Available online at ScienceDirect. Procedia Engineering 127 (2015 )

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1 Available online at ScienceDirect Procedia Engineering 17 (015 ) International Conerence on Computational Heat and Mass Transer-015 MHD Flow o a Nanoluid Embedded with Dust Particles Due to Cone with Volume Fraction o Dust and Nano Particles V Ramana Reddy Janke a, Sandeep Naramgari b, Sugunamma Vangala a, a Department o Mathematics, S.V.University, Tirupati-51750, India. b Fluid dynamics division, VIT University,Vellore-63014, India. Abstract This paper devotedly analyze the momentum and heat transer behaviour o MHD nanoluid low embedded with the conducting dust particles past a cone in the presence o non-uniorm heat source/sink, volume ractions o dust and nano particles. The governing partial dierential equations o the low and heat transer are transormed into nonlinear ordinary dierential equations by using sel similarity transormation and solved numerically. The eects o various non-dimensional governing parameters on velocity and temperature proiles are discussed with the help o graphs. Further, the eects o these parameters on skin riction coeicient and Nusselt number are also discussed and presented through tables. Moreover it is ound that an increase in the mass concentration o dust particles depreciates the velocity proiles o the luid and dust phases. 015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license 015 The Authors. Published by Elsevier B.V. ( Peer-review under under responsibility responsibility o the o organizing the organizing committee committee o ICCHMT o ICCHMT Keywords: MHD, Dusty luid; Nanoluid; Vertical cone; Volume raction; Non-uniorm heat source/sink. 1. Introduction The suspension o nano sized metallic or non-metallic particles in the base luids give the nanoluid. Generally, the base luids like water, kerosene and ethylene glycol have low thermal conductivity. But the suspension o above mentioned metals or non-metals in the orm o nano size, helps to enhance the thermal conductivity o the Corresponding author. Tel.: address: vsugunar@yahoo.co.in The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility o the organizing committee o ICCHMT 015 doi: /j.proeng

2 V Ramana Reddy Janke et al. / Procedia Engineering 17 ( 015 ) baseluids. Masuda et al. [1] was one o the researchers to observe an enhancement in the thermal conductivity o the base luid by the dispersion o ultra-ine particles in the base luids. Further, the researchers like Awad et al. [] and Raju et al. [3] have investigated the eects o dierent physical parameters on the low o nanoluids over dierent channels with the help o their investigations. The dusty luids have wide range o industrial applications as well as in science and technology. So, several researchers analyzed the low characteristics o dusty luids. Among them Ramana Reddy et al. [4] is one to analyze the heat generation or absorption eects on dusty viscous low under the inluence o aligned magnetic ield. Gireesha et al. [5] discussed the MHD heat transer eects on dusty luid low over a stretching sheet. The combined heat and mass transer low along with MHD eects have tremendous applications in power transormer electronics solar energy systems etc. Owing to this act Mohan Krishna et al. [6] analyzed the inluence o radiation and chemical reaction on MHD convective low with suction and heat generation eects. The low o viscous Ag-water and Cu-water nanoluids past a stretching surace has been analyzed by Vajravelu et al. [7]. On the other hand the low o Jerey nanoluid with nano particles was analyzed by Hayat et al. [8]. To understand the mechanism o convective heat transer, it is important to study the low behaviour past an axisymmetric structure such as vertical and horizontal cylinders along with the Cone and spheres. So, Roy et al. [9], Khan and Sultan [10] concentrated their research on the low o a luid through dierent channels with axisymmetric structures. Nadeem and Saleem [11] discussed the unsteady Eyring Powell nanoluid low in a rotating cone. Since the both the nano and dusty luids are useul to enhance the thermal conductivity, Sandeep and Sulochana [1] studied the MHD low o dusty nanoluids over a stretching surace. By making use o all the above cited articles, we make an attempt to analyze the heat transer in MHD dusty nanoluids past a cone under the inluence o non uniorm heat source/sink along with the volume ractions o nano and dust particles. The reduced governing equations o the low are solved numerically. Further, the eects o various parameters involved in the governing equations are discussed through graphs and tables, which are drawn with the help o MATLAB package. Mathematical Analysis Consider a steady, laminar, incompressible boundary layer low o a dusty nanoluid over a vertical cone pointing downward with hal angle γ and radius r. The x -axis varies along the surace o the cone and y -axis is normal to the surace o the cone. The origin is located at the vertex o the cone. The low ield is exposed by the inluence o external magnetic ield strength B 0 along x -axis. The temperature distribution T w will vary along the surace o the cone. The luid has a uniorm ambient temperaturet. The Boundary layer equations, which govern the low o the present problem per the above made assumptions, are given by (See Khan and Sultan [10], Sandeep and Sulochana [1]) ( ru) ( rv) + = 0, x y ( rup) ( rvp) + = 0, x y u u u ρn (1 φd ) u + v = (1 φd ) μn + g( ρβ) ( ) cos ( ), n T T γ + KN up u σb u x y y up up Nmup + vp = KN( u up), x y N x y y τυ T T T 1 ( ρcp) nu + v = kn + ( u ) ''', p u + q The appropriate boundary conditions o the problem are given by (1) () (3) (4) (5)

