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Stagnation point flow of radiative Oldroyd-B nanofluid over a rotating disk
Computer Methods and Programs in Biomedicine 191 (2020) 105342
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Computer Methods and Programs in Biomedicine journal homepage: www.elsevier.com/locate/cmpb
Stagnation point flow of radiative Oldroyd-B nanofluid over a rotating disk Abdul Hafeez a,∗, Masood Khan a, Jawad Ahmed a,b a b
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Department of Basic Sciences, University of Engineering and Technology, Taxila 47050, Pakistan
a r t i c l e
i n f o
Article history: Received 12 December 2019 Revised 14 January 2020 Accepted 15 January 2020
Keywords: Oldroyd-B fluid MHD Stagnation point flow Heat generation/absorption Nonlinear thermal radiation Convective boundary condition
a b s t r a c t Background: Nanofluids are known for better heat transfer characteristics in many heat exchanger devices due to their enhanced heat transfer abilities. Recently, scientists give the idea of nanofluid which is the mixture of base fluid and solid nanoparticles having very small size. For physical phenomenon of conventional fluids by mean of suspensions of nanoparticles in base fluids and prompted produce a new composite known as “nanofluids”. These composite contain the nanoparticles with 1–100 nm sized which are suspended in the base fluids. Here we have considered a subclass of non-Newtonian fluid called Oldroyd-B fluid. The fluid motion over the disk surface is produced due to the rotation as well as radially stretching of disk. Further, the impact of non-linear thermal radiation and heat generation/absorption is introduced to visualize the heat transfer behavior. The convective boundary is also taken into consideration in order to investigate the fluid thermal characteristics. The novel features of thermophoresis and Brownian motion during the nanoparticles movement in fluid motion are studied with revised Buongiorno model. The physical problem is modeled with the concept of classical Fourier’s and Fick’s laws. The von Karman variables are used to convert the partial differential equations (PDEs) into non-dimensional ordinary differential equations (ODEs). Method: The system of governing ordinary differential equations (ODEs) with boundary conditions (BCs) are highly non-linear in nature. To handle these non-linear ODEs, we use a numerical technique called BVP Midrich scheme which uses the midpoint method to acquire the numerical solution of the governing problem. The solutions of the governing problem are obtained by utilizing Maple software. Results: Effect of different involved controlling parameters on the velocity profiles, temperature and concentration distributions are analyzed graphically. Additionally, the numerical data for local Nusselt and Sherwood numbers are also tabulated. The reduction in heat transfer rate at the wall is noticed against thermoporesis and Brownian motion parameters, respectively. The concentration gradient at the wall reduces with an increment in mass transfer parameter. © 2020 Published by Elsevier B.V.
1. Introduction Based on its applications in engineering fields and industries like hot rolling, paper making, wire drawing and much more, the stagnation point flow has been extensively investigated by the researchers. The quality of such products may be checked with the flow field and heat transfer. Initially in 1911, the stagnation point flow was proposed by Hiemenz [1]. He used the similarity variables to solve the two dimensional stagnation point flow and obtained the exact solution of the problem. After that the study of stagnation point flow towards a stretching plate was disclosed by ∗
Corresponding author. E-mail address: [email protected] (A. Hafeez).
https://doi.org/10.1016/j.cmpb.2020.105342 0169-2607/© 2020 Published by Elsevier B.V.
Chiam [2]. Later on, many researchers have been done their work on stagnation point flow of different fluid models. The study of heat transport in a stagnation point flow over a stretching sheet was explored by Chiam [3]. In this study he used regular perturbation technique and obtained analytical closed form solutions upto second order. Furthermore, Mahapatra and Gupta [4] also investigated the stagnation point flow of incompressible fluid over a stretching sheet. The study of two dimensional stagnation point flow of Maxwell fluid over a shrinking sheet is reported by Motsa et al. [5] and obtained the solution by using successive linearisation method. Moreover, the partial slip effect on stagnation point flow of an incompressible fluid by a shrinking sheet is analyzed by Bhattacharyya et al. [6]. This study reveals the conditions of the existence, non-existence uniqueness and duality of the solutions
2
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
of self-similar equations numerically. Recently, Khan et al. [7] explored the stagnation point flow of nanofluid towards a permeable surface with the impact of thermal radiation. In their study, they used a numerical technique called bvp4c technique in MATLAB and obtained dual nature solutions numerically. Some articles based on stagnation point flow were investigated by researchers via [8–12]. For a long time, researchers working in non-Newtonian fluid mechanics in order to predict the characteristics and behavior of fluid. The resulting Navier–Stokes equations with their complexity are noticed due to the presence of nonlinear inertia terms. When the viscosity is variable based on applied force or stress, it is classified as non-Newtonian fluid. The common everyday example is maize starch dissolved in water. Newtonian fluid behavior such as water can be defined by temperature and pressure exclusively. But the non-Newtonian fluid’s physical behavior depends on the forces that operate on it from time to time. A number of nonNewtonian fluids have triggered the perplexing rheology of natural liquids. Non-Newtonian fluids have recently acquired tremendous importance for industrial, commercial and mechanical applications. These liquids are used in chemical processes, material handling, oil storage, food processing and many others. Products such as ketchup, mud, oils, blood, shampoos, and so other thin and dense liquids are classified as non-Newtonian fluids. Generally, there are three types of non-Newtonian fluids that are the rate type, differential type and Integral type. The rate type fluids demonstrate the relaxation and retardation times behavior. Oldroyd-B fluid is a subclass of rate type material which exhibits the behavior of both relaxation and retardation times. This model describes the flow of viscoelastic fluids and was proposed by Oldroyd [13]. After that, the study of MHD flow Oldroyd-B nanofluid induced by a stretching sheet are analyzed by Hayat et al. [14] and obtained analytical solution of the problem. Recently, the flow of magnetized OldroydB fluid over a rotating disk with the impact of thermal radiation is discussed by Khan et al. [15]. In their study, they obtained numerical solution through BVP Midrich scheme in Maple. Subsequently, numerous researchers have been done their work on Oldroyd-B fluid by taking different physical effect shown by Refs. [16–18]. In recent times, the researchers are much engaged for improvement of heat transfer rate in the forced convection phenomenon because of its applications in many engineering disciplines that are, in the fields of chemical production, air conditioning, manufacturing, power generation and many others physical processes. Nanofluids have new properties that potentially make them useful in many heat transfer applications. Some of including are in fuel cells, hybrid powered engines, domestic refrigerator, thermal management in vehicle/engine cooling and so forth. The concept of nanofluids that are the mixture of base fluid and solid nanoparticles, have been recently given by scientists. A new composite called “nanofluids” is developed for the physical phenomenon of conventional fluids. These composites include 1–100 nm-sized nanoparticles embedded in the base fluids. These suspensions was initially planned by Choi and Eastman [19]. In their study, they argued that these suspensions have higher conductive and convective heat transfer relative to the base fluids. After that, the various scientists did their work on the features of nanofluid flow. In order to investigate the thermal properties of the base fluid, Buongiorno [20] proposed a mathematical model using the thermophoresis and Brownain motion impacts to improve the thermal properties in base fluid. He found that the conventional fluid with thermophoresis and Brownain motion plays a significant role in improving the fluid’s thermal conductivity. The study of magnetized three-dimensional flow of viscoelastic nanofluid with nonlinear thermal radiation is investigated by Hayat et al. [21]. However, further the flow of nanofluid due to a rotating disk is explored by Hayat et al. [22] and obtained numerical solution. Furthermore, numerous researchers have been done their work on the study of
such flows with influence of nanoparticle in base fluid through Refs. [23–28]. Having gone through the literature and motivated by the prominent features of stagnation point flow, heat generation/absorption and nonlinear thermal radiation, it is concluded that rotating disk flow of Oldroyd-B nanofluid with these physical situations is not investigated yet. Therefore, the present article focuses on the steady three dimensional stagnation point flow of OldroydB nanofluid over a permeable rotating disk subject to convective boundary condition. The von Karman transformations are used to convert the PDEs in non-dimensional ODEs to acquire the solutions of the governing problem. To obtain the numerical solution of the system of equations, we use BVP Midrich scheme that used the midpoint method through Maple programming. The impact of different physical parameters which are involved in the governing system of equations on different velocity fields, temperature and concentration distributions are discussed in detail.
