Flow and Heat Transfer Analysis of Copper-water Nanofluid with Temperature Dependent Viscosity Past a Riga Plate

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1 Journal o Magnetics (), (017) ISSN (Print) ISSN (Online) Flo and Heat Transer Analysis o Copper-ater Nanoluid ith Temperature Dependent Viscosity Past a Riga Plate A. Ahmad, S. Ahmed, and F. M. Abbasi* Department o Mathematics, COMSATS Institute o Inormation Technology, Islamabad 44000, Pakistan (Received 5 January 017, Received in inal orm 9 March 017, Accepted 14 March 017) Flo o electrically conducting nanoluids is o pivotal importance in countless industrial and medical appliances. Fluctuations in thermophysical properties o such luids due to variations in temperature have not received due attention in the available literature. Present investigation aims to ill this void by analyzing the lo o copper-ater nanoluid ith temperature dependent viscosity past a Riga plate. Strong all suction and viscous dissipation have also been taken into account. Numerical solutions or the resulting nonlinear system have been obtained. Results are presented in the graphical and tabular ormat in order to acilitate the physical analysis. An estimated expression or skin riction coeicient and Nusselt number are obtained by perorming linear regression on numerical data or embedded parameters. Results indicate that the temperature dependent viscosity alters the velocity as ell as the temperature o the nanoluid and is o considerable importance in the processes here high accuracy is desired. Addition o copper nanoparticles makes the momentum boundary layer thinner hereas viscosity parameter does not aect the boundary layer thickness. Moreover, the regression expressions indicate that magnitude o rate o change in eective skin riction coeicient and Nusselt number ith respect to nanoparticles volume raction is prominent hen compared ith the rate o change ith variable viscosity parameter and modiied Hartmann number. Keyords : temperature dependent viscosity, copper-ater nanoluid, riga plate, regression relations 1. Introduction Nanoluids play central role in modern engineering appliances. Due to their unique chemical and thermophysical properties such luids are preerred over conventional luids in industrial processes involving heat transer. Many such processes make use o high eective thermal conductivity o nanoluids to acilitate heat transer and to prevent energy loss. Nanoluids are readily being used in domestic and industrial cooling systems, IT industry, geothermal energy extraction, solar panels, cellular and automobile industries, magnetic drug delivery systems and many other industrial and medical processes to enhance their eiciency. Such ide utilization o nanoluids is the chie reason behind the ever groing interest o motivated researchers in the analysis o such luids. Since the innovative idea o nanoluids proposed by Choi [1], several investigations have been carried out to examine The Korean Magnetics Society. All rights reserved. *Corresponding author: Tel: Fax: , abbasisarkar@gmail.com dierent aspects o such luid, some o hich can be seen through Res. [-1]. Flos governed by all motion and all suction ind large number o applications in industry. In such los, thickness o momentum boundary layer can be controlled by the motion and porosity o the all. On the other hand, electromagnetic orces play central role in controlling boundary layer thickness in hydromagnetic los. Fluids ith high electric thermal conductivities are governed by classical MHD los. Contrary to this, currents produced by externally applied magnetic ields alone are not suicient to generate lo in eakly conducting luid. In order to achieve an eective lo control external electric ield must be added. To generate simultaneous electric and magnetic ields Gailitis and Lielausis [13] introduced the Riga plate hich can produce Lorentz orce parallel to the all hich in turn is helpul in lo control. The term or the all-parallel Lorentz orce named Grinberg-term [14] is negative exponential unction o direction normal to the plate. Recently Ahmad et al. [15] examined the lo o nanoluid past a Riga plate. In this study authors used the Bongiorno s model [] or the analysis o nano- 017 Journal o Magnetics

