Heat Mass Transfer DOI 10.1007/s00231-015-1720-2
ORIGINAL
Mixed convection flow over a horizontal circular cylinder with constant heat flux embedded in a porous medium filled by a nanofluid: Buongiorno–Darcy model Leony Tham1 · Roslinda Nazar2 · Ioan Pop3
Received: 2 April 2015 / Accepted: 9 November 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract The steady laminar mixed convection boundary layer flow from a horizontal circular cylinder in a nanofluid embedded in a porous medium, which is maintained at a constant surface heat flux, has been studied by using the Buongiorno–Darcy nanofluid model for both cases of a heated and cooled cylinder. The resulting system of nonlinear partial differential equations is solved numerically using an implicit finite-difference scheme known as the Keller box method. The solutions for the flow and heat transfer characteristics are evaluated numerically and studied for various values of the governing parameters, namely the Lewis number, Brownian number, mixed convection parameter, buoyancy ratio parameter and thermophoresis parameter. It is also found that the boundary layer separation occurs at the opposing fluid flow, that is when the mixed convection parameter is negative. It is also observed that increasing the mixed convection parameter delays the boundary layer separation and the separation can be completely suppressed for sufficiently large values of the mixed convection parameter. The Brownian and buoyancy ratio parameters appear to affect the fluid flow and heat transfer profiles.
* Roslinda Nazar
[email protected] 1
Faculty of Agro Based Industry, Universiti Malaysia Kelantan, 17600 Jeli, Kelantan, Malaysia
2
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
3
Department of Mathematics, Babes¸-Bolyai University, 400084 Cluj‑Napoca, Romania
List of symbols a Radius of the cylinder (m) C Nanoparticle volume fraction Cf Skin friction coefficient Cw, C∞ Nanoparticle volume fraction at the wall and ambient nanoparticle, respectively DB Brownian diffusion coefficient DT Thermophoretic diffusion coefficient f Dimensionless stream function g Gravitational acceleration (m/s2) K Permeability of the porous medium (m2) kf Effective thermal conductivity of the fluid (W/m K) Le Lewis number Nb Brownian motion parameter Nr Buoyancy ratio parameter Nt Thermophoresis parameter Sh Sherwood number Pe Péclet number qm Mass heat flux (W/m2) qw Constant surface heat flux (W/m2) Ra Modified Rayleigh number for a porous medium T Fluid temperature (°C) Tw, T∞ Temperature at the wall and ambient temperature, respectively (°C) u, v Dimensionless velocity in the x- and y-directions, respectively ue(x) Dimensionless free stream velocity x, y Dimensionless Cartesian coordinates along the surface of the cylinder and normal to it, respectively Greek symbols αm Effective thermal diffusivity of the porous medium (m2/s)
13
β Volumetric volume expansion coefficient of the nanofluid φ Dimensionless nanoparticle volume fraction ε Porosity of porous medium λ Mixed convection parameter μ Dynamic viscosity (m2/s) μf Dynamic viscosity of the fluid (m2/s) θ Dimensionless fluid temperature θw Wall temperature distribution ρ Density (kg/m3) ρf Density of the fluid (kg/m3) ρp Density of nanoparticle mass (kg/m3) τ Shear stress from the surface of the cylinder (Pa) υf Kinematic viscosity of the fluid (m3/s) ψ Stream function
1 Introduction The term nanofluid was introduced by Choi [1] in 1995. It has a higher thermal conductivity compared to other conventional heat transfer fluids (water, ethylene glycol mixture and engine oil) even at low particle concentrations. This feature may have a remarkable impact upon a number of industrial sectors in which heat transfer plays a fundamental role. Nanofluids are produced by suspensioning solid nano-sized particles (smaller than 100 nm) in the conventional heat transfer fluids, and besides enhancing thermal conductivity, nanofluids have a greater viscosity, and enhanced value of critical heat flux [2]. These nanofluids, with typical thermal conductivity enhancements in the range of 15–40 % over the base fluid [3], are due to the existence of nanoparticles such as metals, oxides or carbon nanotubes with unique chemical and physical properties. Eastman et al. [4] reported an increase of 40 % in the effective thermal