2. Mathematical formulation We assume a steady, incompressible three-dimensional boundary layer flow of Oldroyd-B fluid over a permeable disk. The flow is assumed to be generated by a rotating and stretching of the disk. All physical quantities are not depend on ϕ , as the flow is axisymmetric about z direction. The stagnation point is at the surface (z = 0 ) and the liquid flows at (z ≥ 0) the upper half plane. The disk at z = 0 is rotated by uniform angular velocity . Here, the magnetic field acts in the vertical direction. The disk is assumed porous with mass flux velocity w0 with w0 > 0 for injection and w0 < 0 for suction. With the help of classical Fourier’s and Fick’s laws, the heat and mass transfer analysis are performed. Moreover, with the influence of nonlinear radiation and heat generation/absorption, the heat transfer analysis is performed. A physical depiction of the problem is plotted in Fig. 1. The disk is kept up at constant concentration Cw . Here T∞ and C∞ are the ambient temperature and concentration, respectively. Additionally, Tf is the convective fluid temperature. The Buongiorno’s model for nanofluid is used to discuss the thermophoresis and Brownian motion due to nanoparticles. The governing equations for the given non-linear problem are [15]
∂u u ∂w + + = 0, ∂r r ∂z
Fig. 1. A Physical sketch of the problem.
(1)
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
∂ u v2 ∂u ∂u ∂ 2u σ 2 u − +w = ν 2 − B0 u + λ1 w − ue ∂r r ∂z ρ ∂z ∂z ∂ ue ∂ 2u ∂ 2u ∂ 2 u 2uv ∂v +ue − λ1 u2 2 + w2 2 + 2uw − ∂r ∂ r∂ z r ∂r ∂r ∂z 2vw ∂v uv2 v2 ∂ u − + 2 + r ∂z r ∂r r 2 1 ∂u ∂ u ∂ 2w ∂ 3u ∂ u ∂ 2u +νλ2 − −2 + w − r ∂z ∂ z ∂ z2 ∂ z3 ∂ r ∂ z2 ∂ u ∂ 2u ∂ 3u − +u , ∂ z ∂ r∂ z ∂ r ∂ z2 ∂v uv ∂v ∂v ∂ 2v σ 2 u + +w = ν 2 − B0 v + λ1 w ∂r r ∂z ρ ∂z ∂z ∂ 2v ∂ 2v ∂ 2v uv ∂ u vw ∂ u −λ1 u2 2 + w2 2 + 2uw +2 +2 ∂ r∂ z r ∂r r ∂z ∂r ∂z u2 v v3 v2 ∂v −2 2 − 2 + r ∂r r r ∂ 3v ∂v ∂ 2 w ∂ 3 v 1 ∂ u ∂v ∂v ∂ 2 u +νλ2 u −2 +w 3 − − ∂ z ∂ z2 r ∂z ∂z ∂ r ∂ z2 ∂ r ∂ z2 ∂z v ∂ 2 u ∂v ∂ 2 u u ∂ 2 v + − − , r ∂ z2 ∂ z ∂ r ∂ z r ∂ z2 u
u
2 1 ∂ qr ∂T ∂T k ∂ T +w = − ∂r ∂ z ρ c p ∂ z2 ρ cp ∂ z 2 ∂ T ∂ C DT ∂ T Q +τ DB + + (T − T∞ ), ∂ z ∂ z T∞ ∂ z ρ cp ∂C ∂C ∂ 2 C DT ∂ 2 T +w = DB 2 + , ∂r ∂z T∞ ∂ z2 ∂z
Familiarizing the Von Karman [30] transformations by
η= φ=
v = r, w = w0 , −k
u→ue = ar,
(2)
(8)
(9)
F 2 − G2 + F H − F + β1 F H 2 + 2F F H − 2GG H
+β2 2F 2 + 2F H − F H + M F + β1 F H − A − A2 = 0, (10)
2F G + G H − G + β1 G H 2 + 2 F G + F G H
−β2 G H − 2F G − 2G H + M G + β1 G H = 0,
4 θ 1 + Rd 3 4
+ Rd (3)
(11)
2 θ 3 θ + 3θ 2 θ (θw − 1 )3 2 2 3 +3 θ 2 θ + 2θ θ 2 (θw − 1 ) + 3 θ θ + θ (θw − 1 )
− Pr H θ + Pr N bθ φ + Nt θ 2 + δ Pr θ = 0, (12)
φ − ScHφ +
Nt θ = 0, Nb
(13)
The transformed BCs are
(4)
F (η ) = R, G(η ) = 1, H (η ) = s,
(5)
F ( η ) → A , G ( η ) → 0,
θ (η ) = −Bi(1 − θ (η ) ),
φ (η ) = 1 at η = 0, θ ( η ) → 0, φ ( η ) → 0
as
η → ∞, (14)
v→0, w = we = −2az, T → T∞ , C → C∞ as z → ∞. (6)
Here u refers for the velocity in r direction and v, w the velocities in ϕ , z directions, respectively. Furthermore, (ν , τ , ρ , σ , cp , α , k, DB , DT , λ1 , λ2 , hf , T, C) are the kinematic viscosity, the ratio of effective heat capicity of particles to the effective heat capacitiy of base fluid, the density of the fluid, the electrical conductivity, the specific heat capacity, the thermal diffusivity of the fluid, the tharmal conductivity, Brownian diffusion coefficient, thermal diffusion coefficient, the relaxation time, the retardation time, the convective heat transfer coefficient, the temperature and the fluid concentration, respectively. We adopt Rosseland’s diffusion approximation [29] for the radiative heat flux qr and the expression is given by
k∗
C − C∞ . Cw − C∞
H + 2F = 0,
∂T = h f T f − T , C = Cw at z = 0, ∂z
4 σ ∗ ∂T4 16 σ ∗ T 3 ∂ T qr = − =− , 3 k∗ ∂ z 3 k∗ ∂ z
√ T − T∞ z, u = rF , v = rG, w = vH, θ = , ν T f − T∞
Substituting the above mentioned transformations into governing Eqs. (1) to (5), we acquire
The boundary conditions are
u = cr,
3
(7)
where is the Rosseland mean spectral absorption coefficient and σ ∗ the Stefan-Boltzmann constant.
w We designate T = T∞ (1 + (θw − 1 ) )θ , where θw = TT∞ > 1 is the temperature ratio parameter.
here primes describe differentiation w.r.t η. The dimensionless physical parameters involved in the above system are as follow
of equations
σ B2
c M = ρ 0 is the magnetic field parameter, R = the stretching parameter, β1 (= λ1 ) the time relaxation
a parameter, β2 (= λ2 ) the retardation parameter, A = the ve-
locity ratio parameter, s =
ter, Bi = ation
Nb =
ν
parameter,
parameter,
hf k
Nt =
τ DB (Cw −C∞ ) ν
√wo
the mass transfer parame-
ν
the Biot number, Rd =
δ =
Q ρcp
τ DT T f −T∞ ν T∞
3 16σ ∗ T∞ 3k∗ k
the
heat
the
thermophoresis
the radi-
generation/absorption parameter,
the Brownian motion parameter, P r (=
the Prandtl number and Sc = Dν
ν (ρ c p ) k
)
the Schmidt number.
B
3. Physical concern parameters The following physical parameters are described as: (i) The local Nusselt number Nur is defined by
16 σ ∗ T 3 Nur = − 1 + 3 kk∗
r k ∂∂Tz
.
z=0
k T f − T∞
(15)
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A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
Making use the transformations in Eq. (15), gives
1
Re− 2 Nur = − 1 +
4Rd 3 [1 + (θw − 1 )θ (0 )] 3
θ (0 ).
(16)
(ii) The local Sherwood number Shr is
Shr = −
r DB ∂∂Cz
z=0
DB (Cw − C∞ )
.
(17)
After using the similarity variables in Eq. (17), we obtain
Re− 2 Shr = −φ (0 ), 1
r2
where, the local Reynolds number is Re = ν
(18)
.
4. Solution methodology The system of ordinary differential equations in Eqs. (9)–(13) with boundary conditions (14) are highly non-linear in nature. We use a numerical procedure called BVP Midrich scheme for handling these non-linear differential equations. The solutions of the governing problem are obtained by utilizing Maple software. The general procedure for the mid-point approach is described by
Z ∗ (t ∗ ) = F (t ∗ , Z ∗ (t ∗ ) ), Z ∗ (t0∗ ) = Z0∗ .
