Journal of Mechanical Science and Technology 28 (12) (2014) 5089~5094 www.springerlink.com/content/1738-494x
DOI 10.1007/s12206-014-1130-2
Cu-water nanofluid flow and heat transfer over a shrinking sheet† Kalidas Das* Dept of Mathematics, A.B.N.Seal College, Cooch Behar, West Bengal 736101, India (Manuscript Received February 8, 2014; Revised August 19, 2014; Accepted September 9, 2014) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract The influence of an external magnetic field and thermal radiation on Cu-water nanofluid flow and heat transfer over a shrinking sheet with slip surface was studied. The governing partial differential equations for steady two dimensional flows are reduced to self-similar ordinary differential equations by similarity transformation technique and then solved numerically using Runge-Kutta-Fehlberg method with shooting technique. Physical interpretation of various embedding parameters is assigned through graphs and tables for the velocity and temperature profiles as well as for skin friction coefficient and Nusselt number. Keywords: Nanofluid; Shrinking sheet; Slip flow; Thermal radiation; Magnetic field ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction Fluid heating and cooling are important in many industries such as power, manufacturing and transportation. Common heat transfer fluids such as water, ethylene glycol, toluene and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals have higher thermal conductivities than these fluids, so it is desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid but has the higher heat transfer properties. The term nanofluid refers to a liquid suspension containing tiny particles having diameter less than 50 nm. Choi [1] experimentally verified that addition of small amount of nanoparticles appreciably enhances the effective thermal conductivity of the base fluid. The common nanoparticles that have been used are aluminum, copper, iron and titanium or their oxides. A number of recent papers [2-4] have considered the mathematical and numerical modeling of convective heat transfer in nanofluids. These models have some advantages over experimental studies due to many factors that influence nanofluid properties. Nazar et al. [5] discussed the stagnation point flow past a shrinking sheet in a nanofluid. Recently, unsteady boundary layer flow of nanofluid over a stretching/shrinking sheet has been examined by Bachok et al. [6]. One of the core concepts of fluid mechanics is no-slip boundary conditions--the assumption that when a liquid flows over a solid surface, the liquid molecules adjacent to the solid are stationary relative to the solid. Investigations show that *
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[email protected] † Recommended by Associate Editor Jaewon Chung © KSME & Springer 2014
slip flow happens when the characteristic size of the flow system is small or the flow pressure is very low. The condition was first questioned by Navier [7] who suggested the general boundary condition which shows the fluid slip at the solidliquid interface. A number of significant studies [8-10] have been undertaken in this regard. Martin and Boyd [11] analyzed slip flow and heat transfer at constant wall temperature. Noghrehabadi et al. [12] studied nanofluid flow over a stretching sheet in the presence of partial slip. The effect of radiation on flow and heat transfer problems has become more important industrially. Many processes occur at high temperature, and knowledge on radiation heat transfer becomes very important for design of reliable equipment, nuclear plants and various propulsion devices, missiles and space vehicles. On the basis of these applications, Cogley et al. [13] showed that in the optically thin limit, the fluid does not absorb its own emitted radiation, but the fluid does absorb radiation emitted by the boundaries. The effect of radiation on heat transfer problems has been studied by Makinde [14], Das [15], Hayat et al. [16], S.Y. Wu et al. [17], Bhattacharyya and Layek [18]. Researchers have become increasingly interested in understanding the flow behavior and heat transfer mechanisms on boundary layer flow that are cooled by electrically conducting fluids in the presence of magnetic field. The effects of magnetic field on nanofluid flow have important applications in physics and engineering. There have been several studies on the effect of the magnetic field on the flow and heat transfer problems of regular fluid. To the author's knowledge very few studies [19-23] have thus far been communicated with regard to convective heat transfer in a nanofluid over a stretch-
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ing/shrinking sheet in presence of external magnetic field. The objective of the present paper is to extend the work of Das [24] to investigate the slip effects on mixed convection boundary layer flow and heat transfer of nanofluids, namely, Cuwater over a shrinking sheet in the presence of a magnetic field.