3 108 V Ramana Reddy Janke et al. / Procedia Engineering 17 ( 015 ) υ ( ) 1 u = Uw( x) = Gr, v = 0, T = Tw( x) = ax, at y = 0, x (6) u 0, up 0, vp v, T T as y, Where uvare, the velocities o the nanoluid along the x and y directions respectively. Similarly u, v stands or the velocities o dust phase along the x and y directions respectively. r = xsin γ is the local radius o the cone, a is the constant, φd is the volume raction o the dust particles, g is the acceleration due to gravity, ρn is the density o the nanoluid, β n is the coeicient o thermal expansion o the nanoluid due to temperature dierence, T is the temperature, Tw is the temperature o the luid ar away rom the surace o the cone, T ambient temperature o the luid. K is the stokes resistance, N is the number density o dust particles, σ is the electrical conductivity o the luid, m is the mass o the dust particle, ( ρc P ) is the heat capacitance o the nanoluid, k n n conductivity o the nanoluid, N 1 particle. p p is the thermal = Nm is the density o the particle phase, τ υ is the relaxation time o the dust The space and temperature dependent heat generation/absorption (non uniorm heat source/sink) q''' is deined as (See Pal [13]), ( w ) kuw( x, t) q''' = A ( T T ) ' + B ( T T ), xυ (7) In Eqn. (7), A and B are parameters o the space and temperature dependent internal heat generation/absorption. The nanoluid constants are given by (See Kalidas [14], Raju et al. [3]) ( ρβ) n = (1 φ)( ρβ ) + φ( ρβ ) s, ( ρcp) n = (1 φ)( ρcp) + φ( ρcp) s, kn ( ks + k ) φ( k ks) μ =, μn =, ρ (1 ), ( ) ( ).5 n = φ ρ + φρs k ks + k + φ k k s (1 φ) (8) Now, we introduce the ollowing similarity transormations to make the governing Eqs. (1) to (5) subject to the boundary conditions represented in Eq. (6), into coupled non linear ordinary dierential equations. υ υ η 1 y υ ( ) '( η), ( ) 4 '( η) ( η), η ( ) 4, ( ) '( η), u = Gr v= Gr = Gr up = Gr F x x 4 x x cos ( w ) υ η T T β gx γ T T v ( ) 4 p = Gr F'( η) F( η), θ( η), Gr, = = x 4 Tw T υ (9) Now deine the dimensionless stream unction by ru = ψ / y and rv = ψ / x, ψ = vxsin γ Gr ( η). Where ( ) 1/4 Using the equations (7)-(9), then Eqs. (3) to (6) will reduce the ollowing orm, and Eqs. (1) and () identically satisies. ( ).5 ''' (1 φ) ( 1 φ+ φ( ρs / ρ )) + 1 φ+ φ( ( ρβ) s / ( ρβ) ) θ +.5 (1 φ) + ( Λαβ ( F ) M ) = 0, (1 φd ) (10)

