Comp. Appl. Math. DOI 10.1007/s40314-017-0421-5
Finite element solution of MHD power-law fluid with slip velocity effect and non-uniform heat source/sink Minakshi Poonia1 · R. Bhargava1
Received: 15 May 2015 / Revised: 20 December 2016 / Accepted: 25 January 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract The laminar steady boundary layer flow of an incompressible MHD Power-law fluid past a continuously moving surface is investigated numerically. The study involves the influence of surface slip and non-uniform heat source/sink on the flow and heat transfer. The governing boundary layer flow equations are transformed into non-dimensional, non-linear coupled ordinary differential equations with the help of suitable similarity transformations. The Galerkin finite element method is implemented to crack the resulting system. The impact of different involved physical parameters is exhibited on the dimensionless velocity profile, temperature distributions and rate of heat transfer in graphical and tabular forms for pseudoplastic and dilatant fluids. The local Nusselt number is found to be the decreasing function of slip parameter, temperature and space-dependent heat sink parameter whereas it increases with increasing values of temperature and space dependent heat source parameter. The problem has important application in attaining the sustainable heat transfer rate for the cooling of fluids, especially in heat exchangers used frequently in chemical industry, in order to increase the trustworthiness of a system, as it removes high heat loads from these systems. Keywords FEM · MHD · Moving surface · Non-uniform heat source/sink · Power-law fluid · Slip velocity Mathematics Subject Classification Primary 76D10; Secondary 80A20 · 35Q35 · 65H10
Communicated by Abimael Loula.
B 1
Minakshi Poonia
[email protected] Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
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List of symbols Roman B Cp
Magnetic field strength (N m A−1 ) Specific heat at constant pressure (J kg−1 K −1 )
Ec f M n N ux N u∗ Pr q Q Q∗ qw Rex Sf T (u, v) U w1 , w 2 , w 3 (x, y)
Eckert number Dimensionless stream function Dimensionless magnetic field parameter Power-law index Local Nusselt number Nusselt number using similarity transformation Prandtl number Rate of internal heat generation or absorption (J s−1 ) Space-dependent heat source/sink (J) Temperature-dependent heat source/sink (J) Wall heat flux (J s−1 ) or (W) Local Reynolds number Stream function (m2 s−1 ) Fluid temperature (K) Velocity vector (m s−1 ) Velocity of moving surface Test functions Co-ordinate axes (m)
Greek α β η λ λl κ μ ν ψ ρ σ τi j θ
Thermal diffusivity (m2 s−1 ) Dimensionless temperature-dependent heat source/sink parameter Similarity variable Dimensionless slip parameter (m) Slip length (m) Thermal conductivity (W m−1 K −1 ) Coefficient of viscosity (Pa s) Kinematic viscosity (m2 s−1 ) Shape function Density of the fluid (kg m−3 ) Electrical conductivity (A2 s3 kg−1 m−3 ) Stress in ‘j’ direction on ‘i’ plane (N m−2 ) Dimensionless temperature
Subscripts w ∞
Conditions at the surface Conditions far away from the surface
Superscripts
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Differentiation w.r.t. η
Finite element solution of MHD power-law…
1 Introduction The boundary layer flow past a continuously moving semi-infinite plate was explained by Sakiadis (1961) in 1961 and Tsou et al. (1967) discussed the heat transfer characteristics over it. Schlichting (1968) showed that both viscous dissipation and variable surface temperature cannot be taken simultaneously for similarity solution. Then, Soundalgekar and Murty (1980) considered the variable temperature of a moving plate without viscous dissipation in order to obtain a similarity solution to the heat equation. Jacobi (1993) presented a convenient correlation for heat transfer from a continuously moving isothermal surface in fluid which was applicable for all values of Prandtl number. Chen (1999) carried out the analysis of heat transfer of a continuously moving plate in forced convection fluid flow. Kumari and Nath (2001) investigated the boundary-layer flow of MHD non-Newtonian fluid with a parallel free stream over a continuously moving surface. The progressive aspect of non-Newtonian fluids, such as polymer melts, lubricants, molten plastics, emulsions and suspensions, is stimulated on account of their innumerable technological applications, including plastic sheets manufacturing, execution of lubricants, food conservation in chemical industries and motion of biological fluids. The feature of Newtonian fluid is described by the characteristic that throughout the movement of fluid, the stress is linearly proportional to the rate of strain. The fluids that do not follow the Newtonian relationship between the strain rate and stress are known as non-Newtonian. Several models of non-Newtonian fluids (Wilkinson 1960; Rivlin and Ericksen 1955; Bird et al. 1960; Truesdell and Noll 1992; Poonia and Bhargava 2014a, b, 2015) have been proposed due to their dissimilarity with Newtonian fluids. Amongst these the Ostwald–de-Waele model (Astin et al. 1973; Astarita and Marrucci 1974) is the most common and straightforward model which is the Power-law correlation between the stress tensor and strain rate tensor: ∂u i n−1 ∂u i τi j = μ (1) ∂x j ∂x j or, τi j = μeff where
