Rotating Flow of Magnetite-Water Nanofluid over a Stretching Surface Inspired By Non-Linear Thermal Radiation and Mass Transfer

Transcription

1 International Journal o Mathematics Research. ISSN Volume 9, Number (017), pp International Research Publication House Rotating Flow o Magnetite-Water Nanoluid over a Stretching Surace Inspired By Non-Linear Thermal Radiation and Mass Transer Dr. Anuradha, S. and Priya, M., * Proessor & Head in Department o Mathematics, Hindusthan College o Arts & Science, Coimbatore-64108, Tamilnadu, India. *Research Scholar in Department o mathematics, Hindusthan College o Arts & Science, Coimbatore-64108, Tamilnadu, India. Abstract We have investigated MHD three dimensional rotating low o magnetitewater nanoluid over a stretching surace inspired by non-linear thermal radiation and mass transer. In this problem, the Maxwell-Garnett model or eective thermal conductivity and electrical conductivity o nanoluid are taken into account. A mathematical ormulation has designed or velocity, momentum, temperature and concentration proiles. The governing partial dierential equations are reduced to a system o ordinary dierential equations using Shooting technique together with Runge -Kutta sixth order iteration scheme. The numerical results o the low characteristics are presented graphically. Keywords: Nanoluid, Magneto hydrodynamics (MHD), Magnetic parameter (M), ratio rate (λ), Radiation parameter (Rd), Thermal Radiation, Mass transer. 1. INTRODUCTION Research o Nanoluid have more attention in various scientiic and engineering applications. Some actual and potential applications are transportation, deense,

2 90 Dr. Anuradha, S. and Priya, M., nuclear, space, and biomedical. It is being observed that the rotating low o magnetite water nanoluid over a stretching surace inspired by non-linear thermal radiation and mass transer has got remarkable importance in every concern ield. Rotation low theory is helpul in determining the viscosity o the luid. In this study, we have observed that An unsteady MHD boundary layers in a rotating low investigated by Debnath [1]. Wang [] reported three dimensional low due to a stretching sheet being stretched in two lateral directions and rotate like a rigid body. Takhar et al. [3] considered an unsteady low over a stretching surace with a magnetic iled in a rotating luid. Sheikholeslami et al. [4] studied the nanoluid low, erroluid low and heat transer in a rotating system in the presence o magnetic ield. In this study, they ound that an increase in the magnetic ield, volume reaction o nanoparticle decreases the velocity o the luid. Mahmood et al. [5] investigated analytical treatment o three dimensional MHD low due to a stretching surace in a rotating luid. Sheikholeslami and Ganji [6] analyzed Eect o non-uniorm magnetic ield on orced convection heat transer o Fe3O4- water nanoluid. Wahiduzzaman et al. [7] established Viscous Dissipation and Radiation eects on MHD Boundary Layer Flow o a Nanoluid past a Rotating Stretching Sheet. Jawad and Azizah [8] examined MHD three dimensional low o a nanoluid in a rotating channel. Mahanthesh et al. [9] Discussed the iluence o non-linear thermal radiation on three dimensional steady low o a nanoluid past a non linear stretching in the presence o soret and duours eect. Mustaa et al. [10] investigated the Rotating low o Magnetite-Water Nanoluid over a Stretching Surace inspired by Non-linear thermal radiation. Apart rom the existence, eect o mass transer is studied and Shooting technique together with Runge -Kutta sixth order iteration scheme is used to analyze the research problem numerically.. MATHEMATICAL MODEL Consider the three dimensional stretching o a surace in a rotating luid. Let (u, v, w) be the velocity components in the direction o Cartesian axes (x, y, z) respectively, with the axes rotating at an angular velocity in the z- direction. The Magnetic ield B is imposed in the z- direction. Since the low is induced by stretching the surace 0 in the x- direction with rate a. At a constant temperature Tw whereas T denotes the temperature outside the thermal boundary layer (Fig.1). C,C are concentrations at the surace and ar away rom the sheet, the equations embodying the conservation o mass, momentum and energy, species are expressed as below: u v w 0 x y z u u u u v w v u B0 u x y z z w (1) ()