4 V Ramana Reddy Janke et al. / Procedia Engineering 17 ( 015 ) F FF β ( F ) = 0, 1 1 k n 1 θ + αβec F + A + B θ + θ = Pr + k ( 1 φ φ( ( ρcp) s/( ρcp) )) Subject to boundary conditions '' Pr ( ') ( ' ) 0, (11) (1) ( η) = S, ( η) = 1, θ( η) = 1, atη = 0, (13) ( η) = 0, F ( η) = 0, F( η) = ( η), θ( η) = 0, asη, where α is the mass concentration o the dust particles, β is the luid particle interaction parameter or the velocity with τ v, Λ is the luid parameter, M is the magnetic ield parameter, Pr is the Prandtl number, Ec is the Eckert Number, which are given by ( ) 1/ α = ( Nm)/ ρ, β = 1/ τ, τ = m / K, Λ = ( x Re)/( υ Gr), M = ( σb x )/ ρ υ Gr,Pr = υ / k, Ec = ( υ ( Gr) )/( C ) 1/ v v 0 p For engineering interest the local skin riction coeicient C and local Nusselt number 1 1 x x x Nux are given by C Re = (0), Nu Re = θ (0), (14) Where Re x is the local Reynolds number deined by Re x xu w ( x) =. υ 3. Results and discussion In order to solve the coupled non-linear ordinary dierential Eqs. (10) to (1) with the help o boundary conditions o the low represented by Eq. (13), we make use o Runge-Kutta and Newton s method presented by Mallikarjuna et al. [15]. Further the eects o various parameters on velocity and temperature distribution involved in the Eqs. (10) to (1) are studied through graphs. For numerical results we have considered M = 1, φ d = 0.1, α = 0., β = 0.5, Pr = 6., Ec = 0., φ = 0.15, A = B = 0.1. Table 1 Thermophysical properties o water, Cu and AlO 3 nano particles. ρ 3 ( Kg m ) c p ( ) ( Wm K ) K JKg K k β (10 ) HO Cu AlO The inluence o magnetic ield parameter ( M ) can be seen in Fig. 1. It is observed that an increase in the magnetic ield parameter reduces the velocity proiles o the both luid and dust phases. This is due to the act that an increase in the magnetic ield parameter develops the opposite orce to the low, is called the Lorentz orce. Figs. and 3 illustrate the eect o volume raction o nano particles ( ) φ on velocity and temperature proiles respectively. It is observed that an increase in the volume raction o the nano particles helps to improve the velocity o the luid as well as dust phases over the boundary layer o the cone. But reverse action takes place in

5 1030 V Ramana Reddy Janke et al. / Procedia Engineering 17 ( 015 ) temperature proiles due to the act that improvement in the velocity proiles enhances the thermal conductivity. Fig. 4 depicts the velocity proiles o the luid and dust phases or dierent values o volume raction o dust particles. It is noticed that an increase in the volume raction o dust particles reduces the velocity proiles or both luid and dust phases, because the volume occupied by the dust particles per unit volume o the luid is high then the dust concentration in the luid will increase. It is observed rom Fig. 5 that the velocity proiles o luid and dust phases are reduced with an enhancement inα. But rom Fig. 6, we noticed an opposite result in temperature proiles. Fig. 7 depicts the velocity proiles o luid and dust phases or dierent values o luid-particle interaction parameter ( β ). We have observed an interesting result that an increase in the luid-particle interaction parameter enhances the velocity o dust phase, but depreciates the velocity o the luid phase. Fig. 8 shows the eect o space dependent non-uniorm heat source/sink parameter on temperature proiles. It is evident that increase in A causes an increase in the temperature ield. The reason behind this is the thermal boundary layer generates the energy or positive values o A. Table shows the validation o the present results with the existed literature under some special limited cases. We ound an excellent agreement o the present results with the existed results. This proves the validity o the present results along with the accuracy o the numerical technique we used in this study. It can be seen rom Table 3 that an enhancement in the physical parameters M, φd, α, β depreciates the proiles o Skin-riction coeicient, Nusselt and Sherwood numbers. Table comparison o the present results or n = 1 (power o in present problem it is 1), when φ = φd = A = B = α = β = 0. Grosan et al.[16] Yih [17] Present Results Table 3. Variation in riction actor, local Nusselt and Sherwood numbers. M φ φ α β d A Cu HO AlO3 H O C Nux C Nux

6 V Ramana Reddy Janke et al. / Procedia Engineering 17 ( 015 ) Fig.1. Velocity proiles or dierent values o magnetic ield parameter M. Fig.. Velocity proiles or dierent values o volume raction o nanoparticles φ. Fig.3. Temperature proiles or dierent values o volume raction o nanoparticles φ. Fig.4. Velocity proiles or dierent values o volume raction o dust particles. φ d

