Viscous dissipation effect on MHD nanofluid flow on an exponentially radiating stretching sheet thermally stratified medium

Transcription

1 Volume 118 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v118i10.57 ijpam.eu Viscous dissipation effect on MHD nanofluid flow on an exponentially radiating stretching sheet thermally stratified medium G.Vijaya Lakshmi 1 L.Anand Babu 2 K.Srinivas Rao 3 1 Department of Mathematics, RBVRR Women College, Telangana,India. garishevijayalakshmi@gmail.com 2 2Department of Mathematics, Osmania University, Telangana, India 3 Department of Mathematics, Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya, Tamilnadu, India. December 26, 2017 Abstract In this paper, the heat and mass transfer effects were studied on the boundary layers of nanofluid on an exponentially stretched sheet which was embedded in a thermally stratified medium. These studies were carried out in the presence of magnetic parameter, viscous and thermal radiation effects. Governing partial differential equations are converted into ordinary differential equations using different types of similarity transformations and then used Keller boxed method to solve them numerically. The velocity, temperature and nanoparticle volume fraction profiles are expressed graphically for different flow pertinent parame

2 ters, namely the Magnetic parameter(m), Radiation parameter (R), stratification parameter (St), suction parameters, permeability parameter(k), Prandtl number (Pr), Eckert number (Ec), Lewis number (Le), thermophoresis parameter (Nt), Brownian parameter (Nb). Comparative study between the previously published results and the present study is made and they shown good agreement. Key Words and Phrases:MHD, Nanofluid,Viscous dissipation, Thermally stratified medium. 1 Introduction In industrial applications the problem of viscous flow and heat transfer is having an important significance especially over a stretching sheet. It is seen that the characteristics of any stretched sheet depends on the rate of stretching, cooling and the thickness achieved after stretching. The flow properties over a stretching sheet was initially introduced by Crane [1] and later heat and mass transfer analysis was studied in this direction by Gupta and Gupta [2], Dutta et al. [3], Char [4] and Andersson [5]. An attempt to explain the heat and mass transfer in boundary layer was made by Magyari and Keller over an exponentially continuous stretching sheet [6]. The term nanofluid for suspension of liquids was firstly introduced by Choi and Eastman [7] and explained the existence of ultrafine particles. These nano particles are used with various base materials to improve the thermal conductivity. Buongiorno et al. analysed the thermal conductivity of nanofluids with a benchmark studies [8]. Later, Khan and Pop considered the studies of nanofluids flow over a stretching sheet and this was a kind of first attempt in this direction [9]. MHD flow of nanofluids over an exponentially stretched sheet embedded in a stratified medium was considered by Loganathan and Vimala with suction and radiation effects [11]. Similar attempt was made by Bhattacharya and Layek on a permeable sheet [12]. Nadeem and Lee discussed the boundary layer flow of a nanofluid over exponentially stretched surface [13] and in recent time? similar work was carried out by Mustafa et al. on an exponentially stretched sheet [14]. The effects of magnetic parameter, Joule heating, viscous dissipation and heat generation was studied by Aissa and Mohammadein 2 148

