Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T20:12:47.421Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of Dual Solutions and Melting Phenomenon in Unsteady Nanofluid Flow and Heat Transfer over a Stretching Surface

Published online by Cambridge University Press:  14 April 2019

S. Ghosh ,
S. Mukhopadhyay  and
K. Vajravelu
S. Ghosh
Affiliation:
Department of Mathematics The University of Burdwan West Bengal, India
S. Mukhopadhyay
Affiliation:
Department of Mathematics The University of Burdwan West Bengal, India
K. Vajravelu*
Affiliation:
Department of Mathematics University of Central Florida Orlando, USA
*
*Corresponding author (Kuppalapalle.Vajravelu@ucf.edu)
Get access

Abstract

The problem of unsteady boundary layer flow of a nanofluid over a stretching surface is studied. Heat transfer due to melting is analyzed. Using a similarity transformation the governing coupled nonlinear partial differential equations of the model are reduced to a system of nonlinear ordinary differential equations, and then solved numerically by a Runge-Kutta method with a shooting technique. Dual solutions are observed numerically and their characteristics are analyzed. The effects of the pertinent parameters such as the acceleration parameter, the Brownian motion parameter, the thermophoresis parameter, the Prandtl number and the Lewis number on the velocity, temperature and concentration fields are discussed. Also the effects of these parameters on the skin friction coefficient, the Nusselt number and the Sherwood number are analyzed through graphs. It is observed that the melting phenomenon has a significant effect on the flow, heat and mass transfer characteristics.

