Iran J Sci Technol Trans Sci (2017) 41:779–785 DOI 10.1007/s40995-016-0027-6
RESEARCH PAPER
A New Numerical Simulation of MHD Stagnation-Point Flow Over a Permeable Stretching/Shrinking Sheet in Porous Media with Heat Transfer M. M. Bhatti1 • T. Abbas2 • M. M. Rashidi3,4
Received: 25 January 2016 / Accepted: 14 May 2016 / Published online: 8 June 2016 Ó Shiraz University 2016
Abstract In this article, a new numerical method is employed to solve the stagnation-point flow problem over a permeable stretching/shrinking sheet through porous media. The effects of magnetohydrodynamics (MHD) and heat transfer are also taken into account. The governing flow problem is based on momentum equation and energy equation which are further simplified with the help of similarity transformations. The reduced resulting highly non-linear coupled ordinary differential equations are solved using the successive linearization method (SLM) and Chebyshev spectral collocation method. The impact of the physical parameters of interest is sketched for velocity and temperature profiles. The numerical comparison is also presented with the existing literature which shows that the present results are in good agreement and also confirms the validity of SLM for the present problem. Keywords Successive linearization method Chebyshev spectral collocation method Heat transfer MHD Porous media
& M. M. Bhatti
[email protected] 1
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
2
Department of Mathematics, Quaid I Azam University, Islamabad, Pakistan
3
Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Shanghai 201804, China
4
ENN-Tongji Clean Energy Institute of Advanced Studies, Shanghai 200072, China
1 Introduction During the past few years, stagnation-point flow has received a major attention by various researchers due to its numerous applications in oil ships, tips of rockets, and submarines. After an impressive work was performed by (Heimenz 1911), various authors investigated stagnationpoint flow problems in different geometrical aspects with various fluid models (Dinarvand et al. 2010; Hayat and Nawaz 2011; Rashidi et al. 2011, 2014; Alsaedi et al. 2012). Stagnation-point flow problems also occur in a various industrial process, such as glass blowing, spinning of fibers, continues casting of metals, melt spinning process, and all these flow problems occur over a cylindrical fiber or stretching sheet. Paullet and Weidman (2007) analyze the stagnation-point flow towards a stretching sheet. Bhattacharyya et al. (2011) studied the slip effects on boundary layer flow with heat transfer over a shrinking sheet. Akbar et al. (2014a) examined the stagnation-point flow with combined effects of slip and convective boundary conditions over a stretching sheet. Das (2012) considered the influence of MHD and slip in a stagnation-point flow of micropolar fluid towards a vertical shrinking sheet. Mabood et al. (2014) explored the effects of magnetohydrodynamics over an exponential shrinking sheet and found the analytic solution using the homotopy analysis method. On the other hand, magnetohydrodynamics has also a significant importance in different industrial applications, such as plasma confinement, cooling of nuclear reactors, pumps, sensors, accelerators, MHD propulsor, heat exchangers, and power generators. In different microchannel designs, MHD is studied as fluid pump and also very helpful to control the flow. Mabood et al. (2015) analyzed the stagnation-point flow with heat transfer under the influence of magnetohydrodynamics and chemical
