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1 Heriot-Watt University Heriot-Watt University Research Gateway MHD stagnation point low o viscoelastic nanoluid with non-linear radiation eects Farooq, Muhammad; Khan, M. Ijaz; Waqas, M.; Hayat, T.; Alsaedi, A.; Khan, M. Imran Published in: Journal o Molecular Liquids DOI: /j.molliq Publication date: 2016 Document Version Peer reviewed version Link to publication in Heriot-Watt University Research Portal Citation or published version (APA): Farooq, M., Khan, M. I., Waqas, M., Hayat, T., Alsaedi, A., & Khan, M. I. (2016). MHD stagnation point low o viscoelastic nanoluid with non-linear radiation eects. DOI: /j.molliq General rights Copyright and moral rights or the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition o accessing publications that users recognise and abide by the legal requirements associated with these rights. I you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

2 Download date: 25. Sep. 2018

3 Accepted Manuscript MHD stagnation point low o viscoelastic nanoluid with non-linear radiation eects M. Farooq, M. Ijaz Khan, M. Waqas, T. Hayat, A. Alsaedi, M. Imran Khan PII: S (16) DOI: doi: /j.molliq Reerence: MOLLIQ 5984 To appear in: Journal o Molecular Liquids Received date: 16 May 2016 Revised date: 29 May 2016 Accepted date: 22 June 2016 Please cite this article as: M. Farooq, M. Ijaz Khan, M. Waqas, T. Hayat, A. Alsaedi, M. Imran Khan, MHD stagnation point low o viscoelastic nanoluid with non-linear radiation eects, Journal o Molecular Liquids (2016), doi: /j.molliq This is a PDF ile o an unedited manuscript that has been accepted or publication. As a service to our customers we are providing this early version o the manuscript. The manuscript will undergo copyediting, typesetting, and review o the resulting proo beore it is published in its inal orm. Please note that during the production process errors may be discovered which could aect the content, and all legal disclaimers that apply to the journal pertain.

4 MHD stagnation point low o viscoelastic nanoluid with non-linear radiation eects M. Farooq a, M. Ijaz Khan b,1, M. Waqas b, T. Hayat b,c, A. Alsaedi c and M. Imran Khan d a Department o Mathematics, Riphah International University, Islamabad Pakistan b Department o Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan c Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department o Mathematics, Faculty o Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia Heriot Watt University, Edinburgh Campus, Edinburgh EH14 4AS, United Kingdom Abstract: This article addresses MHD stagnation point low o viscoelastic nanoluid towards a stretching surace with nonlinear radiative eects. Nanoluid model consists o Brownian motion and thermophoresis. Heat transer is studied by employing convective condition at the stretching surace. Newly constructed condition or heat transer is imposed. The relevant problems are modeled by considering nonlinear thermal radiation. Similarity transormation is utilized to reduce the nonlinear partial dierential equations into coupled nonlinear ordinary dierential equations. The convergent solutions or velocity, temperature and concentration equations developed. Graphical results are drawn or the velocity, temperature, concentration, skin riction coeicient and Nusselt number. It is noticed that skin riction increases or larger magnetic parameter. Keywords: Non-linear radiative eect; viscoelastic luid; Convective boundary condition 1.Introduction The utilization o nanotechnology has attracted the attention o recent investigators since nanoscale materials owns chemical, electrical and unique optical properties. Recent advancements made it conceivable to diuse nanoparticles in ordinary heat transer liquids including engine oil, ethylene glycol and water to create another class o heat transer liquids with great eectiveness. Liquids shaped o immersed luid and suspended nanoparticles are called nanoluids. These liquids have much higher thermal conductivity when compared with traditional heat transer liquids even at low molecule concentrations [1]. Also the improved thermal conductivity o such liquids is useul or dierent particular applications or instance drug delivery, space cooling, transportation, solar energy absorption, nuclear engineering, power generation and several others. There are two models or nanoluids namely the Tiwari and Das model [2] and the Buongiorno's model [3]. According to [3], nanoluid velocity may be observed as the sum o the relative velocity and base luid (i.e the slip velocity). Also the model provided by [3] is based on the thermophoresis and Brownian diusion mechanisms. In order to analyze the convective transport in nanoluids several investigators considered this model. For instance, Abbas et al. [4] considered heat generation eects in the hydromagnetic low o nanoluid induced by a curved stretching sheet. Alsaedi et al. [5] examined the stagnation point low o nanoluid towards a permeable stretched surace with convective boundary conditions and internal heat generation/absorption. Ziaei-Rad et al. [6] discussed MHD low o nanoluid by a permeable stretched surace. Heat transer in MHD stagnation point low o nanoluid induced by stretching/shrinking surace is analyzed by Nandy and Mahaparta. [7]. Hayat et al. [8] considered magnetohydrodynamics unsteady low o viscous nanoluid with double stratiication. MHD three dimensional low o nanoluid in presence o convective conditions is 1

