MHD NANOFLUID FLOW AND HEAT TRANSFER DUE TO ROTATING DISK WITH BOUNDARY CONDITION OF THIRD KIND

International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN (P): 2249–6890; ISSN (E): 2249–8001 Vol. 10, Issue 3, Jun 2020, 1107-1118 © TJPRC Pvt. Ltd.

MHD NANOFLUID FLOW AND HEAT TRANSFER DUE TO ROTATING DISK WITH BOUNDARY CONDITION OF THIRD KIND D. PRAKASH & NARSU SIVAKUMAR Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, Tamilnadu, India ABSTRACT A novel numerical approach is developed to examine the boundary layer flow, heat and mass transfer of rotating nanofluid with boundary conditions of third kind on the rotating disk. The rotating disk exchanges heat with an external hot fluid. The famous Buongiorno model has been used to subsume Brownian motion and thermophoresis of the nanofluid. Moreover, Passive control of nanoparticle-concentration at the wall of the disk is applied. A suitable similarity transformation is applied to form a system of non-linear ordinary differential equations from the governing partial differential equation. Runge-Kutta-Felberg method with shooting technique is used to solve the resultant equations. The impact of several exciting pertinent parameters on the boundary layer characteristics is examined through graphs.

KEYWORDS :Boundary Conditions of Third Kind, Nanofluid & Rotating Disk

1. INTRODUCTION Nanofluids, which consist of suspended nanoparticles, can considerably change the transport and thermal properties of the base fluid, and thus may enhance thermal conductivity. Choi [1] seems to be the first, who used the term

Original Article

Received: May 02, 2020; Accepted: May 22, 2020; Published: Jun 24, 2020; Paper Id.: IJMPERDJUN202095

nanofluids in his research. Few promising applications of nanofluids include cooling of microelectronics, efficiency of hybrid-powered engines, heating/cooling of home appliances, cooling towers, transportation, vehicle thermal management and drug targeting in cancer patients.” Due to a prominent status of nanofluids in the practical applications, a list of review articles on nanofluids and its applications can be found in [2-4]. Bachok et al. [5] have examined the boundary layer flow and heat transfer over a rotating porousdisk in a nanofluid. Rashidi et al. [6] have investigated the effect of entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid. Later, Turkyilmazoglu [7] analyzed the boundary layer flow and heat transfer characteristics over a rotating disk with different distinct nanofluids.

On the other hand, Von Karman [8], who is the first person to explore the three-dimensional viscous flow stimulated by rotating disk having constant angular velocity. Such flow problem has significance in numerous practical applications in aeronautical science as well as in other engineering branches such as the dynamics of hurricanes and tornadoes, rotor-stator systems in turbines, geothermal extraction, food processing, chemical mixing chambers, designing of nuclear reactors, power generators, computer storage devices, oil recovery process and many others [9].” After the pioneering work [8], the flow due to a rotating disk became an interesting topic for the aforementioned applications. Some other works on this field are, for instance, [10–12] and a later with variety of physical features from the works [13–17] and so on.

Moreover, at high temperature, thermal radiation can significantly affect the heat transfer and the www.tjprc.org

SCOPUS Indexed Journal

editor@tjprc.org

1108

D. Prakash & Narsu Sivakumar

temperature distribution of a nanofluid. Heat transfer by simultaneous free convection and thermal radiation in the case of nanofluid has not gained as much attention. This is unfortunate because, thermal radiation plays an important role in determining the overall surface heat transfer in situations, where convective heat transfer coefficients are small, as is the case in free convection where such situations are common in space technology.” Due to the importance of radiation effects in nanofluid theory, researchers [18-20] have analyzed it with thin plates either vertical or horizontal. “Even though the aforementioned studies have been dealt with nanofluids, the flow and heat transfer characteristics of nanofluid due to rotating disk with the combined effects of Newtonian heating, magnetic and radiation on the rotating disk has not been carried out by the researchers. Motivated by the application of effective cooling with rotating electronic components, we indent to investigate the problem of nanofluid with revised model suggested by Nield and Kuznetsov [21, 22].”