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reaction over a stretching sheet. Rashidi et al. (2015) studied the entropy generation with MHD effects on the third-grade fluid over a stretching sheet. (Narayana and Babu (2015) studied numerically the heat and mass transfer of Jeffrey fluid model over a stretching sheet under the influence of chemical reaction, MHD, and thermal radiation. Some relevant studies on the said topic can be found from (Akbar et al. 2014b; Mabood et al. 2016; Rashidi et al. 2016) and several therein. There has been an increasing interest in studying porous media flow with heat transfer under the influence magnetic and non-magnetic field. Porous media can be described as a material volume that comprises of the solid matrix which is interconnected void. The porous media can also be described by its permeability which depicts the conductivity of the flow in the porous media. Another important phenomenon of fluid and the porous media are tortuosity which describes the hindrance to the flow diffusion by local velocity or local boundaries. Bhukta et al. (2015) studied the dissipation effects on MHD mixed convection flow with heat source/ sink over a stretching porous sheet. Sinha (2015) examined the heat transfer and MHD influence over a porous stretching wall for the third-grade fluid. Agbaje et al. (2016) investigated numerically the stagnation-point flow problem with heat transfer and MHD effects towards a stretching sheet. Some more studies on the present topic are available in (Ene et al. 2015; Mahanta et al. 2015; Khalili et al. 2015; Sheikholeslami 2014a, b; Sheikholeslami et al. 2015, 2016a, b). With the above discussion in mind, the aim of this present investigation is to solve numerically the MHD stagnation flow over a permeable stretching/shrinking sheet through the porous media with heat transfer. The successive linearization method (SLM) and Chebyshev spectral collocation method are employed to solve the governing flow problem. The main advantage of this method is that it converges more rapidly as compared to other similar techniques. Another advantage of this method is that it can directly apply to non-linear differential equations which are transformed into a linear system of equations. Furthermore, this system of equations is solved with the help of the Chebyshev spectral collocation method. The method of the solution of the coupled non-linear equations is quite interesting. The solution is mainly rely on a choice of a function satisfying the boundary conditions, and the unknown functions are obtained by iteratively solving the linearized version of the governing equations. The numerical comparison is also presented, which shows that the present results are in excellent agreement with the existing published studies. Furthermore, the comparison with the existing data is obtained after 4 and 5 iterations, which confirms the validity of the present flow problem. This paper is formulated as, Sect. 2 describes the mathematical formulation of the problem, Sect. 3 illustrates the