(19)
The term used for the modified Euler method is
Zn∗+1 = Zn∗ + h∗ F tn∗ +
h∗ h∗ , Zn∗ + F (tn∗ , Zn∗ ) , 2 2
(20)
where h∗ refers the step size and tn∗ = t0∗ + nh∗ . The implicit approach of the mid-point method strategy is articulated as
Zn∗+1 = Zn∗ + h∗ F tn∗ +
h∗ 1 ∗ ∗ , Xn∗ + Z ,Z , 2 2 n n+1
n = 0, 1, 2, ..... (21)
The local error at each step of the mid-point procedure is of O(h∗3 ) and the global error is of the order O(h∗2 ). With more computational intensive, the mid-point algorithm error declines more rapidly as h∗ → 0 and shall be a more stable solution. The absolute error convergence of this method is upto 1 × 10−6 . 4.1. Validation of numerical code A comparison of numerical values of F (0), −G (0 ) and −θ (0 ) is made for limiting case that is in the absence of time relaxation parameter, retardation parameter, stretching parameter, magnetic field, velocity ratio parameter, radiation parameter, Biot number, heat generation/absorption and nanoparticles. These comparisons of numerical results to those of Turkyilmazoglu [31] and Bachok et al. [32] can be seen in Table 1. This table shows us a validity of our numerical code for the system of governing equations. 5. Results and discussion In the present investigation, our aim is to interpret the characteristics of swirling stagnation point flow of Oldroyd-B nanofluid
Table 1 A comparison of the F (0), −G (0 ) and −θ (0 ) on fixed Prandtl number Pr = 6.2 with those of previously published papers.
F (0) −G (0 ) −θ ( 0 )
Turkyilmazoglu [31]
Bachok et al. [32]
Present study
0.51023262 0.61592201 0.93387794
0.5102 0.6159 0.9337
0.5101162643 0.6158492796 0.9336941128
Fig. 2. Radial (red line), azimuthal (black line), axial (green line) velocities and temperature (blue line) profile in a stationary frame. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
and the physical behavior of the flow and heat transport. In order to describe the flow behavior, heat and mass transfer with respect to various physical involved parameters, results are drawn in Figs. 3–9. The variation of involved physical dimensionless parameters in the above system of ordinary differential equations that are, magnetic field parameter M, stretching parameter R, suction/injection parameter s, relaxation time parameter β 1 , retardation time parameter β 2 , velocity ratio parameter A, thermal radiation parameter Rd, temperature ratio parameter θ w , heat generation/absorption parameter δ , thermophoresis parameter Nt, Brownian motion parameter Nb, Prandtl number Pr and Schmidt number Sc for velocity fields, temperature and concentration distributions are analyzed. For this, we fixed various physical parameters to illustrate physical structure are M = 1.0, R = 1.3, s = 0.1, β1 = 0.05, β2 = 0.2, A = 1.5, θw = 1.1, Rd = 0.03, Bi = 0.8, δ = 0.5, Nt = 0.1, Nb = 0.1, Pr = 5.0 and Sc = 5.0. It should also be mentioned that all plots are approaching asymptotically the far-field boundary conditions. The solution of the system of differential Eqs. (9)–(12) with boundary conditions are plotted for Newtonian case with Pr = 0.71 and in the absence of stagnation point, thermal radiation, heat generation/absorption and nanoparticles interpreted in Fig. 2. This figure shows the temperature θ (η) and laminar velocities in radial F(η), azimuthal G(η) and axial −H (η ) directions, respectively in a stationary frame. This profile is numerically based solution with the help of Maple software. The influence of stretching parameter R on velocities in radial F(η) and azimuthal G(η) directions, temperature θ (η) and concentration φ (η) distributions, respectively are sketched in Fig. 3(a)-(d) with the specified values of other parameters. Fig. 3(a) shows the impact of stretching parameter R on fluid velocity in radial direction. In this figure, the radial velocity F(η) shows a rising behavior of fluid velocity with increase in stretching parameter R in the range 0.1 to 2.0. Basically, the stretching parameter R is the ratio of stretch c to the swirl rate. So, by higher estimation of stretching parameter R(= 0.1, 1.0, 1.5, 2.0 ), the fluid velocity in radial direction F(η) increases due to an increase in the disk stretching rate. On the other hand, the contrary variation on velocity in azimuthal direction is noticed in Fig. 3(b). This is because of the less in the swirl rate of the disk as compare to stretching rate. Moreover, the fluid temperature variation on stretching parameter R is discussed through Fig. 3(c) with the default values of other parameters. It is remarked that the temperature θ (η) declines with the enlargement of stretching parameter R. For the concentration distribution φ (η), the results are quite different.
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
5
Fig. 3. Variation of R on (a): F(η) (b): G(η) (c): θ (η) (d): φ (η).
Fig. 4. Variation of A on (a): F(η) (b): G(η).
Fig. 4 (a-b) are sketched to see the influence of velocity ratio parameter A on radial velocity F(η) and azimuthal velocity G(η) with respect to η. It is seen that radial velocity F(η) grows up with an increase in the values of velocity ratio parameter A(= 0.0, 0.1, 0.3, 0.5 ). As the velocity ratio parameter A is the ratio of free stream velocity to rotation rate of the disk. It can be perceived physically that when the values of A < 1, the rotating velocity of the disk will be dominant than the velocity of the free stream. As rotation of disk is also contributes in the radial velocity of the fluid and the disk rotation accelerates with more speed, the development of centrifugal force will become more stronger which as a result push the fluids particles in the radial direction, so, enhancement of the radial flow velocity is noticed. Furthermore, the reduction of the velocity in azimuthal direction is considered in Fig. 4(b).
This is due to decrease in the rotation rate of the disk. Fig. 5(a)-(c) depict the impact of mass transfer parameter s on velocity F(η), temperature θ (η) and concentration φ (η), respectively against η. In Fig. 5(a), we observed that the fluid velocity increases with an increase in mass transfer parameter s(= 0.0, 0.1, 0.3, 0.5 ) with A = 0.1. Enlargement in temperature field θ (η) is noticed for the case of injection as changes from 0.0 to 0.5 which displayed in Fig. 5(b). Furthermore, the concentration distribution for mass transfer parameter s is pictured in Fig. 5(c). It is scrutinized that the concentration of the fluid enhances as s changes from 0.0 to 0.5. In order to see the impact of fluid temperature distribution on thermal radiation parameter Rd, Biot number Bi and heat generation/absorption parameter δ , respectively, results are sketched via Fig. 6(a)-(c). Here, distributions given in these sketches show that
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A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
Fig. 5. (a): Effect of s on F(η) (b): Effect of s on θ (η) (c): Effect of s on φ (η).
Fig. 6. Variation of θ (η) on (a): Rd (b): Bi (c): δ .
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
7
Fig. 7. (a): Effect of Nt on θ (η) (b): Effect of Nt on φ (η).
all results are satisfied boundary conditions asymptotically. It is sighted that the temperature of the fluid θ (η) is booted up under the action of radiation parameter Rd against η. Basically, the process of radiation describe by dimensionless radiation parameter Rd which rises the fluid temperature due to provided of extra heat to the fluid. Furthermore, the impact of Biot number Bi on temperature distribution is discussed through Fig. 6(b). As we seen that the temperature within the fluid flow are boosted up with an increment in Biot number Bi(= 0.1, 0.3, 0.5, 0.8 ). By the definition of Biot number Bi, the increasing values of Bi implies that the convective heat transfer coefficient increases, thereby enhancing more heat transfer from the surface. This then causes the liquid to heat up and thus enhances the temperature of the Oldroyd-B fluid. Furthermore, the temperature θ (η) is influenced by heat generation/absorption parameter δ (see Fig. 6(c)). This figure highlighted that the temperature of the fluid enhances by enlarging values of heat generation/absorption parameter δ (= 0.0, 0.3, 0.6, 0.9 ). Additionally, the thermal boundary layer thickness becomes thicker with the enlargement of δ in this manner. The effect of Nt (thermophoresis parameter) on temperature and concentration of Oldroyd-B liquid with respect to η are illustrated in Fig. 7(a) and (b). It is concluded that the temperature of Oldroyd-B fluid enhances as Nt increases from 0.1 to 0.7. The same behavior is seen for concentration of nanoparticle as shown in Fig. 7(b). Basically, the thermophoretic force produces in the fluid by taking the higher values of Nt and consequently, heat and mass transport enhance in the fluid. To see the influence of Nb (Brownian motion parameter) on temperature θ (η) and concentration distribution φ (η), respectively, results are drawn in Fig. 8(a) and (b). From Fig. 8(a), it is rendered that the temperature rises for higher values of Nb. Moreover, the concentration distribution φ (η) declines as Nt changes form 0.1 to 0.7 with Bi = 0.2.
