 3, 5, 7.1
1
−0.8
−0.6
−0.4
−0.2
0
η
0.2
0.4
0.6
0.8
0
1
0
0.5
1
1.5
2
2.5
3
η
Fig. 26 Shear stress profiles for different c when Gr ¼ 5, Gc ¼ 3, Pr ¼ 0:72, R ¼ 0:5 and S ¼ 0:5
Fig. 28 Temperature gradient for different Pr when Gr ¼ 5, Gc ¼ 3, R ¼ 0:5 and Sc ¼ 0:24
0.7 1.4
0.6 1.2
R = 0.2, 0.3, 0.4, 0.5, 0.6
1
0.4
S = 0, 0.5, 1, 1.5, 2
0.8
− θ (η)
′
− θ (η)
0.5
′
0.3
0.6
0.2 0.4
0.1 0.2
0
0
1
2
3
4
5
η
6
7
8
9
10
Fig. 27 Temperature gradient for different R when Gr ¼ 5, Gc ¼ 3, Pr ¼ 0:72 and S ¼ 0:5
in the case of smaller Prandtl number as the thermal boundary layer is thicker and the shear stress at the plate is reduced. Figures 23 shows that the wall shear stress f 00 ð0Þ enhances due to the increasing values of c. Figures 27, 28, 29 reveals that the temperature gradient increases near the sheet surface and it decreases away from the sheet with an in increase in either R or Pr or S. For the temperature gradient profiles, there exists intercepts among the curves. The wall temperature gradient (the rate of heat transfer) h0 ð0Þ increases with the increasing values of R, Pr and S. Figure 27 shows that the wall temperature gradient h0 ð0Þ increases with an increase in radiation parameter
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0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
η
Fig. 29 Temperature gradient for different S when Gr ¼ 5, Gc ¼ 3, R ¼ 0:5 and Sc ¼ 0:24
R. This can be attributed to the fact that as thermal radiation increases, the dominance effect of temperature gradient increases, leading to an increase in the rate of heat transfer at the surface. From Fig. 28, the numerical results show that the increasing values of Prandtl number leads to a decrease in the thermal boundary layer and in general, lower average temperature within the boundary layer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivity of the fluid and therefore heat is able to diffuse away from the heated surface more rapidly for higher values of Pr. Hence, in the case of
Int J Adv Eng Sci Appl Math 0.8
0.8
0.7
0.7
0.6
0.6
Gc = 2, 3, 5, 7, 10
S = 0, 0.5, 1, 1.5, 2 0.5
− φ (η)
0.4
′
− φ′(η)
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
4
5
η
6
7
0
8
Fig. 30 Concentration gradient for different Gc when Gr ¼ 5, S ¼ 0:5 and Sc ¼ 0:24
0
1
2
3
4
η
5
6
7
8
Fig. 32 Concentration gradient for different S when S ¼ 0:5 and Gc ¼ 3
Fig. 31, it is seen that the concentration gradient at surface of the sheet /0 ð0Þ increases with increasing value of Schmidt number Sc due to a decrease in the chemical species molecular diffusivity and an increase in the concentration gradient at surface of the sheet.
4 3.5 3
− φ′(η)
2.5
5 Conclusion
Sc = 0.24, 0.62, 0.78, 0.95, 2.62
2 1.5 1 0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
η
Fig. 31 Concentration gradient for different Sc when Gr ¼ 5, Gc ¼ 3 and Sc ¼ 0:24
smaller Prandtl number, the thermal boundary layer is thicker and the wall temperature gradient h0 ð0Þ is reduced. The negative value of h0 ð0Þ means heat transfer from the surface of the sheet to the ambient fluid. In Figs. 30, 31, 32, it is observed that the concentration gradient increases near the surface of the sheet and it decreases away from the surface of the sheet with an in increase in either Gc or Sc or S. For the concentration gradient profiles, there exists intercepts among the curves. The concentration gradient (the rate of mass transfer) at surface of the sheet /0 ð0Þ increases with the increasing values of Gc, Sc and S. From
A numerical study has been performed for the doublediffusive convection boundary layer flow past a vertical isothermal shrinking sheet with a second-order slip in the presence of thermal radiation. Similarity variables have been used to transform the governing equations to a system of non-linear ordinary differential equations and solved numerically by using shooting iteration technique together with fourth order Runge–Kutta integration. From all the numerical computations, the main conclusions are summarized as follows: • • • •
•
The fluid velocity enhances due to the buoyancy forces. The fluid velocity retards when the slip parameters enlarge. The fluid velocity reduces in the presence of thermal radiation. The combination of the first order slip parameter, the second order slip parameter and the mass suction parameter greatly affect the fluid flow and shear stress on the wall as well as in the fluid flow. The effects of the radiation parameter, the Prandtl number, the suction parameter are to decrease the thermal boundary layer thickness.
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