2 18 Flo and Heat Transer Analysis o Copper-ater Nanoluid ith Temperature A. Ahmad, S. Ahmed, and F. M. Abbasi luid. It is ell established act that the temperature o a luid perturbs its thermophysical properties. Such variations in thermophysical properties o a luid ith temperature become additionally important hen high accuracy is desired. Several analysts have used Brinkmann s model [16] or the viscosity o to phase los in their analysis o nanoluids. Although this model takes into account the eect o nanoparticles volume raction on the viscosity, yet the dependence o temperature is ignored at once. Despite o sizeable literature available on the mechanical analysis o nanoluids, not much has been said about lo o nanoluids ith temperature dependent viscosity. This study aims to plug this gap by combining the Brinkman s viscosity model ith the exponential model o the temperature dependent viscosity. Flo o electrically conducting nanoluid past a Rigalate ith strong suction is investigated here. The nanoluid is composed o ater and copper nanoparticles. Viscosity i the nanoluid is taken to be temperature dependent. Viscous dissipation is also taken into account. Resulting nonlinear system is solved numerically. Obtained results are presented graphically or physical analysis. Analytical expressions are developed or skin riction and Nusselt number by perorming linear regression on the obtained numerical data. Comparison or the case o constant and variable viscosity is also presented ith the anticipation that this study ill serve as a reerence or uture research in the ield.. Mathematical Formulation Consider the lo o electrically conducting luid containing nanoparticles induced by a Riga plate placed at y = 0 ith x-axis aligned vertically upard. The Riga plate consists o an alternating array o electrodes and Fig. 1. Sketch o a Riga plate consisting o an alternating array o electrodes and permanent magnets. The Lorentz orce and the mainstream velocity are parallel to the array. permanent magnets mounted on a plane surace (see Figure 1). Employing the Oberbeck-Boussinesq approximation and the assumption that the nanoparticles concentration is dilute, the boundary layer equations or the conservation o total mass, momentum and thermal energy are ritten as: U V = 0, (1) X Y ρ e U U + V U = U μ X Y Y e g( ρβ) Y e ( T T ) + πj om o exp π -- y, () 8 a ( ρc e ) U T + V T T = K, (3) X Y e Y The plate ith suction velocity V o is assumed to move in x direction ith constant velocity. At the sheet the temperature T takes the constant value T. The ambient value o T is given as T. The boundary conditions may be ritten as: U = U, V = V, T = T at Y = 0 UX, ( Y) 0, T = T as Y (4) In Eqs. (1)-(4), (U, V) are the velocity components in (X, Y) directions respectively. j o (A/m ) is the applied current density in the electrodes, M o (Tesla) is the magnetization o the permanent magnets and a is the idth o magnets and electrodes. In Eq. (), the last term knon as Grinberg-term represents the all-parallel Lorentz orce and is ully decoupled rom the lo and decreases exponentially in the direction normal to the plate. Relations or eective heat capacity (ρc) e, eective density ρ e, eective viscosity μ e [16], eective thermal expansion (ρβ) e, and eective thermal conductivity K e o the nanoluid are given as ollos: K e = K (5) K , p+ K φ( K K p ) K p + K + φ( K K p ) μ = μ ( 1 α ( T T )) e ( 1 φ).5 ρ e = ( 1 φ)ρ + φρ p, ( ρc) e = ( 1 φ) ( ρc) + φ( ρc) p, ( ρc) e = ( 1 φ) ( ρβ) + φ( ρβ) p. (6) In these equations ρ, C, β and φ are the density, speciic heat, thermal expansion coeicient, electric conductivity, and nano particle volume raction respectively and the subscripts p and denote the nanoparticles and luid phase respectively. Introducing the non-dimensional parameters