5 studied by Hayat et al. [9]. Mixed convection low o nanoluid with Newtonian heating is investigated by Hayat et al. [10]. Few more studies on nanoluids and reactive lows can be seen in the res. [11-22]. The analysis o non-newtonian materials has engrossed continous consideration o recent investigators. Such consideration is due to their occurrence in geophysics, oil reservoir engineering, bioengineering, chemical and nuclear industries, polymer solution, cosmetic processes, paper production etc. No doubt all non-newtonian materials on the basis o their behavior in shear are not predicted by one constitutive relationship. This act o non-newtonian materials is dierent than the viscous materials. Thus various models o non-newtonian luids have been suggested. Amongst these there is second grade luid model. No doubt the consideration o second grade luid predicts the normal stress eects. Moreover heat transer has awesome part in these luids. Study o heat transer characteristics in the stretched low o non- Newtonian luids is thus a popular area o research. Turkyilmazoglu [23] studied MHD mixed convection low o viscoelastic luid by a permeable stretched surace. Cortell [24] analyzed the low and heat transer o second grade luid with nonlinear radiation and heat generation/absorption. Mukhopadhayay et al. [25] considered the low o Casson luid by an unsteady stretching surace. Animasaun et al. [26] examined the nonlinear thermal radiation eects in low o second grade liquid with homogeneous-heterogeneous reactions. They considered the unequal diusivities case in this attempt. Hayat et al. [27] presented the MHD three-dimensional low o nanoluid with velocity slips and nonlinear thermal radiation. Mushtaq et al. [28] explored the nanoluid low and nonlinear radiative heat transer with solar energy. Kandasamy et al. [29] studied the impact o solar radiations in low o viscous material subject to magneto nanoparticles. Ibrahim et al. [30] studied the magnetohydrodynamic (MHD) stagnation point low o nanoluid with nonlinear radiative heat transer induced by a stretched surace. Hayat et al. [31] analyzed magnetohydrodynamic (MHD) low o second grade nanoluid by a nonlinear stretched surace. Reddy et al. [32] considered the Jerey liquid low with torsionally oscillating disks. Das et al. [33] considered MHD Jerey liquid and radiative low by a stretched sheet with surace slip and melting heat transer. Gao and Jian [34] investigated MHD low o Jerey liquid with circular microchannel. Hayat et al. [35] explored the MHD Jerey nanoluid stagnation point low with Newtonian heating. Narayana and Babu [36] examined the MHD heat and mass transer o Jerey luid by a stretched surace with thermal radiation and chemical reaction. Nallapu and Radhakrishnamacharya [37] described the Jerey liquid low with narrow tubes in the presence o magnetic ield. Farooq et al. [38] presented Jerey liquid low with Newtonian heating and MHD eects. Rao et al. [39] studied the characteristics o heat transer in low o second grade liquid induced by non-isothermal wedge. The objective o present article is to analyze the stretched low o viscoelastic luid with nonlinear radiation and convective boundary conditions. Thus second grade luid has been dealt with the linearized orm o thermal radiation. Homotopy analysis method [40-55] has been implemented or the development o convergent series solution or velocity, temperature and nanoparticle concentration. Graphical results or velocity, temperature and nanoparticle concentration are analyzed or various embedded parameters o interest. 2. Formulation Here we are interested to analyze the stagnation point low o viscoelastic nanoluid. Brownian motion and thermophoresis eects are considered. Fluid is electrically conducting in the 2

