Home Hydromagnetic effects on non-Newtonian Hiemenz flow
Article
Licensed
Unlicensed Requires Authentication

Hydromagnetic effects on non-Newtonian Hiemenz flow

  • Suman Sarkar EMAIL logo and Bikash Sahoo
Published/Copyright: September 3, 2021
Journal of Applied Analysis
From the journal Journal of Applied Analysis Volume 28 Issue 1

Abstract

The stagnation point flow of a non-Newtonian Reiner–Rivlin fluid has been studied in the presence of a uniform magnetic field. The technique of similarity transformation has been used to obtain the self-similar ordinary differential equations. In this paper, an attempt has been made to prove the existence and uniqueness of the solution of the resulting free boundary value problem. Monotonic behavior of the solution is discussed. The numerical results, shown through a table and graphs, elucidate that the flow is significantly affected by the non-Newtonian cross-viscous parameter L and the magnetic parameter M.

Keywords: Stagnation point flow; magnetic field; non-Newtonian fluid; existence and uniqueness; topological shooting argument
MSC 2010: 34B08; 34B60; 34C12; 76D03; 76D05

Funding statement: The first author would like to acknowledge the financial support received from the Ministry of Human Resource Development (MHRD), Government of India, in the form of a PhD sponsorship. This work is part of the PhD thesis of the first author.

A Nomenclature

We use the following notations:

  1. a: Constants, s - 1 .

  2. u: Velocity component along x-axis, ms - 1 .

  3. v: Velocity component along y-axis, ms - 1 .

  4. x: Axial axis, m .

  5. y: Transverse axis, m .

  6. u e : Free stream velocity along x-axis, ms - 1 .

  7. v e : Free stream velocity along y-axis, ms - 1 .

  8. f: Dimensionless velocity.

  9. L: Non-Newtonian interaction parameter.

  10. M: Magnetic interaction parameter.

Greek symbols:

  1. μ: Dynamic viscosity, kgm - 1 ⁢ s - 1 .

  2. ν: Kinematic viscosity, m 2 ⁢ s - 1 .

  3. ρ: Fluid density, kgm - 3 .

  4. σ: Electric conductivity.

  5. ζ: Dimensionless similarity variable.

  6. α: Missing condition parameter.

Subscripts/superscripts:

  1. τ i ⁢ j : Stress tensor.

  2. e i ⁢ j : Strain rate.

  3. μ c : Coefficient of cross-viscosity.

  4. B 0 : Magnetic filed strength.

  5. f ′ : First order derivative with respect to ζ.

  6. f ′′ : Second order derivative with respect to ζ.

  7. f ′′′ : Third order derivative with respect to ζ.

Acknowledgements

The authors are grateful to the referees for their educative and constructive comments that refined the paper.

References

[1] F. T. Akyildiz, S. Tatar and S. Ulusoy, Existence and uniqueness for a nonlinear inverse reaction-diffusion problem with a nonlinear source in higher dimensions, Math. Methods Appl. Sci. 36 (2013), no. 17, 2397–2402. 10.1002/mma.2765Search in Google Scholar

[2] P. D. Ariel, Hiemenz flow in hydromagnetics, Acta Mech. 103 (1994), no. 1–4, 31–43. 10.1007/BF01180216Search in Google Scholar

[3] P. D. Ariel, A new finite-difference algorithm for computing the boundary layer flow of viscoelastic fluids in hydromagnetics, Comput. Methods Appl. Mech. Engrg. 124 (1995), no. 1–2, 1–13. 10.1016/0045-7825(94)00729-7Search in Google Scholar

[4] P. D. Ariel and I. Teipel, On dual solutions of stagnation point flow of a viscoelastic fluid, Z. Angew. Math. Mech. 74 (1994), no. 8, 341–347. 10.1002/zamm.19940740809Search in Google Scholar

[5] W. A. Coppel, On a differential equation of boundary-layer theory, Philos. Trans. Roy. Soc. London Ser. A 253 (1960), 101–136. 10.1098/rsta.1960.0019Search in Google Scholar

[6] L. K. Forbes, Steady flow of a “Reiner–Rivlin” fluid between rotating plates, Phys. Fluids 30 (2018), Article ID 103104. 10.1063/1.5053833Search in Google Scholar

[7] S. P. Hastings and J. B. McLeod, Classical Methods in Ordinary Differential Equations, Grad. Stud. Math. 129, American Mathematical Society, Providence, 2012. 10.1090/gsm/129Search in Google Scholar

[8] K. Hiemenz, Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder, Dinglers Polytech. J. 326 (1911), 321–324. Search in Google Scholar

[9] M. K. Kwong, The shooting method and multiple solutions of two/multi-point BVPs of second-order ODE, Electron. J. Qual. Theory Differ. Equ. 2006 (2006), Paper No. 6. 10.14232/ejqtde.2006.1.6Search in Google Scholar

