RADIATION EFFECTS ON AN UNSTEADY MHD NATURAL CONVECTIVE FLOW OF A NANOFLUID PAST A VERTICAL PLATE

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1 RADIATION EFFECTS ON AN UNSTEADY MHD NATURAL CONVECTIVE FLOW OF A NANOFLUID PAST A VERTICAL PLATE by Loganathan PARASURAMAN a *, Nirmal Chand PEDDISETTY a and Ganesan PERIYANNAGOUNDER a a Department o mathematics, Anna University, Chennai , India Numerical analysis is carried out on an unsteady MHD natural convective boundary layer low o a nanoluid past an isothermal vertical plate in the presence o thermal radiation. The governing partial dierential equations are solved numerically by an eicient, iterative, tri-diagonal, semi-implicit inite-dierence method. In particular, we investigate the eects o radiation, magnetic ield and nanoparticle volume raction on the low and heat transer characteristics. The nanoluids containing nanoparticles o aluminium oxide, copper, titanium oxide and silver with nanoparticle volume raction range less than or equal to 0.04 are considered. The numerical results indicate that in the presence o radiation and magnetic ield, an increase in the nanoparticle volume raction will decrease the velocity boundary layer thickness while increasing the thickness o the thermal boundary layer. Meanwhile, an increase in the magnetic ield or nanoparticle volume raction decreases the average skin-riction at the plate. Excellent validation o the present results has been achieved with the published results in the literature in the absence o the nanoparticle volume raction. Keywords: MHD; natural convection; nanoluid; semi-ininite vertical plate; thermal radiation 1. Introduction The study o magnetohydrodynamic low and heat transer has received considerable attention in recent years due to its wide variety o applications in engineering and technology such as MHD generators, plasma studies, nuclear reactors and geothermal energy extractions. The presence o an external magnetic ield is used as a control mechanism in material manuacturing industry, as the convection currents are suppressed by Lorentz orce which is produced by the magnetic ield. Radiation heat transer is also essential in many engineering areas as the design o pertinent equipment involves processes occurring at high temperatures. Nuclear power plants, gas turbines and various propulsion devices or aircrat, missiles, satellites and space vehicles are examples o such engineering areas. The natural convection heat transer is a vital phenomenon in the cooling mechanism o various engineering systems due to its minimum cost, low noise, smaller size and reliability. Ostrach *Corresponding Author: Nirmal Chand Peddisetty Tel: , address: logu@annauniv.edu,nirmalprasad000@gmail.com 1