6 presence o transversely applied magnetic ield B 0. Magnetic Reynolds number is small. Eect o electric ield is ignored. Heat transer through convective condition is discussed. Newly developed boundary condition or mass transer is imposed. Unlike the classical case, the nonlinear thermal radiation is considered. Physical description o low is shown in Fig. 1. Applying boundary layer approximation ( o( x) o( u) o(1), o( y) o( v) o( ) ) the mathematical problems here are u v 0, x y u u u u u u du 2 x 2 e B0 u xy y u v Ue U u k 2 0, 3 2 x y dx u u u y v 3 y y xy 2 2 T T T C T D T T 1 qr u v D, 2 B x y y y y T y y 2 2 C C C D T T u v DB, 2 2 x y y T y T u Uw x ax, v 0, k h T T at y 0, y C DT T DB 0 at y 0, y T y u U x bx T T C C y e,, as. In the aorementioned equations u and v are the velocity components parallel to the x and y directions respectively, U w the stretching velocity, U e the ree stream velocity, k 0 the viscoelastic parameter, the thermal diusivity, the eective heat capacity o nanoparticles, D the Brownian diusion coeicient, D the thermophoresis diusion B coeicient, T the ambient temperature, C the ambient concentration, the density o the luid, q r the radiative heat lux and a and b the dimensional constants. Note that the dissipation and Joule heating terms are ignored or low luid velocity. In view o transormations [30]: T (1) (2) (3) (4) (5) 3

7 a y, a x, u ax ( ), v a ( ), T T C C,, T T C incompressibility condition given in (1) is automatically satisied. Utilizing Rosseland approximation o thermal radiation we get the ollowing expression 4 4 T 16 3 T qr T. 3k y 3k y In above equation stands Stean Boltzmann constant and k or mean absorption coeicient. The overhead Eq. (7) is nonlinear in T. Now Eq. (3) yields 3 2 T T 16 T T C T D T T u v DB. x y y 3( c) k y y y T y TT We characterize the non-dimensional temperature T T with 1 1 N r T T T T Nr and the temperature ratio parameter. The irst term on the right hand side o the above mentioned equation can be simpliied as T 3 1 Rd 1 Nr 1. y y 3 16 T Here Rd 3kk indicate the radiation parameter and Rd 0 designates no thermal radiation eect. From Eqs.(2)-(4), (6), (8) and (9) we have A M ( A ) 2 0, 3 2 Rd Nr Nb Nt Pr 0, Nt Le Pr 0, Nb 0 0, 0 1, 0 Bi 0 1 at 0, Nb Nt 0 0 0, at 0, A, 0, 0, as. h Here prime indicates dierentiation with respect to. Moreover Bi k a represents Biot number, Pr b the Prandle number, A a the ratio parameter, ( ak ) the viscoelastic 0 parameter, Le D B the Lewis number, M B c DBC p a the magnetic parameter, Nb c the 3 16 T Brownian motion parameter, Rd 3kk c D TT T p the radiation parameter, Nt c T the T thermophoresis parameter and Nr T the temperature ratio parameter. (6) (7) (8) (9) (10) (11 (12) (13) 4

8 Skin riction coeicient and local Nusselt number are w xqw C, Nu, 1 2 x U k T T 2 w where wall shear stress w and wall heat lux q w are represented in the ollowing expressions w u u u u u T w k u v 2, q k. y xy y x y 3k y0 Utilizing Eq. (17) in Eq. (16) we obtain 1 Nux 3 C Rex 1 3 0, 1 RdwNr 0, 2 Re where ax Re 2 x denotes the Reynolds number and Nu x indicates the local Nusselt number. 3. Series solutions and convergence Homotopy analysis technique provides us great reedom and an easy way to adjust and control the convergence region o the series solutions. Convergence domain is in the region parallel to axis. Thereore we have plotted -curves in the Figs The admissible ranges o the auxiliary parameters, and are noted [-1.4, -0.1], [-2.2, -0.65] and [-2.0, -0.1].The initial guesses and linear operators relating to the momentum, energy and concentration equations are taken ( ) A 1 Ae A e, where 0 Bi Bi Nt 0( ) e, 0( ) e, 1Bi 1Bi Nb,,, [ C C e C e ] 0, [ C e C e ] 0, [ C e C e ] 0, C i indicate the arbitrary constants. x (14) (15) (16) (17) (18) (19) 5