∂u i ∂x j
(2)
∂u i n−1 μeff = μ ∂x j
(3)
Here the index ‘i’ identifies the stress component direction and index ‘j’ specifies the surface upon which it acts, ‘μ’ is the consistency of flow, ‘μeff ’ is the apparent or effective viscosity and ‘n’ is called the power-law index. When we consider the case n = 1, the fluid is known as Newtonian fluid and has constant apparent viscosity, but in case n = 1, the fluid is called as Power-law non-Newtonian fluid and apparent viscosity is a function of shear rate. If 0 < n < 1, the Power-law fluid is said to be shear-thinning (pseudoplastic fluid) and if n > 1, then it is known as shear-thickening (dilatant fluid). The property of shear-thinning fluids is that the viscosity decreases with increased shear rate and this is possible when ‘n’ is smaller than unity by the relation (3). Naturally, the fluid becomes more shear thinning as small the value of ‘n’ is taken. Some fluids like polymer melts, e.g. molten polystyrene, some paints and polymer solutions, e.g. polyethylene oxide in the water are considered as Power-law shear-thinning fluids. The range of ‘n’ for these fluids is revealed as 0.3–0.7 in the literature. Also, the smaller value of ‘n’ (0.1–0.15) is
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experienced with fine particle suspensions like kaolin-in-water and bentonite-in-water. In case of shear-thickening fluids, the viscosity grows with increased shear rate. The range for ‘n’ is taken 1 < n < 2 in literature. The examples of Power-law shear-thickening fluids are slurries of clay, corn starch and certain solutions of surfactants and many others. The Power-law fluids exhibit inelasticity, i.e. being deformed by the applied stress and do not come back to their original position even if the stress is released. Acrivos et al. (1960) examined heat transfer and flow to non-Newtonian Power-law fluids over a horizontal flat plate in a numerical way. Mansutti and Ramgopal (1991) studied the boundary layer flow of Power-law dilatant fluids and reflected the fact that the fluid could shear thin. Numerical solutions were presented by Ece and Büyük (2002) for the laminar steady natural convection Power-law fluids on a heated flat plate with mixed boundary conditions over temperature and obtained a transformation which was related to the similarity solutions of velocity and temperature in the boundary layer. Guedda (2005) carried out the problem on the natural convection of a fluid, then this work was extended to the non-Newtonian case by Guedda and Hammouch (2008) and they showed the influence of power-law index on the asymptotic behaviour of solutions. A lot of articles (Yadav et al. 2015, 2013; Yadav and Lee 2015; Yadav et al. 2014) have been devoted in the literature to the study of fluid flow in the presence of a magnetic field. Hartman (1937) proposed initiating electrically charged fluid flow through a magnetic field. The motion of fluids is controlled by the electromagnetic field in many technological applications, such as MHD power generators, in the petroleum industry, in purification of molten metals from non-metallic insertions and in the cooling of nuclear reactors. Watanabe and Pop (1994) investigated the magnetohydrodynamic flow over a flat plate. Soh (2005) classified all the invariant solutions for the boundary-value problem of the MHD Power-law non-Newtonian fluid. The above-mentioned previous studies were confined to fluid flows having a no-slip condition (fluid is considered as sticking on the solid boundary, i.e. the fluid has zero velocity relative to the boundary) at the surface. But, in reality, the no-slip condition does not always exist as few molecules near the surface bounce down along the surface. However, the wall slip is clearly manifested by some of the non-Newtonian fluids such as polymer melts. The wall slip condition is used in many applications such as polishing of internal cavities, artificial heart valves and micro-electromechanical systems (MEMS). The Blasius boundary layer problem studied at the MEMS scale was analyzed by Martin and Boyd (2000) taking into consideration the slip boundary condition. The influence of the slip boundary condition was studied by Rao and Rajagopal (1999) on the flow of fluids in a channel. Roux (1999) described the flow of non-Newtonian fluids with slip effects. Recently, Taamneh and Omari (2013) numerically investigated the slip Power-law fluid flow and transfer of heat in a porous channel. The Slip velocity effect over a moving surface on a non-Newtonian Power-law fluid, in the presence of heat generation, was studied by Mahmoud (2007). The study of heat sources or sinks (Yadav et al. 2015, 2012) is essential in several physical problems that involve exothermic or endothermic processes as in nuclear reactors, and chemical and power industries. The heat sources are used to heat up the system, whereas the heat sinks are used to cool down the system. If the temperature difference between the ambient fluid and the surface is appreciable, the heat sources or sinks produce a strong effect on the heat transfer. The influence of free convection and heat generation/absorption on heat transfer characteristics was shown by Vajravelu and Nayfeh (1992). Abo-Eldahab and Aziz (2004) investigated the influence of heat absorption or generation on hydromagnetic mixed convection heat transfer over an inclined surface. Abel and Mahesha (2008) examined the