2. MATHEMATICAL FORMULATION “A laminar flow of an electrically conducting nanofluid over an impermeable disk, located in the xy plane is considered with the disk that rotates in its own plane with constant angular velocity, which induces flow in the neighbouring layers of nanofluid. Cylindrical coordinate system (r, θ, z) has been considered for the model development (See fig.1). The rotating disk exchanges the heat with an external fluid by convection. Assuming rotational symmetry, the transport equations can be expressed with the revised nanofluid model as follows:”

Figure 1: Physical Geometry.

Conservation of mass 𝜕𝑢 𝜕𝑟

𝑢

+

𝑟

+

𝜕𝑤

=0

𝜕𝑧

(1)

Conservation of momentum 𝜌 (𝑢

𝜕𝑢 𝜕𝑟

𝜌 (𝑢

−

𝜌𝑣 𝜌𝑟

+

𝑣2

+𝑤

𝑟

𝑢𝑣 𝑟

𝜕𝑢 𝜕𝑧

+𝑤

) = 𝜇(

𝜌𝑣 𝜌𝑧

𝜕2 𝑢 𝜕𝑟 2

)= 𝜌 (

+

𝜌2 𝑣 𝜌𝑟 2

1 𝜕𝑢 𝑟 𝜕𝑟

+

−

1 𝜌𝑣 𝑟 𝜌𝑟

𝑢 𝑟2

−

+ 𝑣 𝑟2

𝜕2 𝑢 𝜕𝑧 2

+

) − 𝜎𝐵02 𝑢

𝜌2 𝑣 𝜌𝑧 2

(2)

) − 𝜌𝐵02 𝑣

(3) 𝜌 (𝑢

𝜕𝑤 𝜕𝑟

+𝑤

𝜕𝑤 𝜕𝑧

) = 𝜇(

Impact Factor (JCC): 8.8746

𝜕2 𝑤 𝜕𝑟 2

+

1 𝜕𝑤 𝑟 𝜕𝑟

+

𝜕2 𝑤 𝜕𝑧 2

)

SCOPUS Indexed Journal

(4)

NAAS Rating: 3.11

MHD Nanofluid Flow and Heat Transfer due to Rotating Disk with Boundary Condition of Third Kind

1109

Conservation of Energy 𝑢

𝜕𝑇 𝜕𝑟

+ 𝑤

𝜕𝑇 𝜕𝑧

= 𝛼 (

𝜕2 𝑇 𝜕𝑟 2

+

1 𝜕𝑇 𝑟 𝜕𝑟

+

𝜕2 𝑇 𝜕𝑧 2

)+

(𝜌𝑐𝑝 )

𝑠

(𝜌𝑐𝑝 )𝑓

[𝐷𝐵

𝜕𝑐 𝜕𝑇 𝜕𝑧 𝜕𝑧

+

𝐷𝑇 𝜕𝑇 2 𝑇∞

( ) ]− 𝜕𝑧

1

𝜕𝑞𝑟

(𝜕𝑐𝑝 )

𝑓

𝜕𝑧

(5)

Conservation of volume concentration 𝑢

𝜕𝐶 𝜕𝑟

+𝑤

𝜕𝐶 𝜕𝑧

= [𝐷𝐵 (

𝜕2 𝐶 𝜕𝑟 2

+

1 𝜕𝐶 𝑟 𝜕𝑟

+

𝜕2 𝑐 𝜕𝑧 2

)+

𝐷𝑇 𝜕 2 𝑇 𝑇∞

(

𝜕𝑟 2

+

1 𝜕𝑇 𝑟 𝜕𝑟

+

𝜕2 𝑇 𝜕𝑧 2

)]

(6)

Subject to the boundary conditions

T  h(T  T ) 𝜌𝐶 𝑢 = 0, 𝑣 = 𝑣𝑤 = 𝑟𝜌, 𝑤 = 0, 𝑘 z , 𝐷𝐵 + 𝜌𝑧

𝜌𝑇 𝜌𝑇 𝑇∞ 𝜌𝑧

= 0, 𝑎𝑡 𝑧 = 0

(7)

