Analytical Solution of Transient Flow of Fractional Oldroyd-B Fluid between Oscillating Cylinders

Khadija Shaikh, Fozia Shaikh, Rahim Bux Khokhar, K.N. Memon


This paper investigates the fractional Oldroyd-B fluid flow across two interminable coaxial cylinders, where the fluid’s motion is generated by the oscillatory movement of cylinders and the oscillating pressure gradient. The profile of velocity and shear stress of the flow is derived with the assistance of Caputo fractional derivative utilizing an analytical technique, finite Hankel transform and Laplace transforms. The semi-analytical solution is then displayed as generalized functions of G and R satisfying fundamental constitutive equations and all initial and boundary conditions. To validate the results, some limitations have been imposed on the determined equations and the results have been contrasted against previous results. Moreover, the influence of various parameters on the flow of fractionalized Oldroyd-B fluid is investigated and depicted graphically.

Full Text:



Alam, M. K., Memon, K., Siddiqui, A., Shah, S., Farooq, M., Ayaz, M., Nofal, T. A. and Ahmad, H. [2021], ‘Modeling and analysis of high shear viscoelastic ellis thin liquid film phenomena’, Physica Scripta 96(5), 055201.

Ali Zafar, A., Bilal Riaz, M. and Imran Asjad, M. [2020], ‘Unsteady rotational flow of fractional maxwell fluid in a cylinder subject to shear stress on the boundary’, Punjab University Journal of Mathematics


Armstrong, R. C. and Hassager, O. [1987], Dynamics of polymeric liquids: Fluid mechanics, Wiley.

Debnath, L. and Bhatta, D. [2016], Integral transforms and their applications, Chapman and Hall/CRC.

Fetecau, C., Fetecau, C., Jamil, M. and Mahmood, A. [2011], ‘Retracted article: Flow of fractional maxwell fluid between coaxial cylinders’, Archive of Applied Mechanics 81(8), 1153–1163.

Haitao, Q. and Mingyu, X. [2009], ‘Some unsteady unidirectional flows of a generalized oldroyd-b fluid with fractional derivative’, Applied Mathematical Modelling 33(11), 4184–4191.

Kamran, M., Athar, M. and Imran, M. [2012], ‘Critical study on rotational flow of a fractional oldroyd-b fluid induced by a circular cylinder’, International Scholarly Research Notices 2012.

Kang, S. M., Nazeer, W., Athar, M., Hisham, M. D. and Kwun, Y. C. [2016], ‘Retracted article: Velocity and shear stress for an oldroyd-b fluid within two cylinders’, Boundary Value Problems 2016(1), 1–11.

Khalique, C. M., Safdar, R. and Tahir, M. [2019], ‘First analytic solution for the oscillatory flow of a maxwells fluid with annulus’, Open Journal of Mathematical Sciences 2, 1–9.

Khaskheli, M. A., Memon, K. N., Sheikh, A. H., Siddiqui, A. M. and Shah, S. F. [2020], ‘Tank drainage for an electrically conducting newtonian fluid with the use of the bessel function’, Eng. Technol. Appl. Sci. Res 10(2).

Lorenzo, C. F. and Hartley, T. T. [2008], ‘Generalized functions for the fractional calculus’, Critical Reviews™ in Biomedical Engineering 36(1).

Mathur, V. and Khandelwal, K. [2017], ‘Flow of fractional maxwell fluid in oscillating pipe-like domains’, International Journal of Applied and Computational Mathematics 3(2), 841–858.

Memon, K. N., Alam, M. K., Baili, J., Nawaz, Z., Shiekh, A. H. and Ahmad, H. [2021], ‘Analytical solution of tank drainage flow for electrically conducting newtonian fluid’, Thermal Science 25(Spec. issue 2), 433–439.

Qi, H. and Jin, H. [2009], ‘Unsteady helical flows of a generalized oldroyd-b fluid with fractional derivative’, Nonlinear analysis: real world applications 10(5), 2700–2708.

Qureshi, S. and Kumar, P. [2019], ‘Using shehu integral transform to solve fractional order caputo type initial value problems’, Journal of Applied Mathematics and Computational Mechanics 18(2).

Rauf, A., Rubbab, Q., Vieru, D. and Majeed, A. [2020], ‘Simultaneous flow of two immiscible fractional maxwell fluids with the clear region and homogeneous porous medium’, Sains Malaysiana 49(11), 2871–2880.

Sadiq, N., Imran, M., Safdar, R., Tahir, M., Javaid, M. and Younas, M. [2020], ‘Exact solution for some rotational motions of fractional oldroyd-b fluids between circular cylinders’, Punjab University Journal

of Mathematics 50(4).

Shah, S. A. R., Memon, K., Shah, S., Sheikh, A. and Siddiqui, A. [2022], ‘Delta perturbation method for thin film flow of a third grade fluid on a vertical moving belt’, STATISTICS, COMPUTING AND INTERDISCIPLINARY RESEARCH 4(1), 61–73.

Shaikh, F., Shah, S. F., Siddiqui, A. and Kumar, L. [2022], ‘Application of recursive approach of pseudoplastic fluid flow between rotating coaxial cylinders’, Alexandria Engineering Journal 61(10), 7823–7832.

Syam, M. and Al-Refai, M. [2019], ‘Fractional differential equations with atangana–baleanu fractional derivative: analysis and applications’, Chaos, Solitons & Fractals: X 2, 100013.

Tahir, M., Naeem, M. N., Javaid, M., Younas, M., Imran, M., Sadiq, N. and Safdar, R. [2018], ‘Unsteady flow of fractional oldroyd-b fluids through rotating annulus’, Open Physics 16(1), 193–200.

Wang, F. and Liu, J. [2020], ‘The first solution for the helical flow of a generalized maxwell fluid within annulus of cylinders by new definition of transcendental function’, Mathematical Problems in Engineering 2020.

Wang, F., Shen, W.-C., Liu, J.-L. and Wang, P. [2020], ‘The analytic solutions for the unsteady rotating flows of the generalized maxwell fluid between coaxial cylinders’, Thermal Science 24(6 Part B), 4041– 4048.

Yang, X.-J., Gao, F. and Yang, J. [2020], General fractional derivatives with applications in viscoelasticity, Academic Press.

Zafar, A., Riaz, M., Shah, N. and Imran, M. [2018], ‘Influence of non-integer-order derivatives on unsteady unidirectional motions of an oldroyd-b fluid with generalized boundary conditions’, The European Physical Journal Plus 133(3), 1–13.



  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.