conductivity of ethylene–glycol with 0.3 % volume of copper nanoparticles of 10 nm diameter. Das et al. [5] reported 10–30 % increase of the effective thermal conductivity in alumina/water nanofluids with 1–4 % of alumina. The exact percentage increment of effective thermal conductivity obtained are different for various researches, however, no doubt that nanofluid is a kind of fluid which enhances the heat transfer characteristics. As such, nanofluids are widely used as coolants, lubricants and heat exchangers. Wong and de Leon [6] compiled the current and future applications of nanofluids and divided the nanofluids applications to heat transfer applications, automotive applications, electronic applications and biochemical applications. Buongiorno [7] has conducted a study of convective transport in nanofluids, the main focus of the study is in term of heat transfer enhancements during convective situations. Through his study, several agents are concluded by
13
Heat Mass Transfer
him as insignificant for heat transfer enhancements, which are suspension, particle rotation, dispersion, and turbulence agents. He suggested a new model to describe the nanofluid based on the mechanics of nanoparticles/base-fluid relative velocity. From the seven slip mechanisms which are found to influence the nanofluid (Brownian diffusion, inertia, thermophoresis, diffusophoresis, Magnus effects, fluid drainage and gravity settling), Buongiorno [7] concluded the conservation equations can be derived based on only two of the major mechanisms—Brownian diffusion and thermophoresis. Convection in porous media is important in modern science and engineering applications, from food and chemical processes, rotating machineries, biomechanics to petroleum industry, and more practical applications can be found in the well-known books by Nield and Bejan [8], Ingham and Pop [9], Pop and Ingham [10] and Vadasz [11]. Some studies are also conducted on the usage of nanofluids porous media condition. Nield and Kuznetsov [12, 13] studied Cheng–Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Khan and Aziz [14] considered the double-diffusion natural convection from a vertical plate embedded in a porous medium saturated with a nanofluid. Further, mixed convection flow past cylinders and spheres embedded in a porous medium filled by a nanofluid, also studied by Nazar et al. [15], Tham et al. [16] and Tham and Nazar [17, 18]. To the best of our knowledge, the study with constant heat flux for the present problem has not been considered before. Therefore, the aim of the present paper is to study the steady mixed convection boundary layer flow past a horizontal circular cylinder subjected to a constant surface heat flux in a nanofluid embedded in a porous medium, with the nanofluid equations model proposed by Buongiorno [7]. The formulation follows closely that proposed by Merkin [19] and the transformed nonsimilar boundary layer equations are solved numerically using an effective technique known as the Keller-box method. The effects of the mixed convection parameter, Brownian motion parameter and thermophoresis parameter on the local wall temperature distribution, the local Sherwood number and various profiles around the cylinder are studied and illustrated through tables and figures. It is worth mentioning to this end, that the usual way in which natural and mixed convection flows are modeled is to assume that the flow is driven either by a prescribed temperature or by a prescribed heat flux. A large number of calculations have been performed, especially, for the boundary layer flow limit, for such surface conditions and for various geometrical configurations. This has been done for both viscous (Newtonian) fluids and fluid saturated porous media as can be seen from the valuable books by Gebhart et al. [20] and Nield and Bejan [8].
Heat Mass Transfer
∂ v¯ ∂ u¯ + = 0, ∂ x¯ ∂ y¯
(1)
∂ C¯ µ ∂ u¯ ∂ T¯ x¯ = 1 − C¯ ∞ ρf ∞ − ρp − ρ f ∞ g sin , K ∂ y¯ ∂ y¯ ∂ y¯ a (2) 2 ∂ T¯ DT ∂ 2 T¯ ∂ T¯ ∂ C¯ ∂ T¯ ∂ T¯ , u¯ + v¯ = αm 2 + τ DB + ∂ x¯ ∂ y¯ ∂ y¯ ∂ y¯ ∂ y¯ ∂ y¯ T¯ ∞
(3)
∂ C¯ ∂ 2 C¯ DT ∂ 2 T¯ 1 ∂ C¯ + v¯ u¯ = DB 2 + , ε ∂ x¯ ∂ y¯ ∂ y¯ T¯ ∞ ∂ y¯ 2
(4)
subject to the boundary conditions
Fig. 1 Physical model and coordinate system