9 Table 1: Homotopy solutions convergence when M Le 0.1, Nt = A = =0.2, Nb = Bi = Rd = 0.5, Nr = 0.7 and Pr = 1.4. order o approximations Discussion The main objective o this section is to analyze the behaviors o various involved parameters on the velocity, temperature, concentration, skin riction coeicient and local Nusselt number. 6

10 4.1. Velocity distribution when Nb = Bi = Rd = 0.5, M Le 0.1, Nt = A = =0.2, Nr = 0.7 and Pr = 1.4. when Nb = Bi = Rd = 0.5, M Le 0.1, Nt = =0.2, Nr = 0.7 and Pr =

11 when Nb = Bi = Rd = 0.5, M Le 0.1, Nt = A = =0.2, Nr = 0.7 and Pr = 1.4. Fig. 4 is drawn or dierent values o viscoelastic parameter γ on the velocity proile. For higher values o γ both velocity distribution and boundary layer thickness increase. Beacuse elasticity o the material increases due to which disturbance occur in material. Thus enhances. Fig. 5 is sketched to see the eects o A on velocity distribution. Higher values o A results in enhancement o velocity distribution. It is noted that velocity boundary layer thickness has opposite behavior or A>1 and A<1. For A=1 there exists no boundary layer due to the act that luid and sheet move with the same velocity. The behavior o magnetic parameter M on velocity distribution is disclosed through Fig. 6. Here we see that the applied magnetic ield slows down the motion o luid which decreases the velocity Temperature distribution when Nb = Bi = 0.5, M Le 0.1, Nt = A = =0.2, Nr = 0.7 and Pr = 1.4. when Nb = Bi = Rd = 0.5, M Le 0.1, Nt = A = =0.2 and Pr =

12 when Nb = Bi = Rd = 0.5, M Le 0.1, Nt = A = =0.2 and Nr = 0.7. Fig. 7 delineates the variation o the temperature in response to a change in the values o radiation parameter Rd. Clearly temperature distribution and the associated thermal boundary layer thickness enhances or large values o radiation parameter Rd. This is due to act that the surace heat lux increases under the impact o thermal radiation which results in larger temperature inside the boundary layer region. The eects o the temperature ratio parameter Nr on the thermal boundary layer are depicted in Fig. 8. From this Fig.8, it is clear that enhancement in the temperature ratio parameter corresponds to higher wall temperature when compared with ambient luid. Consequently temperature o the luid enhances. Moreover, we perceive that the thermal boundary layer thickness increases or the large values o the temperature ratio parameter. Fig. 9 discloses the eatures o Prandtl number Pr on temperature distribution. It is demonstrated through this Fig. that the temperature and the associated thermal boundary layer thickness are decreasing unctions o the Prandtl number. This happens because o the way that the luids with higher Prandtl number have low thermal conductivity which reduces the conduction and hence the thermal boundary layer thickness Concentration distribution 9

13 when Nb = Bi = Rd = 0.5, M 0.1, Nt = A = =0.2, Nr = 0.7 and Pr = 1.4. when Nb = Bi = Rd = 0.5, M Le 0.1, Nt = A = =0.2 and Nr = 0.7. when Nb = Bi = Rd = 0.5, M Le 0.1, A = =0.2, Nr = 0.7 and Pr =