3 Journal o Magnetics, Vol., No., June x = X --,, l y = Y L --, u = , U v = V U V ----, θ = T T o T T L = a (7) π --, l = U L , V υ o = υπ a in the Eqs. (1)-(4) and obtain the non-dimensional orm as: v ---- = 0 x y ( 1+( ρ 1)ϕ) u v = αθ x y y ( 1 ϕ).5 y (8) + λ( 1+( ρβ 1)ϕ)θ + Ze y, (9) ( 1+( ρc 1)ϕ) u----- θ + v θ = ---- A θ 1 αθ , (10) x y Pr y ( 1 ϕ).5 y ith boundary conditions u = 1, v = v, θ = 1 at y = 0 (11) uxy (, ) 0, θ = 0 as y In the above equations λ is the Richardson number, Z is the modiied Hartmann number, Pr is the Prandtl s number, υ is the kinematic viscosity o the luid, given as: a ΔT 1 a j ( ) om υρc o λ =,, Pr g β Z = = πυu 8π Uμ K (1) μ Kp + K φ( K Kp) υ =, A = ρ K + K + φ( K K ) p p With assumption o strong suction/injection [17] Eqs. (9)-(11) can be reritten as: v = 0, y ( 1+( ρ 1)ϕ)v = αθ x y ( 1 ϕ).5 y (13) + λ( 1+( ρβ 1)ϕ)θ + Ze y, (14) θ A θ 1 αθ ( 1 + ( ρc 1) ϕ) v = + (15).5, y Pr y (1 ϕ) y Continuity equation and condition v = v at y = 0 implies v = v. Thus the governing equations may be ritten as: ( 1+( ρ 1)ϕ)v = αθ y y ( 1 ϕ).5 y + λ( 1+( ρβ 1)ϕ)θ + Ze y, (16) θ A θ 1 αθ 1 + ( 1) v = +,.5 y Pr y (1 ϕ) y ( ρc ϕ) (17) here or the suction velocity e rite v = s, s > 0. Corresponding boundary conditions are u = 1, θ = 1 at y = 0 uxy (, ) 0, θ = 0 as y 3. Solution Methodology (18) To solve the boundary value problem (16)-(18) numerically, nonlinear shooting method is adopted. Shooting technique is used to guess the initial condition o converted initial value problem such that it satisies the boundary conditions o original boundary value problem. The resulting dierential equations can then be integrated using initial value solver like Runge-Kutta technique. The skin riction coeicient, a physical quantity o practical interest, is deined as: τ C =, (19) ρeu here τ is the shear stress at the surace U τ = μe. (0) Y Y = 0 In dimensionless orm, it yields: τ μou = L 1 αθ (1 ).5 ϕ y y= 0, (1) 1 1 αθ C Re =. () 1 + (.5 ρ 1) ϕ (1 ϕ ) y y= 0 Similarly, Nusselt number (Nu) is deined as ql Nu =, (3) k ( T T ) here the heat lux at the surace q is deine as T q = ke. (4) Y Y = 0 Using similarity variables conveniently e can rite Ke Nu = (0). (5) K θ It must be noted that the eective lo quantities have been taken into account in the evaluation o these quantities. Thereore, it ill be more appropriate to name these quantities as the eective skin riction coeicient and the eective Nusselt number. Linear regression is perormed on the obtained numerical data to develop the correlation expressions or eective skin riction coeicient and the

4 184 Flo and Heat Transer Analysis o Copper-ater Nanoluid ith Temperature A. Ahmad, S. Ahmed, and F. M. Abbasi eective Nusselt number. 4. Results and Discussion This section is explicitly prepared to examine the numerical results ith the aid o graphs. Plots or velocity, temperature, skin riction coeicient and Nusselt number are presented and analyzed. Figures -4 have been plotted to study the behavior o velocity proile or variation in modiied Hartmann number (Z), variable viscosity parameter (α) and nanoparticles volume raction (φ). Figure 3 depicts that the velocity proile increases ith an increase in the modiied Hartman number. Such increase is large near the Riga plate and it reduces smoothly as the distance rom the plate increases. This observation highlights that the direction and magnitude o applied magnetic ield largely alters the velocity o nanoluid. Figure 3 presents the eects o α on velocity proile. This igure shos that velocity o nanoluid increases ith an increase in α. Such increase is large Fig.. (Color online) Velocity o ater based nanoluid containing copper particles or varying Riga plate parameter Z ith s =, λ = 0.5, Pr = 6., φ = 0.1, α = 0.1, n =3. Fig. 4. (Color online) Velocity o ater based nanoluid containing copper particles or varying φ ith s = 1.0, λ =0.5, Pr = 6., α = 0.3, Z =.0, n =3. Fig. 3. (Color online) Velocity o ater based nanoluid containing copper particles or varying α ith s =1.0, λ =0.5, Pr = 6., φ = 0.05, Z =.0, n =3. Fig. 5. (Color online) Temperature proile o ater based nanoluid containing copper particles or varying Riga plate parameter Z ith s =, λ = 0.5, Pr = 6., φ =0.1, α =0.1, n =3.