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Finite element solution of MHD power-law…
effect of slip flow of MHD second-grade fluid on the heat transfer in a porous medium in the presence of a non-uniform heat source or sink. The Power-law fluids with slip effect and non-uniform heat source/sink are used in a number of industries, for example, the heat exchanges in an internal combustion engine in which a circulating fluid, i.e. the engine coolant (polymer melts and solutions) flows through the radiator coils and absorbs heat. After the fluid leaves the engine, it passes through a heat radiator, which transfers the heat from the fluid to the air through the exchanger. Cooling techniques make sure that a thermal boundary layer in contact with the material is maintained below a critical temperature, allowing it a longer life. The Galerkin finite element method is incorporated herein. This is a robust numerical technique to obtain the approximate solutions to boundary value problems. This method employs a variational method in which an error function is minimized and a more desirable approximate solution is generated. The basic notion behind FEM is connecting finite number of tiny elements of small sub-domain to cover a large domain, resulting in a set of simultaneous algebraic equations which can be solved easily. In the present study, the steady, laminar heat transfer and boundary layer flow of Powerlaw fluid past a continuously moving surface is examined. The magnetohydrodynamic effect, slip velocity effect and non-uniform heat source/sink are included in order to investigate their effect over flow and heat transfer. The Galerkin finite element method is implemented to solve momentum and heat equations in non-dimensional form. The results are shown graphically and also in tabulated form. An extremely good correlation is found between the present work and previous work showing the validation of the result. The considerable applications of the problem are found in number of industries, including chemical, food, power and petroleum refinery. For example, polymer melts or solutions are cooled or heated in heat exchangers in order to achieve a heat transfer rate at a certain level.
2 Mathematical formulation A thin semi-infinite solid horizontal plate which is placed in a plane in such a way that along the plate the distance is measured by x coordinate and normal to it by the y coordinate and it is continuously moving in x-direction. The incompressible Power-law non-Newtonian fluid is flowing along the plate and laminar, steady boundary layer flow is considered. The magnetic effect along x-axis is produced by forcing a uniform magnetic field of strength ‘B’ in ydirection. The schematic diagram of coordinate system and flow configuration is displayed in Fig. 1. The non-uniform heat source or sink is taken into consideration in presence of viscous dissipation in the energy equation. The governing boundary-layer equations for the conservation of mass, momentum and energy are given by: ∂u ∂v + =0 (4) ∂x ∂y ∂u ∂u μ ∂ u +v = ∂x ∂y ρ ∂y
∂u n−1 ∂u σ B2u − ∂y ∂y ρ
n+1 ∂u ∂T ∂2T ∂T +v = κ 2 + μ +q ρC p u ∂x ∂y ∂y ∂y
123
(5)
(6)
M. Poonia, R. Bhargava Fig. 1 Sketch of physical flow model and co-ordinates
The corresponding boundary conditions are: ∂u n−1 ∂u u = U + λ1 , v = 0, T = Tw at y = 0 ∂y ∂y u → 0, T → T∞ at y → ∞
(7)
where ‘u’ and ‘v’ are the velocity components in the x and y directions, respectively. ρ, κ and μ are the fluid density, thermal conductivity and consistency parameter, respectively. n is the power-law index, Cp is the specific heat at constant pressure, σ is electrical conductivity, ‘B’ is the strength of the applied magnetic field and ‘q’ is the rate of internal heat generation or absorption. ‘U ’ is the velocity of the moving surface. Tw is the temperature of the surface and T∞ is temperature of fluid outside the boundary layer. λ1 is the slip length which is defined as an extrapolated distance travelled at which fluid velocity becomes equal to the wall velocity. The LHS of momentum equation (5) represents the inertia term of forced flow and the first term of RHS is due to the power-law fluid, and the second term of RHS shows the magnetic effect. In the energy equation (6), LHS denotes thermal convection, and the first term of RHS shows thermal conductivity effect, the second term reflects viscous dissipation effect, and the third term denotes non-uniform heat source or sink effect. For the boundary conditions (7), at the wall, i.e. y = 0, the first expression shows the slip condition on the wall, the second expression is for no penetration, and the third one denotes the wall temperature. Far away from the wall, i.e. y → ∞, the first expression is for free stream velocity and the second is for free stream temperature. The non-uniform heat source or sink is modelled as follows (Abel and Mahesha 2008): q=
ρκU [Q(T − T∞ ) + Q ∗ (Tw − T∞ ) f ] μx
(8)
where Q and Q ∗ are the coefficients of space-dependent and temperature-dependent heat source/sink, respectively. Here, Q > 0, Q ∗ > 0 corresponds to heat source and Q < 0, Q ∗ < 0 corresponds to heat sink. With the help of similarity transformation, the number of independent variables is reduced by using non-dimensional variables. Physically, it describes a flow which looks the same either at all times or at all length scales.