2 [1] reviewed various industrial and engineering applications o natural convection such as thermal insulators or buildings, the electronics industry, solar collectors and cooling systems or nuclear reactors. The study o magneto hydrodynamic ree convection through a viscous luid past a semiininite vertical plate is considered very essential to understand the behavior o the perormance o luid motion in several applications. The problem o natural convection in a regular luid past a vertical plate is a classical problem irst studied theoretically by Pohlhausen []. The similarity solution to this problem was given by Ostrach [1]. Siegel [3] was the irst to study the transient reeconvective low past a semi-ininite vertical plate by integral method. Gebhart [4] have studied this problem by an approximate method. Hellums and Churchill [5] employed an explicit inite dierence technique, which is conditionally stable and convergent. Takhar et al. [6] have considered transient ree convection past a semi-ininite vertical plate with variable surace temperature. Soundalgekar and Ganesan [7] have studied transient ree convection or the isothermal plate by an implicit inite dierence method which is unconditionally stable and convergent. Convective lows with radiation are also encountered in many industrial processes such as heating and cooling o chambers, energy processes, evaporation rom large reservoirs, solar power technology and space vehicle reentry. Thermal radiation eects o an optical thin gray gas bounded by a stationary vertical plate are investigated by England and Emery [8]. Soundalgekar and Takhar [9] have studied the radiation ree convective low o an optically thin gray gas past a semi-ininite vertical plate. Hossain and Takhar [10] have considered the radiation eects on mixed convection along an isothermal vertical plate. Das et al. [11] analyzed radiation eects on low past an impulsively started ininite vertical plate. Fluid cooling and heating plays an important role in many industries and engineering applications particularly in nuclear reactors, thin ilm solar energy collectors, manuacturing, power generation and transportation. In these applications use o common heat transer luids such as water, ethylene glycol and engine oil limits the heat transer capabilities due to their poor heat transer properties. The low thermal conductivity o the conventional heat transer luids is a primary limitation in enhancing the perormance and the compactness o many engineering electronic devices. Nanoluid is the nanotechnology based heat transer luid that can be derived by stably suspending nanometer-sized particles in conventional heat transer luids. Nanoluid is the term irst coined by Choi [1] to describe this new class o nanotechnology based luids that exhibit thermal properties superior to those o their host luids. Nanoluids were ound to be devoid o problems such as sedimentation, erosion and high pressure drop due to the small size o the particles and small volume raction o particles needed to heat transer enhancement as compared with micro particle slurries. The nanoparticles used in nanoluids are made o metals (Al, Cu, carbides, metal oxides, nitrides or nonmetals (Graphite, Carbon nanotubes and the base luid is usually liquid such as water or ethylene glycol. A comprehensive survey o convective transport in nanoluids was made by Buongiorno [13] who had considered seven slip mechanisms that can produce relative velocity between the nanoparticles and the base luid. He showed that Brownian diusion and thermophoresis are important mechanisms in laminar low. Kuznetsov and Nield [14] investigated the natural convective boundary layer low o a nanoluid past an ininite vertical plate by considering thermophoresis and Brownian motion o nanoparticles. Nield and Kuznetsov [15] also have studied Cheng-Minkowycz

3 problem or natural convection boundary layer low in a porous medium saturated by nanoluids. Khan and Pop [16] reported the boundary layer low o nanoluid past a stretching sheet. The mixed convection MHD low o a nanoluid over a stretching sheet with the eects o viscous dissipation and variable magnetic ield is analyzed by Habibimatin et al. [17]. Hamad et al. [18] investigated the magnetic ield eects on the ree convection low o a nanoluid past a semi-ininite vertical lat plate. Steady MHD ree convective low o a nanoluid past a vertical plate is investigated by Chamkha and Aly [19]. Entropy analysis or MHD low over a nonlinear stretching inclined transparent plate embedded in a porous medium due to solar radiation is investigated by Dehsara et al. [0]. MHD mixed convective boundary layer low o a nanoluid through a porous medium due to an exponential stretching sheet is investigated by Ferdows et al. [1]. Ahmed et al. [] studied numerically the mixed convection boundary layer low rom a vertical lat plate embedded in a porous medium illed with nanoluids. Hady et al. [3] have analyzed the radiation eects on viscous low o a nanoluid and heat transer over a nonlinearly stretching sheet. Shakhaoath khan et al. [4] investigated the eects o magnetic ield on radiative low o a nanoluid past a stretching sheet. Recently, Loganathan et al. [5] have reported radiation eects on an unsteady natural convective low o a nanoluid past an ininite vertical plate. To authors knowledge, no studies have been communicated so ar with regard to on an unsteady MHD natural convective low o a nanoluid past an isothermal semi-ininite vertical plate in the presence o radiation. The objective o this paper is to analyze the eects o radiation, magnetic ield and nanoparticle volume raction on transient natural convective low o Al O 3, Cu, TiO and Ag water nanoluids past a semi-ininite vertical plate. The present study is o immediate application to all those processes which are highly aected by heat enhancement concept and magnetic ield. The results are validated against nanoparticle volume raction φ = 0 and are shown graphically.. Mathematical Analysis A two dimensional low o a viscous incompressible nanoluid past a semi ininite vertical plate is considered. Initially the plate and the luid are at the same temperaturet. Then at time t ' 0, the temperature o the plate is suddenly raised to T w and is maintained at the same value. We also assume that x - coordinate to be directed upward along the plate and y - coordinate is taken normal to the plate. The luid is water based nanoluid containing dierent types o nanoparticles: aluminium oxide (Al O 3, copper (Cu, titanium oxide (TiO and silver (Ag. In this study, nanoluids are assumed to behave as single phase luids with local thermal equilibrium between the base luid and the nanoparticles suspended in them so that no slip occurs between them. A schematic representation o physical model and coordinate system is depicted in ig. 1. The thermo physical properties o the nanoparticles (Re. [6] are given in table 1. The basic unsteady momentum and thermal energy equations according to the model or nanoluids given by Tiwari and Das [7] satisying Boussinesq approximation (Schlichting [8] in the presence o radiation and magnetic ield are as ollows: u u 0 x y (1 3

