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Investigation of second grade fluid through temperature dependent thermal conductivity and non-Fourier heat flux
Accepted Manuscript Investigation of second grade fluid through temperature dependent thermal conductivity and non-Fourier heat flux T. Hayat, Salman Ahmad, M. Ijaz Khan, A. Alsaedi, M. Waqas PII: DOI: Reference:
S2211-3797(17)32313-6 https://doi.org/10.1016/j.rinp.2018.03.050 RINP 1361
To appear in:
Results in Physics
Please cite this article as: Hayat, T., Ahmad, S., Khan, M.I., Alsaedi, A., Waqas, M., Investigation of second grade fluid through temperature dependent thermal conductivity and non-Fourier heat flux, Results in Physics (2018), doi: https://doi.org/10.1016/j.rinp.2018.03.050
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Investigation of second grade fluid through temperature dependent thermal conductivity and non-Fourier heat flux T. Hayata,b , Salman Ahmada , M. Ijaz Khana,1 , A. Alsaedib and M. Waqasa a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan b Nonlinear Analysis and Applied Mathematics (N AAM ) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O.Box 80257, Jeddah 21589, Saudi Arabia 1 Corresponding author email: [email protected] (M. Ijaz Khan) March 29, 2018 Abstract Here we investigated stagnation point flow of second grade fluid over a stretchable cylinder. Heat transfer is characterized by non-Fourier law of heat flux and thermal stratification. Temperature dependent thermal conductivity and activation energy are also accounted. Transformations procedure is applying to transform the governing PDE’s into ODE’s. Obtained system of ODE’s are solved analytically by HAM. Influence of flow variables on velocity, temperature, concentration, skin friction and Sherwood number are analyzed. Obtained outcome shows that velocity enhanced through curvature parameter, viscoelastic parameter and velocities ratio variable. Temperature decays for larger Prandtl number, thermal stratification, thermal relaxation and curvature parameter. Sherwood number and concentration field show opposite behavior for higher estimation of activation energy, reaction rate, curvature parameter and Schmidt number.
Keywords: Second grade fluid; Temperature dependent thermal conductivity; Stagnation point flow; Activation energy; Thermal stratification; Non-Fourier heat flux.
1
Introduction
In universe there are various substances having diverse characteristics. Naver-Stokes theory is incapable to address the properties of these substances. These materials are called non-Newtonian materials. Example of non-Newtonian materials personal care products, food stuffs, certain oils, biological fluids, shampoos and many others. Therefore various non-Newtonian models are suggested like Maxwell, Williamson, second 1
grade, third grade, Sutterby, Cross, Casson, Oldroyed-B, Burgers, micropolar, generalized Burgers, Sisko, Jeffeey etc. The characteristic of these liquids models are discussed in Refs. [1 − 10]. In present work we considered second grade fluid model. Which predict normal stress effect. Studies associated to second grade model are mentioned through Refs. [11 − 17]. Activation energy can be defined as the minimum amount of energy that is acquire to activate molecules or atoms to a position in which they can take place physical transport or chemical reaction. Recently, many researchers and scientists are characterized the flow of binary chemical reaction with activation energy. It is due to their wide range applications like cooling of nuclear reacting, chemical engineering chemical recovery of thermal oil and geothermal reservoirs. Influences of activation energy and chemical reaction in flow of nanofluid over a stretchable surface is addressed by Mustafa et al. [18]. Khan et al. [19] studied stagnation point flow of Cross nanoliquid with activation energy and nonlinear radiative heat heat flux. Makinde et al. [20] examined influence of activation energy in unsteady convective flow. Characteristics of activation energy in flow of nanomaterial with convective boundary conditions and chemical reaction is explored by Zeshan et al. [21]. Shafique et al. [22] using numerical approach to investigate flow of viscoelastic liquid with activation energy and chemically reactive species.
Our interest here is to investigate stagnation point flow of second grade fluid over a stretchable cylinder. Heat transport is analyzed through non-Fourier flux model and temperature dependent thermal conductivity. Impacts of activation energy and thermal stratification are also accounted. By transformations procedure transformed the governing PDE’s into ODE’s. ODE’s system are tackled by homotopy analysis method [23 − 32]. Outcomes of flow variables on temperature, velocity, concentration, skin friction and Sherwood number are analyzed and discussed graphically.
2
Formulation
Stagnation point flow of second grade fluid due to stretchable cylinder considered. Heat transport attributes are scrutinized through temperature dependent thermal conductivity and variable temperature property (stratification phenomena) at the cylinder surface. Moreover, activation energy is also accounted. Non-Fourier’s heat model is implemented for the consider flow instead of Fourier law of heat conduction. The governing flow expressions in vector form are:
2.1
Continuity and momentum equations
For considered flow, the continuity and momentum equations are: ∇ • V = 0, dV = ∇ • τ, ρ dt
2
(1) (2)
where V indicates the velocity, ρ density and τ the cauchy stress tensor. For second grade material cauchy stress tensor is defined as τ = −pI + µA1 + k0 A2 + k1 A21 .
(3)
where p represents pressure, µ viscosity, k0 and k1 material constants. The RivlinEricksen tensors Ai (1, 2) are [15]: An =
dAn−1 + An−1 L + LT An−1 , dt
n = 1, 2.
(4)
According to the appropriate condition [17] : k0 ≥ 0, µ ≥ 0, k0 + k1 = 0.
(5)
Invoking Eq. (5) in Eq. (4), one has τ = −pI + µA1 + k0 (A2 − A21 ).