As the resistance for the mass transport phenomenon produced by larger Brownian motion, so the concentration distribution will be decreased. The curves of temperature distribution θ (η) on the impact Prandtl number Pr is elaborated in Fig. 9(a). It reveals that the temperature of the Oldroyd-B fluid declines by taking different values of Prandtl number between 2.0 to 7.0. In the physical point of view that the larger rate Prandtl number Pr (> 1 ) decreases the thermal diffusivity, hence in a result, the reduction in temperature θ (η) is seen. Further more, to notice the influence of Schmidt number Sc on concentration φ (η), result is drawn via Fig. 9(b). It is observed that for higher estimation of Schmidt number Sc causes the reduction in concentration of nanoparticle. To see the variation of local Nusselt number Re−1/2 Nur and local Sherwood number Re−1/2 Shr on different physical parameters, results are drawn in Tables 2 and 3, respectively. In Table 2, we seen that, with the enlargement of mass transfer parameter s, relaxation time parameter β 1 , retardation time parameter β 2 , respectively, the Nusselt number reduces, while a converse results can be noticed for Biot number Bi. Additionally, the heat transfer at the wall shows a decreasing results for radiation Rd, thermophoresis Nt and Brownian motion Nb parameters, respectively. Furthermore, the variation of local Sherwood number Re−1/2 Shr with respect to mass transfer parameter s, relaxation time parameter β 1 , retardation time parameter β 2 , thermophoresis parameter Nt, Brownian motion parameter Nb and Schmidt number Sc, respectively, numerical results of Oldroyd-B fluid over a porous disk are presented in Table 3. It can be observed that, the mass transfer at the well declines on s, β 1 and β 2 , respectively. On the other hand, results show the higher trend of Sherwood number against thermophoresis parameter Nt, Brownian motion parameter Nb and Schmidt number Sc.
Fig. 8. (a): Effect of Nb on θ (η) (b): Effect of Nb on φ (η).
8
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
Fig. 9. (a): Effect of Pr on θ (η) (b): Effect of Sc on φ (η).
6. Conclusions Table 2 Influence of physical parameters s, β 1 , β 2 , Bi, Rd, Nt and Nb on Re−1/2 Nur with R = 1.3, M = 1, A = 0.5, θw = 1.1, Sc = 5 = Pr and δ = 0.5. s
β1
β2
Bi
Rd
Nt
Nb
Re−1/2 Nur
0.1 0.2 0.3 0.3
0.03
0.2
0.5
0.1
0.1
0.1
0.32869670 0.30445296 0.27164688 0.27142563 0.27003383 0.26992818 0.27147688 0.26767199 0.26528126 0.26242469 0.27614758 0.28880341 0.25893435 0.21710155 0.18330227 0.19931250 0.17118192 0.13864581 0.17132248 0.11515603 0.05633411
0.3
0.05 0.08 0.1 0.1
0.3
0.1
0.1 0.3 0.4 0.5
0.3
0.1
0.5
0.5 0.55 0.6 0.7
0.3
0.1
0.5
0.7
0.3 0.5 0.7 0.6
0.3
0.1
0.5
0.7
0.6
0.1 0.2 0.3 0.2
0.1 0.2 0.3
Table 3 Influence of physical parameters s, β 1 , β 2 , Nt, Nb and Sc on Re−1/2 Shr with R = 1.3, M = 1, A = 0.5, Pr = 5, Rd = 0.1, Bi = 0.5 and δ = 0.5. s
β1
β2
Nt
Nb
Sc
Re−1/2 Shr
0.1 0.2 0.3 0.3
0.03
0.2
0.1
0.1
3.0
2.30389026 2.07722109 1.86641287 1.86447322 1.86148902 1.85943551 1.85286187 1.84598114 1.81700298 2.24122884 2.61593126 3.21403998 2.24122884 2.29799179 2.31603409 2.29779179 2.48750201 2.65290729
0.3
0.05 0.08 0.1 0.1
0.3
0.1
0.3 0.4 0.5 0.1
0.3
0.1
0.1
0.3 0.4 0.5 0.3
0.3
0.1
0.1
0.3
0.1 0.3 0.5 0.3
5.0 7.0 9.0
In this survey, we have investigated the numerical solution of stagnation point flow of MHD Oldroyd-B nanofluid over rotating permeable disk. The convective boundary condition is also considered here in this study. The nonlinear thermal radiation is also taken for the enhancement of temperature of the Oldroyd-B fluid. The system of PDEs are transformed into nonlinear dimensionless ODEs by using von Karman transformations. The BVP Midrich technique in Maple software is utilized to acquire the numerical solutions of the governing problem. In the end, there are some important results of the present survey are follow: 1. The velocity profile in radial direction shows an increasing behavior for stronger rate of stretching parameter and reduction is noticed in the azimuthal direction. 2. With the enlargement of velocity ratio parameter, the radial velocity enhances while declines in azimuthal velocity field. 3. Significant variation is observed in velocity and temperature of the fluid against mass transfer parameter. 4. Stronger rate of thermal radiation and Biot number, respectively show an increasing behavior of temperature of the Oldroyed-B nanofluid. 5. It is observed that for higher estimation of Schmidt number Sc causes to reduce in concentration of nanoparticle. 6. Increasing of relaxation time parameter and retardation time parameter, respectively, the reduction in rate of heat transfer is noticed. Declaration of Competing Interest The authors declare that there is no conflict of interest. References [1] K. Hiemenz, Die grenzschicht an einem in den gleichformigen flussigkeitsstrom eingetauchten geraden kreiszylinder, Dinglers Polytech. J. 326 (1911) 321–324. [2] T.C. Chaim, Stagnation-point flow towards a stretching plate, J. Phys. Soci. Jpn. 63 (1994) 2443–2444. [3] T.C. Chiam, Heat transfer with variable conductivity in a stagnation-point flow towards a stretching sheet, Int. Commun. Heat Mass Transf. 23 (1996) 239–248. [4] T.R. Mahapatra, A.S. Gupta, Heat transfer in stagnation point flow towards a stretching sheet, Heat Mass Transf. 38 (2002) 517–521. [5] S.S. Motsa, Y. Khan, S. Shateyi, A new numerical solution of maxwell fluid over a shrinking sheet in the region of a stagnation point, Math. Prob. Eng. 2012 (2012). [6] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet, Int. J. Heat Mass Transf. 54 (2011) 308–313. [7] M. Khan, Hashim, A. Hafeez, A review on slip-flow and heat transfer performance of nanofluids from a permeable shrinking surface with thermal radiation: dual solutions, Chem. Eng. Sci. 173 (2017) 1–11.
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342 [8] T.R. Mahapatra, A.S. Gupta, Heat transfer in stagnation-point flow towards a stretching sheet, Heat Mass Transf. 38 (2002) 517–521. [9] A. Ishak, K. Jafar, R. Nazar, I. Pop, MHD stagnation point flow towards a stretching sheet, Phys. A 388 (17) (2009) 3377–3383. [10] A. Malvandi, F. Hedayati, D.D. Ganji, Slip effects on unsteady stagnation point flow of a nanofluid over a stretching sheet, Powder Technol. 253 (2014) 377–384. [11] Hashim, A. Hafeez, A.S. Alshomrani, M. Khan, Multiple physical aspects during the flow and heat transfer analysis of Carreau fluid with nanoparticles, Sci. Rep. 8 (2018) 17402. [12] A. Hamid, Hashim, M. Khan, A. Hafeez, Unsteady stagnation-point flow of williamson fluid generated by stretching/shrinking sheet with ohmic heating, Int. J. Heat Mass Transf. 126 (2018) 933–940. [13] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. Ser.A 200 (1950) 523–541. [14] T. Hayat, T. Muhammad, S.A. Shehzad, A. Alsaedi, An analytical solution for magnetohydrodynamic oldroyd-b nanofluid flow induced by a stretching sheet with heat generation/absorption, Int. J. Ther. Sci. 111 (2017) 274–288. [15] M. Khan, A. Hafeez, J. Ahmed, Impacts of non-linear radiation and activation energy on the axisymmetric rotating flow of oldroyd-b fluid, Phys. A (2020) 124085. [16] B. Mahanthesh, B.J. Gireesha, S.A. Shehzad, F.M. Abbasi, R.S.R. Gorla, Nonlinear three-dimensional stretched flow of an oldroyd-b fluid with convective condition, thermal radiation, and mixed convection, Appl. Math. Mech. Eng. Edi. 38 (2017) 969–998. [17] K.G. Kumar, G.K. Ramesh, B.J. Gireesha, R.S.R. Gorla, Characteristics of joule heating and viscous dissipation on three-dimensional flow of oldroyd b nanofluid with thermal radiation, Alex. Eng. J. 57 (2018) 2139–2149. [18] Y. Zhang, B. Yuan, Y. Bai, Y. Cao, Y. Shen, Unsteady cattaneo-christov double diffusion of oldroyd-b fluid thin film with relaxation-retardation viscous dissipation and relaxation chemical reaction, Powder Technol. 338 (2018) 975–982. [19] S. Choi, J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, ASME Int. Mech. Eng. Congr. Exposition, San Francisco, 1995.