Keywords

Melting phenomenonDual solutionsNanofluidUnsteady boundary layer
Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 5 , October 2019 , pp. 705 - 717
Copyright
© The Society of Theoretical and Applied Mechanics 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Vajravelu, K. and Mukhopadhayay, S., “Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes: Numerical Solutions,” Elsevier, Academic Press, Oxford (2015), ISBN .Google Scholar
Crane, L. J., “Flow past a Stretching Plate,” Zeitschrift fur Angewandte Mathematik und Physik ZAMP, 21, pp. 645–647 (1970).CrossRefGoogle Scholar
Magyari, E. and Keller, B., “Exact Solutions for Self-Similar Boundary-Layer Flows Induced by Permeable Stretching Walls,” Europian Journal of Mechanics B-Fluids, 19, pp. 109–122 (2000).CrossRefGoogle Scholar
Liao, S. J., “On the Analytic Solution of Magnetohydrodynamic Flows of Non-Newtonian Fluids over a Stretching Sheet,” Journal of Fluid Mechanics, 488, pp. 189–212 (2003).CrossRefGoogle Scholar
Cortell, R., “A Note on Flow and Heat Transfer of a Viscoelastic Fluid over a Stretching Sheet,” International Journal of Nonlinear Mechanics, 41, pp. 78–85 (2006).CrossRefGoogle Scholar
Hayat, T., Abbas, Z. and Said, M., “Heat and Mass Transfer Analysis on the Flow of a Second Grade Fluid in the Presence of Chemical Reaction,” Physics Letter A, 372, pp. 2400–2408 (2008).CrossRefGoogle Scholar
Sajid, M. and Hayat, T., “Influence of Thermal Radiation on the Boundary Layer Flow Due to an Exponentially Stretching Sheet,” International Communications in Heat and Mass Transfer, 35, pp. 347–356 (2008).CrossRefGoogle Scholar
Mukhopadhyay, S., “Heat Transfer Analysis for Unsteady MHD Flow past a Non-Isothermal Stretching Surface,” Nuclear Engineering and Design, 241, pp. 4835–4839 (2011).CrossRefGoogle Scholar
Choi, S. U. S. and Eastman, J. A., “Enhancing Thermal Conductivities of Fluids with Nanoparticles,” in: Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, 1995.Google Scholar
Khan, W. A. and Pop, P., “Boundary Layer Flow of a Nanofluid past a Stretching Sheet,” International Journal of Heat and Mass Transfer, 53, pp. 2477–2483 (2010).Google Scholar
Makinde, O. D. and Aziz, A., “Boundary Layer Flow of a Nanofluid past a Stretching Sheet with a Convective Boundary Condition,” International Journal of Thermal Sciences, 50, pp. 1326–1332 (2011).CrossRefGoogle Scholar
Buongiorno, J., “Convective Transport in Nanofluids,” ASME Journal of Heat Transfer, 128, pp. 240–250 (2006).CrossRefGoogle Scholar
Kuznetsov, A. V. and Nield, D. A., “Natural Con-vective Boundary Layer Flow of a Nonofluid past a Vertical Plate,” International Journal of Thermal Sciences, 49, pp. 243–247 (2010).CrossRefGoogle Scholar
Bachok, N., Ishak, A. and Pop, I., “Boundary Layer Flow of Nanofluids over a Moving Surface in a Flowing Fluid,” International Journal of Thermal Sciences, 49, pp. 1663–1668 (2010).CrossRefGoogle Scholar
Vajravelu, K., et al., “Convective Heat Transfer in the Flow of Viscous Ag-Water and Cu-Water Nanofluids over a Stretching Surface,” International Journal of Thermal Sciences, 50, pp. 843–851 (2011).Google Scholar
Rana, P. and Bhargava, R., “Finite Element Simulation of Transport Phenomena of Viscoelastic Nanofluid over a Stretching Sheet with Energy Dissipation,” Communications in Nonlinear Science and Numerical Simulation, 17, pp. 212–226 (2012).CrossRefGoogle Scholar
Rashidi, M. M., Vishnu Ganesh, N., Abdul Hakeem, A. K. and Ganga, B., “Buoyancy Effect on MHD Flow of Nanofluid over a Stretching Sheet in the Presence of Thermal Radiation,” Journal of Molecular Liquids, 198, pp. 234–238 (2014).CrossRefGoogle Scholar
Hayat, T. and Shehzad, S. A., “Thermally Stratified Radiative Flow of Third Grade Fluid over a Stretching Surface, “ Journal of Thermophysics and Heat Transfer, 28, pp. 155–161 (2014).CrossRefGoogle Scholar
Ghosh, S. and Mukhopadhyay, S., “Nanofluid Flow past an Exponentially Porous Stretching Sheet with Heat and Mass Fluxes,” Acta Technica, 61, pp. 17–29 (2016).Google Scholar
Chamkha, A. J., “Unsteady MHD Convective Heat and Mass Transfer past a Semi-Infinite Vertical Permeable Moving Plate with Heat Absorption,” International Journal of Engineering Science, 42, pp. 217–230 (2004).CrossRefGoogle Scholar
Ishak, A., Nazar, R. and Pop, I., “Boundary Layer Flow and Heat Transfer over an Unsteady Stretching Vertical Surface,” Meccanica, 44, pp. 369–375 (2009).CrossRefGoogle Scholar
Mahdy, A., “Unsteady Mixed Convection Boundary Layer Flow and Heat Transfer of Nanofluids Due to Stretching Sheet,” Nuclear Engineering and Design, 249, pp. 248–255 (2012).CrossRefGoogle Scholar
Makinde, O. D., “Chemically Reacting Hydromagnetic Unsteady Flow of a Radiating Fluid past a Vertical Plate with Constant Heat Flux,” Z Nature Forsch A, 67a, pp. 239–247 (2012).Google Scholar