𝑢 → 0, 𝑣 → 0, 𝑇 → 𝑇∞ , 𝐶 → 𝐶∞ , 𝑎𝑠 𝑧 → ∞ where u, v and w denote the velocities along the radial, azimuthal and vertical directions, respectively, T denotes the local nanofluid temperature, C the local concentration of nanoparticles, B 0 the magnetic field strength, μ the dynamic viscosity of nanofluid, (ρcp)s and (ρcp)f are the heat capacities of the nanoparticle material and the base fluid, respectively, α is the thermal diffusivity of nanofluid, h is the interphase heat transfer co-efficient, DB the coefficient of Brownian diffusion, and DT the thermophoretic diffusion coefficient.”

With the help of following dimensionless variables Ω

𝜂 = √ 𝑧, [𝑢, 𝑣, 𝑤] = [𝑟Ω𝐹(𝜂), 𝑟Ω𝐺(𝜂), √𝑣𝑓 Ω𝐻(𝜂)], 𝑣

(8)

𝑓

[𝑝, 𝑇, 𝐶] = [2Ω𝜇𝑃(𝜂), 𝑇∞ + (𝑇𝑤 − 𝑇∞ )𝜃(𝜂), 𝐶∞ (1 + 𝜙(𝜂))] The governing equations (1)-(6) with the boundary conditions (7) can be written as 𝐻′ + 2𝐹 = 0

(9)

𝐹 ′′ − 𝐻𝐹 ′ − 𝐹 2 + 𝐺 2 − 𝑀𝐹 = 0

(10)

𝐺 ′′ − 𝐻𝐺 ′ − 2𝐹𝐺 − 𝑀𝐺 = 0

(11) 2

𝜃 ′′ (1 + 𝑅𝑑) − Pr 𝐻𝜃 ′ + Pr 𝑁𝑏𝜃 ′ 𝜙 ′ + Pr 𝑁𝑡𝜃 ′ = 0 𝜙 ′′ − 𝑆𝑐𝐻𝜙 ′ +

𝑁𝑡 𝑁𝑏

(12)

𝜃 ′′ = 0

(13)

And the boundary conditions are 𝐻(0) = 0, 𝐹(0) = 1, 𝐺(0) = 1, 𝜃′(0) = 𝐵𝑖 ( (0)  1) , 𝑁𝑏𝜙 ′ (0) + 𝑁𝑡𝜃(0) = 0

(14)

𝐹(∞) = 0, 𝐺(∞) = 0, 𝜃(∞) = 0, 𝜙(∞) = 0

(15)

In which, the dimensionless physical parameters are magnetic number𝑀 =

𝜎𝑓 𝐵02 ⁄ , Prandtl number 𝑃𝑟 = 𝑣 ⁄𝛼 , 𝜌Ω

the Reynolds number 𝑅𝑒 = 𝑣𝑤 𝑟 ⁄𝑣, Brownian motion parameter 𝑁𝑏 = 𝜏𝐷𝐵 𝐶∞ ⁄𝑣, thermophoretic diffusion parameter www.tjprc.org

Bi 

h

 k 

SCOPUS Indexed Journal

editor@tjprc.org

1110

D. Prakash & Narsu Sivakumar

𝑁𝑡 = 𝜏𝐷𝑇 (𝑇𝑤 − 𝑇∞ )⁄𝑣𝑇∞ , the Schmidt number 𝑆𝑐 = 𝑣 ⁄𝐷𝐵 , the radiation parameter

Rd  16 1T

, Biot number

3

3K1k

The local skin friction co-efficient and the rate of heat transfer can be defined as follows” 𝑣

𝐶𝑓 = √ 2 (𝐹 ′ (0)2 + 𝐺 ′ (0)2 )1⁄2 , 𝑁𝑢 = −√ 𝑟 Ω

𝑟 2Ω 𝑣

𝜃 ′ (0), 𝑆ℎ = −√

𝑟 2Ω 𝑣

𝛷 ′ (0)

3. NUMERICAL METHODOLOGY In order to understand the physical nature of the problem, the coupled system of equations (9)-(13) along with the boundary conditions (14)-(15) was solved numerically by Runge-Kutta Fehlberg method along with shooting technique.

