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Exact solutions for the unsteady rotational flow of an Oldroyd-B fluid with fractional derivatives induced by a circular cylinder

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Abstract

In this research article, the unsteady rotational flow of an Oldroyd-B fluid with fractional derivative model through an infinite circular cylinder is studied by means of the finite Hankel and Laplace transforms. The motion is produced by the cylinder, that after time t=0+, begins to rotate about its axis with an angular velocity Ωt p. The solutions that have been obtained, presented under series form in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The corresponding solutions that have been obtained can be easily particularized to give the similar solutions for Maxwell and Second grade fluids with fractional derivatives and for ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian fluids) performing the same motion, are obtained as limiting cases of general solutions.

The most important things regarding this paper to mention are that (1) we extracted the expressions for the velocity field and the shear stress corresponding to the motion of Second grade fluid with fractional derivatives as a limiting case of our general solutions corresponding to the Oldroyd-B fluid with fractional derivatives, this is not previously done in the literature to the best of our knowledge, and (2) the expressions for the velocity field and the shear stress are in the most simplified form, and the point worth mentioning is that these expressions are free from convolution product and the integral of the product of the generalized G-functions.

Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.

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Acknowledgements

The authors would like to express their gratitude to the referees for their careful assessment and constructive comments and corrections.

The author M. Kamran is thankful and grateful to Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan; and especially to Higher Education Commission of Pakistan, for supporting and facilitating the research work.

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Authors and Affiliations

  1. Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan

    M. Kamran

  2. Department of Mathematics, GC University, Faisalabad, Pakistan

    M. Imran

  3. Department of Mathematics, University of Education, Lahore, Pakistan

    M. Athar

Authors
  1. M. Kamran
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  2. M. Imran
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  3. M. Athar
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Corresponding author

Correspondence to M. Kamran.

Appendix A

Appendix A

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Kamran, M., Imran, M. & Athar, M. Exact solutions for the unsteady rotational flow of an Oldroyd-B fluid with fractional derivatives induced by a circular cylinder. Meccanica 48, 1215–1226 (2013). https://doi.org/10.1007/s11012-012-9662-y

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  • DOI: https://doi.org/10.1007/s11012-012-9662-y

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