(6)
By using Eq. (6), continuity and momentum equations taken the form ∂(ru) ∂(rv) + = 0, (7) ∂x ∂r ∂ 2 u 1 ∂u ∂u ∂u k0 ∂ 3u ∂ 3u ∂u ∂ 2 v ∂u ∂u u +v =ν + + u + u − + ∂x ∂r ∂r2 r ∂r ρ ∂r3 ∂x∂r2 ∂r ∂r2 ∂x ∂r2 1 ∂ 2u ∂v ∂u ∂u ∂u ∂ 2u due (x) + − + , (8) v 2 +u + ue (x) r ∂r ∂x∂r ∂r ∂r ∂x ∂r dx with u(x, R) = uw (x) =
u0 x u∞ x , v(x, R) = 0, u(x, r) → ue (x) = as r → ∞, l l
(9)
where u and v represents are velocity components, l characteristic length, ν kinematic viscosity, R cylinder radius and ue (x) free stream velocity. Implementing the following transformations r 2 r u0 r − R 2 u0 x ′ R u0 ν η= f (η) u = f (η), , v=− νl 2R r l l
(10)
Continuity equation verified trivially and momentum equation with boundary conditions becomes (1 + 2αη)f ′′′ + f f ′′ + 2αf ′′ − f ′2 + 4Kα(f ′ f ′′ − f f ′′′ ) +K(2αη + 1)(f ′′2 + 2f ′ f ′′′ − f f ′′′′ ) + A2 = 0,
(11)
f (0) = 0, f ′ (0) = 1, f ′ (∞) = A, (12) k0 u 0 u∞ where A = u0 denotes velocities ratio, K = ρνl viscoelastic parameter and q νl α = u0 R2 curvature parameter. 3
2.2
Energy equation
Mathematically it is defined as dT −1 ∇ • q, = dt ρcp
(13)
where T (x, r) stands for temperature, cp specific heat and q heat flux. According to generalized Fourier’s law heat flux we have ∂q + V • ∇ q − q • ∇ V + (∇ • V)q = −k(T )∇T, (14) q+λ ∂t here λ indicates thermal relaxation time and k(T ) thermal conductivity. For incompressible flow Eq. (14) reduced to
q + λ(V • ∇q − q • ∇v) = −k(T )∇T.
(15)
Using Eq. 15 in Eq. 13, one has ∂T λ ∂T 1 ∂ ∂ 2T ∂ 2T ∂u ∂T ∂T k(T )r − +v = u2 2 + v 2 2 + u u ∂x ∂r ρcp r ∂r ∂r ρcp ∂x ∂r ∂x ∂x 2 ∂ T ∂v ∂T ∂v ∂T ∂u ∂T + 2uv +u +v +v , ∂r∂x ∂x ∂r ∂r ∂r ∂r ∂x
(16)
with x T (x, R) = Tw (x) = T0 + a , T (x, r) → T∞ as r → ∞. l
(17)
Temperature dependent thermal conductivity (variable thermal conductivity) can be defined as [33] : k(T ) = k∞ (1 + ǫθ),
(18)
where θ shows dimensionless temperature, k∞ ambient thermal conductivity and ǫ small scalar parameter. In dimensionless form the temperature equation is (2αη + 1)θ′′ + P rf θ′ + 2αθ′ + ǫ{(1 + 2αη)(θ′2 + θθ′ ) + 2αθθ′ } − P rγf {f ′ θ′ + f θ′′ } = 0,(19) θ(0) = 1 − S, θ(∞) = 0,
µcp k∞
indicates Prandtl number, S where P r = tion and γ = λul 0 thermal relaxation parameter. 4
(20) =
l(T∞ −T0 ) ax
thermal stratifica-
2.3
Concentration equation
Mathematically n dC T −Ea 2 2 = D∇ C − kr △C Exp , dt T∞ k⋆T
(21)
In expended form concentration equation becomes 2 n ∂C T ∂ C 1 ∂C −Ea ∂C 2 − kr (C − C∞ ) , +v =D + Exp u ∂x ∂r ∂r2 r ∂r T∞ k⋆T
(22)
with C(x, R) = Cw ,
C(x, r) → C∞
as r → ∞,
(23)
here C stands for concentration, kr reaction rate, n fitted rate constant, Ea activation energy and k ⋆ Boltzmann constant. Implementing equation (10), the Eq. 22 the following form E ′′ ′ ′ (1 + 2αη)φ + 2αφ + Scf φ − Scβ(1 + nTc θ) 1 − . Tc θ + 1
ν D
Tw −T0 T∞
Note that Sc = represents Schmidt number, Tc = kr2 l Ea ratio, β = u0 , reaction rate and E = k⋆ T∞ activation energy.
2.4 2.4.1
(24)
temperature
Physical quantity Skin friction
Mathematically skin friction can be defined as Cf =
−2τw , ρu2w
here wall shear stress (τw ) at r = R is given by 2 ∂u ∂ u ∂ 2 v ∂u ∂u ∂v ∂u τw = µ + k0 u 2 + v 2 + − . ∂r ∂r ∂r ∂r ∂x r ∂r r=R Invoking Eq. (26) in Eq. (25), we get p Cf Rex = −2(1 + 3K)f ′′ (0), where Rex represents the local Reynolds number. 5
(25)
(26)
(27)
2.4.2
Sherwood number
Mathematically it is defined as Shx =
xJw , D(Cw − C∞ )
(28)
where mass flux (Jw ) at r = R is
∂C Jw = −D ∂r
.
(29)
r=R
Substituting Eq. (29) in Eq. (28), we get Sh √ x = −φ′ (0). Rex
3
(30)
Solution methodology
The obtained system of nonlinear partial differential equations are solved analytically by homotopic technique. For such type of solution appropriate initial guesses are define as fo (η) = Aη + (1 − A)(1 − e−η ), θ0 (η) = (1 − S)e−η , φ0 (η) = e−η ,
(31)
and linear operator are Lf =
∂ ∂2 ∂2 ∂3 − , L = − 1, L = − 1, θ φ ∂η 3 ∂η ∂η 2 ∂η 2
(32)
with Lf (b0 + b1 e−η + b2 eη ) = 0, Lθ (b5 e−η + b6 eη ) = 0, Lφ (b7 e−η + b8 eη ) = 0,
(33)
where bi (i = 0, 1, ..., 8) indicate the arbitrary constants.