3 for MHD micropolar fluids which passed over stretching sheets [15]. However effects of viscous dissipation on mixed convection heat transfer on exponentially stretched surface was studied by Partha et al. [16]. Boundary layer slip flow and heat transfer of nanofluids due to permeable stretching sheets were studied by Malvandi et al. under convective boundary conditions [17]. Mukhopadhyay conducted a study on MHD boundary layer flow and heat transfer over the exponentially stretched sheets embedded in thermally stratified medium [18] and this work was later extended by Shekar [19]. Later, Lakshmi et al. studied the effects of Joule heating, heat transfer flow and viscous dissipation on MHD boundary layer flow over an exponentially stretched sheet which embedded in thermally stratified medium. In a slight modified way, heat and mass transfer in MHD stagnation point nanofluid flow over stretching sheet in porous medium under the influence of prescribed surface heat flux and chemical reactions were discussed in [20]. 2 Formulation of the assumed problem Now consider a steady MHD boundary layer flow of nanofluid over an exponentially stretching sheet in a porous medium embedded in a thermally stratified medium. Here the magnetic field is B = is is assumed to be perpendicular to x-axis and here B 0 is a constant. The value of Re is considered to be very small by neglecting the induced magnetic field. Similarly, T w and C w at the stretching surfaces are assumed to be constant and with a constant B 0 e x 2L ambient temperature of T and nanoparticle volume friction as C. On the other hand, the stretched sheet is embedded in the thermally stratified medium with variable ambient temperature of T under a condition where,t w (x) > T (x). With a reference temperature of T 0, it is assumed that the value of T w (x) = T 0 + ce x 2L and T (x) = T 0 + be x 2L by considering b > 0 and c 0.It is also assumed that the host fluid and nanoparticles are under equilibrium and with no slip occurrence between them. Physical model and coordinate system for the assumed set up is shown in Fig.1 to write the basic continuity, momentum and energy equations for the defined problem with assumed boundary layer approximations

4 Figure 1: Physical model and coordinate system for defined problem u T T +ν x y = u x + v y = 0 (1) u u x + v u y = ϑ 2 u y 2 σb2 (x) ρ f u ν k u (2) κ ρc p 2 T y 2 1 ρc p q r y + (ρc p) p (ρc p ) f C T D B y y + D T T ( T y ) 2 + µ ρc p (3) u C x + ϑ C y = D 2 C B y + D T 2 T (4) 2 T y 2 The boundary conditions for velocity, temperature and concentration fields are given by u = U(x), v = V w (x), T = T w (x), C = C w (x) v 0, T T, C C asy (5) The radiative heat flux q r is given by using the Rosseland approximation q r = (4σ S) T 4 (6) (3K e ) y where, Stefan Boltzmann constant is given by σ S and mean absorption coefficient by K e respectively. In this work the analysis is limited to the optically thick fluids by using the Rosseland approximation.however, if temperature difference within the flow is 4 ( ) 2 u y 150

5 smaller means the (5) can be linearized by expanding T 4 into the Taylor series about T and by neglecting higher order terms will be giving the following form as T 4 4T 4 T 3T 4 (7) Now, by substituting the Equations (6) 7)and into Equation (3) can be written as follows u T x +ϑ T y = T T 3 2 T C T α 2 2 y +16σ +τ[d 3k y 2 B y y +D T ( T T y )2 ]+ µ ( u ρc ρ (8) The following similarity transformations are used to get (1) to (4) along with the condition in (5), so that y )2 ( ) 1 ( ) 1 U = U 0 e x L f U0 2 x (η), ν = e 2L (f(η) + f U0 2 x (η)), η = e 2L.y 2νL 2νL θ(η) = T T, ϕ(η) = C C, T w (x) = T 0 +be ( x T w T 0 C W C 0 2L ), T (x) = T 0 +ce x 2L (9) here, the characteristic length is given by L,parameter of temperature distribution in stretching surface is given by T 0 and variable ambient temperature is given by T. f + f.f 2f 2 + (M + K)f = 0 (10) 1 P r ( R)θ +fθ f θ f St+Nbθ φ +Ntθ 2 +Ecf 2 = 0 (11) φ + Le(fφ f φ) + Nt Nb θ = 0 (12) Then the transformed boundary conditions can be written by f = S, f = 1, θ = 1 St, φ = 1 at η = 0 f 0, θ 0, φ 0 at η (13) where the magnetic field parameter is given by M = 2σB2 0 L U 0ρ f, Prandtl number is given by P r = ν, the radiation parameter is R = 4σ T 3, α κ κ 5 151