3 Rotating Flow o Magnetite-Water Nanoluid over a Stretching Surace Inspired 91 v v v u v w u v B0 v x y z z T T T T 1 qr Q u v w T T x y z z z C C C C u v w D k(c C ) x y z z cp cp (3) (4) (5) With the boundary conditions u uw ax v 0 w 0 T T at z 0 w u 0 v 0 T T as z (6) In which u and v are the velocity components along the x * 4 respectively, 4 T q r 3k * z is the Roseland radiative heat lux in which and y directions * is the * Stean- Boltzmann constant and k is the mean absorption coeicient. The dynamic viscosity o nanoluid as 1.5 (7) The eective density and eective heat capacity c p 1 s are expressed as (8) cp 1 cp cp (9) s We take into account the Maxwell-Garnett model or eective thermal conductivity o nanoluid k given below ks k k ks k k k k k k s s (10)

4 9 Dr. Anuradha, S. and Priya, M., Moreover the electrical conductivity o nanoluid 3 s 1 s s In equations (7)-(11), denotes the nanoparticle volume raction and the subscripts s and correspond to the solid and luid phases respectively. We look or similarity solution o equations (1)-(5) in the ollowing orms is (11) a ' z u ax v axg v w av T T Tw T c c c c w (1) In view o the above quantities, the continuity Equation (1) is identically satisied while Equations ()-(5) become 1.5 s s ''' ' '' ' g M '' ' ' ' g g g Mg 0.5 s s '' Q ' k k Rd 1 w 1 0 c Pr k p 1 s cp '' ' Sc Sc 0 (13) (14) (15) (16) ' 0 0 g ' 0 g 0 ( ) 0 (17) Where, cp is the Prandtl number o the base luid, Pr the Radiation parameter, k B0 M Rd is the magnetic ield parameter and 3 16 T denotes 3kk a is the

5 Rotating Flow o Magnetite-Water Nanoluid over a Stretching Surace Inspired 93 ratio o rotation rate to the stretching rate. chemical reaction parameter. Sc v is Schmidt Number and k D is a 3. NUMERICAL ANALYSIS In this study the set o Non-dimensional Non-linear couple boundary layer equations with boundary conditions does not possess a closed orm analytical solution. The governing partial dierential equations can be converted to closed orm equations by using Shooting method then it has been solved numerically by Runge-Kutta sixth order integration technique. The entire numerical analysis is done by using Mathematica computer language. From the process o numerical computation the luid velocity, the temperature, the concentration, the Skin riction coeicient, the Nusselt number and Sherwood number are proportional to ' ( ), ( ), ( ), " ( ), ' ( ), ' ( ). 4. RESULTS AND DISCUSSION The eect o Magnetic ield parameter (M) on the velocity, temperature and concentration proiles are shown in ig.4.1, 4., 4.3.It is observed rom these igures the velocity, concentration ield decreases while the temperature ield increase with the increase o Magnetic parameter (M).The eects o various parameters on the components o Skin riction ( ), the Nusselt number (Nu) and the Sherwood number (Sh) are shown in ig.4.4, 4.5, 4.6. It is observed rom these igures the Skin riction ( ), Nusselt number (Nu) increases while the Sherwood number (Sh) decrease with the increase o Magnetic parameter (M). The eect o Ratio rate ( ) on the velocity, temperature and concentration proiles are shown in ig.4.7, 4.8, 4.9.It is observed rom these igures the velocity, concentration ield decreases while the temperature ield increase with the increase o ratio rate (λ).the eects o various parameters on the components o Skin riction ( ), the Nusselt number (Nu) and the Sherwood number (Sh) are shown in ig.4.10, 4.11, 4.1. It is observed rom these igures the Skin riction coeicient ( ), Nusselt number (Nu) increases while the Sherwood number (Sh) decrease with the increase o ratio rate (λ). The eect o Radiation parameter (Rd) on the velocity, temperature and concentration proiles are shown in ig.4.13, 4.14, 4.15.It is observed rom these igures the velocity, concentration ield decreases while the temperature ield increase with the increase o Radiation parameter (Rd).The eects o various parameters on the components o Skin riction coeicient ( ), the Nusselt number (Nu) and the