Table 1. Thermophysical properties of regular fluid and nanoparticles. Physical properties
Regular fluid
Cu
Cp (J/kg K)
4179
385
Ρ (kg/m3)
997.1
8933
k (W/mK)
0.613
400
7
α ×10 (m /s)
1.47
1163.1
β×10-5 (1/K)
21
1.67
2. Mathematical analysis Consider the steady two-dimensional mixed convection stagnation point flow of a viscous incompressible electrically conducting Cu-water nanofluid over a heated shrinking sheet in the region y>0. Keeping the origin fixed, two equal and opposite forces are applied along the x-axis, which results in shrinking of the sheet. A uniform magnetic field of strength B0 is imposed along the y-axis which is normal to the flow direction. It is assumed that the velocity of the sheet is uw ( x) = bx , where b is a real number. The shrinking surface is held at prescribed surface temperature Tw ( T¥ + cx , c is a positive constant), which is higher than the ambient temperature T¥ . It is further assumed that the magnetic Reynolds number is small so that the induced magnetic field is negligible in comparison to the applied magnetic field. Under the above assumptions, the governing equations for the problem can be written as ¶u ¶v + = 0, ¶x ¶y u
dU ¶u ¶u +v =U + dx ¶x ¶y
1 é ¶ 2u 2 ù ê m nf 2 + ( gβ) nf (T - T¥ ) - s B0 u ú , ρ nf ë ¶y û
u
¶T ¶T ¶ 2T ¶q r 1 +v = a nf , 2 ¶x ¶y ¶y ( r c p ) nf ¶y
(1)
4s * ¶T 4 , 3k * ¶y
T 4 = 4T¥3 T - 3T¥4 .
(5)
Thus, ¶qr 16T¥3 s * ¶ 2T . =¶y 3k * ¶y 2
(6)
The physical properties characterizing the base fluid and the nanoparticles, namely, effective density, thermal diffusivity, heat capacitance, thermal conductivity, thermal expansion coefficient and dynamic viscosity, are assumed to be constants while those of the nanofluid are assumed to be functions of the solid volume fraction of nanoparticles f are given by r nf = (1 - f ) r f + fr s , a nf =
k nf ( r C p ) nf
,
( r C p ) nf = (1 - f )( r C p ) f + f ( r C p ) s ,
(2)
(3)
where u, v are velocity components along x, y-axis, respectively, g is acceleration due to gravity, b nf is the thermal expansion coefficient of the nanofluid, U ( x) = ax is the straining velocity of the stagnation point flow, T is the temperature of the fluid within the boundary layer, a nf is the thermal diffusivity of the nanofluid, r nf is the effective density of the nanofluid and m nf is the effective viscosity of the nanofluid. The radiative heat flux term by using the Rosseland approximation is given by qr = -
2
(4)
where s * is the Stefan-Boltzmann constant and k * is the mean absorption coefficient. Assuming that the differences in temperature within the flow are such that T 4 may be expanded in Taylor's series about T¥ and neglecting higher order terms, one may get
é k s + 2k f - 2f (k f - k s ) ù mf , ú , m nf = 2.5 + + k 2 k 2 f ( k k ) ú (1 - f ) f f s û ëê s
k nf = k f ê
where the subscripts nf, f and s represent the thermophysical properties of the nanofluid, base fluid and the nanoparticles, respectively. The associated boundary conditions are ¶u ü , v = 0, T = Tw for y=0, ï ¶y ý, ï u ® U ( x), T ® T¥ as y ® ¥ þ u = bx + l
(7)
where l is the slip length as a proportional constant of the slip velocity. The mathematical analysis of the problem can be simplified by introducing the following dimensionless variables: 1
ü ï ïï ý, 1 ï T T ¥ ï v = -(a v f ) 2 f (h ), q = Tw - T¥ ïþ
æ a h =ç ç vf è
ö2 ÷ y, u = axf ¢(h ), ÷ ø
(8)
where f (h ) is the dimensionless stream function and q (h )
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is the dimensionless temperature. Now substituting Eq. (8) in Eqs. (2) and (3) and using Eq. (6), the following ordinary differential equations are obtained: é ïì ù æ r ö ïü ê í1 - f + f çç s ÷÷ ý ( f ¢2 - ff ¢¢ - 1) ú êï ú è r s ø þï 2.5 î ú=0 f ¢¢¢ - (1 - f ) ê ìï ê æ ( rb ) s ö üïú 2 ÷ ú ê + M f ¢ - zq í1 - f + f ç ç ( rb ) f ÷ ýú êë ïî è ø ïþû
k (1 + N r )q ¢¢ - Pr f k nf
éì æ (rC p )s ê ïí1 - f + f ç ç (rC p ) f êï è ëî
(9) 0.1
(10) The boundary conditions Eq. (7) then turn into (11)
Here prime denotes differentiation with respect to h , l = b / a(< 0) is the shrinking parameter, k = l
a
nf
φ 0.0
ù ö üï ÷ ý ( f ¢q - f q ¢ )ú = 0 . ÷ ú ø þï û
f = 0, f ¢ = l + kf ¢¢, θ = 1 at h = 0 ü ý. f ¢ ® 1, θ ® 0 as h ® ¥ þ
Table 2. Comparison of f ''(0) for various values of λ and φ.