5 Journal o Magnetics, Vol., No., June near the Riga plate. Moreover, it is also deduced rom this igure that the velocity o nanoluid ith constant viscosity is less than that o nanoluid ith temperature dependent viscosity. Copper-ater nanoluid moves sloly in comparison to pure ater (see Fig. 4). Velocity o nanoluid decreases by increasing φ. This act indicates that the addition o copper nanoparticles not only ill inluence the thermodynamics but also the mechanics o lo. Further it is noted that the boundary layer thickness decreases ith an increase in φ. Hence copper nanoparticles can be used to control the boundary layer thickness. Variations in temperature proile are studied through Figs Temperature o nanoluid increases by increasing modiied Hartmann number (see Fig. 5). Whereas this observation is reversed as the distance rom the plate increases. Hence an increase in the strength o applied magnetic ield inluences the temperature o nanoluid. Variable viscosity also inluences the temperature o nanoluid (see Fig. 6). This Fig. indicates that the temperature o nanoluid increases by a small amount hen α is increased. Figure 6 depicts that the temperature o copperater nanoluid decreases considerably hen the nanoparticles volume raction is increased. This igure also indicates that the ordinary ater has higher temperature compared ith that o copper-ater nanoluid. This act highlights the use o copper nanoparticles as coolants. The eective skin riction coeicient and the eective Nusselt number are plotted in Figs. 8 and 9 or variations in embedded parameters. It is observed through Fig. 8 Fig. 7. (Color online) Temperature proile o ater based nanoluid containing copper particles or varying φ ith s = 1.0, λ = 0.5, Pr = 6., α =0.3, Z=.0, n =3. Fig. 8. (Color online) Skin riction ith s = 1.0, λ = 0.5, Pr = 6., n =3. Fig. 6. (Color online) Temperature proile o ater based nanoluid containing copper particles or varying α ith s =1.0, λ = 0.5, Pr = 6., φ =0.05, Z=.0, n = 3. Encircled part o proile is magniied in small sub-igure or detailed analysis. that the eective skin riction coeicient increases ith an increase in the nanoparticles volume raction. Also the ater ree o copper nanoparticles possesses loer values o eective skin riction coeicient hen compared ith that o copper-ater nanoluid. Moreover, an increase in the modiied Hartmann number results in an increase in the eective skin riction coeicient. This igure also indicates that the eective skin riction coeicient is small or nanoluid ith constant viscosity and it increases ith an increase in α. Such increase is uniorm and linear. An analytical expression in the orm o Riga plate parameter (Z), nanoparticles volume raction (φ) and variable thermal properties parameter (α) to estimate the skin

6 186 Flo and Heat Transer Analysis o Copper-ater Nanoluid ith Temperature A. Ahmad, S. Ahmed, and F. M. Abbasi Table. Correlation o Nusselt number (Nu = C + C Nb N b + C Nt N t ) and maximum percentage error deine as [ Nu est Nu / Nu]100 or dierent values o λ and suction parameter here the values o α, φ is consider in the interval (0.0, 0.1) and Z in the interval ( 5.0, 5.0). λ s C C a C p C Z Max. % error Fig. 9. (Color online) Nusselt number or s = 1.0, λ =0.5, Pr = 6., n =3. riction or copper-ater nanoluid ith s = 1.0, λ =0.5 and n = 3 is given by: S = α φ Z The above expression is obtained perorming linear regression on numerical data or 1900 dierent values or α, φ and Z. The maximum percentage error beteen numerical and estimated values is less than 5 % or α, φ in the interval (0.0, 0.1) and Z in the interval ( 5.0, 5.0). This correlated expression exhibit that the rate o