6 94 Dr. Anuradha, S. and Priya, M., Sherwood number (Sh) are shown in ig.4.16, 4.17, It is observed rom these igures the Skin-Friction coeicient ( ), Nusselt number (Nu) increases while the Sherwood number (Sh) decrease with the increase o radiation parameter (Rd). The eect o Magnetic ield parameter (M): The eect o Ratio rate ( ):

7 Rotating Flow o Magnetite-Water Nanoluid over a Stretching Surace Inspired 95 The eect o Radiation parameter (Rd): 5. CONCLUSION In this study, the Rotating low o Magnetite-water Nanoluid over a Stretching Surace inspired by Non-linear thermal Radiation and Mass transer.the governing equations and boundary conditions are reduced to ordinary dierential equations using Shooting technique together with Runge-Kutta sixth order iteration scheme. The velocity, concentration ield decreases while the temperature ield increase with the increase o Magnetic parameter (M).

8 96 Dr. Anuradha, S. and Priya, M., The Skin riction ( ), Nusselt number (Nu) increases while the Sherwood number (Sh) decrease with the increase o Magnetic parameter (M). The velocity, concentration ield decreases while the temperature ield increase with the increase o ratio rate (λ). The Skin riction coeicient ( ), Nusselt number (Nu) increases while the Sherwood number (Sh) decrease with the increase o ratio rate (λ). The velocity, concentration ield decreases while the temperature ield increase with the increase o Radiation parameter (Rd). The Skin riction coeicient ( ), Nusselt number (Nu) increases while the Sherwood number (Sh) decrease with the increase o radiation parameter (Rd). REFERENCES [1] Debnath, L., (1975), On unsteady magneto hydrodynamics boundary layers in a rotating low, Z.Angew.Math.Mechanics, 5(10), pp [] Wang, C.Y., (1984), The three dimensional low due to a stretching lat surace, Phy. Fluids, 7, pp [3] Takhar, H. S., and Nath, G., (1998), Unsteady low over a stretching surace with a magnetic iled in a rotating luid, Math. Physics, 49(6), pp [4] Sheikholeslami, M., Hatami, M., and Ganji, D., (014), Nanoluid low and heat transer in a rotating system in the presence o a magnetic ield, J. Mol. Liq, 190, pp [5] Mahmood, T., Ali, S., and Khan, M. A., (014), Magnetohydrodynamic Flow due to a Stretching Surace in Rotating Fluid, J. Math, 46, pp [6] Sheikholeslami, M., Rashidi, M. M., and Ganji, D., (015), Eect o nonuniorm magnetic ield on orced convection heat transer o Fe3O4- water nanoluid, Comput. Methods. Appl. Mech. Engg, 94, pp [7] Wahiduzzaman, M., Khan, M. S., Karim, I., Biswas,P., and Uddin, M. S., (015), Viscous Dissipation and Radiation eects on MHD Boundary Layer Flow o a Nanoluid past a Rotating Stretching Sheet, Appl. Math, 6(3), pp [8] Jawad Raza., Azizah Mohd Rohni., Zurni Omar., and Muhammad Awais.,(016), Heat and mass transer analysis o MHD nanoluid low in a

9 Rotating Flow o Magnetite-Water Nanoluid over a Stretching Surace Inspired 97 rotating channel with slip eects, J. mol. liq,19, pp [9] Mahanthesh., Gireesha,B.J., Gorla., and Rama Subba Reddy.,(016), Nanoparticles Eect on 3D Flow, Heat and Mass Transer o Nanoluid with Nonlinear Radiation, Thermal-Diusion and Diusion-Thermo Eects, J. Nano,5(5), pp [10] Mustaa, M., Mushtaq, A., Hayat, T., and Alsaedi, A., (016), Rotating low o Magnetite- water nanoluid over a stretching surace inspired by nonlinear thermal radiation, PLOSONE 11():e

10 98 Dr. Anuradha, S. and Priya, M.,