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Finite element solution of MHD power-law…
Now, the similarity transformation is used to introduce the non-dimensional variables as follows: η=y
U 2−n νx
1 n+1
1
, Sf (η) = (νxU 2n−1 ) n+1 f (η), θ (η) =
T − T∞ Tw − T∞
(9)
∂ Sf −∂ Sf and v = ∂y ∂x which identically satisfies continuity equation (3), ‘ f ’ is dimensionless stream function, θ is the dimensionless temperature. Now, with the help of relation (9), the following dimensionless quantities are obtained: where η is similarity variable, Sf is the stream function defined as u =
U Pr = αx
U 2−n νx
κ Q∗ , β= μC p
−2 n+1
,
Ec =
U2 , C p (Tw − T∞ )
σ B2 , M= ρ
κQ , μC p n λ1 U 3 n + 1 λ= U νx
γ =
(10)
Here ‘α’ is thermal diffusivity, ‘Pr ’ is the Prandtl number,‘Ec’ is the Eckert number, ‘γ ’ is the space-dependent heat source/sink parameter, ‘β’ is the temperature-dependent heat source/sink parameter, ‘M’ is magnetic field parameter and ‘λ’ is dimensionless slip parameter. Using relations (9) and (10), Eqs. (5) and (6) can be rewritten as: n(n + 1) f | f |n−1 + f f − (n + 1)M f = 0 θ +
Pr f θ + Pr Ec| f |n+1 + Pr (γ f + βθ ) = 0 n+1
(11) (12)
The boundary conditions (7) transform as: f (0) = 0, f (0) = 1 + λ f | f |n−1 , θ (0) = 1 f (∞) = 0, θ (∞) = 0
(13)
Here prime denotes differentiation with respect to η. The physical quantity of interest is the local Nussult number N u x which can be expressed as follows: xqw N ux = , (14) κ(Tw − T∞ ) ∂ T is wall heat flux. where qw = −κ ∂ y y=0 Now, using the similarity variables (9),we have the following expression for N u x as follows: 1
N u x = −(Rex ) n+1 θ (0)
(15)
1
N u ∗ = N u x /(Rex ) n+1 = −θ (0) where Rex =
(16)
ρU 2−n x n is the local Reynolds Number. μ
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M. Poonia, R. Bhargava
3 Galerkin finite element method Herein the Galerkin finite element method (GFEM) (Reddy 1985) is employed for solving the system of non-linear ordinary differential equations (11) and (12) with boundary conditions (13). We assume: f = Z (17) Then, the system of Eqs. (11) and (12) becomes: n(n + 1)Z |Z |n−1 + f Z − (n + 1)M Z = 0 θ +
Pr f θ + Pr Ec|Z |n+1 + Pr (γ Z + βθ ) = 0 n+1
(18) (19)
and the corresponding boundary conditions (13) now reduces to: f (0) = 0, Z (0) = 1 + λZ |Z |n−1 , θ (0) = 0 Z (η) = 0, θ (η) = 0 as η → ∞
(20)
The numerical procedure is as follows: – First, the semi-infinite domain is set in order to acquire the approximate solutions. Here, we replace the semi-infinite domain η[0,∞) by a finite domain η[0,D) and the boundary conditions at infinity are imposed at η = D. The length D is selected to be 20 in such a way that no sufficient changes are observed in the solution after increasing the value more than 20. – This entire domain is discretized into a number of finite elements. Here, the domain is divided into 8000 line elements of equal size with step size h e = 0.0025. The collection of all the finite elements is named as mesh. The mesh is nothing but the assembly of all these finite elements. – A random element e is separated from the mesh and the weighted integral formulation of the system over this element is written as: w1 ( f − Z )dη = 0 (21) e
e
w2 (n(n + 1)Z |Z |n−1 + f Z − (n + 1)M Z )dη = 0
Pr w3 θ + f θ + Pr Ec|Z |n+1 + Pr (γ Z + βθ ) dη = 0 n+1 e
(22)
(23)
where, w1 , w2 , and w3 are arbitrary test functions, which may be viewed as variations in f , Z and θ , respectively. – Each element is assumed as 3-noded. The two nodes are at both ends and one is in the middle of the line element. We assume an approximate solution of the following form: f =
3 j=1
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f jψj,
Z=
3 j=1
Z jψj, θ =
3 j=1
θjψj
(24)
Finite element solution of MHD power-law…
where, ψ j are the shape functions for an element which are interpreted as: ηe − η ηe − η ψ1 (e) = 1 − 1−2 ηe+1 − ηe ηe+1− ηe ηe − η ηe − η (e) 1− ψ2 = 4 ηe+1 − ηe ηe+1 − ηe ηe − η ηe − η (e) ψ3 = − 1−2 ηe+1 − ηe ηe+1 − ηe
(25)