6 the permeable parameter of the porous medium is given by K = 2νL the permeable parameter of the porous medium is given by αk e ( L x ) S = v0 αν 2L for S > 0 corresponds to suction and s < 0 corresponds to blowing, the Lewis number is Le = ν D B, the Eckert number is E c = U 0e ( x L ) C p, the Brownian motion parameter is by Nb = DBτ (Cw C ) ν the thermophoresis parameter is by Nt = D T and the T τ (Tw T ) ν stratification parameter is given by St = c. The stable stratified b environment is for St > 0 and for St = 0 corresponds to unstratified environment. 3 Numerical Method The equation (10), (11) and (12) along with the boundary conditions given in (13) are solved by converting them to initial value problems and by considering f = p, p = q, q = 2p 2 + (M + K)p fq θ = t, t 1 = (ft + pθ pst + Nb.θϕ + Ntt 2 + ECq 2 ) 1 + 4R 3 φ = g = Le(pφ fg) Nt Nb t The step size considered here is h = 0.05 and obtained the numerical solution and also solved algebraic system of equation using implicit finite difference scheme i.e., using Keller box method. Table 1: Comparison of results for -f (0) when S=K=R=M=0 Magyari and Keller Bhattacharya et al. Lavanya et al. Present Results f (0)

7 Table 2: Nusselt Number for different values of Thermal stratification parameter, Brownian motion parameter, thermophoresis parameter and Lewis number St Nb Nt Le. θ (0) Results and Discussions In this work the parameters velocity, temperature and concentrations were discussed in detail for different numerical values obtained from governing equations. Later, they were compared with previous research outcomes of Magyari and Keller [6], Bhattacharyya and Layek [18] and Lavanya et al. [11] and found to have an excellent agreement as compared with obtained results as shown in Fig. 2 and Table 1. For different values of governing parameters such as M, K, Pr, R, St, Nt, Nb, Ec and Le the numerical results were plotted which represents velocity, temperature and nanoparticle volume fraction profiles with a varying magnetic parameter (M) between Fig. (av). Effect of magnetic parameter (M) on the velocity profile in a flow is shown in Fig. a and it will be reduced with the increasing value of M. The reason for such reduction in velocity profile was explained by Lorentz force, where the magnetic field will introduce retarding body force and this force acts transverse to the actual direction of applied field. Therefore the boundary layer flow will decelerates so that the boundary layer thickness will be decreasing under the influence of magnetic parameter. Eckert Number (Ec) is an expression to establish the relationship 7 153

8 (a) Variation of velocity f (η) with(b) Variation of temperature θ(η) η for M with η for M (c) Nanoparticle volume fraction(d) Variation of temperature θ(η) with η for M with η for E c (e) Variation of temperature θ(η) (f) Variation of temperature θ(η) with η with S t with suction with η with S t without suction 8 154

9 (g) Velocity profiles for different val-(hues of K ent values of Temperature profiles for differ- K (i) Nanoparticle volume fraction for(j) Velocity profiles for different values of different values of K S (k) Temperature profiles for differ-(lent values of S different values of Nanoparticle volume fraction for S 9 155

10 (m) Variation of θ(η) with η for dif-(nfernt values of P r different values of P Nanoparticle volume fraction for r (o) Temperature profiles for differ-(pent values of R profiles for different values of Nanoparticle volume fraction R (q) Temperature profiles for differ-(rent values of N t different values of N Nanoparticle volume fraction for t