4 T T T 1 T qr u v kn t ' x y cp y y n u u u u n B0 u u v n g( T T t ' x y y n n ( (3 X Vertical plate Temperature proile Nanoluid Magnetic ield Velocity proile g u Y v The initial and boundary conditions are: Figure 1 Physical model and coordinate system t ' 0, u 0, v 0, T T or all x and y t ' 0, u 0, v 0, T T at x 0 u 0, v 0, T T at y 0 w u 0, v 0, T T as y (4 Here u and v are the velocity components in the x - and y- directions; t ' is the time; g is the acceleration due to gravity; T Temperature o the luid; T temperature o the luid ar away rom the plate; T w Temperature o the plate; q r is the radiative heat lux and B 0 is the strength o applied magnetic ield. For nanoluids the expressions o density n, thermal expansion coeicient (ρβ n and heat capacitance are given by c p n 4

5 n 1 s 1 n s cp 1 cp cp n s (5 Table 1 Thermo-physical properties o water and nanoparticles ρ (Kgm -3 C p (JKg -1 K -1 k(wm -1 K -1 βx10-5 (K -1 H O Al O Cu TiO Ag model [9] is given by The eective thermal conductivity o the nanoluid according to Hamilton and Crosser ke ks n 1 k n 1 ( k ks k k n 1 k ( k k s s Where n is the empirical shape actor or the nanoparticle. In particular, n = 3, or spherical shaped nanoparticles and n = 3/ or cylindrical ones. φ is the solid volume raction o nanoparticles, µ is the dynamic viscosity, υ is the kinematic viscosity, β is the thermal expansion coeicient, ρ is the density and k is the thermal conductivity. Here the subscripts n, and s represent respectively the thermo physical properties o the nanoluids, base luid and the solid nanoparticles respectively. Assuming the Rosseland approximation (Re. 30, leads to the radiative heat lux: (6 q r 4 * 3k * 4 T y (7 respectively. Where σ* and k*are Stean-Boltzmann constant and the mean absorption coeicient q r * * 4 4 4a ( T T y I the temperature dierences within the low are suiciently small such that T 4 may be expressed as a linear unction o temperature, then expanding T 4 in Taylor series about T and neglecting higher order terms, we get ( T T T T (9 We now introduce the ollowing non-dimensional quantities in Eq. (1 to Eq. (3: 5

6 X x, Y L ygr L 1/4, U ulgr 1/, V vlgr 1/4 t ' Gr, t L 1/, T T T T w 3 3 1/ gl ( Tw T B0 L 16 a* T, Pr, Gr, L Gr M and R (10 1/ Gr k Where Gr is the Grasho number and Pr is the Prandtl number Now Eq. (1 to Eq. (3 in view o the above Eq. (5 to Eq. (9 are: U V X Y 0 (11 1 kn 1 R U V t X Y k Pr Pr cp Y 1 s cp (1 U U U U V t X Y 1 1 U s 1 MU.5 Y s 1 (13 1 The transormed boundary conditions are: t 0, U 0, V 0, 0 or all X and Y t 0, U 0, V 0, 0 at X 0 U 0, V 0, 1 at Y 0 U 0, V 0, 0 as Y (14 Let k E1 k n 1 1 Pr c 1 c p s.5 p, E 1 1, 1 s 1 6