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[20] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transf. 128 (2006) 240–250. [21] T. Hayat, T. Muhammad, A. Alsaedi, M.S. Alhuthali, Magnetohydrodynamic three-dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation, J. Mag. Mag. Mater. 385 (2015) 222–229. [22] T. Hayat, T. Muhammad, S.A. Shehzad, A. Alsaedi, On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip effect: a numerical study, Comp. Meth. Appl. Mech. Eng. 315 (2017) 467–477. [23] T. Muhammad, A. Alsaedi, T. Hayat, S.A. Shehzad, A revised model for Darcy– Forchheimer three-dimensional flow of nanofluid subject to convective boundary condition, Res. Phys. 7 (2017) 2791–2797. [24] T. Muhammad, A. Alsaedi, S.A. Shehzad, T. Hayat, A revised model for Darcy– Forchheimer flow of maxwell nanofluid subject to convective boundary condition, Chin. J. Phys. 55 (3) (2017) 963–976. [25] M. Sheikholeslami, Magnetic field influence on CuO–H2 O nanofluid convective flow in a permeable cavity considering various shapes for nanoparticles, Int. J. Hyd. Energy 42 (2017) 19611–19621. [26] M. Sheikholeslami, Finite element method for PCM solidification in existence of cuo nanoparticles, J. Molecular Liq. 265 (2018) 347–355. [27] M. Sheikholeslami, Numerical approach for MHD Al2 O3 -water nanofluid transportation inside a permeable medium using innovative computer method, Comp. Meth. Appl. Mech. Eng. 344 (2019) 306–318. [28] M. Sheikholeslami, New computational approach for exergy and entropy analysis of nanofluid under the impact of lorentz force through a porous media, Comp. Meth. Appl. Mech. Eng. 344 (2019) 319–333. [29] S. Rosseland, Astrophysik und Atom-Theoretische Grundlagen, Springer Verlag, Berlin, 1931. 41–44 [30] T.V. Karman, Uberlaminare und turbulente reibung, Z. Angew Math. Mech. 1 (1921) 233–252. [31] M. Turkyilmazoglu, Nanofluid flow and heat transfer due to a rotating disk, Comput. Fluids 94 (2014) 139–146. [32] N. Bachok, A. Ishak, I. Pop, Flow and heat transfer over a rotating porous disk in a nanofluid, Phys. B 406 (2011) 1767–1772.
Contents lists available at ScienceDirect
Computer Methods and Programs in Biomedicine journal homepage: www.elsevier.com/locate/cmpb
Stagnation point flow of radiative Oldroyd-B nanofluid over a rotating disk Abdul Hafeez a,∗, Masood Khan a, Jawad Ahmed a,b a b
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan Department of Basic Sciences, University of Engineering and Technology, Taxila 47050, Pakistan
a r t i c l e
i n f o
Article history: Received 12 December 2019 Revised 14 January 2020 Accepted 15 January 2020
Keywords: Oldroyd-B fluid MHD Stagnation point flow Heat generation/absorption Nonlinear thermal radiation Convective boundary condition
a b s t r a c t Background: Nanofluids are known for better heat transfer characteristics in many heat exchanger devices due to their enhanced heat transfer abilities. Recently, scientists give the idea of nanofluid which is the mixture of base fluid and solid nanoparticles having very small size. For physical phenomenon of conventional fluids by mean of suspensions of nanoparticles in base fluids and prompted produce a new composite known as “nanofluids”. These composite contain the nanoparticles with 1–100 nm sized which are suspended in the base fluids. Here we have considered a subclass of non-Newtonian fluid called Oldroyd-B fluid. The fluid motion over the disk surface is produced due to the rotation as well as radially stretching of disk. Further, the impact of non-linear thermal radiation and heat generation/absorption is introduced to visualize the heat transfer behavior. The convective boundary is also taken into consideration in order to investigate the fluid thermal characteristics. The novel features of thermophoresis and Brownian motion during the nanoparticles movement in fluid motion are studied with revised Buongiorno model. The physical problem is modeled with the concept of classical Fourier’s and Fick’s laws. The von Karman variables are used to convert the partial differential equations (PDEs) into non-dimensional ordinary differential equations (ODEs). Method: The system of governing ordinary differential equations (ODEs) with boundary conditions (BCs) are highly non-linear in nature. To handle these non-linear ODEs, we use a numerical technique called BVP Midrich scheme which uses the midpoint method to acquire the numerical solution of the governing problem. The solutions of the governing problem are obtained by utilizing Maple software. Results: Effect of different involved controlling parameters on the velocity profiles, temperature and concentration distributions are analyzed graphically. Additionally, the numerical data for local Nusselt and Sherwood numbers are also tabulated. The reduction in heat transfer rate at the wall is noticed against thermoporesis and Brownian motion parameters, respectively. The concentration gradient at the wall reduces with an increment in mass transfer parameter. © 2020 Published by Elsevier B.V.
1. Introduction Based on its applications in engineering fields and industries like hot rolling, paper making, wire drawing and much more, the stagnation point flow has been extensively investigated by the researchers. The quality of such products may be checked with the flow field and heat transfer. Initially in 1911, the stagnation point flow was proposed by Hiemenz [1]. He used the similarity variables to solve the two dimensional stagnation point flow and obtained the exact solution of the problem. After that the study of stagnation point flow towards a stretching plate was disclosed by ∗
Corresponding author. E-mail address: [email protected] (A. Hafeez).
https://doi.org/10.1016/j.cmpb.2020.105342 0169-2607/© 2020 Published by Elsevier B.V.
Chiam [2]. Later on, many researchers have been done their work on stagnation point flow of different fluid models. The study of heat transport in a stagnation point flow over a stretching sheet was explored by Chiam [3]. In this study he used regular perturbation technique and obtained analytical closed form solutions upto second order. Furthermore, Mahapatra and Gupta [4] also investigated the stagnation point flow of incompressible fluid over a stretching sheet. The study of two dimensional stagnation point flow of Maxwell fluid over a shrinking sheet is reported by Motsa et al. [5] and obtained the solution by using successive linearisation method. Moreover, the partial slip effect on stagnation point flow of an incompressible fluid by a shrinking sheet is analyzed by Bhattacharyya et al. [6]. This study reveals the conditions of the existence, non-existence uniqueness and duality of the solutions
2
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
of self-similar equations numerically. Recently, Khan et al. [7] explored the stagnation point flow of nanofluid towards a permeable surface with the impact of thermal radiation. In their study, they used a numerical technique called bvp4c technique in MATLAB and obtained dual nature solutions numerically. Some articles based on stagnation point flow were investigated by researchers via [8–12]. For a long time, researchers working in non-Newtonian fluid mechanics in order to predict the characteristics and behavior of fluid. The resulting Navier–Stokes equations with their complexity are noticed due to the presence of nonlinear inertia terms. When the viscosity is variable based on applied force or stress, it is classified as non-Newtonian fluid. The common everyday example is maize starch dissolved in water. Newtonian fluid behavior such as water can be defined by temperature and pressure exclusively. But the non-Newtonian fluid’s physical behavior depends on the forces that operate on it from time to time. A number of nonNewtonian fluids have triggered the perplexing rheology of natural liquids. Non-Newtonian fluids have recently acquired tremendous importance for industrial, commercial and mechanical applications. These liquids are used in chemical processes, material handling, oil storage, food processing and many others. Products such as ketchup, mud, oils, blood, shampoos, and so other thin and dense liquids are classified as non-Newtonian fluids. Generally, there are three types of non-Newtonian fluids that are the rate type, differential type and Integral type. The rate type fluids demonstrate the relaxation and retardation times behavior. Oldroyd-B fluid is a subclass of rate type material which exhibits the behavior of both relaxation and retardation times. This model describes the flow of viscoelastic fluids and was proposed by Oldroyd [13]. After that, the study of MHD flow Oldroyd-B nanofluid induced by a stretching sheet are analyzed by Hayat et al. [14] and obtained analytical solution of the problem. Recently, the flow of magnetized OldroydB fluid over a rotating disk with the impact of thermal radiation is discussed by Khan et al. [15]. In their study, they obtained numerical solution through BVP Midrich scheme in Maple. Subsequently, numerous researchers have been done their work on Oldroyd-B fluid by taking different physical effect shown by Refs. [16–18]. In recent times, the researchers are much engaged for improvement of heat transfer rate in the forced convection phenomenon because of its applications in many engineering disciplines that are, in the fields of chemical production, air conditioning, manufacturing, power generation and many others physical processes. Nanofluids have new properties that potentially make them useful in many heat transfer applications. Some of including are in fuel cells, hybrid powered engines, domestic refrigerator, thermal management in vehicle/engine cooling and so forth. The concept of nanofluids that are the mixture of base fluid and solid nanoparticles, have been recently given by scientists. A new composite called “nanofluids” is developed for the physical phenomenon of conventional fluids. These composites include 1–100 nm-sized nanoparticles embedded in the base fluids. These suspensions was initially planned by Choi and Eastman [19]. In their study, they argued that these suspensions have higher conductive and convective heat transfer relative to the base fluids. After that, the various scientists did their work on the features of nanofluid flow. In order to investigate the thermal properties of the base fluid, Buongiorno [20] proposed a mathematical model using the thermophoresis and Brownain motion impacts to improve the thermal properties in base fluid. He found that the conventional fluid with thermophoresis and Brownain motion plays a significant role in improving the fluid’s thermal conductivity. The study of magnetized three-dimensional flow of viscoelastic nanofluid with nonlinear thermal radiation is investigated by Hayat et al. [21]. However, further the flow of nanofluid due to a rotating disk is explored by Hayat et al. [22] and obtained numerical solution. Furthermore, numerous researchers have been done their work on the study of
such flows with influence of nanoparticle in base fluid through Refs. [23–28]. Having gone through the literature and motivated by the prominent features of stagnation point flow, heat generation/absorption and nonlinear thermal radiation, it is concluded that rotating disk flow of Oldroyd-B nanofluid with these physical situations is not investigated yet. Therefore, the present article focuses on the steady three dimensional stagnation point flow of OldroydB nanofluid over a permeable rotating disk subject to convective boundary condition. The von Karman transformations are used to convert the PDEs in non-dimensional ODEs to acquire the solutions of the governing problem. To obtain the numerical solution of the system of equations, we use BVP Midrich scheme that used the midpoint method through Maple programming. The impact of different physical parameters which are involved in the governing system of equations on different velocity fields, temperature and concentration distributions are discussed in detail.