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solution methodology, and finally Sect. 4 is devoted to numerical results and discussion. 1.1 Mathematical Formulation Let us consider the two-dimensional laminar forced convection on MHD Hiemenz flow through the porous media. The fluid having incompressible, irrotational, and constant density, and it is electrically conducting by an external magnetic field. The governing equations of continuity, momentum, and energy equations can be written as (Rashidi et al. 2011) o~ u o~ v þ ¼ 0; ox oy
ð1Þ
u~
o~ u o~ v 1 o~ p o2 u~ m~ u rB2 u~ þ v~ þ ¼m 2 0 ; ox ox q ox ox q k~
ð2Þ
u~
oT~ oT~ o2 T~ þ v~ ¼ a 2 ; oy ox oy
ð3Þ
where x and y are the coordinate axis, u~; v~ are the velocity components along x and y direction, p~ is the pressure, q is the fluid density, m is the kinematic viscosity, k~ is porosity parameter, r is the electrical conductivity, T~ is the temperature, a is the thermal diffusivity, and B0 is the applied magnetic field along y direction. The magnetic Reynolds number is very small, such that the thickness of the magnetic boundary layer is very large and the induced magnetic field is neglected. The corresponding for the present flow problem are y ¼ 0 : v~ ¼ vw ; u ¼ 0; T~ ¼ T~w ¼ T~1 þ Axk ;
ð4Þ
y ! 1 : u~ ¼ U1 ¼ Cx; T~ ¼ T~1 ;
ð5Þ
where A is constant, vw is the suction/injection of the uniform surface, k is the exponent of the wall temperature which is considered to be 0 or 1, U1 is the free stream velocity, and C is the positive number. With the help of free stream, Eq. (2) takes the new form U1
dU1 1 dp~ mU1 rB20 U1 þ þ : þ q dx dx q k~
ð6Þ
Solving Eqs. (2) and (6), we get u~
o~ u o~ v dU1 o2 u~ mðu~ U1 Þ þ v~ ¼ U1 þm 2 ox ox ox dx k~ rB20 ðu~ U1 Þ : q
ð7Þ
The steam function satisfying Eq. (1) is defined as u~ ¼ ou=oy and v~ ¼ ou=ox. Defining the following similarity variables p T~ T~1 C ux f ¼ pffiffiffi y; f 0 ðfÞ ¼ p ; h ¼ ; ð8Þ C a T~w T~1 a
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781 000
and using Eq. (8) into Eq. (7) and Eq. (3), we get 0
000
00
2
0
0
Pr f þ 1 f þ ff þ kð1 f Þ þ M ð1 f Þ ¼ 0; 00
0
0
h þ f h ¼ kf h:
ð9Þ ð10Þ
Their corresponding boundary conditions are f ð0Þ ¼ S;
f 0 ð0Þ ¼ a;
hð0Þ ¼ 1;
hð1Þ ¼ 0:
ð12Þ
fI ð0Þ ¼ 0;
fI0 ð1Þ ¼ 0:
ð21Þ
2fN ; Pr 0 2fN 2fN ðM þ K Þ ; ¼ Pr
A0;I1 ¼
ð22Þ
A1;I1
ð23Þ
00
A2;I1 ¼
2fN : Pr
ð24Þ
We solve Eq. (20) numerically by a well-known method, namely, Chebyshev spectral collocation method. For numerical implementation, the physical region ð0; 1Þ is truncated to ð0; CÞ; we can take C to be sufficient large. With the help of the following transformations, this region is further transformed into ½1; 1, we have X ¼ 1 þ 2C1 f:
ð25Þ
We define the following discretization between the interval ½1; 1. Now, we can apply Gause–lobatto collocation points to define the nodes between ½1; 1 by
We apply the successive linearization method (SLM) to Eq. (9) with their boundary conditions in Eq. (11), by setting ðI ¼ 1; 2; 3; . . .Þ;
ð14Þ
N¼0
where fI are unknown functions which are obtained by iteratively solving the linearised version of the governing equation and assuming that fI ð0 N I 1Þ are known from the previous iterations. Our algorithm starts with an initial approximation f0 which satisfy the given boundary conditions in Eq. (7) according to SLM. The suitable initial guess for the governing flow problem is f0 ¼
0
fI ð0Þ ¼ 0;
ð13Þ
2 Numerical Method
fN ðfÞ;
ð20Þ
the corresponding boundary conditions become
ð11Þ