7 s E3, E4, E5 s s cp 1 1 s 1 cp (15 Model Table Thermal conductivity and dynamic viscosity or various shapes o nanoparticles Shape o nanoparticles Thermal conductivity Dynamic viscosity I Spherical kn ks k ( k ks k k k ( k k s s / (1 n.5 II Spherical kn ks k ( k ks k k k ( k k s s µ µ n ( III Cylindrical (nano tubes k k n 1 1 ks k ( k ks 1 ks k ( k ks / (1 n.5 IV Cylindrical (nano tubes k k n 1 1 ks k ( k ks 1 ks k ( k ks µ µ n ( Numerical procedure The above set o two dimensional, coupled, nonlinear partial dierential equations (11 (13 under the initial and boundary conditions (14 is solved using a semi-implicit inite dierence method. So we express them by inite-dierence scheme o Crank-Nicolson type, which converges aster and also stable unconditionally. The region o integration is considered as a rectangle as shown in ig..the inite dierence equations at every internal nodal point or a particular i-level constitute a tridiagonal system, which is solved by applying Thomas algorithm. Computations are carried out or all the time levels until the steady state is reached. The steady state solution is assumed to have been reached, when the absolute dierence between the values o U, as well as temperature T at two consecutive time steps are less than 10-5 at all grid points. The temperatures at two time levels are shown in the table 3 below. The steady state is reached at t = when the absolute dierence between temperatures is less than

8 Table 3 Comparison o temperatures at two dierent time levels Y Time T Time T t 0.05 T1-T E-05 1E-05 1E-05 1E Figure Discretization o domain 4. Stability and convergence o the inite dierence scheme The inite dierence equations corresponding to U and θ are as ollows: U U n1 n i, j i, j n Ui, j t U U U U n1 n1 n n i, j i1, j i, j i1, j X V n i, j U U U U n1 n1 n n i, j1 i, j1 i, j1 i, j1 4Y n1 n1 n1 n n n Ui, j1 U i, j Ui, j1 Ui, j1 U i, j U i, j1 E ( Y n1 n n1 n i, j i, j Ui, j U i, j E3 ME 4 (16 n1 n n1 n1 n n n1 n1 n n i, j i, j n i, j i 1, j i, j i1, j n i, j1 i, j1 i, j1 i, j1 U i, j V t X i, j 4Y n 1 n 1 n 1 n n n i, j 1 i, j n1 n i, j1 i, j1 i, j i, j1 R i, j i, j E1 E ( Y 5 Pr (17 The stability criterion o the scheme or constant mesh sizes is proved as ollows: The general ix iy term o the Fourier expansion or U, at time t = 0 are all e e apart rom a constant i 1. At a later time t, these terms will become 8

9 U t e e ( i X i Y ( i t e X e i Y (18 (19 Substituting U and in Eq. (0 and Eq. (1, regarding the coeicients U and V as constants over any one time step and denoting the values ater a time step by ', ': ix ' U ( ' (1 e V ( ' (i sin Y t X 4 Y ( ' (cos Y E Y E ME ( ' ( ' (0 ix ' U ( ' (1 e V ( ' (i sin Y t X 4 Y ( ' (cos Y 1 E 1 Y (1 Let (1 U t ix V t t M A (1 e ( isin Y E (cos Y 1 E t X Y Y 4 U t ix V t t R B (1 e ( isin Y E (cos Y 1 E t 1 X Y Y 5 Pr ( (3 Substituting A and B in Eq. (5 and Eq. (6, we have t (1 A ' (1 A E3 ( ' (4 (1 B ' (1 B (5 Eq. (4 and Eq. (5 can be written in the matrix orm as ollows: 1 A E3t ' 1 A (1 A(1 B ' 1 B 0 1 B (6 9