2 Basic equations Consider the problem of a steady two-dimensional mixed convection flow of a nanofluid past a horizontal circular cylinder of radius a that is maintained at a constant surface heat flux qw and is placed in a constant free stream temperature T∞, as shown in Fig. 1. It is assumed that nanoparticles are size independent towards the effects on fluid flow and heat transfer, and these nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. This prevents the particles from agglomeration and deposition on the porous matrix. It is assumed that a uniform free stream (1/2)U∞ is flowing vertically upwards over a cylinder, so that the free stream velocity u¯ e (¯x ) is given by u¯ e (¯x ) = U∞ sin(¯x /a), where the velocity of fluid increases from zero velocity on the stationary boundary to free stream velocity of the fluid in the direction normal to the surface of cylinder. It is also assumed that the free stream velocity is directed vertically upward with the mixed convection parameter, λ > 0 (qw > 0) for the assisting flow and λ < 0 (qw < 0) for the opposing flow. The cylinder is considered to be long enough so that the end effects can be neglected and accordingly, the flow field can be assumed two-dimensional. Under these assumptions along with the Oberbeck–Boussinesq approximation, the basic steady conservation of mass, momentum, energy and nanoparticle volume fraction equations for a nanofluid in Cartesian coordinates x¯ and y¯ are (see Buongiorno [7])
v¯ (¯x , y¯ ) = 0, knf (∂ T¯ /∂ y¯ ) = −qw , ¯ x , y¯ ) = C¯ w at y¯ = 0, 0 ≤ x¯ ≤ π, C(¯ u¯ (¯x , y¯ ) → u¯ e (¯x ), T¯ (¯x , y¯ ) → T¯ ∞ , ¯ x , y¯ ) → C¯ ∞ as y¯ → ∞, C(¯
(5)
0 ≤ x¯ ≤ π.
Here u¯ and v¯ are the velocity components along the x¯ and y¯ axes, respectively, μ is the dynamic viscosity of the fluid, K is the permeability of the porous medium, C¯ is nanoparticle volume fraction, ρf is the density of the fluid, T¯ is the temperature, ρp is the nanoparticle mass density, g is the gravity, αm is the thermal diffusivity of the porous medium, τ is the shear stress, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, ε is the porosity of the porous medium, knf is the effective thermal conductivity of the nanofluid and qw is the constant surface heat flux. Subscript w representing condition at the wall and ∞ representing ambient condition. We introduce now the non-dimensional variables defined as x = x¯ /a, y = Pe1/2 (¯y/a), u = u¯ /U∞ , v = Pe1/2 (¯v/U∞ ), ¯ θ = (kf /qw a)Pe1/2 (T¯ − T¯ ∞ ), C = (C¯ − C¯ ∞ )/�C,
(6)
where Pe = U∞a/αm is the Péclet number, α is the effective thermal diffusivity of the fluid, C¯ = C¯ w − C¯ ∞ is the characteristic of nanoparticle volume fraction. Considering the boundary layer approximation (Pe → ∞), we obtain the following boundary layer equations for the problem under consideration:
∂u ∂v + = 0, ∂x ∂y ∂C ∂T ∂u = − Nr sin x, ∂y ∂y ∂y
(7) (8)
13
u
Heat Mass Transfer
∂ 2θ
∂θ ∂θ ∂C ∂θ ∂θ +v = 2 + Nb + Nt ∂x ∂y ∂y ∂y ∂y ∂y
2
,
∂C ∂ 2C Nt ∂ 2 θ ∂C +v = + , Le u 2 ∂x ∂ y¯ ∂y Nb ∂y2
(9)
subject to boundary conditions (10)
subject to boundary conditions
v(x, y) = 0, (∂θ/∂y) = −1, C(x, y) = 1 at y = 0, 0 ≤ x ≤ π, u(x, y) → ue (x), θ(x, y) → 0, C(x, y) → 0 as y → ∞, 0 ≤ x ≤ π,
(11)
Nb =
Ra , Pe
Le =
τ DB �C¯ , αm
αm , ε DB Nt =
Nr =
(12)
with Ra = (1 − C∞ )gKρf ∞ β(qw a/kf )a/(µαm ) being the modified Rayleigh number for the porous medium filled by a nanofluid. It is worth mentioning that mixed convection parameter, namely the ratio of the modified Rayleigh number to Peclet number, which sometimes known as the buoyancy parameter, provides the condition of the fluid flow: when λ > 0, it corresponds to the assisting fluid flow; when λ < 0, it corresponds to the opposing fluid flow; and when λ = 0, buoyancy effects are not present and fluid flow is purely in the form of forced convection flow. The range of λ for the current study will depend on the existing of boundary layer solution. Following Merkin [19], we look for a solution of Eqs. (7)–(10) of the form:
ψ = xf (x, y),
θ = θ(x, y),
C = φ(x, y),
(13)
where ψ is the stream function, which is defined in the usual way as u = ∂ ψ/∂ y and v = − ∂ ψ/∂ x, that automatically satisfies Eq. (7). Substituting Eq. (13) into Eqs. (8)– (10), and taking into account that ue(x) = sin x, we get
sin x ∂f , = 1 + (θ − Nrφ) ∂y x
(14)
2 ∂θ ∂φ ∂f ∂θ ∂ 2θ ∂θ ∂θ ∂f ∂θ + Nb + Nt − , = x + f ∂y ∂y ∂y ∂y ∂y ∂x ∂x ∂y ∂y2
(15)
13
(17)
0 ≤ x ≤ π.