11 (s) Temperature profiles for differ-(tent values of N b different values of N Nanoparticle volume fraction for b (u) Nanoparticle volume fraction for(v) Effect of Nb and Nt on Heat different values of L e Transfer Coefficient between kinetic energy (K.E) and enthalpy. It embodies conservation of K.E. into internal energy due to the work done against the viscous fluids stress relationship between K.E. in the flow and enthalpy. It means that the positive Ec is nothing but cooling of sheet. That means the loss of heat in a sheet will be transferred to the fluid. In the experimental results it can be seen from Fig. 5 that Ec will be increasing with the increasing value of temperature. At this juncture it is important to understand the effect of St on θ(η) with and without suction at boundary layer. For various values of St with and without suction the temperature profiles θ(η) are shown in Fig. e and f respectively. From the results it can be seen that increase in the value of St decreases the temperature profiles. That means the free-stream temperature will increase and surface temperature will decrease with increasing value of St. On the other hand, thermal boundary layer thickness also decreases with the increasing value of St. Profiles will decay from higher value at the walls to zero values in free stream. That means the profiles will converge at outer edges of boundary layers

12 Figure 2: Effect of Nb and Nt on Mass Transfer Coefficient Now it is very important to see how the permeability parameter (K) affects the velocity, temperature and nanoparticle volume friction. With the raise in the value of K it is observed that the resistance to the fluid motion increases, which influence the velocity and temperature to decrease as shown in Fig. gh Influence of the magnetic parameter (M) on temperature parameter and nanoparticles volume friction is shown Fig. b and c. The increasing magnetic field allows the temperature and concentration profiles to increase. However, due to resistive force according to Lorentz force, the opposition for fluid motion generates the heat so that thermal boundary layer thickness will be thicker when magnetic field is very much stronger. Now by varying suction parameter (S) values in an increasing order resulted to affect the velocity, temperature and nanoparticle volume fraction as they tend to decrease in boundary layer region. All these parameters are inversely proportional to the suction parameter. Under the influence of mass suction fluids will be getting closer to the sheet and velocity boundary layer will become thinner and the similar effect will be there on thermal and nanoparticle volume boundary layers thickness. However, in case of mass injection (i.e. when fluid is taken away from the sheet) the boundary layer thickness of the velocity, thermal and nanoparticles will be broader as shown in Fig Influence of suction / injection parameter (S) is very much regime towards velocity, temperature and concentration over the stretching sheet boundary layer. With the suction at S = 0.0, 0.5 and 1.0, the flow was observed to be strongly decelerated and with the injection at S = 1.0, 0.5 and 0.25 the flow was accelerating strongly due to increase in blowing. That means, suction is causing boundary layer to be closer to the wall and by

13 which the momentum leading to a serious drop in the velocity. From the above it can be summarized that with suction momentum boundary layer thickness will be decreasing and with injection of nanofluids via lateral mass flux over the sheet increases the momentum by which the velocity and momentum boundary layer thickness will be increasing. Flow reversal velocity is not due to suction and will remain positive for the complete boundary layer so that the flow control will be excellent in nano fluids due to suction parameters. Also due to suction both temperature and concentration profiles will decreases and gradual decays will be observed from wall to free stream. However, temperature and nano particle concentrations will be increasing due to the injection process. For the injection process both profiles seem to be having greater thickness as compared to suction. On the other hand suction will get stronger suppression of nano-particles and due to which the diffusion takes place which later helps to regulate diffusion of heat at the boundary layers. Such a response of suction and injection processes will be having a significant impact on the nanofluids to achieve flow control, cooling and distribution of nano-particles. Now considering the Prandtl number (Pr), it is seen that there is a sensitive effect on temperature and nanoparticle volume friction. With increasing Pr values thickness of thermal boundary layer will reduce and this is due to the decreased thermal diffusivity. For the higher values of Pr nanoparticle volume overshoots near the sheet even when there is a reduction in nanoparticle volume boundary layer thickness. Effect of Pr on temperature profile θ(η) and nanoparticle volume fractions ϕ(θ) is shown in Fig. m and n respectively. It is observed that for higher Pr values the nanoparticle volume fraction will increase in fluids near the sheet as compared to the wall in a uniform thermophoretic particle deposition. It can be seen in Fig. m, that the increase in Pr values decreases the temperature distribution and thermal diffusion will be leading towards depletion in temperature. Now the effect of radiation parameter (R) on temperature θ(η) is having a proportional relation and nanoparticle volume fraction ϕ(η) will be having an inverse proportional relation with R, i.e. with the increased temperature φ(η) found to be decreasing when the value of R is increasing in steps of 0.5, 1.0, 1.5 and 2.0. The reason for this is due to reduction in boundary layer thickness and