The detailed procedure adopted to solve the governing equations has been explained in our earlier publications [23-25].In this method, it is most important to choose the appropriate finite values of 

  . To select   , we begin with some

initial guess value and solve the problem with some particular set of parameters to obtain F ' (0), G ' (0),  (0),  (0). The solution process is repeated with another large values of η∞ until two successive values of F ' (0), G ' (0),  (0),  (0) differ only by the specified significant digit. The last value of η ∞ is chosenas appropriate value of the limit η → ∞ for that particular set of parameters.”

The coupled non-linear boundary value problem of second orders in F, G, θ, φ and firstorder in H has been reduced to a system of simultaneous equations of first order for nine unknowns. Thus, we set y1 = H, y2 = F, y3 = F′′, y4 = G, y5 = G′,y6 = θ, y7 = θ′, y8 = φ, y9 = φ′. The reducedsystem of equations is as follows:

And the corresponding initials are

Where δ1, δ2, δ3, δ4are determined such that it satisfies y2(∞) = 0, y4(∞) = 0, y6(∞) = 0, y8(∞) = 0. The shooting method is used to guess δ1, δ2, δ3, δ4 until the boundary conditions y2(∞) = 0, y4(∞) = 0, y6(∞) = 0, y8(∞) = 0 are satisfied.

Then the resulting differential equations can be integrated by fifth-order Runge-Kutta-Fehlberg integration scheme with step size h = 0.01. The above procedure is repeated until we get the results up to the desired degree of accuracy, 10−5. The detailed flowchart has been presented in APPENDIX -1 to visualize the sequence of steps and decisions needed to perform

Impact Factor (JCC): 8.8746

SCOPUS Indexed Journal

NAAS Rating: 3.11

MHD Nanofluid Flow and Heat Transfer due to Rotating Disk with Boundary Condition of Third Kind

1111

a numerical process.

4. RESULTS AND DISCUSSIONS Here, we discuss the numerical solution to Eq. (9)-(13) subject to boundary conditions Eq. (14)-(15) for selected values of the parameters. “In all the numerical results reported below, the default values to the physical parameters

( M  Rd  1.0, Pr  7.0, Nb  Nt  0.5, Sc  3.0, Bi  1) have been considered unless otherwise specified. For the sake of brevity, the flow and heat transfer characteristics with the Prandtl number has been neglected for the discussion.”

Figure 2: Transverse velocity for different values of magnetic parameter M.

(b) Figure 3: Axial velocity for different values of magnetic parameter M The influence of magnetic effect on the transverse and axial velocity distributions has been shown in the fig.2 and fig.3. The negative values of H(η) with various values of magneticparameter indicate that the downward fluid flow in www.tjprc.org

SCOPUS Indexed Journal

editor@tjprc.org

1112

D. Prakash & Narsu Sivakumar

verticaldirection. An increase in the magnetic parameter leads to increase the axial velocity due to the inclusion of magnetic force. On the other hand, the transverse velocity distribution decreases with an increase in the magneticparameter near to the disk, whereas the reverse trend has been found away from the disk.”

Fig. 4 and fig.5 illustrate that the effect of magneticparameter on the temperature and concentration distributions.

The temperature and concentration distribution increases with an increment in the magneticparameter due to anelectricallyconducting fluid flows through transverse magneticfield, which leads to reduce the convection and drop the heat transferrate at the disk.

Figures 6 and fig.7 portray that the temperature and concentration distribution with the thermal radiation. An increase in the thermalradiation is to increase the thermal boundary layerthickness and decrease in the temperature distribution. On the other hand, the non-uniform behaviour in the concentrationprofile has been observed with the increase in the radiation parameter.

Figure 4: Temperature profile for different values of magnetic parameter M.