~ is the where Pr ¼ m= a is the Prandtl number, k ¼ m=kC 2 porosity parameter, M ¼ B0 r=Cq is the Hartmann number, að¼ b=aÞ is a stretching parameter, such that a [ 0 corresponds to the stretching sheet case, a\0 correspond to shrinking sheets case, and for a ¼ 0; planar stagnation flow towards a stationary sheet case occurs, and for a ¼ 1, the p flow having no boundary layer.S ¼ vw = Ca is the suction injection parameter, whereas S ¼ 0 corresponds to impermeable surface.
I1 X
0
where AP;I1 ðP ¼ 0; 1; 2; 3; . . .Þ; we have
f 0 ð1Þ ¼ 1;
The velocity components are defined as pffiffiffiffiffiffi u ¼ U1 f 0 ðfÞ; v ¼ f C a;
f ð fÞ ¼ f I ð fÞ þ
00
fI þ A0;I1 fI þ A1;I1 fI þ A2;I1 fI ¼ rI1 ;
fef þ aef þ Sef a þ 1 ef : ef
ð15Þ
ðJ ¼ 0; 1; 2; 3. . .N Þ;
ð26Þ
with ðN þ 1Þ number of collocation points. The Chebyshev spectral collocation method based on the concept of differentiation matrix D. This differentiation matrix maps a vector of the function values G ¼ ½f ðX0 Þ; . . .; f ðXN ÞT ; the collocation points to a vector G0 are defined as G0 ¼
N X
DKJ f ðXK Þ ¼ DG:
ð27Þ
K0
The derivative of p order for the function f ðXÞ can be written as f p ðXÞ ¼ Dp G:
ð28Þ
ð17Þ
ð18Þ
AI1 GI ¼ RI1 :
where Lðf ; f 0 ; f 00 ; f 000 Þ ¼ f 000 ;
pJ ; N
The entries of matrix D can be computed by the method proposed by (Bhatti et al., 2016). Now, applying the spectral method, with derivative matrices on linearized equation Eq. (20) and Eq. (21), we get the following linearized matrix system:
We write the equation in general form as Lðf ; f 0 ; f 00 ; f 000 Þ þ Nðf ; f 0 ; f 00 ; f 000 Þ ¼ 0;
XJ ¼ cos
and
ð29Þ
The boundary conditions take the following form 0
00
000
0
2
00
0
0
Nðf ; f ; f ; f Þ ¼ 1 f þ ff þ kð1 f Þ þ M ð1 f Þ;
ð19Þ
where L and N are the linear and non-linear part of Eq. (9), respectively. By substituting Eq. (14) in Eq. (9) and taking the linear terms only, we get
fI ðXN Þ ¼ 0; ¼ 0;
N X K0 N X
DNK fI ðXK Þ ¼ 0;
N X
D0K fI ðXK Þ
K0
D20K fI ðXK Þ ¼ 0;
ð30Þ
K0
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where AI1 ¼ D3 þ A0;I1 D2 þ A1;I1 D þ A2;I1 :
ð31Þ
In the above equation, As;I1 ðs ¼ 0; 1; . . .3Þ are ðN þ 1Þ ðN þ 1Þ diagonal matrices with As;I1 ðXJ Þ on the main diagonal and GI ¼ fI ðXJ Þ;
RI ¼ rI ðXJ Þ: ðJ ¼ 0; 1; 2; 3; . . .N Þ: ð32Þ
The solutions for fI are obtained with the help of Eqs. (29) and (30). We obtain the solution for f ðfÞ from solving Eq. (30), and, now, Eq. (10) is linear; therefore, we will apply the Chebyshev pseudospectral method directly, and we get BH ¼ S;
M
Present results
Kechil and Hashim (2009)
Yih (1998)
0
1.8893
1.8846
1.8893
0.5
2.0534
1 2
2.2029 2.9201
2.2029 2.9201
2.2029 2.9201
hðX0 Þ ¼ 0;
ð34Þ
Rashidi et al. (2011)
2.9201
Table 2 Numerical comparison of f00 (0) with the existing literature with different values of M when S = 0, k = 0, and Pr = l M
Present results
Kechil and Hashim (2009)
Yih (1998)
0
0.7565
0.7559
0.7565
0.5
0.9487
1 2
1.1164 1.8776
1.1164 1.8776
1.1164 1.8776
ð33Þ
with their corresponding boundary conditions boundary conditions hðXN Þ ¼ 1;
Table 1 Numerical comparison of f00 (0) with the existing literature with different values of M when S = 0, k = 0, and Pr = l
where B ¼ D2 þ f D kf 0 ;
ð35Þ
where H ¼ hðXJ Þ; S is a vector of zeros, and all vectors in Eq. (35) are converted to diagonal matrix. We imposed the boundary conditions Eq. (34) on the first and last rows of B and S, respectively.