2. Mathematical formulation We assume a steady, incompressible three-dimensional boundary layer flow of Oldroyd-B fluid over a permeable disk. The flow is assumed to be generated by a rotating and stretching of the disk. All physical quantities are not depend on ϕ , as the flow is axisymmetric about z direction. The stagnation point is at the surface (z = 0 ) and the liquid flows at (z ≥ 0) the upper half plane. The disk at z = 0 is rotated by uniform angular velocity . Here, the magnetic field acts in the vertical direction. The disk is assumed porous with mass flux velocity w0 with w0 > 0 for injection and w0 < 0 for suction. With the help of classical Fourier’s and Fick’s laws, the heat and mass transfer analysis are performed. Moreover, with the influence of nonlinear radiation and heat generation/absorption, the heat transfer analysis is performed. A physical depiction of the problem is plotted in Fig. 1. The disk is kept up at constant concentration Cw . Here T∞ and C∞ are the ambient temperature and concentration, respectively. Additionally, Tf is the convective fluid temperature. The Buongiorno’s model for nanofluid is used to discuss the thermophoresis and Brownian motion due to nanoparticles. The governing equations for the given non-linear problem are [15]
∂u u ∂w + + = 0, ∂r r ∂z
Fig. 1. A Physical sketch of the problem.
(1)
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
∂ u v2 ∂u ∂u ∂ 2u σ 2 u − +w = ν 2 − B0 u + λ1 w − ue ∂r r ∂z ρ ∂z ∂z ∂ ue ∂ 2u ∂ 2u ∂ 2 u 2uv ∂v +ue − λ1 u2 2 + w2 2 + 2uw − ∂r ∂ r∂ z r ∂r ∂r ∂z 2vw ∂v uv2 v2 ∂ u − + 2 + r ∂z r ∂r r 2 1 ∂u ∂ u ∂ 2w ∂ 3u ∂ u ∂ 2u +νλ2 − −2 + w − r ∂z ∂ z ∂ z2 ∂ z3 ∂ r ∂ z2 ∂ u ∂ 2u ∂ 3u − +u , ∂ z ∂ r∂ z ∂ r ∂ z2 ∂v uv ∂v ∂v ∂ 2v σ 2 u + +w = ν 2 − B0 v + λ1 w ∂r r ∂z ρ ∂z ∂z ∂ 2v ∂ 2v ∂ 2v uv ∂ u vw ∂ u −λ1 u2 2 + w2 2 + 2uw +2 +2 ∂ r∂ z r ∂r r ∂z ∂r ∂z u2 v v3 v2 ∂v −2 2 − 2 + r ∂r r r ∂ 3v ∂v ∂ 2 w ∂ 3 v 1 ∂ u ∂v ∂v ∂ 2 u +νλ2 u −2 +w 3 − − ∂ z ∂ z2 r ∂z ∂z ∂ r ∂ z2 ∂ r ∂ z2 ∂z v ∂ 2 u ∂v ∂ 2 u u ∂ 2 v + − − , r ∂ z2 ∂ z ∂ r ∂ z r ∂ z2 u
u
2 1 ∂ qr ∂T ∂T k ∂ T +w = − ∂r ∂ z ρ c p ∂ z2 ρ cp ∂ z 2 ∂ T ∂ C DT ∂ T Q +τ DB + + (T − T∞ ), ∂ z ∂ z T∞ ∂ z ρ cp ∂C ∂C ∂ 2 C DT ∂ 2 T +w = DB 2 + , ∂r ∂z T∞ ∂ z2 ∂z
Familiarizing the Von Karman [30] transformations by
η= φ=
v = r, w = w0 , −k
u→ue = ar,
(2)
(8)
(9)
F 2 − G2 + F H − F + β1 F H 2 + 2F F H − 2GG H
+β2 2F 2 + 2F H − F H + M F + β1 F H − A − A2 = 0, (10)
2F G + G H − G + β1 G H 2 + 2 F G + F G H
−β2 G H − 2F G − 2G H + M G + β1 G H = 0,
4 θ 1 + Rd 3 4
+ Rd (3)
(11)
2 θ 3 θ + 3θ 2 θ (θw − 1 )3 2 2 3 +3 θ 2 θ + 2θ θ 2 (θw − 1 ) + 3 θ θ + θ (θw − 1 )
− Pr H θ + Pr N bθ φ + Nt θ 2 + δ Pr θ = 0, (12)
φ − ScHφ +
Nt θ = 0, Nb
(13)
The transformed BCs are
(4)
F (η ) = R, G(η ) = 1, H (η ) = s,
(5)
F ( η ) → A , G ( η ) → 0,
θ (η ) = −Bi(1 − θ (η ) ),
φ (η ) = 1 at η = 0, θ ( η ) → 0, φ ( η ) → 0
as
η → ∞, (14)
v→0, w = we = −2az, T → T∞ , C → C∞ as z → ∞. (6)
Here u refers for the velocity in r direction and v, w the velocities in ϕ , z directions, respectively. Furthermore, (ν , τ , ρ , σ , cp , α , k, DB , DT , λ1 , λ2 , hf , T, C) are the kinematic viscosity, the ratio of effective heat capicity of particles to the effective heat capacitiy of base fluid, the density of the fluid, the electrical conductivity, the specific heat capacity, the thermal diffusivity of the fluid, the tharmal conductivity, Brownian diffusion coefficient, thermal diffusion coefficient, the relaxation time, the retardation time, the convective heat transfer coefficient, the temperature and the fluid concentration, respectively. We adopt Rosseland’s diffusion approximation [29] for the radiative heat flux qr and the expression is given by
k∗
C − C∞ . Cw − C∞
H + 2F = 0,
∂T = h f T f − T , C = Cw at z = 0, ∂z
4 σ ∗ ∂T4 16 σ ∗ T 3 ∂ T qr = − =− , 3 k∗ ∂ z 3 k∗ ∂ z
√ T − T∞ z, u = rF , v = rG, w = vH, θ = , ν T f − T∞
Substituting the above mentioned transformations into governing Eqs. (1) to (5), we acquire
The boundary conditions are
u = cr,
3
(7)
where is the Rosseland mean spectral absorption coefficient and σ ∗ the Stefan-Boltzmann constant.
w We designate T = T∞ (1 + (θw − 1 ) )θ , where θw = TT∞ > 1 is the temperature ratio parameter.
here primes describe differentiation w.r.t η. The dimensionless physical parameters involved in the above system are as follow
of equations
σ B2
c M = ρ 0 is the magnetic field parameter, R = the stretching parameter, β1 (= λ1 ) the time relaxation
a parameter, β2 (= λ2 ) the retardation parameter, A = the ve-
locity ratio parameter, s =
ter, Bi = ation
Nb =
ν
parameter,
parameter,
hf k
Nt =
τ DB (Cw −C∞ ) ν
√wo
the mass transfer parame-
ν
the Biot number, Rd =
δ =
Q ρcp
τ DT T f −T∞ ν T∞
3 16σ ∗ T∞ 3k∗ k
the
heat
the
thermophoresis
the radi-
generation/absorption parameter,
the Brownian motion parameter, P r (=
the Prandtl number and Sc = Dν
ν (ρ c p ) k
)
the Schmidt number.