– As we are using the Galerkin scheme, so the weight function is chosen in such a way that these are the same as we have shape functions defined, i.e. wi = ψi (i = 1, 2, 3). – Now, this approximate solution (24) is substituted in Eqs. (21), (22) and (23) and we construct the finite element model of the equations as follows: [K e ]{Y e } = {F e }
(26)
where, [K e ] is the elemental stiffness matrix, {Y e } denotes vector of elemental nodal unknowns, and {F e } indicates the force vector. These have the following form: ⎤ ⎡ ⎡ ⎡ 1 ⎤ ⎤ [K 11 ] [K 12 ] [K 13 ] {f } {f} [K e ] = ⎣ [K 21 ] [K 22 ] [K 23 ] ⎦ , {Y e } = ⎣ {Z } ⎦ , {F e } = ⎣ { f 2 } ⎦ {θ } [K 31 ] [K 32 ] [K 33 ] { f 3} dψ j 21 dη, K i12 K i11 ψi ψi ψ j dη, K i13 j = j =− j = 0, K i j = 0 dη e e dψ j dψi dψ j 22 Ki j = − | Z¯ |n−2 −n(n +1) n(n 2 −1)ψi dη dη dη e dψ j f¯ − (n + 1)Mψi ψ j dη + ψi dη dψ j n 31 32 | Z¯ | +(n +1)γ Pr ψi ψ j dη = 0, K = 0, K = (n +1)Pr Ecψ K i23 i j ij ij dη e dψ j dψi dψ j ¯ + Pr ψ −(n + 1) dη = ψ f + (n + 1)β Pr ψ K i33 i i j j dη dη dη e dZ n−1 dZ dθ 3 f i1 = 0, f i2 = −n(n + 1)ψi , f = −2ψ i i dη dη dη e e
f¯ =
3 j=1
f¯j ψ j ,
Z¯ =
3
Z¯ j ψ j
j=1
In order to make system linearized, the functions f¯ and Z¯ are incorporated which are assumed to be known. An iterative scheme named quasilinearization method is used for it, even to deal with the non-linear boundary condition coming from Eq. (20). – After the assembly of elemental system, we acquire the following global system: [K ]{Y } = {F}
(27)
After assembly we get a system of algebraic equations. Since the entire domain is split into 8000 line elements maintaining uniformity and all are of same size and each element is 3-noded, then the full domain accommodates 16,001 nodes. At each node, we have to evaluate three functions. Thus, after assembly, a set of
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M. Poonia, R. Bhargava
Fig. 2 Grid independence test: showing every fiftieth element of mesh for temperature profile 1/n+1
Table 1 Comparison of N u x /Rex
for n = 1, M = γ = β = Ec = λ = 0 for various values of Pr
Pr
Soundalgekar and Murty (1980)
Jacobi (1993)
Chen (1999)
Present
0.7
0.3508
0.3492
0.34925
0.34924
1.0
–
0.4438
0.44375
0.44375
2.0
0.6831
–
–
0.68309
7.0
–
1.387
1.3870
1.38699
10.0
1.6808
1.679
1.6802
1.67996
100.0
–
5.448
5.5442
5.54481
48,003 algebraic equations is obtained. After imposing the boundary conditions, 47,998 equations remain to be solved by applying the Gauss elimination method undergoing the computational procedure of an accuracy of 10−5 order. The integrations are solved by the Gaussian quadrature. The results converge excellently. The grid independence test (grid convergence test or grid invariance test) is carried out to assist the four-decimal-point accuracy. The results are improved by this test using successively smaller step sizes for the calculations. Initially, a coarse mesh of 200 elements is taken with step size h e = 0.1000. After that, a medium mesh of 2000 elements is prepared by increasing the elements ten times with step size h e = 0.0100. Then, by fourfold the elements, we have a fine mesh of 8000 elements with step size h e = 0.0025. The temperature values for pseudoplastic and dilatant fluids are noted for all the meshes as shown in Table 2. It can be noticed from the same table that after increasing the number of elements more than 8000 (here we have taken 1.25-fold of elements, i.e. 10,000), accuracy is not affected. Hence, the step size is fixed as h e = 0.0025. The representation of the coarse, medium, fine and finer meshes is shown for every fiftieth element of the mesh in Fig. 2. In order to assess the validity of the present numerical solutions, the comparison with 1/n+1 is compared previous results available in literature has been shown. The value of N u/Rex with Soundalgekar and Murty (1980), Jacobi (1993) and Chen (1999) for n = 1, M = Ec = γ = β = λ = 0 for various values of ‘Pr ’ as displayed in Table 1 and observed in fine agreement with all of these known results.