Physical quantities of interest are the local wall temperature distribution θw(x) and the local Sherwood number Shx, which the local Serwood number is defined as
x¯ q¯ m DB C¯
(18)
where q¯ m = − DB
∂ C¯ ∂ y¯ y¯ =0
is the wall mass flux. Using
variables (6) and (13), we obtain 1/2 Pe− Shx = − x
∂φ (x, 0), and θw (x) = θ (x, 0), ∂y
(19)
where Pex = u¯ e (¯x ) x¯ /αm is the local Péclet number.
(ρp − ρf ∞ )�C¯ , ρf ∞ (1 − C¯ ∞ )β(qw a/kf )
τ DT (qw a/kf ) , αm T¯ ∞
f (x, y) = 0, (∂θ/∂y) = −1, φ(x, y) = 1 at y = 0, 0 ≤ x ≤ π, θ (x, y) → 0, φ(x, y) → 0 as y → ∞,
Shx =
where the (qwa/kf) is used following Bejan [21], λ is the constant mixed convection parameter, Nr is the buoyancy ratio parameter, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter and Le is the Lewis number, which are defined as =
Nt ∂ 2 θ ∂f ∂φ ∂φ ∂f ∂φ ∂ 2φ , (16) + − + Lef = xLe ∂y2 ∂y Nb ∂y2 ∂y ∂x ∂x ∂y
3 Results and discussion Equations (14)–(16) subject to the boundary conditions (17) have been solved numerically for different values of parameters λ, Nb, Nr and Nt, and at some streamwise of positions x, using a very efficient implicit finite-difference scheme known as the Keller-box method along with the Newton’s linearization technique as described by Cebeci and Bradshaw [22]. The solution is obtained by the following four steps: 1. Reduce Eqs. (14)–(16) to a first-order system, 2. Write the difference equations using central differences, 3. Linearize the resulting algebraic equations by Newton’s method, and write them in the matrix–vector form, 4. Solve the linear system by the block tridiagonal elimination technique. The edge of boundary layer thickness, y∞ has been adjusted for different range of parameters with the step sizes of Δy = 0.02 and Δx = 0.02 have been used for the computations, as the computed values are sufficient to generate accurate numerical results. The iterations were continued until an accuracy of 10–7 was achieved. Several values of the local wall temperature, θw(x) and the local Sherwood number Pe−1/2 Sh are given in Tables 1, 2, 3 and 4. These tables show that
Heat Mass Transfer Table 1 Values of the local wall temperature θw(x) for Le = 5, Nb = 0.1, Nr = 0.1, Nt = 0.1, and various values of λ x
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
λ −0.8
−0.75
−0.73
−0.698
−0.697
−0.5
0
0.5
1
2
5
−0.5514 −0.5493
−0.5915 −0.5889 −0.5815 −0.5694 −0.5529 −0.5324 −0.5075
−0.6071 −0.6043 −0.5965 −0.5837 −0.5662 −0.5444 −0.5189 −0.4906 −0.4606 −0.4304 −0.4026
−0.6315 −0.6282 −0.6194 −0.6057 −0.5867 −0.5630 −0.5331 −0.5015 −0.4677 −0.4327 −0.3990 −0.3690 −0.3448 −0.3254 −0.3028
−0.6322 −0.6293 −0.6208 −0.6069 −0.5879 −0.5642 −0.5362 −0.5048 −0.4709 −0.4362 −0.4029 −0.3735 −0.3506 −0.3345 −0.3238 −0.3179
−0.7688 −0.7649 −0.7537 −0.7352 −0.7098 −0.6776 −0.6392 −0.5950 −0.5457 −0.4922 −0.4357 −0.3780 −0.3220 −0.2724 −0.2355 −0.2156
−1.0437 −1.0383 −1.0225 −0.9966 −0.9608 −0.9154 −0.8611 −0.7982 −0.7275 −0.6497 −0.5656 −0.4762 −0.3825 −0.2858 −0.1879 −0.0927