B
3. Physical concern parameters The following physical parameters are described as: (i) The local Nusselt number Nur is defined by
16 σ ∗ T 3 Nur = − 1 + 3 kk∗
r k ∂∂Tz
.
z=0
k T f − T∞
(15)
4
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
Making use the transformations in Eq. (15), gives
1
Re− 2 Nur = − 1 +
4Rd 3 [1 + (θw − 1 )θ (0 )] 3
θ (0 ).
(16)
(ii) The local Sherwood number Shr is
Shr = −
r DB ∂∂Cz
z=0
DB (Cw − C∞ )
.
(17)
After using the similarity variables in Eq. (17), we obtain
Re− 2 Shr = −φ (0 ), 1
r2
where, the local Reynolds number is Re = ν
(18)
.
4. Solution methodology The system of ordinary differential equations in Eqs. (9)–(13) with boundary conditions (14) are highly non-linear in nature. We use a numerical procedure called BVP Midrich scheme for handling these non-linear differential equations. The solutions of the governing problem are obtained by utilizing Maple software. The general procedure for the mid-point approach is described by
Z ∗ (t ∗ ) = F (t ∗ , Z ∗ (t ∗ ) ), Z ∗ (t0∗ ) = Z0∗ .
(19)
The term used for the modified Euler method is
Zn∗+1 = Zn∗ + h∗ F tn∗ +
h∗ h∗ , Zn∗ + F (tn∗ , Zn∗ ) , 2 2
(20)
where h∗ refers the step size and tn∗ = t0∗ + nh∗ . The implicit approach of the mid-point method strategy is articulated as
Zn∗+1 = Zn∗ + h∗ F tn∗ +
h∗ 1 ∗ ∗ , Xn∗ + Z ,Z , 2 2 n n+1
n = 0, 1, 2, ..... (21)
The local error at each step of the mid-point procedure is of O(h∗3 ) and the global error is of the order O(h∗2 ). With more computational intensive, the mid-point algorithm error declines more rapidly as h∗ → 0 and shall be a more stable solution. The absolute error convergence of this method is upto 1 × 10−6 . 4.1. Validation of numerical code A comparison of numerical values of F (0), −G (0 ) and −θ (0 ) is made for limiting case that is in the absence of time relaxation parameter, retardation parameter, stretching parameter, magnetic field, velocity ratio parameter, radiation parameter, Biot number, heat generation/absorption and nanoparticles. These comparisons of numerical results to those of Turkyilmazoglu [31] and Bachok et al. [32] can be seen in Table 1. This table shows us a validity of our numerical code for the system of governing equations. 5. Results and discussion In the present investigation, our aim is to interpret the characteristics of swirling stagnation point flow of Oldroyd-B nanofluid
Table 1 A comparison of the F (0), −G (0 ) and −θ (0 ) on fixed Prandtl number Pr = 6.2 with those of previously published papers.
F (0) −G (0 ) −θ ( 0 )
Turkyilmazoglu [31]
Bachok et al. [32]
Present study
0.51023262 0.61592201 0.93387794
0.5102 0.6159 0.9337
0.5101162643 0.6158492796 0.9336941128
Fig. 2. Radial (red line), azimuthal (black line), axial (green line) velocities and temperature (blue line) profile in a stationary frame. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
and the physical behavior of the flow and heat transport. In order to describe the flow behavior, heat and mass transfer with respect to various physical involved parameters, results are drawn in Figs. 3–9. The variation of involved physical dimensionless parameters in the above system of ordinary differential equations that are, magnetic field parameter M, stretching parameter R, suction/injection parameter s, relaxation time parameter β 1 , retardation time parameter β 2 , velocity ratio parameter A, thermal radiation parameter Rd, temperature ratio parameter θ w , heat generation/absorption parameter δ , thermophoresis parameter Nt, Brownian motion parameter Nb, Prandtl number Pr and Schmidt number Sc for velocity fields, temperature and concentration distributions are analyzed. For this, we fixed various physical parameters to illustrate physical structure are M = 1.0, R = 1.3, s = 0.1, β1 = 0.05, β2 = 0.2, A = 1.5, θw = 1.1, Rd = 0.03, Bi = 0.8, δ = 0.5, Nt = 0.1, Nb = 0.1, Pr = 5.0 and Sc = 5.0. It should also be mentioned that all plots are approaching asymptotically the far-field boundary conditions. The solution of the system of differential Eqs. (9)–(12) with boundary conditions are plotted for Newtonian case with Pr = 0.71 and in the absence of stagnation point, thermal radiation, heat generation/absorption and nanoparticles interpreted in Fig. 2. This figure shows the temperature θ (η) and laminar velocities in radial F(η), azimuthal G(η) and axial −H (η ) directions, respectively in a stationary frame. This profile is numerically based solution with the help of Maple software. The influence of stretching parameter R on velocities in radial F(η) and azimuthal G(η) directions, temperature θ (η) and concentration φ (η) distributions, respectively are sketched in Fig. 3(a)-(d) with the specified values of other parameters. Fig. 3(a) shows the impact of stretching parameter R on fluid velocity in radial direction. In this figure, the radial velocity F(η) shows a rising behavior of fluid velocity with increase in stretching parameter R in the range 0.1 to 2.0. Basically, the stretching parameter R is the ratio of stretch c to the swirl rate. So, by higher estimation of stretching parameter R(= 0.1, 1.0, 1.5, 2.0 ), the fluid velocity in radial direction F(η) increases due to an increase in the disk stretching rate. On the other hand, the contrary variation on velocity in azimuthal direction is noticed in Fig. 3(b). This is because of the less in the swirl rate of the disk as compare to stretching rate. Moreover, the fluid temperature variation on stretching parameter R is discussed through Fig. 3(c) with the default values of other parameters. It is remarked that the temperature θ (η) declines with the enlargement of stretching parameter R. For the concentration distribution φ (η), the results are quite different.
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
5
Fig. 3. Variation of R on (a): F(η) (b): G(η) (c): θ (η) (d): φ (η).
Fig. 4. Variation of A on (a): F(η) (b): G(η).
Fig. 4 (a-b) are sketched to see the influence of velocity ratio parameter A on radial velocity F(η) and azimuthal velocity G(η) with respect to η. It is seen that radial velocity F(η) grows up with an increase in the values of velocity ratio parameter A(= 0.0, 0.1, 0.3, 0.5 ). As the velocity ratio parameter A is the ratio of free stream velocity to rotation rate of the disk. It can be perceived physically that when the values of A < 1, the rotating velocity of the disk will be dominant than the velocity of the free stream. As rotation of disk is also contributes in the radial velocity of the fluid and the disk rotation accelerates with more speed, the development of centrifugal force will become more stronger which as a result push the fluids particles in the radial direction, so, enhancement of the radial flow velocity is noticed. Furthermore, the reduction of the velocity in azimuthal direction is considered in Fig. 4(b).
This is due to decrease in the rotation rate of the disk. Fig. 5(a)-(c) depict the impact of mass transfer parameter s on velocity F(η), temperature θ (η) and concentration φ (η), respectively against η. In Fig. 5(a), we observed that the fluid velocity increases with an increase in mass transfer parameter s(= 0.0, 0.1, 0.3, 0.5 ) with A = 0.1. Enlargement in temperature field θ (η) is noticed for the case of injection as changes from 0.0 to 0.5 which displayed in Fig. 5(b). Furthermore, the concentration distribution for mass transfer parameter s is pictured in Fig. 5(c). It is scrutinized that the concentration of the fluid enhances as s changes from 0.0 to 0.5. In order to see the impact of fluid temperature distribution on thermal radiation parameter Rd, Biot number Bi and heat generation/absorption parameter δ , respectively, results are sketched via Fig. 6(a)-(c). Here, distributions given in these sketches show that
6
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
Fig. 5. (a): Effect of s on F(η) (b): Effect of s on θ (η) (c): Effect of s on φ (η).
Fig. 6. Variation of θ (η) on (a): Rd (b): Bi (c): δ .