4
Convergence analysis
Convergence of series solution is depend upon the values of auxiliary parameter ~f , ~θ and ~φ . For acceptable values of these parameter we have captured ~−curves in F ig. 1. From F ig. 1 convergence intervals are −1.3 ≤ ~f ≤ −0.6, −1.5 ≤ ~θ ≤ −0.7 and −1.2 ≤ ~φ ≤ −0.5. T able 1 shows that respectively momentum, energy and concentration equations are converge at 16th, 30th and 25th order of approximations.
6
Table 1: Convergence of series solution when α = A = β = E = 0.1, K = S = 0.2, γ = P r = 1 and Sc = Tc Order of approximation −f ′′ (0) 1 −0.82872 4 −0.83919 8 −0.83894 12 −0.83895 16 −0.83895 20 −0.83895 24 −0.83895 28 −0.83895 −0.83895 32 36 −0.83895
5
= n = 0.5. −θ′ (0) −0.98978 −1.03530 −1.04050 −1.04052 −1.04053 −1.04054 −1.04055 −1.04056 −1.04056 −1.04056
−φ′ (0) −0.73845 −0.73229 −0.82971 −0.82981 −0.82991 −0.82992 −0.82993 −0.82993 −0.82993 −0.82993
Discussion
Here we focused to describe the influences of different flow variables on velocity (f ′ (η)), temperature (θ(η)), concentration (φ(η)), skin friction (Cfx ) and Sherwood number (Shx ). Immersion is specifically given to the upshot of viscoelastic parameter (K), velocities ratio (A), curvature parameter (α), Prandtl number (P r), thermal relaxation (γ), Schmidt number (Sc), reaction rate (β) and activation energy (E). Characteristics of viscoelastic parameter (K), curvature parameter (α) and velocities ratio parameter (A) on velocity (f ′ (η)) is portrayed in F igs. (2 − 4). F ig. 2 captured effect of curvature parameter (α) on velocity (f ′ (η)). It is noted that velocity enhanced for larger curvature parameter (α). For higher estimation of curvature parameter (α) radius of cylinder is reduced. It offer small resistance to motion of liquid particles and that is why velocity increased. Influence of viscoelastic parameter (K) on velocity (f ′ (η)) is described in F ig. 3. Clearly velocity (f ′ (η)) is increasing function of viscoelastic parameter. Physically for larger estimation of viscoelastic parameter viscosity of fluid reduces and thus velocity enhanced. F ig. 4 sketched behavior of velocities ratio parameter (A) on velocity (f ′ (η)). Velocity (f ′ (η)) boosts via A. Influences of viscoelastic parameter (α), thermal stratification (S), thermal relaxation (γ), small scalar parameter (ǫ) and Prandtl number (P r) on temperature (θ(η)) is examined in F igs. (5 − 9). F ig. 5 revealed impact of curvature parameter (α) on temperature (θ(η)). Temperature (θ(η)) abate through larger values of curvature parameter (α). F ig. 6 shows impact of thermal stratification (S) on temperature (θ(η)). It is noticed that temperature (θ(η)) is reduced through thermal stratification (S). It is due to the reason for higher values of thermal stratification (S) convective flow between heated cylinder and ambient fluid is reduces. F ig. 7 elucidates characteristics of thermal relaxation (γ) on temperature (θ(η)). For larger values of thermal relaxation (γ) material particles required more time to transfer energy to adjacent liquid particles and therefore velocity decays. F ig. 8 investigates the influence of ǫ on temperature (θ(η)). Clearly temperature (θ(η)) show increasing behavior against ǫ. Higher values of ǫ correspond to higher thermal conductivity i.e large amount heat transfer from heated cylinder to fluid and so temperature (θ(η)) enhanced. F ig. 9 described outcome of temperature (θ(η)) through variation in Prandtl number (P r). It is noticed 7
that temperature (θ(η)) show decreasing behavior for larger Prandtl number (P r). Thermal diffusivity reduced for rising estimation of Prandtl number (P r). Therefore temperature (θ(η)) decays. Impacts of curvature parameter (α), Schmidt number (Sc), reaction rate (β) and activation energy (E) on concentration (φ(η)) is described in F igs. (10 − 13). F ig. 10 demonstrates effect of curvature parameter (α) on concentration (φ(η)). Higher estimation of curvature parameter (α) correspond to lower concentration field (φ(η)). Impact of Schmidt number Sc on concentration (φ(η)) is portrayed in F ig. 11. Concentration (φ(η)) decays through larger Schmidt number (Sc). Physically Schmidt number is the ratio of momentum to mass diffusivity. Here mass diffusivity decays for larger Schmidt number and therefore concentration field decays. F ig. 12 evaluated behavior of reaction rate (β) on concentration (φ(η)). It is noticed that concentration (φ(η)) decline through reaction rate (β). Larger variation of β correspond to higher destructive chemical reaction rate which terminates or dissolves effectively specie of liquid. Therefore concentration (φ(η)) reduces. Variation of concentration (φ(η)) through activation energy (E) is explored in F ig. 13. Concentration (φ(η)) boosts via activation energy (E). It is due to the fact that larger values of activation energy (E) promote constrictive chemical reaction rate and thus concentration enhanced. Effects of viscoelastic parameter (K), curvature parameter (α), Schmidt number (Sc), activation energy (E) and reaction rate (β) on skin friction (Cfx ) and Sherwood number (Shx ) is evaluated in F igs. (14 − 16). F ig. 14 demonstrates influences of curvature parameter (α) and viscoelastic parameter (K) on skin friction ia portrayed in F ig. 14. Clearly skin friction (Cfx ) is increasing function of both curvature parameter (α) and viscoelastic parameter (K). Variation of Sherwood number (Shx ) through various estimation of Schmidt number and curvature parameter (α) is explored in F ig. 15. It is noted that Sherwood number (Shx ) is enhanced for larger estimation of curvature parameter (α) as well as Schmidt number (Sc). F ig. 16 explored characteristics of reaction rate (β) and activation energy (E) on Sherwood number (Shx ). Sherwood number boosts via reaction rate (β) while decays with activation energy (E).