−1.2606 −1.2540 −1.2348 −1.2034 −1.1600 −1.1050 −1.0390 −0.9626 −0.8767 −0.7821 −0.6797 −0.5706 −0.4559 −0.3366 −0.2140 −0.0891
−1.4449 −1.4373 −1.4154 −1.3793 −1.3294 −1.2663 −1.1906 −1.1030 −1.0044 −0.8957 −0.7782 −0.6529 −0.5210 −0.3839 −0.2429 −0.0992
−1.7556 −1.7464 −1.7197 −1.6759 −1.6153 −1.5386 −1.4465 −1.3400 −1.2200 −1.0880 −0.9450 −0.7926 −0.6323 −0.4656 −0.2941 −0.1197
−2.4612 −2.4483 −2.4109 −2.3494 −2.2645 −2.1569 −2.0278 −1.8784 −1.7102 −1.5250 −1.3245 −1.1108 −0.8859 −0.6522 −0.4119 −0.1676
−0.3165
π
−0.2115
−0.0372
−0.0009
0.0023
0.0030
0.0042
Table 2 Values of the local wall temperature θw(x) for Le = 5, Nb = 0.1, Nr = 0.5, Nt = 0.1, and various values of λ x
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 π
λ −1.19
−1.08
−1.05
−1.02
−1.01
−0.5
0
0.50
1.0
2.0
5.0
−0.4795 −0.4778
−0.5362 −0.5344 −0.5291 −0.5206 0.5078
−0.5522 −0.5501 −0.5443 −0.5348 −0.5220 −0.5064 −0.4887 −0.4699 −0.4516 −0.4305 −0.4447 0.0082
−0.5683 −0.5661 −0.5596 −0.5492 −0.5351 −0.5177 −0.4979 −0.4764 −0.4544 −0.4331 −0.4136 −0.3963 −0.3689 −0.3350 −0.2098
−0.5737 −0.5714 −0.5648 −0.5541 −0.5395 −0.5217 −0.5011 −0.4788 −0.4559 −0.4336 −0.4131 −0.3950 −0.3791 −0.3651 −0.3521 −0.3382
−0.8344 −0.8301 −0.8178 −0.7975 −0.7694 −0.7340 −0.6915 −0.6426 −0.5879 −0.5280 −0.4641 −0.3975 −0.3302 −0.2654 −0.2078 −0.1626
−1.0437 −1.0383 −1.0225 −0.9966 −0.9608 −0.9154 −0.8611 −0.7982 −0.7275 −0.6497 −0.5656 −0.4762 −0.3825 −0.2858 −0.1879 −0.0927
−1.2197 −1.2133 −1.1948 −1.1644 −1.1224 −1.0692 −1.0054 −0.9315 −0.8484 −0.7569 −0.6579 −0.5524 −0.4415 −0.3263 −0.2079 −0.0876
−1.3737 −1.3664 −1.3456 −1.3113 −1.2639 −1.2039 −1.1320 −1.0487 −0.9550 −0.8517 −0.7400 −0.6209 −0.4957 −0.3654 −0.2314 −0.0949
−1.6388 −1.6302 −1.6053 −1.5643 −1.5078 −1.4362 −1.3502 −1.2508 −1.1389 −1.0156 −0.8822 −0.7400 −0.5904 −0.4348 −0.2749 −0.1120
−2.2536 −2.2417 −2.2075 −2.1512 −2.0734 −1.9749 −1.8567 −1.7199 −1.5659 −1.3964 −1.2128 −1.0172 −0.8114 −0.5975 −0.3775 −0.1537
−0.3136
the boundary layer separates from the cylinder for some values of λ < 0 (opposing flow). Increasing λ delays separation and the separation can be completely suppressed in the range 0 ≤ x < π for sufficiently large values of λ. A value of = 0 (< 0) is found below which the
−0.1397
−0.0372
−0.0046
0.0014
0.0028
0.0038
boundary layer solution is not possible. However, there is a value of = S (< 0) for which the boundary layer starts to separate. It is seen further from Tables 1 and 2 that the value of S which first gives no separation lies between −0.697 and −0.698 when Nr = 0.1 and −1.01
13
Heat Mass Transfer −1/2
Table 3 Values of the local Sherwood number Pex x
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Shx for Le = 5, Nb = 0.1, Nr = 0.1, Nt = 0.1, and various values of λ
λ −0.8
−0.75
−0.73
−0.698
−0.697
−0.5
0
0.5909 0.5885
0.6757 0.6726 0.6636 0.6490 0.6290 0.6040 0.5696
0.7081 0.7047 0.6951 0.6793 0.6576 0.6305 0.5985 0.5624 0.5232 0.4823 0.4417
0.7581 0.7541 0.7429 0.7255 0.7015 0.6712 0.6332 0.5920 0.5470 0.4990 0.4499 0.4014 0.3546 0.3077 0.2538