Impact Factor (JCC): 8.8746

SCOPUS Indexed Journal

NAAS Rating: 3.11

MHD Nanofluid Flow and Heat Transfer due to Rotating Disk with Boundary Condition of Third Kind

1113

(b) Figure 5: Concentration profile for different values of magnetic parameter M

Figure 6: Temperature profile for different values of radiation parameter

(b) Figure 7: Concentration profile for different values of radiation parameter The influence of nanofluid parameters and convective heating parameter on the temperature and concentration www.tjprc.org

SCOPUS Indexed Journal

editor@tjprc.org

1114

D. Prakash & Narsu Sivakumar

profile has been plotted in fig.8 and fig.9. It is seen that with the fixed Biot number, an increase in the nanofluid parameters such as Brownian motion and thermophoresis parameters is to increase the thickness of the temperature profile. Further, the temperature distribution decreases which enhance the rate of heat transfer with the fixed value of nanofluid parameter.

From fig.9, the thickness of the concentration profile is decreased with an increase in the nanofluid parameters, whereas the reverse trend has been observed away from the disk.”Moreover, due to external hot fluid, the thermal and concentration profile gets accelerated near to disk, which is expected.

Figure 8. Temperature profile for different values of nanofluid parameters and Biot numbers

Figure 9: Concentration profile for different values of nanofluid parameters and Biot numbers

Impact Factor (JCC): 8.8746

SCOPUS Indexed Journal

NAAS Rating: 3.11

MHD Nanofluid Flow and Heat Transfer due to Rotating Disk with Boundary Condition of Third Kind

1115

5. CONCLUSIONS A numerical analysis is made to portray the boundary layer rotating flow of magneto-nanofluid with the boundary conditions of third kind. With the inclusion of novel Brownian diffusion and thermophoresis effects, it brings out that magnetic and thermal radiation fields have the key role to modify flow and heat transfer characteristics. The important remarks from this study are outlined below: An increase in the magnetic parameter is to increase the axial velocity, thermal and concentration profile thickness, whereas decrease the transverse velocity profile thickness. 

An increase in the combined Brownian motion and thermophoresis parameter is to increase the thermal boundary layer thickness and decrease the concentration layer thickness.”



External hot fluid enhances the thermal and concentration profile near to the disk.

REFERENCES 1.

Choi, S. U. S. (1995). Enhancing thermal conductivity of fluids with nanoparticles. In D. A. Siginer & H. P. Wang (Eds.), Developments applications ofnon-Newtonian flows (pp. 99–105). ASME Fluids Engineering Division.

2.

SK Das, SUS Choi, W Yu, T Pradeep (2007). Nanofluids: Science and Technology. New Jersey, Wiley.

3.

XQ Wang, AS Mujumdar (2008). A review on nanofluids - part I: theoretical and numerical investigations Braz J Chem Eng., 25, 613-630

4.

S Kakac, A Pramuanjaroenkij (2009). Review of convective heat transfer enhancement with nanofluids, Int. J. Heat and Mass Transfer. 52, 3187-3196.

5.

N. Bachok, A Ishak, I Pop (2011). Flow and heat transfer over a rotating porous disk in a nanofluid, Physica B: Condensed Matter. 406, 1767-1772

6.

M.M. Rashidi, S. Abelman, N. Freidooni Mehr, (2013). Entropy generation in steady MHD flow due to a rotating porous disk in a nanofluidInternational Journal of Heat and Mass Transfer.62, 515-525.

7.

M. Turkyilmazoglu, I. Pop (2013), Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect, International Journal of Heat and Mass Transfer. 59, 167-171

8.

Von Kármán T. Z Angew Math Mech. 1, 233–252 (1921).

9.

M. Turkyilmazoglu (2014),Nanofluid flow and heat transfer due to a rotating diskComputers & Fluids 94, 139–146

10. W.G. Cochran (1934),The flow due to a rotating discProc Camb Phil Soc. 30, 365–375.

11. M.G. Roger and G.N. Lance(1960),The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk, J Fluid Mech. 7, 617–631.

12. E.T. Benton(1966),On the flow due to a rotating disk J Fluid Mech. 24, 781–800.

13. R.J. Lingwood (1995),Absolute instability of the boundary layer on a rotating disk J Fluid Mech 299, 17–33 14. P.D. Ariel (2001),On Computation of MHD Flow Near a Rotating Disk Z Angew Math Mech 82, 235–246 15. T. Fang (2007),Flow over a stretchable disk Phys Fluids 19, 128105 16. M. Turkyilmazoglu (2010),Purely analytic solutions of magnetohydrodynamic swirling boundary layer flow over a porous