6
Conclusions
In this paper we addressed stagnation point flow of second grade material over a stretchable cylinder. The major results are listed below: ⋆ Velocity (f ′ (η)) boosts via curvature parameter (α), viscoelastic parameter (K) and velocities ratio parameter (A). ⋆ Temperature (θ(η)) decays through curvature parameter (α), thermal stratification (S), thermal relaxation (γ) and Prandtl number (P r) while it reduces with ǫ. ⋆ For larger estimation of curvature parameter (α), Schmidt number (Sc) and reaction rate (β) concentration (φ(η)) reduce however it is enhanced for activation energy (E). ⋆ Skin friction (Cfx ) shows increasing behavior against curvature parameter (α) and viscoelastic parameter (K). ⋆ Sherwood number and concentration (φ(η)) highlight opposite behavior for activation energy (E), Schmidt number (Sc), curvature parameter (α) and reaction rate (β).
8
References [1] M. M. Cross, Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems, J. Colloid Sci. 20 (1965) 417437. [2] M. J. Babu and N. Sandeep, UCM flow across a melting surface in the presence of double stratification and cross-diffusion effects, J. Mol. Liq. 232 (2017) 27-35. [3] N. B. Khan and Z. Ibrahim, Numerical investigation of vortex-induced vibration of an elastically mounted circular cylinder with One-degree of freedom at high Reynolds number using different turbulent models, Proceed. Instit. Mech. Eng. Part M, Doi.org/10.1177/1475090217751992, (2017). [4] T. Hayat, S. Ahmad, M. I. Khan and A. Alsaedi, Simulation of ferromagnetic nanomaterial flow of Maxwell fluid, Results Phys. 8 (2018) 34-40. [5] N. B. Khan, Z. Ibrahim, M. F. Javed and M. Jameel, Numerical investigation of the vortex-induced vibration of an elastically mounted circular cylinder at high Reynolds number (Re = 104) and low mass ratio using the RANS code, Plos One, 12 (2017) e0185832. [6] A. K. A. Hakeem, S. Saranya and B. Ganga, Comparative study on Newtonian/non-Newtonian base fluids with magnetic/non-magnetic nanoparticles over a flat plate with uniform heat flux, J. Mol. Liq. 230 (2017) 445-452. [7] T. Hayat, S. Ahmad, M. I. Khan and A. Alsaedi, Exploring magnetic dipole contribution on radiative flow of ferromagnetic Williamson fluid, Results Phys. 8 (2018) 545-551. [8] T. Hayat, S. Ahmad, M. I. Khan and A. Alsaedi, Modeling and analyzing flow of third grade nanofluid due to rotating stretchable disk with chemical reaction and heat source, Physica B: Cond. Matt. 537 (2018) 116-126. [9] N. B. Khan, Z. Ibrahim, M. I. Khan, T. Hayat and M. F. Javed, VIV study of an elastically mounted cylinder having low mass-damping ratio using RANS model, Int. J. Heat Mass Transf. 121 (2018) 309-314. [10] N. B. Khan, Z. Ibrahim and M. F. Javed, Numerical investigation of the vortexinduced vibration of an elastically mounted circular cylinder having low mass rati using the RANS code, The 2017 Word congress on Advances in Structural Engineering and Mechanics, (ASEM17) ILSAN (Seoul), Korea. [11] N. Chemetov and F. Cipriano, Well-posedness of stochastic second grade fluids, J. Math. Anal. Appl. 454 (2017) 585616. [12] A. Ahmad and S. Asghar, Flow of a second grade fluid over a sheet stretching with arbitrary velocities subject to a transverse magnetic field, Appl. Math. Lett. 24 (2011) 1905-1909. [13] S. Shang, Anticipating stochastic equation of two-dimensional second grade fluids, J. Math. Anal. Appl. 461 (2018) 11381152. 9
[14] T. Hayat, S. Ahmad, M. I. Khan and A. Alsaedi, Non-Darcy Forchheimer flow of ferromagnetic second grade fluid, Results Phys. 7 (2017) 3419-3424. [15] R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rat. Mech. Analysis 4 (1955) 323-425. [16] J. E. Dunn and R. L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rat. Mech. Analysis 56.3 (1974) 191-252. [17] J. E. Dunn and K. R. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, Int. J. Eng. Sci. 33.5 (1995) 689-729. [18] M. Mustafa, J. A. Khan, T. Hayat and A. Alsaedi, Buoyancy effects on the MHD nanofluid flow past a vertical surface with chemical reaction and activation energy, Int. J. Heat Mass Transf. 108 (2017) 13401346. [19] M. I. Khan, T. Hayat, M. I. Khan and A. Alsaedi, Activation energy impact in nonlinear radiative stagnation point flow of Cross nanofluid, Int. Comm. Heat Mass Transf. 