is the
is the magnetic field parameslip parameter, M = B0 s ar f b gc ter, z = 2 is the mixed convection parameter, a
f ''(0)
λ Nazar et al. [5]
Das [24]
Present result
-0.25
1.40224
1.40649
1.40648
-0.75
1.48930
1.48930
1.48930
-1.00
1.32882
1.32882
1.32882
-0.25
2.14368
2.14271
2.14275
-0.75
2.27677
2.27673
2.27677
-1.00
2.03143
2.03141
2.03141
Table 3. Values of C f and - Nu for various values of M, Nr and k. M
Nr
k
Regular fluid
Cu-Water
Cf
-Nu
Cf
-Nu
0.0 0.3 0.1
0.09015
3.15872
0.02432
1.61169
0.5
0.18538
0.00253
0.05156
1.54879
1.0
0.29245
0.00002
0.18872
1.50993
0.4 0.0
0.15284
2.84864
0.03248
1.65981
0.4
0.15285
3.29797
0.02755
1.28846
0.8
0.15344
3.49505
0.01382
0.98365
0.3 0.0
0.16916
3.72572
0.03378
3.71231
0.1
0.15284
3.23237
0.03042
3.37263
0.2
0.13844
1.75244
0.02762
1.76738
Nazar et al. [5] and recent work of Das [24] and, hence, justify the use of the present code for the current model.
Nr = 16T¥3s * / 3k *k is the thermal radiation parameter and Pr = u f / a f is the Prandtl number.
4. Numerical results and discussion
The skin friction coefficient Cf, and the Nusselt Number, Nu, are important physical parameters that can be obtained from the following expressions:
Due to the large number of parameters, the number has been restricted to three new entrants, say M, k and Nr along with the nanoparticle volume fraction parameter φ. The default values of the parameters are considered as ζ = 0.2, λ = 1.1, M = 0.4, k = 0.1, Pr = 6.7 and Nr = 0.3 unless otherwise specified. From Table 3 the thermal radiation parameter Nr reduces the skin friction at the wall for Cu-water nanofluid, but the effect is not prominent for a regular fluid. On the other hand, the heat transfer rate at the plate increases with increasing the values of Nr for a regular fluid, but the effect is opposite for Cu-water nanofluid. One can see that the magnetic field parameter M enhances the skin friction coefficient for both types of fluid, but the Nusselt number decreases with the increase of M. It is evident from Table 3 that the skin friction coefficient decreases on increasing slip parameter k for both fluids, but the reverse effect occurs on the Nusselt number. Table 4 shows that as the nanoparticle volume fraction φ increases, the skin friction coefficient decreases, whereas heat transfer performance of the nanofluids is enhanced with the increase of particle volume fraction. Fig. 1 shows that the velocity component f ¢(h ) decreases with the increase of the magnetic field parameter M for both regular fluid and nanofluid. From Fig. 2, the fluid velocity
Cf = -
1
(1 - f )
Nu = -
k nf kf
2.5
f ¢¢(0) ,
(12)
Re x
Re x (1 + Nr )q ¢(0) .
(13)
3. Method of solution The set of non-linear and locally similar ordinary differential Eqs. (9), (10) with the boundary conditions Eq. (11) have been solved numerically by using the Runge-Kutta-Fehlberg method along with shooting technique. A step size of Dh = 0.01 was selected to be satisfactory for a convergence criterion of 10-6 in all cases. For numerical computation infinity condition has been considered for a large but finite value of h where no considerable variation in velocity, temperature etc., occurs. To assess the accuracy of the present code, the skin friction coefficient f ''(0) for different values of λ and φ is obtained in Table 2 when ζ = k = M = Nr = 0. This table indicates that the present results are compatible with the previous works of
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Table 4. Values of C f and - Nu for various values of M and φ. Cu-Water
M
φ Cf
-Nu
0.0
0.0
0.0902419
3.158720
0.03
0.0445787
2.634428
0.06
0.0154244
0.680811
0.00
0.1528451
3.232371
0.03
0.0955914
2.897232
0.06
0.0606229
2.017333
0.4
Fig. 3. Velocity profiles for various values of Nr.