4 Results and discussion The current numerical study is conducted with an objective to examine the behavior of different parameters in boundary layer regime on dimensionless temperature and velocity fields.
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Finite element solution of MHD power-law… Table 2 Grid Independence Test for Temperature distributions when λ = 0.15, Pr = 5.0, M = 1.75, Ec = 0.15, γ = 0.1, β = 0.1 |θ (η)| n
0.5
2.0
η
Coarse mesh of 200 elements (h e = 0.1000)
Medium mesh of 2000 elements (h e = 0.0100)
Fine mesh of 8000 elements (h e = 0.0025)
Finer mesh of 10,000 elements (h e = 0.0020)
0.50
1.29843
1.57834
2.11835
2.11836
2.50
1.35392
1.62604
2.38727
2.38728
4.50
0.85772
1.02984
1.52564
1.52565
6.50
0.39543
0.52728
0.78683
0.78684
8.50
0.10789
0.20787
0.31347
0.31348
10.50
0.06542
0.06545
0.06548
0.06549
0.50
1.00345
1.22935
1.55548
1.55549
2.50
0.45499
0.75794
1.10936
1.10937
4.50
0.10029
0.23128
0.33547
0.33548
6.50
0.08083
0.08085
0.08087
0.08088
8.50
0.02023
0.02025
0.02027
0.02028
10.50
0.00590
0.00601
0.00603
0.00603
Table 3 Absolute Values of 1
N u x /Rexn+1 at different parameters
λ
γ
β
Ec
Pr
M
n = 0.5
n = 2.0
0.1
0.1
0.1
0.1
5
2.5
6.1799
3.1147
0.3
0.1
0.1
0.1
5
2.5
3.8127
2.7453 2.4654
0.5
0.1
0.1
0.1
5
2.5
0.1231
0.05
0.3
0.1
0.1
5
2.5
5.0936
3.5576
0.05
0.7
0.1
0.1
5
2.5
5.2694
4.2211 2.5624
0.05
−0.3
0.1
0.1
5
2.5
4.8300
0.05
−0.7
0.1
0.1
5
2.5
4.6542
1.8990
0.05
0.1
−0.3
0.1
5
2.5
2.7110
0.8054 0.0637
0.05
0.1
−0.7
0.1
5
2.5
1.6726
0.05
0.1
0.3
0.1
5
2.5
0.2298
6.0894
0.05
0.1
0.7
0.1
5
2.5
7.7686
24.3375
0.05
0.1
0.1
0.2
5
2.5
10.5484
6.8034
0.05
0.1
0.1
0.3
5
2.5
16.0910
10.3808
0.05
0.1
0.1
0.4
5
2.5
21.6337
13.9583
0.05
0.1
0.1
0.1
7
2.5
7.9465
4.3406
0.05
0.1
0.1
0.1
10
2.5
10.1149
5.9635
0.05
0.1
0.1
0.1
13
2.5
12.5036
7.5292
0.05
0.1
0.1
0.1
5
2.0
5.9941
1.2497
0.05
0.1
0.1
0.1
5
1.5
6.0990
1.9694
0.05
0.1
0.1
0.1
5
1.0
6.1405
2.6233
The parameters describing the flow characteristics are Prandtl number, Slip parameter, spacedependent heat sink/source parameter, magnetic field parameter and temperature-dependent heat sink/source parameter. The results are exhibited through graphs and tables. The selected
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Fig. 3 Velocity and Temperature Distributions when n ≤ 1 and n > 1, while λ = 0.05, Pr = 2.0, M = 0.5, Ec = 0.1, γ = 0.1, β = 0.1
computations through graphs are exhibited in Figs. 2, 9, Tables 2 and 3. The parameters range is set as 0.2 ≤ n ≤ 2.0, 0.3 ≤ M ≤ 1.5, 0.5 ≤ Pr ≤ 5.0, 0.01 ≤ Ec ≤ 0.25, −0.1 ≤ γ ≤ 0.9, −0.05 ≤ β ≤ 0.6 and 0.1 ≤ λ ≤ 10. – Effect of Power-law index: Figure 3 depicts the temperature and velocity distributions for different values of Power-law index (n). Figures 3a, b show the effect of ‘n’ on dimensionless velocity and temperature field, respectively, for the pseudoplastic fluids (n < 1). It is noticed that the velocity grows, while the temperature falls with the increasing values of ‘n’. In both the figures, we have also shown the case of Newtonian fluid (n = 1). Figure 3c, d shows the dimensionless velocity and temperature field, respectively, for the dilatant fluids (n > 1). It is observed from the figures that velocity first increases and then decreases after a point, while the temperature is observed to be decreased with the increasing values of ‘n’. This is because, with increasing values of ‘n’ (but n < 1), the fluid becomes more shear thinning and hence, the viscosity decreases. Due to decreased viscosity, the fluid becomes thin and can move easily so velocity increases. As, the value of ‘n’ is increased more than unity, it is observed that the velocity grows near the plate, while it falls as we go far away from the plate. This shows that the shear thinning property of fluid tends to shear thickening property of fluid, so the viscosity increases and as a result, the velocity decreases. As, the values of ‘n’ increases, heat transfer rate decreases (from relation (16)) then, the temperature decreases. The direction of heat transfer is reversed in case of pseudoplastic fluids. – Effect of Slip parameter:
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Fig. 4 Velocity and temperature profiles for various values of λ for Pr = 2.0, M = 0.5, Ec = 0.1, γ = 0.1, β = 0.1