A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
7
Fig. 7. (a): Effect of Nt on θ (η) (b): Effect of Nt on φ (η).
all results are satisfied boundary conditions asymptotically. It is sighted that the temperature of the fluid θ (η) is booted up under the action of radiation parameter Rd against η. Basically, the process of radiation describe by dimensionless radiation parameter Rd which rises the fluid temperature due to provided of extra heat to the fluid. Furthermore, the impact of Biot number Bi on temperature distribution is discussed through Fig. 6(b). As we seen that the temperature within the fluid flow are boosted up with an increment in Biot number Bi(= 0.1, 0.3, 0.5, 0.8 ). By the definition of Biot number Bi, the increasing values of Bi implies that the convective heat transfer coefficient increases, thereby enhancing more heat transfer from the surface. This then causes the liquid to heat up and thus enhances the temperature of the Oldroyd-B fluid. Furthermore, the temperature θ (η) is influenced by heat generation/absorption parameter δ (see Fig. 6(c)). This figure highlighted that the temperature of the fluid enhances by enlarging values of heat generation/absorption parameter δ (= 0.0, 0.3, 0.6, 0.9 ). Additionally, the thermal boundary layer thickness becomes thicker with the enlargement of δ in this manner. The effect of Nt (thermophoresis parameter) on temperature and concentration of Oldroyd-B liquid with respect to η are illustrated in Fig. 7(a) and (b). It is concluded that the temperature of Oldroyd-B fluid enhances as Nt increases from 0.1 to 0.7. The same behavior is seen for concentration of nanoparticle as shown in Fig. 7(b). Basically, the thermophoretic force produces in the fluid by taking the higher values of Nt and consequently, heat and mass transport enhance in the fluid. To see the influence of Nb (Brownian motion parameter) on temperature θ (η) and concentration distribution φ (η), respectively, results are drawn in Fig. 8(a) and (b). From Fig. 8(a), it is rendered that the temperature rises for higher values of Nb. Moreover, the concentration distribution φ (η) declines as Nt changes form 0.1 to 0.7 with Bi = 0.2.
As the resistance for the mass transport phenomenon produced by larger Brownian motion, so the concentration distribution will be decreased. The curves of temperature distribution θ (η) on the impact Prandtl number Pr is elaborated in Fig. 9(a). It reveals that the temperature of the Oldroyd-B fluid declines by taking different values of Prandtl number between 2.0 to 7.0. In the physical point of view that the larger rate Prandtl number Pr (> 1 ) decreases the thermal diffusivity, hence in a result, the reduction in temperature θ (η) is seen. Further more, to notice the influence of Schmidt number Sc on concentration φ (η), result is drawn via Fig. 9(b). It is observed that for higher estimation of Schmidt number Sc causes the reduction in concentration of nanoparticle. To see the variation of local Nusselt number Re−1/2 Nur and local Sherwood number Re−1/2 Shr on different physical parameters, results are drawn in Tables 2 and 3, respectively. In Table 2, we seen that, with the enlargement of mass transfer parameter s, relaxation time parameter β 1 , retardation time parameter β 2 , respectively, the Nusselt number reduces, while a converse results can be noticed for Biot number Bi. Additionally, the heat transfer at the wall shows a decreasing results for radiation Rd, thermophoresis Nt and Brownian motion Nb parameters, respectively. Furthermore, the variation of local Sherwood number Re−1/2 Shr with respect to mass transfer parameter s, relaxation time parameter β 1 , retardation time parameter β 2 , thermophoresis parameter Nt, Brownian motion parameter Nb and Schmidt number Sc, respectively, numerical results of Oldroyd-B fluid over a porous disk are presented in Table 3. It can be observed that, the mass transfer at the well declines on s, β 1 and β 2 , respectively. On the other hand, results show the higher trend of Sherwood number against thermophoresis parameter Nt, Brownian motion parameter Nb and Schmidt number Sc.
Fig. 8. (a): Effect of Nb on θ (η) (b): Effect of Nb on φ (η).
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A. Hafeez, M. Khan and J. Ahmed / Computer Methods and Programs in Biomedicine 191 (2020) 105342
Fig. 9. (a): Effect of Pr on θ (η) (b): Effect of Sc on φ (η).
6. Conclusions Table 2 Influence of physical parameters s, β 1 , β 2 , Bi, Rd, Nt and Nb on Re−1/2 Nur with R = 1.3, M = 1, A = 0.5, θw = 1.1, Sc = 5 = Pr and δ = 0.5. s
β1
β2
Bi
Rd
Nt
Nb
Re−1/2 Nur
0.1 0.2 0.3 0.3
0.03
0.2
0.5
0.1
0.1
0.1
0.32869670 0.30445296 0.27164688 0.27142563 0.27003383 0.26992818 0.27147688 0.26767199 0.26528126 0.26242469 0.27614758 0.28880341 0.25893435 0.21710155 0.18330227 0.19931250 0.17118192 0.13864581 0.17132248 0.11515603 0.05633411
0.3
0.05 0.08 0.1 0.1
0.3
0.1
0.1 0.3 0.4 0.5
0.3
0.1
0.5
0.5 0.55 0.6 0.7
0.3
0.1
0.5
0.7
0.3 0.5 0.7 0.6
0.3
0.1
0.5
0.7
0.6
0.1 0.2 0.3 0.2
0.1 0.2 0.3
Table 3 Influence of physical parameters s, β 1 , β 2 , Nt, Nb and Sc on Re−1/2 Shr with R = 1.3, M = 1, A = 0.5, Pr = 5, Rd = 0.1, Bi = 0.5 and δ = 0.5. s
β1
β2
Nt
Nb
Sc
Re−1/2 Shr
0.1 0.2 0.3 0.3
0.03
0.2
0.1
0.1
3.0
2.30389026 2.07722109 1.86641287 1.86447322 1.86148902 1.85943551 1.85286187 1.84598114 1.81700298 2.24122884 2.61593126 3.21403998 2.24122884 2.29799179 2.31603409 2.29779179 2.48750201 2.65290729
0.3
0.05 0.08 0.1 0.1
0.3
0.1
0.3 0.4 0.5 0.1
0.3
0.1
0.1
0.3 0.4 0.5 0.3
0.3
0.1
0.1
0.3
0.1 0.3 0.5 0.3
5.0 7.0 9.0
In this survey, we have investigated the numerical solution of stagnation point flow of MHD Oldroyd-B nanofluid over rotating permeable disk. The convective boundary condition is also considered here in this study. The nonlinear thermal radiation is also taken for the enhancement of temperature of the Oldroyd-B fluid. The system of PDEs are transformed into nonlinear dimensionless ODEs by using von Karman transformations. The BVP Midrich technique in Maple software is utilized to acquire the numerical solutions of the governing problem. In the end, there are some important results of the present survey are follow: 1. The velocity profile in radial direction shows an increasing behavior for stronger rate of stretching parameter and reduction is noticed in the azimuthal direction. 2. With the enlargement of velocity ratio parameter, the radial velocity enhances while declines in azimuthal velocity field. 3. Significant variation is observed in velocity and temperature of the fluid against mass transfer parameter. 4. Stronger rate of thermal radiation and Biot number, respectively show an increasing behavior of temperature of the Oldroyed-B nanofluid. 5. It is observed that for higher estimation of Schmidt number Sc causes to reduce in concentration of nanoparticle. 6. Increasing of relaxation time parameter and retardation time parameter, respectively, the reduction in rate of heat transfer is noticed. Declaration of Competing Interest The authors declare that there is no conflict of interest. References [1] K. Hiemenz, Die grenzschicht an einem in den gleichformigen flussigkeitsstrom eingetauchten geraden kreiszylinder, Dinglers Polytech. J. 326 (1911) 321–324. [2] T.C. Chaim, Stagnation-point flow towards a stretching plate, J. Phys. Soci. Jpn. 63 (1994) 2443–2444. [3] T.C. Chiam, Heat transfer with variable conductivity in a stagnation-point flow towards a stretching sheet, Int. Commun. Heat Mass Transf. 23 (1996) 239–248. [4] T.R. Mahapatra, A.S. Gupta, Heat transfer in stagnation point flow towards a stretching sheet, Heat Mass Transf. 38 (2002) 517–521. [5] S.S. Motsa, Y. Khan, S. Shateyi, A new numerical solution of maxwell fluid over a shrinking sheet in the region of a stagnation point, Math. Prob. Eng. 2012 (2012). [6] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet, Int. J. Heat Mass Transf. 54 (2011) 308–313. [7] M. Khan, Hashim, A. Hafeez, A review on slip-flow and heat transfer performance of nanofluids from a permeable shrinking surface with thermal radiation: dual solutions, Chem. Eng. Sci. 173 (2017) 1–11.
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