Fig. 1. Velocity profiles for various values of M. Fig. 4. Velocity profiles for various values of φ.
Fig. 2. Velocity profiles for various values of k. Fig. 5. Temperature profiles for various values of M.
within the boundary layer is enhanced with the increase of slip parameter k for both nanofluid and regular fluid. This yields a decrease in the boundary layer thickness. Note that all velocity profiles exhibit the expected reverse flow region, i.e., the values of f ¢(h ) are initially negative and increase as it comes significantly close to the wall, but for large h it starts to increase and, ultimately, it becomes positive. It can be easily seen from Fig. 3 that the velocity distribution across the boundary layer increases with increasing the values of Nr for regular fluid, but the effect is not significant for nanofluid. It is noticed from Fig. 4 that the fluid velocity increases with increase in φ within the boundary layer region for Cu-water nanofluid. This leads to a decrease in the momentum boundary layer thickness. Fig. 5 has the temperature profiles within the boundary
layer for various values of magnetic field parameter M in the presence/absence of nanoparticles. The nanofluid temperature increases on increasing the strength of the magnetic field in the boundary layer region for both regular fluid and Cu-water and, as a consequence, the thickness of the thermal boundary layer decreases. Further, for the particular values of M, the temperature increases in certain initial range of h and is maximum at h = 0.4 (not precisely determined), but it decreases after that and tends asymptotically to zero as the distance increases from the boundary. The variations of temperature profiles for different values of the slip parameter k are demonstrated in Fig. 6. As slip parameter k increases, the temperature distribution for heated shrinking sheet decreases within the boundary layer region. Fig. 7 illustrates the effect of
K. Das / Journal of Mechanical Science and Technology 28 (12) (2014) 5089~5094
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5. Conclusions
Fig. 6. Temperature profiles for various values of k.
Fig. 7. Temperature profiles for various values of Nr.
A mathematical model has been presented for the boundary layer flow of Cu-water nanofluid over a heated shrinking sheet in presence of a transverse magnetic field, including thermal radiation and slip effects. The model has been transformed and rendered into non-dimensional form and then solved numerically by employing an extensively validated and highly efficient Runge-Kutta-Fehlberg method with shooting technique. Some of the important findings of the investigations are listed as follows: (1) Increase in the strength of magnetic field leads to deceleration of the fluid. But, the fluid velocity is enhanced by increasing the nanoparticle volume fraction parameter and slip parameter. (2) The thermal boundary layer thickness increases for increasing slip parameter and nanoparticle volume fraction parameter, whereas the reverse effect occurs for magnetic field. (3) The influence of a magnetic field and slip flow is to reduce the wall heat transfer rate. Also, the rate of heat transfer from the wall increases with the increase of thermal radiation parameter Nr for regular fluid, but the effect is opposite for nanofluid flow. (4) The addition of nanoparticles produces a remarkable enhancement on heat transfer with respect to that of a pure fluid. This is due to the high conductivity of the solid particles Cu.
Acknowledgment The author wishes to express his cordial thanks to reviewers for valuable suggestions to improve the presentation of this article.
References
Fig. 8. Temperature profiles for various values of φ.
thermal radiation Nr on temperature distribution. One can see that the temperature distribution increases uniformly with the increase of thermal radiation parameter for regular fluid within the region h < 0.8 (not precisely determined), but outside this region these profiles overlap and decrease with the increase of Nr. On the other hand, the thermal radiation parameter decreases the temperature of nanofluid within the boundary layer region. This leads to an increase in the thermal boundary layer thickness. Fig. 8 shows the variation of the nanoparticles volume fraction parameter φ on temperature distribution. The presence of nanoparticle volume fraction leads to an increase in the thickness of the thermal boundary layer, and this rate of increase is high in certain initial range of h but it decreases after that.
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Kalidas Das’s Ph.D. in Fluid Mechanics (MHD) is from the University of Kalyani (1997). He is now an Assistant Professor (senior) of Mathematics, A.B.N.Seal college, West Bengal, India. So far he had 69 research papers published in National and International journals to his credit in the fields of fluid mechanics and bio-mechanics. Two students have completed their research work and three students are still being guided by him at the moment. He is the author of many books on graduate level. His interests include the fields of fluid mechanics and bio-mechanics.