3 Results and Discussion In this section, the graphical results of all the physical parameters are sketched and discussed with the help of graphs and tables. To verify the validity of our present methodology, we have compared our results with the existing published literature. Tables 1, 2, 3 and 4 show the numerical comparison with different published studies. From these tables, we can observe that the present methodology is in excellent agreement with the existing literature. We also observed that the present methodology converges very expeditiously, as we reached to our results after 4 or 5 iterations. These verify the validity of our methodology for solving the present problem. Figure 1 shows the velocity profile for different values of Hartmann number ðM Þ. It can be noticed from this figure that velocity profile increases due to the increment in the Hartmann number ðM Þ: It depicts from Fig. 2 that due to the influence of Prandtl number ðPr Þ, the velocity profile decreases. Here, we can also observe that for large values of Prandtl number, momentum diffusivity is more dominating as compared to thermal diffusivity. It is also analyzed that thermal boundary layer for liquid metals is very much higher than velocity boundary layer. In various heat
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Table 3 Numerical comparison of f00 (0) with the existing literature with different values of M when S = 1, k = 0, and Pr = 1 M
Present results
Kechil and Hashim (2009)
Yih (1998)
0
1.2325
1.2306
1.2325
0.5
1.4197
1 2
1.5853 2.3466
1.5852 2.3466
1.5853 2.3466
Table 4 Numerical comparison of f’’ (0) with the existing literature with different values of M when S = 1, k = 0, Pr = 1, and k = 0 M
Present results
Kechil and Hashim (2009)
Yih (1998)
Rashidi et al. (2011)
0
1.3236
1.3236
1.3236
1.3237
0.5
1.3314
1
1.3380
1.3380
1.3380
1.3380
2
1.3644
1.3644
1.3644
1.3644
transfer problems, the Prandtl number is very much favorable to control the thickness of thermal and momentum boundary layers. Moreover, we can observed that when the Prandtl number is small, velocity tends to move near the walls and the free stream velocity is observed throughout the boundary layer. From Fig. 3, we can analyze that when the porosity parameter ðkÞ increases, then the velocity profile behaves as an increasing function, and the same behavior of velocity is observed for suction/injection parameter ðSÞ, as shown in Fig. 4. Figures 5 and 6 are plotted for different values of stretching parameter ðaÞ.
Iran J Sci Technol Trans Sci (2017) 41:779–785
Fig. 1 Velocity profile for different values of M when k = 1, Pr = 1, S = 1, and a = -1
Fig. 2 Velocity profile for different values of Pr when k = 1, S = 1, a = -1, and M = 1
Fig. 3 Velocity profile for different values of k when Pr = 1, S = 1, a = -1, and M = 1
Fig. 4 Velocity profile for different values of S when k = 1, Pr = 1, a = -1, and M = 1
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Fig. 5 Velocity profile for different values of a when k = 1, Pr = 1, S = 1, and M = 1
Fig. 6 Velocity profile for different values of a when k = 1, Pr = 1, S = 1, and M = 1
Fig. 7 Temperature profile for different values of k when k = 1, Pr = 1, S = 1, and M = 1
It can be seen from Fig. 5 that when the stretching parameter ðaÞ increases, then the velocity profile diminishes which shows that boundary layer thickness decreases. Physically, this phenomena can be described as, for different fixed values of stretching sheet, an increment in b depicts an increment in straining motion close to the stagnation region, which implies an increment on external stream and this tends to lead the thinning of boundary layer. It can be examined from Fig. 6 that when stretching parameter ðaÞ decreases, which leads to decelerate the fluid flow. It can be scrutinized from Fig. 7 that due to the
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problem illustrate the reliability of the algorithm and provides a wider applicability to solving any non-linear differential equation.
References
Fig. 8 Temperature profile for different values of Pr when k = 1, S = 1, and M = 1
Fig. 9 Temperature profile for different values of S when k = 1, Pr = 1, and M = 1
increment in k, temperature profile decreases. Figure 8 shows the influence of Prandtl number on temperature profile. It can be viewed from this figure that when the Prandtl number increases, then temperature profile increases. Physically, when the Prandtl number is small, then the free stream velocity remains continues throughout the wall and the velocity profile will be closer to the wall. These results are important for those fluids, whose have a high range of Prandtl number, such as ammonia, R-12 refrigerant, seawater, n-butanol, and oils. From Fig. 9, we can observe that when the suction parameter increases, then temperature profile diminishes.
4 Conclusion In this article, the successive linearization method (SLM) and Chebyshev spectral collocation method are employed to solve the MHD stagnation-point flow over a permeable stretching/shrinking sheet with heat transfer. This method found very much promising to solve the boundary value problem and analyze that it converges early as compared to other similar methods. Numerical comparison depicts that the present results are in good agreement with the existing literature. The results obtained for the governing flow
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