10 ' E Where η is a column vector o elements ψ and Φ. Now, or stability the modulus o each eigenvalue o the ampliication 1 Amatrix 1 BE should not exceed unity. The eigenvalues o E are: 1 A, 1 B Assume that U is everywhere positive and V is everywhere negative and let a U t X, b V t t, c d t Y Y, Substituting in Eq. (7, we have: X Y M A asin csin i( asinx bsin Y E4 d Since the real part o A is greater than or equal to zero, we have 1 A 1. Similarly, 1 B 1 1 A 1 B Hence the scheme is unconditionally stable which is also shown by Soundalgekar and Ganesan [7] or ordinary luids. The local truncation error is o( t X Y and it tends to zero as t, X and Y tend to zero. Hence the scheme is compatible. The stability and compatibility ensures the convergence o the inite dierence scheme. 5. Nusselt Number The local as well as average values o Nusselt number in dimensionless orm are as ollows: Nu X k n 1/4 XGr k Y Y 0 (7 k 1 n 1/4 Nu Gr dx k Y 0 Y 0 (8 The derivatives involved in Eqs. (7 - (8 are evaluated by using a ive-point approximation ormula and the integrals are evaluated by Newton-Cotes closed integration ormula 6. Results and discussion In order to get the physical insight into the low problem, numerical computations are carried out or various values o parameters that describe the low characteristics and the results are illustrated graphically. The dimensionless partial dierential equations together with the appropriate boundary conditions are solved by a semi- implicit inite-dierence method o Crank Nicolson type. The step size η = 0.05 is used to obtain the numerical solution. We consider our dierent types o nanoluids containing aluminium oxide (Al O 3, titanium oxide (TiO, copper (Cu and silver (Ag nanoparticles with water as a base luid. The nanoparticle volume raction is considered in the range o , as sedimentation takes place when the nanoparticle volume raction exceeds 8%. In this study, we 10

11 have considered spherical nanoparticles with thermal conductivity and dynamic viscosity shown in Model I in Table. The Prandtl number, Pr o the base luid is kept constant at 6.. The eects o nanoparticle volume raction, magnetic ield and radiation on the transient velocity and temperature proiles o the nanoluids are plotted. When φ = 0 this study reduces the governing equations to those o a regular luid. In order to veriy the accuracy o the numerical method, the present results are compared with those o Takhar et al. [6] in the literature and the comparisons are in excellent agreement as shown in ig. 3. Figure 3 Comparison o temperature proiles o Cu water nanoluid with Takhar et al. [6] Figure 4 Comparison o velocity proiles o cu water nanoluid or dierent values o M & R Fig. 4 depicts the velocity proiles o copper water nanoluid or dierent values o magnetic and radiation parameters. The velocity o the copper water nanoluid decreases with an increase in magnetic ield and radiation. The applied magnetic ield has the tendency to slow down the movement o the luid as a result o increased retarding orce. Hence, the presence o a magnetic ield decreases the momentum boundary layer thickness. This causes the velocity proiles o the nanoluid to have distinctive peaks near the immediate vicinity o the plate and as M increases these peaks decrease and move gradually downstream. In addition, this decrease in the low movement is also accompanied by an increase in the radiation parameter. Figure 5 Velocity proiles o Cu water nanoluid or dierent values o M and R Figure 6 Transient velocity proiles o silver water nanoluid 11