change o skin riction ith respect to φ is much prominent hen compared ith that or other to parameters. Also rate o change in skin riction ith respect to α, φ and Z is positive, hich can also be seen rom Fig. 7. In Table 1 correlation o skin riction or dierent values o λ and s are presented ith corresponding maximum percentage error. Similarly, an expression or eective Nusselt number or the same range o parameters α, φ and Z ith s =1.0, λ = 0.5 and n =3 is Table 1. Correlation o skin riction (S = C + C a α + C p φ + C z Z) and maximum percentage error deine as [ S est S / S ]100 or dierent values o λ and suction parameter here the values o α, φ is consider in the interval (0.0, 0.1) and Z in the interval ( 5.0, 5.0). λ s C C a C p C Z Max. % error Nu = α φ + 0.7Z, ith maximum percentage error less than 1. The above expression depicts that the rate o change o Nusselt number ith respect to α and Z is positive hereas the rate o change ith respect to φ is negative. The same behavior o Nusselt number can be observed rom Fig. 9. In Table the constants o the correlation expression or Nusselt number dierent values o λ and s are given. 5. Concluding Remarks Electro-magnetic mixed convection boundary layer lo o copper-ater nanoluid ith eakly conducting base luid induced by a Riga-plate is studied. Key indings o this study are summarized belo: Velocity o the copper-ater nanoluid increases near the Riga plate ith an increase in the value o modiied Hartmann number. Temperature dependent viscosity alters the velocity as ell as the temperature o the nanoluid and is o considerable importance in the processes here high accuracy is desired. Increase in the eective skin riction coeicient and eective Nusselt number is noted or large values o viscosity parameter. Velocity and temperature o the nanoluid decrease ith an increase in the nanoparticles volume raction. Addition o copper nanoparticles makes the momentum boundary layer thinner hereas viscosity parameter does not aect the boundary layer thickness. It is observed that rate o change in eective skin riction coeicient ith respect to α, φ and Z is positive and rate o change ith respect to nanoparticles raction is much prominent compared to others. Moreover, rate o change in eective Nusselt number ith respect to α and Z is positive hereas it is negative ith respect to φ ith prominent magnitude as compare to change ith respect to α and Z.

7 Journal o Magnetics, Vol., No., June Reerences [1] S. U. S. Choi, J. Heat Transer 66, 99 (1995). [] J. Buongiorno, J. Heat Transer 18, 40 (006). [3] S. Dong, L. Zheng, X. Zhang, and P. Lin, Microluid Nanoluidics 17, 53 (014). [4] M. Sheikholeslami and D. D. Ganji, J. Mol. Liq. 4, 56 (016). [5] M. Sheikholeslami, Soheil Soleimani, and D. D. Ganji, J. Mol. Liq. 13, 153 (016). [6] M. J. Uddin, O. A. Beg, and A. Ismail, International Journal o Numerical Methods or Heat & Fluid Flo 6, 5 (016). [7] G. K. Ramesh, B. J. Gireesha, and C. S. Bageadi, Int. J. Heat. Mass. Trans. 55, 4900 (015). [8] J. Sui, L. Zheng, and X. Zhang, Int. J. Therm. Sci. 104, 461 (016). [9] F. M. Abbasi, T. Hayat, and B. Ahmad, Physica E 67, 47 (015). [10] F. M. Abbasi, T. Hayat, and A. Alsaedi, Physica E 68, 13 (015). [11] Mohsen Sheikholeslami, J. Magn. Magn. Mater. 5, 903 (017). [1] Mohsen Sheikholeslami, J. Phys. A, 381, 494 (017). [13] A. Gailitis and O. Lielausis, Appl. Magnetohydrodyn. 1, 143 (1961). [14] E. Grinberg, Appl. Magnetohydrodyn. Rep. Phys. Inst. Riga. 1, 147 (1961). [15] A. Ahmed, S. Asghar, and S. Azal, J. Magn. Magn. Mater. 40, 44 (016). [16] H. C. Brinkman, J. Chem. Phys. 0, 571 (195). [17] A. Pantokratoras, Mathematical Problems in Engineering (008) 1497.