14 increasing heat transfer rated due to the increasing value of R as shown in Fig. 0 and p. Here effect of thermophoresis parameter (Nt) was studied on the temperature parameter θ(η) and nanoparticle volume fraction ϕ(η) profiles are studied as it is considered to be a key parameter to analyse temperature distribution and nanoparticle volume fraction ϕ(η) in nanofluids. Both these parameters (i.e. θ(η) and ϕ(η)) are having a proportional relation with Nt as shown in Fig. q and r. This is due to the movement of nanoparticles from hot area to cold area and thereby the magnitude of θ(η) and ϕ(η) increases significantly. Therefore the thickness of nanoparticle volume boundary layer will increase with a slight change in the value of Nt. Now effect of Brownian motion parameter (Nb) on temperature θ(η) and concentration distributions ϕ(η) are in a nanofluid plays a critical role because the Brownian motion depends on the size of nanoparticle. The size of nanoparticle here plays a key role since the particle motion and its effect on the fluid have a critical impact on the heat transfer. It means that the effective motion of nanoparticles will increases within the flow when the value of Nb increases. Such a chaotic motion increases the intensity of the flow and thereby K.E. of nanoparticles due to which the temperature levels of nanofluids increases rapidly as shown in Fig. S.On the other hand any kind of minor decrease in the value of ϕ(η) is accounted only away from the sheet leads to a strong Brownian motion. That means any kind of increment in the value of Nb will reduces the value of nanoparticle volume fraction profiles as shown in Fig. t. Therefore it is clear that the increasing value of Nb will increase the temperature of the fluid and decreases the nanoparticle volume fraction. Also with the increase of Nb decreases the nanoparticle volume boundary layer thickness. Lewis number (Le) is another important parameter which influences the nanoparticle volume friction ϕ(η). The value of ϕ(η) will increase nearer to the sheet at the initial stage and decreases significantly when it is away from the sheet as shown in Fig. u. The means Le = 0.1, 1.0, and 5 the value of ϕ(η) will increase rapidly at the initial stages and decreases rapidly for the larger values of Le. It is also evident that with the increase in the value of Le, the ϕ(η) decreases and nanoparticle volume boundary layer thickness will reduce. Now the influence of Nb and Nt will be examined on the local Nusselt number θ (0) and Sherwood number ϕ (0) since both the parameters

15 are purely influences the heat transfer rate on the surface of sheet in a negative direction. There will be an increment in the resistance to the diffusion of mass with the increase in Nt and due to which reduction of concentration gradient takes place as shown in Fig. v. Similarly, variation of local Sharewood number ϕ (0) will be changing with respect to the value of Nb as shown in F ig.2. The value of -ϕ (0) increases with an increment in the value of Nb and decreases with the increment of Nt. 5 Conclusion In this study, the numerical solutions for steady state viscous dissipation and thermal radiation of MHD boundary layer flow were studied with constant parameter values. These studies were considered in thermally stratified medium for the nanofluids passing over the exponentially stretched sheet. Effects of suction parameter with increasing values of radiation and magnetic parameters along with Eckert number were considered. It is found that the increasing stratification parameter will reduces the temperature and the obtained results shows a good agreement. Increasing magnetic parameter decreased the velocity profiles and increases the temperature and concentration profiles. Increasing value of S results to decrease the value of velocity, temperature and concentration. As Prandtl number increases, temperature decreases. As Nb and Nt increases, temperature decreases, concentration decreases & increases. As increase in R, temperature increases and concentration is decreased up to certain level and then increased. As magnetic parameter and stretching parameter increases, skin friction increases. As increase in Le, concentration decrease