[10] T. R. Mahapatra and S. Sidui, Non-axisymmetric Homann stagnation-point flow of a viscoelastic fluid towards a fixed plate, Eur. J. Mech. B Fluids 79 (2020), 38–43. 10.1016/j.euromechflu.2019.08.010Search in Google Scholar

[11] T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Math. Sci. Eng. 145, Academic Press, New York, 1979. Search in Google Scholar

[12] J. E. Paullet, Analysis of stagnation point flow of an upper-convected Maxwell fluid, Electron. J. Differential Equations 2017 (2017), Paper No. 302. Search in Google Scholar

[13] M. Reiner, A mathematical theory of dilatancy, Amer. J. Math. 67 (1945), 350–362. 10.2307/2371950Search in Google Scholar

[14] B. Sahoo, Hiemenz flow and heat transfer of a third grade fluid, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 811–826. 10.1016/j.cnsns.2007.12.002Search in Google Scholar

[15] B. Sahoo and F. Labropulu, Steady Homann flow and heat transfer of an electrically conducting second grade fluid, Comput. Math. Appl. 63 (2012), 1244–1255. 10.1016/j.camwa.2011.12.041Search in Google Scholar

[16] B. Sahoo and I. V. Shevchuk, Heat transfer due to revolving flow of ‘Reiner–Rivlin” fluid over a stretchable surface, Thermal Sci. Eng. Progr. 10 (2019), 327–336. 10.1016/j.tsep.2019.03.004Search in Google Scholar

[17] P. Samanta and C. Srinivasa Rao, Existence and uniqueness of a non-negative monotonic solution of a nonlinear ordinary differential equation, Differ. Equ. Dyn. Syst. (2019), 10.1007/s12591-019-00483-x. 10.1007/s12591-019-00483-xSearch in Google Scholar

[18] M. Sheikholeslami and M. M. Bhatti, Influence of external magnetic source on nanofluid treatment in a porous cavity, J. Porous Media 22 (2019), 1475–1491. 10.1615/JPorMedia.2019024518Search in Google Scholar

[19] I. Tlili, M. M. Bhatti, S. M. Hamad, A. A. Barzinjy, M. Sheikholeslami and A. Shafee, Macroscopic modeling for convection of hybrid nanofluid with magnetic effects, Phys. A 534 (2019), Article ID 122136. 10.1016/j.physa.2019.122136Search in Google Scholar

[20] R. A. Van Gorder and K. Vajravelu, Hydromagnetic stagnation point flow of a second grade fluid over a stretching sheet, Mech. Res. Comm. 37 (2010), no. 1, 113–118. 10.1016/j.mechrescom.2009.09.009Search in Google Scholar

[21] H. Weyl, On the differential equations of the simplest boundary-layer problems, Ann. of Math. (2) 43 (1942), 381–407. 10.2307/1968875Search in Google Scholar
Received: 2020-05-30
Revised: 2020-09-14
Accepted: 2020-10-30
Published Online: 2021-09-03
Published in Print: 2022-06-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative
  3. On tangential approximations of the solution set of set-valued inclusions
  4. Stabilization of the wave equation with a nonlinear delay term in the boundary conditions
  5. Fixed point to fixed circle and activation function in partial metric space
  6. A note on the validity of the Schrödinger approximation for the Helmholtz equation
  7. Certain classes of analytic functions defined by Hurwitz–Lerch zeta function
  8. A new factor theorem on absolute matrix summability method
  9. On a solution to a functional equation
  10. Hydromagnetic effects on non-Newtonian Hiemenz flow
  11. Stability of a class of entropies based on fractional calculus
  12. Asymptotic behavior of solution of Whitham–Broer–Kaup type equations with negative dispersion
  13. Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts
  14. Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control
  15. On the asymptotic formulas for perturbations in the eigenvalues of the Stokes equations due to the presence of small deformable inclusions
  16. Extended homogeneous balance conditions in the sub-equation method
https://doi.org/10.1515/jaa-2021-2063
Keywords for this article
Stagnation point flow; magnetic field; non-Newtonian fluid; existence and uniqueness; topological shooting argument

Articles in the same Issue

  1. Frontmatter
  2. L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative
  3. On tangential approximations of the solution set of set-valued inclusions
  4. Stabilization of the wave equation with a nonlinear delay term in the boundary conditions
  5. Fixed point to fixed circle and activation function in partial metric space
  6. A note on the validity of the Schrödinger approximation for the Helmholtz equation
  7. Certain classes of analytic functions defined by Hurwitz–Lerch zeta function
  8. A new factor theorem on absolute matrix summability method
  9. On a solution to a functional equation
  10. Hydromagnetic effects on non-Newtonian Hiemenz flow
  11. Stability of a class of entropies based on fractional calculus
  12. Asymptotic behavior of solution of Whitham–Broer–Kaup type equations with negative dispersion
  13. Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts
  14. Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control
  15. On the asymptotic formulas for perturbations in the eigenvalues of the Stokes equations due to the presence of small deformable inclusions
  16. Extended homogeneous balance conditions in the sub-equation method
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 21.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jaa-2021-2063/html
Scroll to top button