0.7596 0.7559 0.7452 0.7277 0.7036 0.6733 0.6374 0.5965 0.5516 0.5040 0.4553 0.4074 0.3616 0.3170 0.2701 0.2127
1.0272 1.0219 1.0065 0.9813 0.9464 0.9023 0.8494 0.7885 0.7201 0.6451 0.5646 0.4798 0.3925 0.3049 0.2191 0.1339
1.5278 1.5198 1.4966 1.4584 1.4057 1.3389 1.2589 1.1663 1.0621 0.9475 0.8235 0.6914 0.5525 0.4083 0.2603 0.1106
1.9063 1.8963 1.8672 1.8195 1.7536 1.6701 1.5700 1.4543 1.3241 1.1807 1.0256 0.8604 0.6867 0.5060 0.3203 0.1311
2.2224 2.2107 2.1768 2.1211 2.0441 1.9468 1.8299 1.6949 1.5430 1.3758 1.1949 1.0023 0.7996 0.5890 0.3725 0.1519
2.7489 2.7344 2.6924 2.6234 2.5280 2.4074 2.2627 2.0954 1.9074 1.7005 1.4767 1.2384 0.9879 0.7276 0.4599 0.1874
3.9296 3.9093 3.8489 3.7496 3.6125 3.4392 3.2316 2.9918 2.7225 2.4264 2.1065 1.7660 1.4084 1.0370 0.6553 0.2670
0.1153
0.0596
0.0073
−0.0025
−0.0037
−0.0047
−0.0066
π
Table 4 Values of the local −1/2 Sherwood number Pex Shx for Le = 5, Nb = 0.1, Nr = 0.5, Nt = 0.1, and various values of λ
x
1
2
5
λ −1.19 −1.08 −1.05 −1.02 −1.01 −0.5
0
0.50
1.0
2.0
5.0
0.0 0.4436 0.5733 0.6087 0.6438 0.6555 1.1608 1.5278 0.2 0.4435 0.5712 0.6063 0.6411 0.6526 1.1548 1.5198 0.4 0.5652 0.5993 0.6332 0.6445 1.1373 1.4966 0.6 0.5554 0.5880 0.6204 0.6312 1.1085 1.4584 0.8 0.5419 0.5727 0.6030 0.6131 1.0688 1.4057 1.0 0.5538 0.5814 0.5907 1.0186 1.3389 1.2 0.5322 0.5562 0.5645 0.9584 1.2589 1.4 0.5088 0.5284 0.5355 0.8888 1.1663
1.8268 1.8171 1.7893 1.7436 1.6804 1.6005 1.5045 1.3936
2.0845 2.0735 2.0418 1.9895 1.9173 1.8260 1.7165 1.5898
2.5237 2.5104 2.4718 2.4085 2.3209 2.2102 2.0774 1.9239
3.5314 3.5126 3.4584 3.3692 3.2462 3.0906 2.9042 2.6888
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
1.2689 1.1316 0.9830 0.8247 0.6582 0.4852 0.3072 0.1260
1.4474 1.2906 1.1210 0.9403 0.7503 0.5528 0.3496 0.1427
1.7514 1.5614 1.3560 1.1373 0.9073 0.6683 0.4225 0.1723
2.4469 2.1809 1.8935 1.5876 1.2662 0.9325 0.5894 0.2403
0.4878 0.4632 0.5104 0.1815
0.4989 0.4686 0.4383 0.4078 0.3640 0.2954 0.1082
π
and −1.02 when Nr = 0.5. Tables 3 and 4 illustrate that when Nb increases, the local Sherwood number Pe−1/2 Sh increases. Opposite results are observed for the Nr case. It is worth mentioning that boundary layer separation occurs when the skin friction vanishes [23] and a minimum value of the Sherwood number is observed near the
13
0.5
0.5046 0.4728 0.4410 0.4093 0.3767 0.3416 0.3007 0.2436
0.8106 0.7246 0.6320 0.5338 0.4314 0.3266 0.2211 0.1157
1.0621 0.9475 0.8235 0.6914 0.5525 0.4083 0.2603 0.1106
0.1130 0.0359 0.0073 −0.0014 −0.0033 −0.0043 −0.0060
boundary layer separation point [24], which is shown in Tables 3 and 4. Figures 2 and 3 show the effects of Lewis number Le and Brownian motion parameter Nb (Fig. 2) or buoyancy ratio parameter Nr (Fig. 3) on the velocity profiles. For the Nb case, the velocity profile increases at the surface, while
Heat Mass Transfer
Fig. 2 Variation of the velocity profiles f ′ (y) for assisting flow with different values of Le and Nb
Fig. 3 Variation of the velocity profiles f ′ (y) for assisting flow with different values of Le and Nr