Mukhopadhyay, S. and Bhattacharyya, K., “Unsteady Flow of a Maxwell Fluid over a Stretching Surface in Presence of Chemical Reaction,” Journal of Egyptian Mathematical Society, 20, pp. 229–234 (2012).CrossRefGoogle Scholar
Bhattacharyya, K., Mukhopadhyay, S. and Layek, G. C., “Unsteady MHD Boundary Layer Flow with Diffusion and First-Order Chemical Reaction over a Permeable Stretching Sheet with Suction or Blowing,” Chemical Engineering Communication, 200, pp. 379–397 (2013).CrossRefGoogle Scholar
Mukhopadhyay, S., De, P. R., Bhattacharyya, K. and Layek, G. C., “Casson Fluid Flow over an Unsteady Stretching Surface,” Ain Shams Engineering Journal, 4, pp. 933–938 (2013).CrossRefGoogle Scholar
Makinde, O. D. and Tshehla, M. S., “Unsteady Hydromagnetic Flow of Radiating Fluid past a Convectively Heated Vertical Plate with the Navier Slip,” Advances in Mathematical Physics, 973593(2014).Google Scholar
Daba, M. and Devaraj, P., “Unsteady Hydromagnetic Chemically Reacting Mixed Convection Flow over a Permeable Stretching Surface with Slip and Thermal Radiation,” Journal of the Nigerian Mathematical Society, 35, pp. 245–256 (2016).CrossRefGoogle Scholar
Mansur, S. and Ishak, A., “Unsteady Boundary Layer Flow of a Nanofluid over a Stretching/Shrinking Sheet with a Convective Boundary Condition,” Journal of Egyptian Mathematical Society, 24, pp. 650–655 (2016).CrossRefGoogle Scholar
Hayat, T., Shafiq, A. and Alsaedi, A., “Characteristics of Magnetic Field and Melting Heat Transfer in Stagnation Point Flow of Tangent Hyperbolic Liquid,” Journal of Magnetism and Magnetic Materials (2016), http://dx.doi.org/10.1016/jjrrmim.2015.10.080 CrossRefGoogle Scholar
Cheng, W. T. and Lin, C. H., “Melting Effect on Mixed Convective Heat Transfer with Aiding and Opposing External Flows from the Vertical Plate in a Liquid-Saturated Porous Medium,” International Journal of Heat Mass Transfer, 50, pp. 3026–3034 (2007).CrossRefGoogle Scholar
Ishak, A., Nazar, R., Bachok, N. and Pop, I., “Melting Heat Transfer in Steady Laminar Flow over a Moving Surface,” Heat and Mass Transfer, 46, pp. 463–468 (2010).CrossRefGoogle Scholar
Yacob, N. A., Ishak, A. and Pop, I., “Melting Heat Transfer in Boundary Layer Stagnation-Point Flow Towards a Stretching/Shrinking Sheet in a Micropolar Fluid,” Computer & Fluids, 47, pp. 16–21 (2011).Google Scholar
Hayat, T., Mustafa, M., Iqbal, Z. and Alsaedi, A., “Stagnation-Point Flow of Couple Stress Fluid with Melting Heat Transfer,” Applied Mathematics and Mechanics -English Edition, 34, pp. 167–176 (2013).CrossRefGoogle Scholar
Nawaz, M., Hayat, T. and Zeeshan, A., “Melting Heat Transfer in an Axi-Symmetric Stagnation-Point Flow of the Jeffrey Fluid,” Journal of Applied Mechanics and Technical Physics, 57, pp. 308–316 (2016).CrossRefGoogle Scholar
Kuznetsov, A. V. and Nield, D. A., “The Cheng-Minkowycz Problem for Natural Convective Boundary Layer Flow in a Porous Medium Saturated by a Nanofluid: a Revised Model,” International Journal of Heat and Mass Transfer, 65, pp. 682–685 (2013).CrossRefGoogle Scholar
Kuznetsov, A. V. and Nield, D. A., “Natural Con-vective Boundary-Layer Flow of a Nanofluid past a Vertical Plate: A Revised Model,” International Journal of Thermal Sciences, 77, pp. 126–129 (2014).CrossRefGoogle Scholar
Nield, D. A. and Kuznetsov, A. V., “Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: A Revised Model,” International Journal of Heat and Mass Transfer, 68, pp. 211–214 (2014).CrossRefGoogle Scholar
Nield, D. A. and Kuznetsov, A. V., “The Onset of Convection in a Horizontal Nanofluid Layer of Finite Depth: A Revised Model,” International Journal of Heat and Mass Transfer, 77, pp. 915–918 (2014).CrossRefGoogle Scholar
Bachok, N., Ishak, A. and Pop, I., “Unsteady Boundary-Layer Flow and Heat Transfer of a Nanofluid over a Permeable Stretching/Shrinking Sheet, “ International Journal of Heat and Mass Transfer, 55, pp. 2102–2109 (2012).CrossRefGoogle Scholar
Malvandi, A., Hedayeti, F., Ganji, D. D. and Rostamiyan, Y., “Unsteady Boundary-Layer Flow of Nanofluid past a Permeable Stretching/Shrinking Sheet with Convective Heat Transfer,” Proceedings IMechE Part C: Journal of Mechanical Engineering and Science, pp. 1–10 (2013).Google Scholar
Weidman, P. D., Kubitschek, D. G. and Davis, A. M. J., “The Effect of Transpiration on Self-Similar Boundary Layer Flow over Moving Surfaces,” International Journal of Engineering Science, 44, pp. 730–737 (2006).CrossRefGoogle Scholar
Postelnicu, A. and Pop, I., “Falkner-Skan Boundary Layer Flow of a Power-Law Fluid past a Stretching Wedge,” Applied Mathematics and Computation, 217, pp. 4359–4368 (2011).CrossRefGoogle Scholar