www.tjprc.org

SCOPUS Indexed Journal

editor@tjprc.org

1116

D. Prakash & Narsu Sivakumar rotating disk Comput Fluids 39, 793–799

17. M. Turkyilmazoglu (2012),MHD fluid flow and heat transfer due to a stretching rotating disk, Int J Therm Sci 51,195–201 18. M. Sheikholeslami, D. D. Ganji, M. Younus Javed, R. Ellahi, (2015) Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model, J. Mag. Magnetic Materials. 374, 36–43 19. M. Sheikholeslami, D. D. Ganji, M.M. Rashidi (2016), Magnetic field effect on unsteady nanofluid flow and heat transfer using Buongiorno model, J. Mag. Magnetic Materials, 416, pp 164–173.

20. A.V Kuznetsov, D.A. Nield, (2013)The Cheng–Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid: A revised model, Int J Heat Mass Transf 65, 682–685 21. D.A. Nield, A.V. Kuznetsov, (2014)Thermal instability in a porous medium layer saturated by a nanofluid: A revised model, Int J Heat Mass Transf 68, 211–214 22. D. Prakash, M. Muthtamilsevlan, D.H. Doh(2014).,Unsteady MHD non-Darcian flow over a vertical stretching plate embedded in a porous medium with non-uniform heat generation, Appl. Math. Computation, 236, 480–492 23. M. Muthtamilselvan, D. Prakash, D.H. Doh(2014).,Effect of thermal non-equilibrium on transient hydromagnetic flow over a moving surface in a nanofluid saturated porous media, J. Mech. Sci. Tech., 28, 3709–3718 24. M. Muthtamilselvan, D. Prakash (2015),Unsteady MHD Non-Darcian Flow Over a Vertical Stretching Plate Embedded in a Porous Medium with Thermal Non-Equilibrium Model, Proc. Inst. Mech. Eng., Part C: J. Mech. Engg. Sci., 229, 703–715.

25. D. Prakash, M. Muthtamilselvan, X.D. Niu (2016), Unsteady MHD Non-Darcian Flow Over a Vertical Stretching Plate Embedded in a Porous Medium with Thermal Non-Equilibrium Model, Adv. Appl. Math. Mech. 8, 52–66.

26. V. Murali Krishna, “Heat Transfer Enhancement by using CuO-Water Nanofluid in a Concentric Tube Heat Exchanger- an Experimental Study”, International Journal of Mechanical Engineering (IJME), Vol. 6, Issue 1, pp. 11-20 27. R. K. Dhal, Vorticity on MHD Free Convection Flow of Fluid with Heat Transfer through Porous Medium by an Oscillating Porous Plate in Slip Flow Region”, BEST: International Journal of Humanities, Arts, Medicine and Sciences (BEST: IJHAMS), Vol. 3, Issue 7, pp 77-82 28. Ezeh, J. C, Ibearugbulem, Owus M & Ebirim, Stanley I, “Fundamental Natural Frequency for Isotopic Rectangular Plate Simply Supported on Three Edges with One Edge Free of Support (SSSF Plate)”, IMPACT: International Journal of Research in Engineering & Technology (IMPACT: IJRET), Vol. 2, Issue 2, pp. 67-74 29. Amir Qashqaei & Ramin Ghasemi Asl, “Numerical Modeling and Simulation of Copper Oxide nanofluids Used in Compact Heat Exchangers”, International Journal of Mechanical Engineering (IJME), Vol. 4, Issue 2, pp. 1-8

APPENDIX-1 FLOWCHART FOR NUMERICAL METHODOLOGY

Impact Factor (JCC): 8.8746

SCOPUS Indexed Journal

NAAS Rating: 3.11

MHD Nanofluid Flow and Heat Transfer due to Rotating Disk with Boundary Condition of Third Kind

www.tjprc.org

SCOPUS Indexed Journal

1117

editor@tjprc.org