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Dual Solutions in Hydromagnetic Viscous Fluid Flow Past a Shrinking Curved Surface

  • Research Article - Mechanical Engineering
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Abstract

An electrically directing viscous fluid’s boundary layer flow on a curved shrinking sheet with magnetic field is considered. Curvilinear coordinates system is used for mathematical modeling of the flow equations. By making use of similarity variables, the developed partial differential equations are reduced into sets of differential equations. Then, the attained differential equations describing the flow phenomena are resolved numerically through employing shooting scheme. The impacts of different pertinent factors, namely curvature, magnetic and suction parameters on velocity and pressure distribution, are shown graphically and are observed that dual-type solutions occur on a specific range of physical parameters. It is also noticed from these results that the flow velocity and the pressure within the boundary layer region are considerably affected by the shrinking and the mass transfer parameter since the pressure is no more constant in curved shrinking surface, as noticeable from the flat shrinking surface. Comparison of the numerical solution for the dual solution between the present studies with the existing solution is noticed in excellent agreement.

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Abbreviations

b :

Constant

\(B_0 \) :

Applied magnetic field

f :

Dimensionless fluid velocity in r-directions

\({f}'\) :

Dimensionless fluid velocity in s-directions

p :

Pressure of the fluid

P :

Dimensionless pressure of the fluid

M :

Dimensionless magnetic parameter

\(v_w\) :

Suction velocity

R :

Radius of the semicircle

S :

Dimensionless suction parameter

\(S_c\) :

Critical values of the suction parameter

\(U_w\) :

Uniform velocity

u :

Velocity component in the s-direction

v :

Velocity component in the r-direction

\(\alpha \) :

Constant

\(\beta \) :

Shrinking parameter

\(\rho \) :

Density of the fluid

\(\mu \) :

Viscosity of the fluid

\(\nu \) :

Kinematics viscosity

\(\eta \) :

Dimensionless variable

\(\sigma \) :

Electrical conductivity

\(\kappa \) :

Dimensionless radius of curvature

\(\tau _{rs}\) :

Shear stress at the surface

References

  1. Miklavcic, M.; Wang, C.Y.: Viscous flow due to a shrinking sheet. Q. Appl. Math. 64, 283–290 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Yao, B.; Wu, G.: Approximate analytical multiple solutions of the boundary layer flow over a shrinking sheet with power law velocity. Int. J. Heat Mass Transf. 85, 615–619 (2015)

    Article  Google Scholar 

  3. Bhatti, M.M.; Shahid, A.; Rashidi, M.M.: Numerical simulation of fluid flow over a shrinking porous sheet by successive linearization method. Alex. Eng. J. 55(1), 51–56 (2016)

    Article  Google Scholar 

  4. Bhattacharyya, K.: Effects of heat source/sink on MHD flow and heat transfer over a shrinking sheet with mass suction. Chem. Eng. Res. Bull. 15(1), 12–17 (2011)

    Google Scholar 

  5. Rohni, A.M.; Ahmad, S.; Ismail, AIMd; Pop, I.: Boundary layer flow and heat transfer over an exponentially shrinking vertical sheet with suction. Int. J. Therm. Sci. 64, 264–272 (2013)

    Article  Google Scholar 

  6. Singh, G.; Chamkha, A.J.: Dual solution for second order slip flow and heat transfer on a vertical permeable shrinking sheet. Ain Shams Eng. J. 4, 911–917 (2013)

    Article  Google Scholar 

  7. Saleh, S.H.M.; Arifin, N.M.; Nazar, R.; Ali, F.M.; Pop, I.: Mixed convection stagnation flow towards a vertical shrinking sheet. Int. J. Heat Mass Transf. 73, 839–848 (2014)

    Article  Google Scholar 

  8. Rosca, N.C.; Pop, I.: Boundary layer flow past a permeable shrinking sheet in a micropolar fluid with second order slip flow. Eur. J. Mech. B/Fluids 48, 115–122 (2014)

    Article  MathSciNet  Google Scholar 

  9. Shafique, M.; Abbas, F.: Numerical solutions of MHD viscous flow of Newtonian fluid due to a shrinking sheet by SOR iterative procedure. Univ. J. App. Math. 3(1), 1–6 (2015)

    Google Scholar 

  10. Mahapatra, T.R.; Sidui, S.: An analytical solution of MHD flow of two viscoelelastic fluid over a sheet shrinking with quadratic velocity. Alex. Eng. J. 55, 163–168 (2016)

    Article  Google Scholar 

  11. Rosca, A.: MHD boundary layer flow over a permeable shrinking surface. Acta Univ. Apulensis 36, 31–38 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Nandy, S.K.; Pop, I.: Effect of magnetic field and thermal radiation on stagnation flow and heat transfer of nanofluid over a shrinking sheet. Int. Commun. Heat Mass Transf. 53, 50–55 (2014)

    Article  Google Scholar 

  13. Mishra, S.R.; Jena, S.: Numerical Solution of Boundary Layer MHD Flow with Viscous Dissipation, p. 756498. Hindawi Publishing Corporation, Cairo (2014)