12 Fig. 5 illustrates the velocity proiles o copper water nanoluid or various values o the nanoparticle volume raction in simultaneous absence and presence o radiation and magnetic ields. It is observed that in the presence o radiation and magnetic ields, the velocity o the nanoluid increases with nanoparticle volume raction, whereas, the converse is true in the absence o radiation and magnetic ields. The above igure ig. 6 shows the velocity proiles o silver water nanoluid in the presence o radiation and magnetic ield with respect to time. It is observed that the velocity o the nanoluid increases with respect to time and a steady state is reached when t = 9.. The velocity o the nanoluid reaches a temporal maximum, which is observed at t = Figure 7 Velocity proiles o various nanoluids Figure 8 Transient temperature proiles o Cu water nanoluid or dierent values o M& R It is observed rom the ig. 8 that the temperature proiles o copper water nanoluid increases in the presence o both magnetic ield and radiation. Radiation increases the rate o energy transport to the luid and hence the thermal boundary layer thickness increases. The eect o a transverse magnetic ield on the luid give rise to a resistive type orce called the Lorentz orce. This orce has the tendency to slow down the motion o the luid to increase the temperature boundary layer. The presence o a magnetic ield increases the temperature inside the boundary layer as shown ig. 8. The steady state temperature proiles o copper water nanoluid with respect to the nanoparticle volume raction φ are shown in ig. 9. As the nanoparticle volume raction increases the temperature o the nanoluid decreases in the presence o radiation and magnetic parameter. This is possible physically because an increase in the nanoparticle volume raction leads to an increase in the thermal conductivity o the nanoluid and hence the thickness o the thermal boundary layer increases. The time taken to reach steady state decreases with an increase in nanoparticle volume raction φ. 1

13 Figure 9 Temperature proiles o Cu water nanoluid or dierent values o φ Figure 10 Transient temperature proiles o silver water nanoluid The temperature proiles o silver water nanoluid increases with time when R = 1 and M = 1. The temporal maximum or silver water nanoluid is observed at time t = 5.6 and steady state is reached at t = 9. as shown in ig. 10 Table 4 Local Nusselt number o various nanoluids in the presence o magnetic ield and radiation Pr=6., R=1, M=1, φ=0.04 X Ag Cu Al O TiO Computations o local Nusselt number Nu X or various nanoluids are shown in the tab. 3. The eects o thermal radiation and magnetic ield on a local Nusselt number o silver water nanoluid are depicted in ig. 11. It is observed that local Nusselt number increases with an increase in the thermal radiation parameter. The magnetic ield retards the heat transer process by decreasing the local Nusselt number. Enhancing magnetic ield tends to decrease the luid velocity and hence decrease the local Nusselt number. Fig. 1 illustrates that the average Nusselt number o copper water nanoluid decreases with an increase in the nanoparticle volume raction when R = 1 and M = 1. This concludes that changes in heat transer rates are associated with nanoparticle volume raction, which indicates the possible use o nanoluids in heat transer processes. 13

14 Figure 11 Local Nusselt number o silver water nanoluid or dierent values o R & M Figure 1 Average Nusselt number o copper water nanoluid Figure 13 Average Nusselt number o silver water nanoluid or dierent values o R & M Figure 14 Local skin-riction o Ag water nanoluid and water or dierent values o M and R From Fig.13 it is observed that the presence o a magnetic ield and radiation decreases the heat transer rates o nanoluids. Fig. 14 shows that an increase in radiation parameter increases the local skin riction in the absence o magnetic ield and in the presence o magnetic ield an increase in radiation decreases the local skin riction. It is also observed that an increase in the nanoparticle volume raction increases the local skin riction. Fig. 15 shows that an increase in the nanoparticle volume raction decreases the average skinriction in the presence o radiation and magnetic ield. This suggests that the presence o nanoparticles decreases the average skin-riction when the radiation and magnetic ields are present. Fig. 16 concludes that an increase in radiation and magnetic ield decreases the average skin-riction o silver water nanoluid. Thus the presence o the magnetic ield helps in reducing the rictional drag on the surace and an increase in the radiation parameter tends to reduce the velocity as shown in ig