decreases for the Nr case. In general, increases in the value of Nr have the tendency to cause more induced flow along the cylinder surface. Opposite situation occurs for the Nb case: near the surface of the cylinder, increasing the Brownian motion parameter Nb, will increase the velocity, while the velocity decreases outside the boundary layer or far from the surface of the cylinder (inviscid flow region). From visual view between both figures, buoyancy ratio parameter Nr appears to have more pronounced effect towards the velocity profiles compared to the Brownian motion parameter Nb. Figures 4 and 5 show the effects of Lewis number Le and Brownian motion parameter Nb (Fig. 4) or buoyancy ratio parameter Nr (Fig. 5) on the temperature profiles. Brownian motion parameter Nb leads to a decrease in the magnitude of the wall temperature gradient, as such it increases the thermal boundary layer thickness. The effects of Nb and Nr are found to be insensitive to the variation of Nb (Brownian motion parameter) and Nr (buoyancy ratio parameter), which is consistent with the finding by Khan and Pop [25].
Fig. 4 Variation of the temperature profiles θ(y) for assisting flow with different values of Le and Nb
Fig. 5 Variation of the temperature profiles θ(y) for assisting flow with different values of Le and Nr
Fig. 6 Variation of the nanoparticle volume fraction profiles φ(y) for assisting flow with different values of Le and Nb
Figures 6 and 7 illustrate the effects of Lewis number Le and Brownian motion parameter Nb (Fig. 6) or buoyancy ratio parameter Nr (Fig. 7) on the nanoparticle volume fraction profile. As the value of Le increases, it is observed that concentration of nanoparticles decreases, and this is due
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Heat Mass Transfer
of nanoparticles volume fraction profiles and local Sherwood number; • An increase in the value of Nr led to the increment of the absolute value of = s (>0) which first gives no separation; • An increase in the value of Nr led to the increment of the value of = 0 (<0) below which a boundary layer solution does not exist. Acknowledgments The first and second authors would like to acknowledge the financial supports received from the Ministry of Higher Education, Malaysia, in the form of research fundings with project codes: RAGS/2013/UMK/SG04/1 and FRGSTOPDOWN/2014/SG04/UKM/01/1, respectively. Fig. 7 Variation of the nanoparticle volume fraction profiles φ(y) for assisting flow with different values of Le and Nr
References to the fact that a higher value of Lewis number Le causes the reduction of Brownian diffusion, with forces the concentration to decrease. The local concentration is found to increase markedly as the Brownian motion increases from 0.1 to 0.5, but local concentration decreases as thermophoresis parameter increases from 0.1 to 0.5. Based on the profiles generated from Figs. 2, 3, 4, 5, 6 and 7, it is observed that these profiles satisfy the far field boundary conditions asymptotically, which support the obtained numerical results.