Figure 4 exhibits the influence of Slip parameter (λ) on temperature and velocity profiles considering both the cases, first n = 0.5, i.e. pseudoplastic fluid, and second n = 2.0, i.e. dilatant fluid. In case of pseudoplastic fluid (n = 0.5), the reduction in velocity is noted as the slip parameter value ‘λ’ is increased (Fig. 4a), while the temperature first increases then decreases with the increasing values of λ (Fig. 4b). For the case of dilatant fluids (n = 2.0), velocity decreases but, the temperature increases as, the values of λ are increased (Fig. 4c, d). Due to slip effect of the fluid, some of the molecules move away from the surface after hitting it so, the velocity of flow at the wall is not the same as the plate velocity. Hence, the velocity decreases due to slip effect. As slip parameter is increased, velocity of the fluid is noticed to decrease. As the movement of the molecules increases because of movement in the first layer, so the temperature rises. – Effect of space-dependent heat sink/source parameter: The influence of space-dependent heat sink parameter (γ < 0) and space-dependent heat source parameter (γ > 0) on temperature in boundary layer regime is depicted in Fig. 5, but these two parameters have no effect on the velocity profile. Figure 5a, b corresponds to pseudoplastic fluids, while Fig. 5c, d correlates with dilatant fluids. For pseudoplastic fluids: when γ < 0, the absolute value of temperature falls as ‘|γ |’ values rise, but a reverse effect is observed when γ > 0. For dilatant fluids: the absolute value of temperature rises as ‘|γ |’ increases. Since, the heat source (γ > 0) in the boundary layer adds energy, consequently, the fluid temperature rises. A peculiar peak in temperature distribution near the surface is
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Fig. 5 Temperature distributions for various values of γ , while λ = 0.05, Pr = 2.0, M = 0.5, Ec = 0.1, β = 0.1
observed as, the heat source effect is made more substantial. Then, the fluid temperature near the surface is at high level than the temperature of the surface. So, the transfer of heat is towards the surface from the fluid. In contrast, the heat sink (γ < 0) absorbs energy and the temperature decreases. – Effect of temperature-dependent heat sink/source parameter: The effect of the temperature-dependent heat sink parameter (β < 0) and temperaturedependent heat source parameter (β > 0) on the temperature distribution is illustrated in Fig. 6. No influence of these parameters is noted on velocity distribution. The Fig. 6a, b reveals that when β < 0, the temperature decreases as ‘|β|’ values are increased, while in the case of β > 0, the temperature near the surface increases as the value of ‘β’ is increased. For dilatant fluids, when β < 0, the temperature decreases as ‘β’ increases, while in the case of β > 0, the temperature is higher near the surface as the value of ‘β’ is increased. This is because energy is absorbed when the heat sink effect is considered so the temperature decreases and the energy is released during the heat source effect, then the temperature increases. – Effect of magnetic field parameter: Figure 7 shows the influence of magnetic field parameter (M) on temperature and velocity distributions within the boundary layer. When pseudoplastic fluids are considered, it is found that velocity decreases with increasing ‘M’, while the temperature increases near the wall but, thereafter, it starts decreasing. For dilatant fluids, velocity decreases but temperature increases. This is because, when magnetic field is applied, a restraining force is exerted by
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Fig. 6 Temperature distributions for various values of β, while λ = 0.05, Pr = 2.0, M = 0.5, Ec = 0.1, γ = 0.1
the fluid due to that the motion of the fluid becomes slow so, the velocity decreases and the temperature increases. – Effect of Eckert number: Figure 8 illustrates the influence of Eckert number (Ec) on the temperature distribution. On velocity profile, for both type of fluids (pseudoplastic and dilatant), the absolute value of temperature increases with increasing values of ‘Ec’. Ec expresses the relationship between the kinetic energy and total internal energy of flow. On increasing Ec, the movement of particles increases due to increase in kinetic energy which leads to increase in temperature. – Effect of Prandtl number: The response of the temperature distribution to varying values of the Prandtl number (Pr ) is shown in Fig. 9. ‘Pr ’ has no influence on velocity profile. As ‘Pr ’ increases, the temperature increases near the surface for pseudoplastic fluids (n < 1) but for dilatant fluids (n > 1) the temperature decreases in the boundary layer regime. This is because, for pseudoplastic fluids, the fluid is shear thinning, then the viscosity decreases and the movement of particles become faster, so the interaction among them increases and hence, the temperature rises. For dilatant fluids, the reverse effect is seen as the fluid is shear thickening. – Effect of different parameters over local Nusselt number: From Table 3, it can be followed that the local Nusselt number decreases as the value of slip parameter is increased. This is because, as the slip effect grows, the tendency of the