16 References [1] Crane, L. J. Flow past a stretching plate. Zeitschrift fr angewandte Mathematik und Physik (ZAMP), 21(4),(1970) [2] Gupta, P. S., & Gupta, A. S. Heat and mass transfer on a stretching sheet with suction or blowing. The Canadian Journal of Chemical Engineering, 55(6),(1977) [3] ElAziz, M. A. Viscous dissipation effect on mixed convection flow of a micropolar fluid over an exponentially stretching sheet. Canadian Journal of Physics 87, (2009) l. [4] Char, M. I. Heat transfer of a continuous, stretching surface with suction or blowing. Journal of Mathematical Analysis and Applications, 135(2), (1988) [5] Andersson, H. I. Slip flow past a stretching surface. Acta Mechanica, 158(1-2), (2002) [6] Magyari, E., & Keller, B. Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. Journal of Physics D: Applied Physics, 32(5), (1999) [7] Choi, S. U., & Eastman, J. A. Enhancing thermal conductivity of fluids with nanoparticles (No. ANL/MSD/CP 84938; CONF ). Argonne National Lab., IL (United States) (1995). [8] Buongiorno, J. Convective transport in nanofluids. Journal of Heat Transfer, 128(3), (2006) [9] Khan, W. A., & Pop, I. Boundary-layer flow of a nanofluid past a stretching sheet. International journal of heat and mass transfer, 53(11), (2010), [10] Loganthan, P., & Vimala, C. MHD Flow of Nanofluids over an Exponentially Stretching Sheet Embedded in a Stratified Medium with Suction and Radiation Effects. Journal of Applied Fluid Mechanics, (2015) 8(1)

17 [11] Bhattacharyya, K., & Layek, G. C. Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet. Physics Research International, (2014). [12] Nadeem, S., & Lee, C. Boundary layer flow of nanofluid over an exponentially stretching surface. Nanoscale Research Letters, 7(1),(2012) 94. [13] Mustafaa, M., Hayat, T., & Obaidat, S. Boundary layer flow of a nanofluid over an exponentially stretching sheet with convective boundary conditions. International Journal of Numerical Methods for Heat & Fluid Flow, 23(6),(2013) [14] Aissa, W. A., & Mohammadein, A. A. Joule heating effects on a micropolar fluid past a stretching sheet with variable electric conductivity. ıj. Comput. Appl. Mech, 6(1),(2005) [15] Partha, M. K., Murthy, P. V. S. N., & Rajasekhar, G. P. Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. Heat and Mass Transfer, 41(4), (2005), [16] Malvandi, A., Hedayati, F., & Ganji, D. D. Boundary Layer Slip Flow and Heat Transfer of Nanofluid Induced by a Permeable Stretching Sheet with Convective Boundary Condition. Journal of Applied Fluid Mechanics, 8(1), (2015). [17] Mukhopadhyay, S. MHD boundary layer flow and heat transfer over an exponentially stretching sheet embedded in a thermally stratified medium Alexandria Engineering Journal, 52(3), (2013) [18] Sandeep, N. Sulochana, C., & Isaac Lare, A.. Stagnationpoint flow of a Jeffrey nanofluid over a stretching surface with induced magnetic field and chemical reaction. In International Journal of Engineering Research in Africa,(2016) (Vol. 20, pp ). Trans Tech Publications. [19] Lare, A. I. Casson fluid flow with variable viscosity and thermal conductivity along exponentially stretching sheet embedded in

18 a thermally stratified medium with exponentially heat generation.journal of Heat and Mass Transfer Research (JHMTR), 2(2),(2015), [20] Lakshmi, G. V., Babu, L. A., & Rao, K. S. Joule Heating and Viscous Dissipation Effects on MHD Boundary Layer Flow and Heat Transfer over an Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium. IJAETMAS, Vol.3, (12), pp

19 165

20 166