14 when Bi = Rd = 0.5, M Le 0.1, Nt = A = =0.2, Nr = 0.7 and Pr = 1.4. Figures 10 and 11 portray characteristics o the Lewis and Prandtl number on the concentration proile. It is evident rom igures that intensiying the Prandtl number and Lewis number diminishes concentration proile and associated concentration boundary layer. Moreover, the concentration ield is reduced when we increase the values o the Lewis number as it is inversely proportional to the Brownian diusion coeicient. As Brownian diusion coeicient is weaker or higher Lewis number and this Brownian diusion coeicient creates a reduction in the concentration ield. Figures 12 and 13 are prepared to put into action, the Brownian motion and thermophoresis parameters on the concentration proile. It is straightorwardly appeared rom these igures that the concentration proile and the associated concentration boundary layer thickness builds up or higher thermophoresis parameter while the opposite behavior is observed or the Brownian motion parameter. Physically, the thermophoresis orce increments with the increase in thermophoresis parameter which tends to move nanoparticles rom hot to cold areas and hence increases the magnitude o nanoparticles volume riction proile. Ultimately the concentration boundary layer thickness turns out to be signiicantly large or slightly increased value o the thermophoresis parameter. Additionally, higher values o the Brownian motion parameter stile the diusion o nanoparticles into the luid regime away rom the surace which as a result decreases the nanoparticles concentration in the boundary layer. 11

15 4.4. Skin riction and local Nusselt number when Nb = Bi = Rd = 0.5, Le 0.1, Nt = =0.2, Nr = 0.7 and Pr = 1.4. when Nb = Bi = Rd = 0.5, M Le 0.1, Nt = =0.2, Nr = 0.7 and Pr =

16 when Nb = Bi = Rd = 0.5, Le 0.1, Nt = =0.2, Nr = 0.7 and Pr = 1.4. when Nb = Bi = Rd = 0.5, M Le 0.1, Nt = =0.2 and Nr = 0.7. Skin riction coeicient is plotted or dierent values o M, A and in Figs. (14 and 15). Skin riction enhances or higher values o M, A and γ. Figs. (16 and 17) are sketched or Nusselt number with dierent values o M, A and Pr. It is analyzed that Nusselt number enhances or larger values o M, A and Pr. Table 1 shows the convergence o series solutions o momentum, energy and concentration equations. It is noted that 25 th and 30 th order o approximations are suicient or the convergence o momentum, energy and concentration equations, respectively. 5. Conclusions Here we explored the impact o passive control o nanoparticles in the stagnation point low o viscoelastic luid. The key points are summarized as ollows: Nanoparticles concentration distribution decreases or an increase in Nb while it enhances or 13

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20 50. M. Waqas, M. I. Khan, M. Farooq, A. Alsaedi, T. Hayat and T. Yasmeen, Magnetohydrodynamic (MHD) mixed convection low o micropolar liquid due to nonlinear stretched sheet with convective condition. Int. J. Heat Mass Trans. (2016) (in press). 51. T. Hayat, M. Rashid, M. Imtiaz and A. Alsaedi, Magnetohydrodynamic (MHD) stretched low o nanoluid with power-law velocity and chemical reaction, AIP Advances. 5 (2015) T. Hayat, M. Imtiaz and A. Alsaedi, Eects o homogeneous-heterogeneous reactions in low o Powell-Eyring luid, J. Cent. South Univ. 22 (2015) T. Hayat, M. Waqas, S. A. Shehzad and A. Alsaedi, MHD stagnation point low o Jerey luid by a radially stretching surace with viscous dissipation and Joule heating, J. Hydrology Hydromech. 63 (2016) T. Hayat, S. Farooq, B. Ahmad and A. Alsaedi, Characteristics o convective heat transer in the MHD peristalsis o Carreau luid with Joule heating, AIP Advances 6 (2016) T. Hayat, S. Qayyum, M. Imtiaz and A. Alsaedi, Three-dimensional rotating low o Jerey luid or Cattaneo-Christov heat lux model, AIP Advances 6 (2016) Highlights Non-linear thermal radiation eect is observed with viscoelastic nanoluid. Characteristics o heat transer are explored in the presence convective boundary condition. Skin riction and Nusselt number have been numerically analyzed. 17