7 103 V Ramana Reddy Janke et al. / Procedia Engineering 17 ( 015 ) Fig.5. Velocity proiles or dierent values o mass concentration o dust particles. α Fig. 6. Temperature proiles or dierent values o mass concentration o dust particles α. Fig.7. Velocity proiles or dierent values o luid-particle interaction parameter β. Fig.8. Temperature proiles or dierent values o non-uniorm heat source/sink parameter A

8 V Ramana Reddy Janke et al. / Procedia Engineering 17 ( 015 ) Conclusions An increase in luid particle interaction parameter enhances the heat transer rate. An increase in volume raction o nano particles enhances the thermal conductivity o the low. A raise in the volume raction o dust particles reduces the momentum boundary layer thickness. Cu-water dusty nanoluid shows better heat transer perormance compared with Al O 3 -water dusty nanoluid. Embedding the conducting dust particles in to nanoluids enhances the heat transer rate. Reerences [1] H. Masuda, A. Ebata, K. Teramae, N. Hishinuma, Alteration o thermal conductivity and viscosity o liquid by dispersing ultra-ine particles, Nets. Buss. 7 (1993) [] F.G. Awad, P. Sibanda, A. A. Khidir, Thermo diusion eects on magneto-nanoluid low over a stretching sheet, Bound. Val. Prob.136 (013) [3] C.S.K. Raju, M. Jayachandrababu, N. Sandeep, V. Sugunamma, J.V. Ramana Reddy, Radiation and soret eects on MHD nanoluid low over a moving vertical plate in a porous medium, Chem. Proc. Eng. Res. 30 (015) 9-3. [4] J.V. Ramana Reddy, V. Sugunamma, P. Mohan Krishna, N. Sandeep, Aligned magnetic ield, radiation and chemical reaction eects on unsteady dusty viscous low with heat generation/ absorption, Chem. Proc. Eng. Res. 7 (014) [5] B.J. Gireesha, G.S. Roopa, H.J. Lokesh, C.S. Bajewadi, MHD low and heat transer o a dusty luid over a stretching sheet, Int. J. Phys. Math. Sci. 3 (01) [6] P. Mohan Krishna, N. Sandeep, V. Sugunamma, Eects o radiation and chemical reaction on MHD convective low over a permeable stretching surace with suction and heat generation, Walailak J. Sci. Tech. 1 (015) [7] K. Vajravelu, K.V. Prasad, J. Lee, C. Lee, I. Pop, R.A. Van Gorder, Convective heat transer in the low o viscous Ag-water and Cu-water nanoluids over a stretching surace, Int. J. Ther. Sci. 50 (011) [8] T. Hayat, S. Asad, A. Alsaedi, Analysis or low o Jerey luid with nanoparticles, Chin. Phys. B. 4 (015) [9] S. Roy, P. Datta, R. Ravindran R, E. Momoniat, Non- Uniorm double slot injection (Suction) on a orced loe over a slender cylinder, Int. J. Heat Mass Tran. 50 (007) [10] N. A. Khan, F. Sultan, On the double diusive convection low o Eyring-Powell luid due to cone through a porous medium with soret and duour eects, AIP Adv. 5 (015) [11] S. Nadeem, S. Saleem, Series solution o unsteady Eyring powell Nanoluid low on a rotating cone, Ind. J. Pure Appl. Phys. 5 (014) [1] N. Sandeep, C. Sulochana, MHD low o dusty nanoluid over a stretching surace with volume raction o dust particles, Ain Shams Eng. J. (015) In Press. [13] D. Pal, Combined eects o non-uniorm heat source/sink and thermal radiation on heat transer over an unsteady stretching permeable surace, Comm. Nonlin. Sci. and Nume. Simu. 16 (011) [14] D. Kalidas, Flow and heat transer characteristics o nanoluids in a rotating rame, Alex. Eng. J. 53 (014) [15] B. Mallikarjuna, A.M. Rashad, A.J. Chamka, S.H.P. Raju, Chemical reaction eects on MHD convective heat and mass transer low past a rotating vertical cone embedded in a variable porosity regime, Ar. Mate. (015) 1-1. [16] T. Grosan, A. Postelnicu, I. Pop, Free convection boundary layer over a vertical cone in a non-newtonian luid saturated porous medium with internal heat generation, Tech. Mech. 4 (004) [17] K.A. Yih, Uniorm lateral mass lux eect on natural convection low o non-newtonian luid over a cone in porous media, Int. Comm. Heat Mass Tran. 5 (1998)