4 Conclusions In this paper, we have studied the problem of mixed convection flow over a horizontal circular cylinder with constant heat flux embedded in a porous medium filled by a nanofluid via the Buongiorno–Darcy model. We have looked into the effects of the mixed convection parameter, λ, Lewis number, Le, Brownian number, Nb, buoyancy ratio parameter, Nr, and thermophoresis parameter, Nt, on the flow and heat transfer characteristics. The governing non-similar boundary layer equations were solved numerically using the Keller-box method. From this study, we could draw the following conclusions that (limited to assisting flow case): • Increasing value of Nb contributes to the increase of the velocity profiles, temperature profiles, nanoparticles volume fraction profiles and local Sherwood number, while it decreases the local wall temperature; • Increasing value of Nr contributes the same effect as Nb, except it gives an opposite effect for nanoparticles volume fraction profiles and local Sherwood number case, where increase of Nr contributes to the decrease
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1. Choi SUS (1995) Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer DA, Wang HP (eds) Development and applications of non-Newtonian flows. ASME MD-vol 231 and FED-vol 66, pp 99–105 2. Masuda H, Ebata A, Teramae K, Hishinuma N (1993) Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7:227–233 3. Choi SUS (1999) Nanofluid technology: current status and future research, energy technology division. Argonne National Laboratory, Argonne 4. Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ (2001) Anomalously increased effective thermal conductivity of ethylene glycol-based nanofluids containing copper nanoparticles. J Appl Phys Lett 78:718–720 5. Das SK, Putra N, Thiesen P, Roetzel W (2003) Temperature dependence of thermal conductivity enhancement for nanofluids. J Heat Transf 125:567–574 6. Wong KV, Leon O (2010) Applications of nanofluids: current and future. Adv Mech Eng 2010:1–11 7. Buongiorno J (2006) Convective transport in nanofluids. ASME J Heat Transf 128:240–250 8. Nield DA, Bejan A (2013) Convection in porous media, 4th edn. Springer, New York 9. Ingham DB, Pop I (2005) Transport phenomena in porous media, vol III. Elsevier, Oxford 10. Pop I, Ingham DB (2001) Convective heat transfer: Mathematical and computational modeling of viscous fluids and porous media. Pergamon, Oxford 11. Vadasz P (2008) Emerging topics in heat and mass transfer in porous media. Springer, New York 12. Nield DA, Kuznetsov AV (2009) The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Transf 52:5792–5795 13. Nield DA, Kuznetsov AV (2011) The Cheng-Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid. Int J Heat Mass Transf 54:374–378 14. Khan WA, Aziz A (2011) Double-diffusive natural convective boundary layer flow in a porous medium saturated with a nanofluid over a vertical plate: prescribed surface heat, solute and nanoparticle fluxes. Int J Therm Sci 50:2154–2160 15. Nazar R, Tham L, Pop I, Ingham DB (2011) Mixed convection boundary layer flow from a horizontal circular cylinder
Heat Mass Transfer embedded in a porous medium filled with a nanofluid. Transp Porous Med 86:517–536 16. Tham L, Nazar R, Pop I (2013) Mixed convection bound ary layer flow past a horizontal circular cylinder embedded in a porous medium saturated by a nanofluid: Brinkman model. J Porous Med 16:445–457 17. Tham L, Nazar R (2012) Mixed convection flow about a solid sphere embedded in a porous medium filled with a nanofluid. Sains Malays 41:1643–1649 18. Tham L, Nazar R (2012) Mixed convection flow about a sphere in a porous medium saturated by a nanofluid: Brinkman model. J Sci Technol 4:35–46 19. Merkin JH (1977) Mixed convection from a horizontal circular cylinder. Int J Heat Mass Transf 20:73–77 20. Gebhart B, Jaluria Y, Mahajan RL, Sammakia B (1988) Buoyancy-induced flows and transport. Hemisphere, New York
21. Bejan A, Dincer I, Lorente S, Miguel AF, Reis AH (2004) Porous and complex flow structures in modern technologies. Springer, New York 22. Cebeci T, Bradshaw P (1984) Physical and computational aspects of convective heat transfer. Springer, New York 23. Alim AM, Rahman MM, Karim MM (2008) Separation points of magneto-hydrodynamic boundary layer flow along a vertical plate with exponentially decreasing free stream velocity. J Nav Archit Mar Eng. doi:10.3329/jname.v5i1.1868 24. Bronzino JD (2000) Biomedical engineering handbook, vol 2. Springer, Heidelberg 25. Khan WA, Pop I (2010) Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transf 53:2477–2483
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