Heat transfer of magnetohydrodynamic flow over a convectively heated cylinder with porous medium

A. Mahesh1, P. Durga Prasad2, C.S.K. Raju3*, P. Prakash4, S.V.K. Varma5

1Research Scholar, Department of Mathematics, S.V. University, Tirupati.

2BS&H (Mathematics), Sree Vidyanikethan Engineering College, A. Rangampet, Tirupati, A.P.

3Department of Mathematics, GITAM University, Bangalore Campus, K.A.

4Research Scholar, Department of Mechanical Engineering, NIT Warangal, Warangal (Telangana),India

5Professor, Department of Mathematics, S.V. University, Tirupati.

*Corresponding Author E-mail:  sivaphd90@gmail.com

Abstract:

A theoretical analysis performed for investigating steady boundary layer flow of convective conditions on magnetohydrodynamic flow over stretched cylinder with time dependent heat source/sink. Proposed mathematical model has a tendency to characterize the effect of magnetohydrodynamic flow over stretched cylinder wit time dependent heat source/sink. The non-linear ordinary differential equations are solved using the Runge-Kutta method. The characteristics of velocity and temperature boundary layers for different physical parameters such as Biot number, Reynolds number, space dependent and temperature dependent heat source/sink, Weissennberg numberover the non-dimensional velocity and temperature profiles are presented graphically in absence and presence of porosity parameter. Moreover, the local friction factor coefficient, Nusselt numbers are also estimated and discussed for aforesaid physical parameters.

KEY WORDS: Stretching cylinder, magnetohydrodynamic, Reynolds number, Biot number, MHD.

1.             INTRODUCTION:

Non-Newtonian fluids are imperative in boundary layer flows because of their engineering and technology related applications. Paints, ketchup, polymeric liquids, apple sauce, jellies, tomato sauce, glues, soaps, blood, cosmetic products are examples of non-Newtonian fluids. As there are diversity of non-Newtonian fluids so it is reasonably complicated to form an equation expressing the viscous and elastic properties of these fluids. In comparison to viscous fluids, mathematical modeling of non-Newtonian fluids is considerably complex and challenging. The Navier-Stokes expressions are proved inadequate for examining rheological characteristics of non-Newtonian materials. Generally the differential, integral and rate types classification have been made in this direction. Further, the phenomenon of stratification is very important in many engineering and manufacturing processes involving non-Newtonian materials. It arises in the flow fields due to change in concentration differences and temperature or fluids with different densities. Double stratification occurs through simultaneous effects of heat and mass transfer. The relevant examples include stratification of lakes and oceans, salinity stratification in rivers, ground water reservoirs, industrial, heterogeneous mixtures in atmosphere, manufacturing food processes and several others. Density differences through gravity are useful for dynamics and involvement of heterogeneous fluid. For instance in lakes, thermal stratification diminishes the blending of oxygen to the base water to develop anoxic from the effort of biological methods. The biomedical aspects are pathology, ophthalmological, public health and nursing etc. The Stratification has a key role in ponds and lakes since it influences the difference between concentration and temperature of oxygen and hydrogen.

Crane [1] was first to introduce the boundary layer flow of viscous fluid through a stretching sheet. Further, Cortell [2] proposed the effects of suction or blowing and heat generation or absorption through porous a stretching surface. Ibrahim and Makinde [3] explained the MHD stagnation point flow and heat transfer of Casson nanofluid past a stretching sheet with partial slip and convective boundary layer condition. Ibrahim and Makinde [4] illustrated that the MHD stagnation point flow of a power-law nanofluid towards a convectively heated stretching sheet with slip. Nadeem and Akram [5] studied theperistaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. In another paper Nadeem and Akram [6] have presented the effects of partial slip on the peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. Nadeem et al. [7] examined the boundary layer flow of second grade fluid in a cylinder with heat transfer. Nadeem et al. [8] studied the axisymmetric stagnation flow of a micropolar nanofluid in a moving cylinder.

Many researchers are contributed on these fluids highlighting the different aspects Waqas et al. [9] explained the MHD mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition.Dulal Pal and Mandal [10] portrayed the magnetohydrodynamic heat transfer of nanofluids past a stretching cylinder with non-uniform heat source/sink and chemical reaction. Hayat et al. [11] investigated the magnetohydrodynamic flow of Burgers fluid with heat source and law heat flux. Hayat et al. [12] studied the MHD three-dimensional nonlinear convective flow of viscoelastic nanofluid with heat and mass flux conditions.Vijaya Lakshmi and Suryanarayana Reddy [13] discussed the effect of radiation on mixed convection flow of a non-Newtonian nanofluid over a non-linearly stretching sheet with heat source/sink. Raju et al. [14] analysed the transpiration effects on MHD Flow over a stretched cylinder with Cattaneo-Christov heat flux with suction or injection.Naseer et al. [15] reported the hyperbolic tangent fluid is used extensively for different laboratory experiments. Naseer et al. [16] have explained the boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder.