91 (2018) 216-224. [20] O. D. Makinde, P. O. Olanrewaju and W. M. Charles, Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture, Africka Matematika, 22 (2011) 6578. [21] A. Zeeshan, N. Shehzad and R. Ellahi, Analysis of activation energy in CouettePoiseuille flow of nanofluid in the presence of chemical reaction and convective boundary conditions, Results Phys. 8 (2018) 502-512. [22] Z. Shafique, M. Mustafa, A. Mushtaq, Boundary layer flow of Maxwell fluid in rotating frame with binary chemical reaction and activation energy, Results Phys. 6 (2016) 627633. [23] A. Shahzad, R. Ali and M. Khan, On the exact solution for axisymmetric flow and heat transfer over a nonlinear radially stretching sheet, Chin. Phys. Lett. 29 (2012) 084705. [24] T. Hayat, M. I. Khan, M. Imtiaz, A. Alsead and M. Waqas, Similarity transformation approach for ferromagnetic mixed convection flow in the presence of chemically reactive magnetic dipole, AIP Phys. Fluid, 28 (2016) 102003. [25] T. Hayat, M. Waqas, M. I. Khan, A. Alsaedi and S. A. Shehzad, Magnetohydrodynamic flow of Burgers fluid with heat source and power law heat flux, Chin. J. Phys. 55 (2017) 318330. [26] T. Hayat, M. I. Khan, M. Farooq, T. Yasmeen and A. Alsaedi, Water-carbon nanofluid flow with variable heat flux by a thin needle, J. Mol. Liq. 224 (2016) 786791. [27] M. I. Khan, M. Waqas, T. Hayat and A. Alsaedi, A comparative study of Casson fluid with homogeneous-heterogeneous reactions, J. Colloid Interface Sci. 498 (2017) 85-90. 10
[28] M. Farooq, M. I. Khan, M. Waqas, T. Hayat, A. Alsaedi and M. I. Khan, MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects, J. Mol. Liq. 221 (2016) 1097-1103. [29] T. Hayat, M. I. Khan, M. Farooq, A. Alsaedi and T. Yasmeen, Impact of Marangoni convection in the flow of carbon–water nanofluid with thermal radiation, Int. J. Heat Mass Transf. 106 (2017) 810-815. [30] M. I. Khan, T. Hayat, M. I. Khan and A. Alsaedi, A modified homogeneousheterogeneous reactions for MHD stagnation flow with viscous dissipation and Joule heating, Int. J. Heat Mass Transf. 113 (2017) 310-317. [31] T. Hayat, M. I. Khan, M. Farooq, A. Alsaedi and M. I. Khan, Thermally stratified stretching flow with Cattaneo-Christov heat flux, Int. J. Heat Mass Transf. 106 (2017) 289-294. T. Hayat, M. I. Khan, M. Farooq, A. Alsaedi, M. Waqas and T. Yasmeen, Impact of Cattaneo-Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface, Int. J. Heat Mass Transf. 99 (2016) 702-710. [32] T. Hayat, M. I. Khan, M. Farooq, T. Yasmeen and A. Alsaedi, Stagnation point flow with Cattaneo-Christov heat flux and homogeneous-heterogeneous reactions, J. Mol. Liq. 220 (2016) 49-55. [33] L. Miao, W. T. Wu, N. Aubry and M. Massoudi, Heat transfer and flow of a slag-type non-linear fluid: Effects of variable thermal conductivity, Appl. Math. Comput. 227 (2014) 77-91.
Fig. 1. h-curves for f (η), θ(η) and φ(η). Fig. 2. f ′ (η) via α. Fig. 3. f ′ (η) via K. Fig. 4. f ′ (η) via A. Fig. 5. θ(η) via α. Fig. 6. θ(η) via S. Fig. 7. θ(η) via γ. Fig. 8. θ(η) via ǫ. Fig. 9. θ(η) via P r. Fig. 10. φ(η) via α. Fig. 11. φ(η) via Sc. Fig. 12. φ(η) via β. Fig. 13. φ(η) via E. Fig. 14. Cfx via α and K. Fig. 15. Shx via α and Sc. Fig. 16. Shx via E and β.
11
S2211-3797(17)32313-6 https://doi.org/10.1016/j.rinp.2018.03.050 RINP 1361
To appear in:
Results in Physics
Please cite this article as: Hayat, T., Ahmad, S., Khan, M.I., Alsaedi, A., Waqas, M., Investigation of second grade fluid through temperature dependent thermal conductivity and non-Fourier heat flux, Results in Physics (2018), doi: https://doi.org/10.1016/j.rinp.2018.03.050
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Investigation of second grade fluid through temperature dependent thermal conductivity and non-Fourier heat flux T. Hayata,b , Salman Ahmada , M. Ijaz Khana,1 , A. Alsaedib and M. Waqasa a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan b Nonlinear Analysis and Applied Mathematics (N AAM ) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O.Box 80257, Jeddah 21589, Saudi Arabia 1 Corresponding author email: [email protected] (M. Ijaz Khan) March 29, 2018 Abstract Here we investigated stagnation point flow of second grade fluid over a stretchable cylinder. Heat transfer is characterized by non-Fourier law of heat flux and thermal stratification. Temperature dependent thermal conductivity and activation energy are also accounted. Transformations procedure is applying to transform the governing PDE’s into ODE’s. Obtained system of ODE’s are solved analytically by HAM. Influence of flow variables on velocity, temperature, concentration, skin friction and Sherwood number are analyzed. Obtained outcome shows that velocity enhanced through curvature parameter, viscoelastic parameter and velocities ratio variable. Temperature decays for larger Prandtl number, thermal stratification, thermal relaxation and curvature parameter. Sherwood number and concentration field show opposite behavior for higher estimation of activation energy, reaction rate, curvature parameter and Schmidt number.
Keywords: Second grade fluid; Temperature dependent thermal conductivity; Stagnation point flow; Activation energy; Thermal stratification; Non-Fourier heat flux.