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Fig. 7 Velocity and temperature profiles for different values of M, while λ = 0.05, Pr = 2.0, Ec = 0.1, γ = 0.1, β = 0.1
Fig. 8 Temperature profiles for different values of Ec, while λ = 0.05, Pr = 2.0, M = 0.5, γ = 0.1, β = 0.1, λ = 0.05
molecules, moving away from the surface after striking over it, increases. Then, the heat transfer carriers are less, hence the rate of heat transfer reduces. With the increasing values of temperature-dependent and space-dependent heat source, the value of the local Nusselt number increases. This happens because the presence of the heat source generates energy and increases the temperature of the fluid, consequently, the rate of heat transfer increases. In contrast, rise in temperature and space-dependent heat sink causes the reduction in the local Nusselt number. Since the presence of the heat sink absorbs energy and decreases the
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Finite element solution of MHD power-law…
Fig. 9 Temperature profiles for different values of Pr while λ = 0.05, Ec = 0.1, M = 0.5, γ = 0.1, β = 0.1
temperature of the fluid, consequently the rate of heat transfer decreases. It is also clear that, the local Nusselt number increases with increasing values of Eckert number and Prandtl number. In case of increasing Pr , the fluid viscosity increases but thermal conductivity decreases. Then, thermal conductivity effect is the dominating effect. As a result, the heat transfer rate at the surface increases. Increasing the values of Ec causes the increase in kinetic energy of particles and hence, the local Nusselt number increases. The magnetic field reduces the local Nusselt number, as magnetic field exerts the restraining force and slows down the fluid motion. This happens because, when magnetic field is applied, a restraining force is exerted by the fluid due to which the motion of the fluid becomes slow and heat transfer rate decreases.
5 Conclusions On the basis of the present study, the results can be written in short as follows: – For pseudoplastic fluids (n < 1), the fluid becomes more shear thinning as Power-law index increases hence the viscosity decreases and the fluid can move easily so velocity increases but in case of dilatant fluids (n > 1) fluid becomes more shear thickening with increasing values of Power-law index hence the viscosity increases so the velocity decreases and the temperature decreases in boundary layer regime for both type of fluids. – The velocity decreases due to slip effect. As, some of the molecules move away from the surface after hitting it, so the velocity of the flow at the plate is not the same as the plate velocity. The temperature also rises because, the movement of the molecules increases in the first layer. – The fluid temperature increases due to the presence of the space-dependent or temperature-dependent heat source in the boundary layer since the heat source generates energy. – The presence of the space-dependent or temperature-dependent heat sink causes the temperature fall in the boundary layer, since the energy is absorbed during the heat sink effect. – The magnetic field applies a restricting force on the fluid, so the fluid velocity reduces and the temperature grows. – On increasing Eckert number, the movement of particles increases due to increase in kinetic energy, hence the temperature increases.
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– For pseudoplastic fluids, the viscosity decreases and the movement of particles becomes faster so the interaction among them increases and hence the temperature rises. For dilatant fluids, the opposite outcome is noticed. – It is observed that the local Nusselt number shows reduction in values with growth of slip effect, magnetic field effect, temperature-dependent and space-dependent heat sink effect, while temperature-dependent and space-dependent heat source effect, Eckert number and Prandtl number promote the local Nusselt number. Acknowledgements One of the authors (Minakshi Poonia) would like to thank Council of Scientific and Industrial Research (CSIR), Government of India, for its financial support through the award of a research grant.
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