In view of these facts the present study focuses on the numerical investigation of two dimensional boundary layer flows of convective conditions on magnetohydrodynamic flow over stretched cylinder with time dependent heat source/sink. The boundary layer equations given as a set of partial differential equations (PDEs) are first changed into non-linear ordinary differential equations (ODEs) ahead being solved numerically via Runge-Kutta-Fehlberg integration method. The effects of the governing flow parameters on the velocity and temperature profiles are discussed and presented in table and graphs.

2. FLUID MODEL:

3. FORMULATION:

4. SOLUTION OF THE PROBLEM:

5. RESULTS AND DISCUSSION:

6. CLOSING REMARKS:

The present computational results characterize the effects of steady boundary layer flow steady boundary layer flow of convective conditions on magnetohydrodynamic flow over stretched cylinder with time dependent heat source/sink. Proposed mathematical model has a tendency to characterize the effect of magnetohydrodynamic flow over stretched cylinder heat source/sink. The resultant non-linear governing partial differential equations (PDEs) are solved using the robust Rune-Kutta method. Based on the present computational investigation the following observations are made:

·      It is observed that heat transfer rate increases with increase in Biot number.

·      The Weissennberg number enhances the skin friction factor.

·      The space and temperature dependent parameter improves the local skin friction factor.

7. REFERENCES:

1.         Crane L. Flow past a stretching plate, Z. Angew. Math .Phy, 1970; 21:p.645-647.

2.         Cortell R.  Flow and Heat transfer of fluid through a pours medium over a stretching sheet with internal heat generation/absorption suction/blowing, Fluid Dyn. Res. 2005; 37:p.231-245.

3.         Ibrahim W.  Makinde O. D.  Magnetohydrodynamic stagnation point flow and heat transfer of Casson nanofluid past a stretching sheet with slip and convective boundary condition. Journal of Aerospace Engineering, 2016; 29: 04015037.

4.         Ibrahim W.  Makinde O.D.  Magnetohydrodynamic stagnation point flow of a power-law nanofluid towards a convectively heated stretching sheet with slip. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering. 2016; 230: p. 345-354.

5.         Nadeem S. Akram S.  Peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel, Z. Naturforsch. 2009;64: p. 559–567.

6.         Nadeem S. Akram S.  Effects of partial slip on the peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel, Int. J. Numer. Methods Fluids. 2010; 63:p. 374-394.

7.         Nadeem S. Rehman A. Lee C.  Lee J. Boundary layer flow of second grade fluid in a cylinder with heat transfer, Math. Prob. Eng. 2012;212:doi.org/10.1155/2012/640289.

8.         Nadeem S. Rehman A. Vajravelu K . Lee J. Lee C. Axisymmetric stagnation flow of a micropolar nanofluid in a moving cylinder, Math. Prob. Eng. 2012: 2012: doi.org/10.1155/2012/378259.

9.         Waqas M. Farooq M. Khan MI.  Alsaedi A. Hayat T. Yasmeen T. Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition. International Journal of Heat and Mass Transfer.2016; 102:p. 766–772

10.       Pal D. Mandal G. Magnetohydrodynamic Heat Transfer of Nanofluids Past a Stretching Cylinder with Non-Uniform Heat Source/Sink and Chemical Reaction, Int. J. Appl. Comput. Math. DOI 10.1007/s40819-016-0241-0.

11.       Hayat T. Waqas M. Ijaz Khan M. Alsaedi A.  Shehzad S.A. Magnetohydrodynamic flow of Burgers fluid with heat source and law heat flux. 10.1016/j.cjph.2017.02.00.

12.       Hayat T. Qayyum S.  Shehzad SA.  Alsaedi A.  Magnetohydrodynamic three-dimensional nonlinear convective flow of viscoelastic nanofluid with heat and mass flux conditions, Neural Comput and Applic. DOI 10.1007/s00521-017-3129-y.

13.       Vijaya Lakshmi S. Suryanarayana Reddy. Effect of Radiation on Mixed Convection Flow of a Non-Newtonian Nanofluid over a Non-Linearly Stretching Sheet with Heat Source/Sink, International Journal of Modern Engineering Research. 2013;3(5): pp-2675-2696.

14.       Raju CSK. Kiran Kumar RVMSS. Varma SVK.. Madaki AG. Durga Prasad P. Transpiration Effects on MHD Flow Over a Stretched Cylinder with Cattaneo-Christov Heat Flux with Suction or Injection. Arab J Sci Eng, DOI 10.1007/s13369-017-2687-8.

15.       Naseer M.  Malik MY, Rehman A. Numerical study of convective heat transfer on the power law fluid over a vertical exponentially stretching cylinder, Applied and Computational Mathematics. 2015; 4(5): p.346-350. doi: 10.11648/j.acm.20150405.13.

16.       Naseer M. Yousaf Malik M.  Nadeem S. Rehman A. The boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder, Alexandria Engineering Journal. 2014; 53:p. 747-750.

 Received on 08.09.2017       Modified on 14.10.2017 Accepted on 05.12.2017      ©A&V Publications All right reserved Research J. Science and Tech. 2018; 10(1):83-90. DOI: 10.5958/2349-2988.2018.00012.8