    Google Scholar 

  14. Jain, S.; Choudhary, R.: Effect of MHD on boundary layer flow in porous medium due to exponentially shrinking sheet with slip. Proc. Eng. 127, 1203–1210 (2015)

  15. Moparthi, A.; Das, R.; Uppaluri, R.; Mishra, S.C.: Optimization of heat fluxes on the heater and the design surfaces of a radiating–conducting medium. Numer. Heat Transf. Part A Appl. 56(10), 846–860 (2009)

  16. Das, R.: Forward and inverse solutions of a conductive, convective and radiative cylindrical porous fin. Energy Conver. Manag. 87, 96–106 (2014)

    Article  Google Scholar 

  17. Das, R.; Mishra, S.C.; Ajith, M.; Uppaluri, R.: An inverse analysis of a transient 2-D conduction-radiation problem using the lattice Boltzmann method and the finite volume method coupled with the genetic algorithm. J. Quant. Spectrosc. Radiat. Transf. 109(11), 2060–2077 (2008)

    Article  Google Scholar 

  18. Sajid, M.; Iqbal, S.A.; Naveed, M.; Abbas, Z.: Joule heating and magnetohydrodynamic effects on ferrofluid flow in a semi porous curved channel. J. Mol. Liquds 222, 1115–1120 (2016)

    Article  Google Scholar 

  19. Patel, D.K.; Das, M.K.; Roy, S.: LES of incompressible turbulent flow inside a cubical cavity driven by two parallel lids moving in opposite direction. Int. J. Heat Mass Transf. 67, 1039–1053 (2013)

    Article  Google Scholar 

  20. Ajith, M.; Das, R.; Uppaluri, R.; Mishra, S.C.: Boundary surface heat fluxes in a square enclosure with an embedded design element. J. Thermophys. Heat Transf. 24(4), 845–849 (2010)

    Article  Google Scholar 

  21. Das, R.: Feasibility study of different materials for attaining similar temperature distributions in a fin with variable properties. Proc. Inst. Mech. Eng. Part E J. Process Mech. Eng. 230(4), 292–303 (2016)

    Article  Google Scholar 

  22. Sajid, M.; Ali, N.; Javed, T.; Abbas, Z.: Stretching a curved surface in a viscous fluid. Chin. Phys. Lett. 27(2), 024703 (2010)

    Article  Google Scholar 

  23. Abbas, Z.; Naveed, M.; Sajid, M.: Heat transfer analysis for stretching flow over curved surface with magnetic field. J. Eng. Thermophys. 22(4), 337–345 (2013)

    Article  Google Scholar 

  24. Abbas, Z.; Naveed, M.; Sajid, M.: Hydromagnetic slip flow of nanofluid over a curved stretching surface with heat generation and thermal radiation. J. Mol. Liq. 215, 756–762 (2016)

    Article  Google Scholar 

  25. Naveed, M.; Abbas, Z.; Sajid, M.: MHD flow of a micropolar fluid due to a curved stretching sheet with thermal radiation. J. Appl. Fluid Mech. 9(1), 131–138 (2016)

    Article  Google Scholar 

  26. Naveed, M.; Abbas, Z.; Sajid, M.: Hydromagnetic flow over an unsteady curved stretching surface. Eng. Sci. Technol. Int. J. 19, 841–845 (2016)

    Article  Google Scholar 

  27. Rosca, N.C.; Pop, I.: Unsteady boundary layer flow over a permeable curved stretching/shrinking surface. Eur. J. Mech. B/Fluids 51, 61–67 (2015)

    Article  MathSciNet  Google Scholar 

  28. Turkyilmazoglu, M.: Multiple solutions of hydromagnetic permeable flow and heat for viscoelastic fluid. J. Thermophys. Heat Transf. 4, 595–605 (2011)

    Article  Google Scholar 

  29. Zaimi, K.; Ishak, A.; Pop, I.: Flow past a permeable stretching/shrinking sheet in a nano fluid using two phase model. PLoS ONE 3 (2014)

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Authors and Affiliations

  1. Department of Basic Sciences and Humanities, Khwaja Fareed University of Engineering and Information Technology Rahim Yar Khan, Rahim Yar Khan, 62400, Pakistan

    M. Naveed

  2. Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan

    Z. Abbas

  3. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    M. Sajid

  4. Department of Computer Science, Bahria University Islamabad Campus, Islamabad, 44000, Pakistan

    J. Hasnain

Authors
  1. M. Naveed
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  2. Z. Abbas
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  3. M. Sajid
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  4. J. Hasnain
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Correspondence to M. Naveed.

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Naveed, M., Abbas, Z., Sajid, M. et al. Dual Solutions in Hydromagnetic Viscous Fluid Flow Past a Shrinking Curved Surface. Arab J Sci Eng 43, 1189–1194 (2018). https://doi.org/10.1007/s13369-017-2772-z

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  • DOI: https://doi.org/10.1007/s13369-017-2772-z

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