1
Introduction
In universe there are various substances having diverse characteristics. Naver-Stokes theory is incapable to address the properties of these substances. These materials are called non-Newtonian materials. Example of non-Newtonian materials personal care products, food stuffs, certain oils, biological fluids, shampoos and many others. Therefore various non-Newtonian models are suggested like Maxwell, Williamson, second 1
grade, third grade, Sutterby, Cross, Casson, Oldroyed-B, Burgers, micropolar, generalized Burgers, Sisko, Jeffeey etc. The characteristic of these liquids models are discussed in Refs. [1 − 10]. In present work we considered second grade fluid model. Which predict normal stress effect. Studies associated to second grade model are mentioned through Refs. [11 − 17]. Activation energy can be defined as the minimum amount of energy that is acquire to activate molecules or atoms to a position in which they can take place physical transport or chemical reaction. Recently, many researchers and scientists are characterized the flow of binary chemical reaction with activation energy. It is due to their wide range applications like cooling of nuclear reacting, chemical engineering chemical recovery of thermal oil and geothermal reservoirs. Influences of activation energy and chemical reaction in flow of nanofluid over a stretchable surface is addressed by Mustafa et al. [18]. Khan et al. [19] studied stagnation point flow of Cross nanoliquid with activation energy and nonlinear radiative heat heat flux. Makinde et al. [20] examined influence of activation energy in unsteady convective flow. Characteristics of activation energy in flow of nanomaterial with convective boundary conditions and chemical reaction is explored by Zeshan et al. [21]. Shafique et al. [22] using numerical approach to investigate flow of viscoelastic liquid with activation energy and chemically reactive species.
Our interest here is to investigate stagnation point flow of second grade fluid over a stretchable cylinder. Heat transport is analyzed through non-Fourier flux model and temperature dependent thermal conductivity. Impacts of activation energy and thermal stratification are also accounted. By transformations procedure transformed the governing PDE’s into ODE’s. ODE’s system are tackled by homotopy analysis method [23 − 32]. Outcomes of flow variables on temperature, velocity, concentration, skin friction and Sherwood number are analyzed and discussed graphically.
2
Formulation
Stagnation point flow of second grade fluid due to stretchable cylinder considered. Heat transport attributes are scrutinized through temperature dependent thermal conductivity and variable temperature property (stratification phenomena) at the cylinder surface. Moreover, activation energy is also accounted. Non-Fourier’s heat model is implemented for the consider flow instead of Fourier law of heat conduction. The governing flow expressions in vector form are:
2.1
Continuity and momentum equations
For considered flow, the continuity and momentum equations are: ∇ • V = 0, dV = ∇ • τ, ρ dt
2
(1) (2)
where V indicates the velocity, ρ density and τ the cauchy stress tensor. For second grade material cauchy stress tensor is defined as τ = −pI + µA1 + k0 A2 + k1 A21 .
(3)
where p represents pressure, µ viscosity, k0 and k1 material constants. The RivlinEricksen tensors Ai (1, 2) are [15]: An =
dAn−1 + An−1 L + LT An−1 , dt
n = 1, 2.
(4)
According to the appropriate condition [17] : k0 ≥ 0, µ ≥ 0, k0 + k1 = 0.
(5)
Invoking Eq. (5) in Eq. (4), one has τ = −pI + µA1 + k0 (A2 − A21 ).
(6)
By using Eq. (6), continuity and momentum equations taken the form ∂(ru) ∂(rv) + = 0, (7) ∂x ∂r ∂ 2 u 1 ∂u ∂u ∂u k0 ∂ 3u ∂ 3u ∂u ∂ 2 v ∂u ∂u u +v =ν + + u + u − + ∂x ∂r ∂r2 r ∂r ρ ∂r3 ∂x∂r2 ∂r ∂r2 ∂x ∂r2 1 ∂ 2u ∂v ∂u ∂u ∂u ∂ 2u due (x) + − + , (8) v 2 +u + ue (x) r ∂r ∂x∂r ∂r ∂r ∂x ∂r dx with u(x, R) = uw (x) =
u0 x u∞ x , v(x, R) = 0, u(x, r) → ue (x) = as r → ∞, l l
(9)
where u and v represents are velocity components, l characteristic length, ν kinematic viscosity, R cylinder radius and ue (x) free stream velocity. Implementing the following transformations r 2 r u0 r − R 2 u0 x ′ R u0 ν η= f (η) u = f (η), , v=− νl 2R r l l
(10)
Continuity equation verified trivially and momentum equation with boundary conditions becomes (1 + 2αη)f ′′′ + f f ′′ + 2αf ′′ − f ′2 + 4Kα(f ′ f ′′ − f f ′′′ ) +K(2αη + 1)(f ′′2 + 2f ′ f ′′′ − f f ′′′′ ) + A2 = 0,
(11)
f (0) = 0, f ′ (0) = 1, f ′ (∞) = A, (12) k0 u 0 u∞ where A = u0 denotes velocities ratio, K = ρνl viscoelastic parameter and q νl α = u0 R2 curvature parameter. 3
2.2
Energy equation
Mathematically it is defined as dT −1 ∇ • q, = dt ρcp
(13)
where T (x, r) stands for temperature, cp specific heat and q heat flux. According to generalized Fourier’s law heat flux we have ∂q + V • ∇ q − q • ∇ V + (∇ • V)q = −k(T )∇T, (14) q+λ ∂t here λ indicates thermal relaxation time and k(T ) thermal conductivity. For incompressible flow Eq. (14) reduced to
q + λ(V • ∇q − q • ∇v) = −k(T )∇T.