15 Figure 15 Average Nusselt number o silver water nanoluid or dierent values o R & M Figure 16 Local skin-riction o Ag water nanoluid and water or dierent values o R and M 7. Conclusions In this paper, the problem o radiation eects on an unsteady MHD low o a nanoluid past a semi-ininite vertical plate is considered. The eects o radiation, magnetic parameter and a nanoparticle volume raction o the low and heat transer characteristics are determined or our kinds o nanoluids: aluminium oxide, copper, titanium oxide and silver. The dimensionless partial dierential equations with the appropriate boundary conditions are solved numerically by an implicit inite dierence method which is unconditionally stable and convergent. The conclusions o this study are as ollows: 1. In the presence o radiation and magnetic ield, an increase in nanoparticle volume raction φ decreases the nanoluid temperature, which leads to a decrease in the heat transer rates.. Increase in nanoparticle volume raction increases the dimensionless surace velocity o the luid in the presence o magnetic ield and radiation, which in turn decreases the average skinriction. 3. The presence o radiation and magnetic ield increases the temperature o the nanoluid with time. 4. The rate o heat transer at the plate decreases with an increase in the radiation and magnetic ield. 5. The velocity o the nanoluid decreases with an increase in the radiation and magnetic ield, whereas, the temperature o the nanoluid increases. 6. The presence o radiation and magnetic ield increases transient velocity and temperature proiles o nanoluid. They reach a steady state and temporal maximum is observed in each case. 7. The average Nusselt number is independent o time when it is large and decreases sharply or smaller values o t. 8. Time required reaching the steady state decreases with an increase in the nanoparticle volume raction. 9. Average skin-riction increases or smaller values o t. The unsteady MHD natural convective low o nanoluids past a semi-ininite vertical plate has immediate applications in all those processes which are highly aected by heat transer concept. 15

16 One o the technological applications o nanoparticles is the use o heat transer luids containing suspensions o nanoparticles to conront cooling problems in the thermal systems. Acknowledgment The authors wish to express their sincere thanks to the reviewers or their valuable comments and suggestions, which have led to deinite improvement in the paper. Reerences 1. Ostrach, S., An analysis o laminar ree-convection low and heat transer about a lat plate parallel to the direction o the generating body orce, NACA Report, 1111 (1953, pp Pohlhausen, E., Der Warmeaustausch zwichen esten korpen und Flussingkeiten mit kleiner Reibung und kleiner Warmeleitung, Z. Angew. Math. Mech., 1(191, pp Siegel, R. Transient ree convection rom a vertical lat plate. Transactions o American Society o Mech. Eng., 80 (1958, pp Gebhart, B., Pera, L., The nature o vertical natural convection lows resulting rom the combined buoyancy eects o thermal and mass diusion, Int. J. Heat Mass Trans., 14 (1971, 1, pp Hellums, J. D., Churchill, S.W., Transient and steady state, ree and natural convection, numerical solutions. Part 1. The isothermal vertical plate, American Institute o Chem. Engineers Journal, 8 (196, 5, pp Takhar, H. S., Ganesan, P., Ekambavanan, V., Soundalgekar, V. M., Transient ree convection past a semi-ininite vertical plate with variable surace temperature, Int. J. Numerical Methods or Heat and Fluid Flow, 7 (1996, 4, pp Soundalgekar, V.M., Ganesan, P., The inite dierence analysis o transient ree convection with mass transer o an isothermal vertical lat plate, Int. J. Engineering Sciences, 19 (1981, 6, pp England, W. G., Emery, A. F., Thermal radiation eects on the laminar ree convection boundary layer o an absorbing gas, J. Heat Trans., 91 (1969, 1, pp Soundalgekar, V. M., Takhar, H. S., Radiation Eects on Free Convection Flow past a Semi- Ininite Vertical Plate, Modelling, Measurement and Control, B51 (1993, pp Hossain, M. A., Takhar, H. S., Radiation eect on mixed convection along a vertical plate with uniorm surace temperature, J. Heat and Mass trans., 31 (1996, pp Das, U. N., Deka, R. K., Soundalgekar, V. M., Radiation eects on low past an impulsively started vertical ininite plate, J. Theoretical Mech., 1 (1996, pp Choi, S. U. S., Enhancing thermal conductivity o luids with nano- particles, ASME Fluids Engineering Division, 31 (1995, pp Buongiorno, J., Convective transport in nanoluids, J. Heat Trans., 18 (006, pp Kuznetsov, A. V., Nield, D. A., Natural convective boundary-layer low o a nanoluid past a vertical plate, Int. J. Therm. Sci., 49 (010,, pp

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