(15)
Using Eq. 15 in Eq. 13, one has ∂T λ ∂T 1 ∂ ∂ 2T ∂ 2T ∂u ∂T ∂T k(T )r − +v = u2 2 + v 2 2 + u u ∂x ∂r ρcp r ∂r ∂r ρcp ∂x ∂r ∂x ∂x 2 ∂ T ∂v ∂T ∂v ∂T ∂u ∂T + 2uv +u +v +v , ∂r∂x ∂x ∂r ∂r ∂r ∂r ∂x
(16)
with x T (x, R) = Tw (x) = T0 + a , T (x, r) → T∞ as r → ∞. l
(17)
Temperature dependent thermal conductivity (variable thermal conductivity) can be defined as [33] : k(T ) = k∞ (1 + ǫθ),
(18)
where θ shows dimensionless temperature, k∞ ambient thermal conductivity and ǫ small scalar parameter. In dimensionless form the temperature equation is (2αη + 1)θ′′ + P rf θ′ + 2αθ′ + ǫ{(1 + 2αη)(θ′2 + θθ′ ) + 2αθθ′ } − P rγf {f ′ θ′ + f θ′′ } = 0,(19) θ(0) = 1 − S, θ(∞) = 0,
µcp k∞
indicates Prandtl number, S where P r = tion and γ = λul 0 thermal relaxation parameter. 4
(20) =
l(T∞ −T0 ) ax
thermal stratifica-
2.3
Concentration equation
Mathematically n dC T −Ea 2 2 = D∇ C − kr △C Exp , dt T∞ k⋆T
(21)
In expended form concentration equation becomes 2 n ∂C T ∂ C 1 ∂C −Ea ∂C 2 − kr (C − C∞ ) , +v =D + Exp u ∂x ∂r ∂r2 r ∂r T∞ k⋆T
(22)
with C(x, R) = Cw ,
C(x, r) → C∞
as r → ∞,
(23)
here C stands for concentration, kr reaction rate, n fitted rate constant, Ea activation energy and k ⋆ Boltzmann constant. Implementing equation (10), the Eq. 22 the following form E ′′ ′ ′ (1 + 2αη)φ + 2αφ + Scf φ − Scβ(1 + nTc θ) 1 − . Tc θ + 1
ν D
Tw −T0 T∞
Note that Sc = represents Schmidt number, Tc = kr2 l Ea ratio, β = u0 , reaction rate and E = k⋆ T∞ activation energy.
2.4 2.4.1
(24)
temperature
Physical quantity Skin friction
Mathematically skin friction can be defined as Cf =
−2τw , ρu2w
here wall shear stress (τw ) at r = R is given by 2 ∂u ∂ u ∂ 2 v ∂u ∂u ∂v ∂u τw = µ + k0 u 2 + v 2 + − . ∂r ∂r ∂r ∂r ∂x r ∂r r=R Invoking Eq. (26) in Eq. (25), we get p Cf Rex = −2(1 + 3K)f ′′ (0), where Rex represents the local Reynolds number. 5
(25)
(26)
(27)
2.4.2
Sherwood number
Mathematically it is defined as Shx =
xJw , D(Cw − C∞ )
(28)
where mass flux (Jw ) at r = R is
∂C Jw = −D ∂r
.
(29)
r=R
Substituting Eq. (29) in Eq. (28), we get Sh √ x = −φ′ (0). Rex
3
(30)
Solution methodology
The obtained system of nonlinear partial differential equations are solved analytically by homotopic technique. For such type of solution appropriate initial guesses are define as fo (η) = Aη + (1 − A)(1 − e−η ), θ0 (η) = (1 − S)e−η , φ0 (η) = e−η ,
(31)
and linear operator are Lf =
∂ ∂2 ∂2 ∂3 − , L = − 1, L = − 1, θ φ ∂η 3 ∂η ∂η 2 ∂η 2
(32)
with Lf (b0 + b1 e−η + b2 eη ) = 0, Lθ (b5 e−η + b6 eη ) = 0, Lφ (b7 e−η + b8 eη ) = 0,
(33)
where bi (i = 0, 1, ..., 8) indicate the arbitrary constants.
4
Convergence analysis
Convergence of series solution is depend upon the values of auxiliary parameter ~f , ~θ and ~φ . For acceptable values of these parameter we have captured ~−curves in F ig. 1. From F ig. 1 convergence intervals are −1.3 ≤ ~f ≤ −0.6, −1.5 ≤ ~θ ≤ −0.7 and −1.2 ≤ ~φ ≤ −0.5. T able 1 shows that respectively momentum, energy and concentration equations are converge at 16th, 30th and 25th order of approximations.
6
Table 1: Convergence of series solution when α = A = β = E = 0.1, K = S = 0.2, γ = P r = 1 and Sc = Tc Order of approximation −f ′′ (0) 1 −0.82872 4 −0.83919 8 −0.83894 12 −0.83895 16 −0.83895 20 −0.83895 24 −0.83895 28 −0.83895 −0.83895 32 36 −0.83895
5
= n = 0.5. −θ′ (0) −0.98978 −1.03530 −1.04050 −1.04052 −1.04053 −1.04054 −1.04055 −1.04056 −1.04056 −1.04056
−φ′ (0) −0.73845 −0.73229 −0.82971 −0.82981 −0.82991 −0.82992 −0.82993 −0.82993 −0.82993 −0.82993
Discussion
Here we focused to describe the influences of different flow variables on velocity (f ′ (η)), temperature (θ(η)), concentration (φ(η)), skin friction (Cfx ) and Sherwood number (Shx ). Immersion is specifically given to the upshot of viscoelastic parameter (K), velocities ratio (A), curvature parameter (α), Prandtl number (P r), thermal relaxation (γ), Schmidt number (Sc), reaction rate (β) and activation energy (E). Characteristics of viscoelastic parameter (K), curvature parameter (α) and velocities ratio parameter (A) on velocity (f ′ (η)) is portrayed in F igs. (2 − 4). F ig. 2 captured effect of curvature parameter (α) on velocity (f ′ (η)). It is noted that velocity enhanced for larger curvature parameter (α). For higher estimation of curvature parameter (α) radius of cylinder is reduced. It offer small resistance to motion of liquid particles and that is why velocity increased. Influence of viscoelastic parameter (K) on velocity (f ′ (η)) is described in F ig. 3. Clearly velocity (f ′ (η)) is increasing function of viscoelastic parameter. Physically for larger estimation of viscoelastic parameter viscosity of fluid reduces and thus velocity enhanced. F ig. 4 sketched behavior of velocities ratio parameter (A) on velocity (f ′ (η)). Velocity (f ′ (η)) boosts via A. Influences of viscoelastic parameter (α), thermal stratification (S), thermal relaxation (γ), small scalar parameter (ǫ) and Prandtl number (P r) on temperature (θ(η)) is examined in F igs. (5 − 9). F ig. 5 revealed impact of curvature parameter (α) on temperature (θ(η)). Temperature (θ(η)) abate through larger values of curvature parameter (α). F ig. 6 shows impact of thermal stratification (S) on temperature (θ(η)). It is noticed that temperature (θ(η)) is reduced through thermal stratification (S). It is due to the reason for higher values of thermal stratification (S) convective flow between heated cylinder and ambient fluid is reduces. F ig. 7 elucidates characteristics of thermal relaxation (γ) on temperature (θ(η)). For larger values of thermal relaxation (γ) material particles required more time to transfer energy to adjacent liquid particles and therefore velocity decays. F ig. 8 investigates the influence of ǫ on temperature (θ(η)). Clearly temperature (θ(η)) show increasing behavior against ǫ. Higher values of ǫ correspond to higher thermal conductivity i.e large amount heat transfer from heated cylinder to fluid and so temperature (θ(η)) enhanced. F ig. 9 described outcome of temperature (θ(η)) through variation in Prandtl number (P r). It is noticed 7
that temperature (θ(η)) show decreasing behavior for larger Prandtl number (P r). Thermal diffusivity reduced for rising estimation of Prandtl number (P r). Therefore temperature (θ(η)) decays. Impacts of curvature parameter (α), Schmidt number (Sc), reaction rate (β) and activation energy (E) on concentration (φ(η)) is described in F igs. (10 − 13). F ig. 10 demonstrates effect of curvature parameter (α) on concentration (φ(η)). Higher estimation of curvature parameter (α) correspond to lower concentration field (φ(η)). Impact of Schmidt number Sc on concentration (φ(η)) is portrayed in F ig. 11. Concentration (φ(η)) decays through larger Schmidt number (Sc). Physically Schmidt number is the ratio of momentum to mass diffusivity. Here mass diffusivity decays for larger Schmidt number and therefore concentration field decays. F ig. 12 evaluated behavior of reaction rate (β) on concentration (φ(η)). It is noticed that concentration (φ(η)) decline through reaction rate (β). Larger variation of β correspond to higher destructive chemical reaction rate which terminates or dissolves effectively specie of liquid. Therefore concentration (φ(η)) reduces. Variation of concentration (φ(η)) through activation energy (E) is explored in F ig. 13. Concentration (φ(η)) boosts via activation energy (E). It is due to the fact that larger values of activation energy (E) promote constrictive chemical reaction rate and thus concentration enhanced. Effects of viscoelastic parameter (K), curvature parameter (α), Schmidt number (Sc), activation energy (E) and reaction rate (β) on skin friction (Cfx ) and Sherwood number (Shx ) is evaluated in F igs. (14 − 16). F ig. 14 demonstrates influences of curvature parameter (α) and viscoelastic parameter (K) on skin friction ia portrayed in F ig. 14. Clearly skin friction (Cfx ) is increasing function of both curvature parameter (α) and viscoelastic parameter (K). Variation of Sherwood number (Shx ) through various estimation of Schmidt number and curvature parameter (α) is explored in F ig. 15. It is noted that Sherwood number (Shx ) is enhanced for larger estimation of curvature parameter (α) as well as Schmidt number (Sc). F ig. 16 explored characteristics of reaction rate (β) and activation energy (E) on Sherwood number (Shx ). Sherwood number boosts via reaction rate (β) while decays with activation energy (E).
6
Conclusions
In this paper we addressed stagnation point flow of second grade material over a stretchable cylinder. The major results are listed below: ⋆ Velocity (f ′ (η)) boosts via curvature parameter (α), viscoelastic parameter (K) and velocities ratio parameter (A). ⋆ Temperature (θ(η)) decays through curvature parameter (α), thermal stratification (S), thermal relaxation (γ) and Prandtl number (P r) while it reduces with ǫ. ⋆ For larger estimation of curvature parameter (α), Schmidt number (Sc) and reaction rate (β) concentration (φ(η)) reduce however it is enhanced for activation energy (E). ⋆ Skin friction (Cfx ) shows increasing behavior against curvature parameter (α) and viscoelastic parameter (K). ⋆ Sherwood number and concentration (φ(η)) highlight opposite behavior for activation energy (E), Schmidt number (Sc), curvature parameter (α) and reaction rate (β).
8
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Fig. 1. h-curves for f (η), θ(η) and φ(η). Fig. 2. f ′ (η) via α. Fig. 3. f ′ (η) via K. Fig. 4. f ′ (η) via A. Fig. 5. θ(η) via α. Fig. 6. θ(η) via S. Fig. 7. θ(η) via γ. Fig. 8. θ(η) via ǫ. Fig. 9. θ(η) via P r. Fig. 10. φ(η) via α. Fig. 11. φ(η) via Sc. Fig. 12. φ(η) via β. Fig. 13. φ(η) via E. Fig. 14. Cfx via α and K. Fig. 15. Shx via α and Sc. Fig. 16. Shx via E and β.
11