## THERMAL SCIENCE

International Scientific Journal

### FRACTIONAL MAGNETOHYDRODYNAMICS OLDROYD-B FLUID OVER AN OSCILLATING PLATE

**ABSTRACT**

This paper presents some new exact solutions corresponding to the oscillating flows of a MHD Oldroyd-B fluid with fractional derivatives. The fractional calculus approach in the governing equations is used. The exact solutions for the oscillating motions of a fractional MHD Oldroyd-B fluid due to sine and cosine oscillations of an infinite plate are established with the help of discrete Laplace transform. The expressions for velocity field and the associated shear stress that have been obtained, presented in series form in terms of Fox H functions, satisfy all imposed initial and boundary conditions. Similar solutions for ordinary MHD Oldroyd-B, fractional and ordinary MHD Maxwell, fractional and ordinary MHD Second grade and MHD Newtonian fluid as well as those for hydrodynamic fluids are obtained as special cases of general solutions. Finally, the obtained solutions are graphically analyzed through various parameters of interest.

**KEYWORDS**

PAPER SUBMITTED: 2011-07-31

PAPER REVISED: 2011-10-22

PAPER ACCEPTED: 2011-10-26

**THERMAL SCIENCE** YEAR

**2013**, VOLUME

**17**, ISSUE

**Issue 4**, PAGES [997 - 1011]

- Penton, R., The transient for stokes' oscillating plane: A solution in terms of tabulated functions, J. Fluid. Mech., 31 (1968), pp. 810-825.
- Puri, P., Kythe, P. K., Thermal effect in Stokes' second problem, Acta Mech., 112 (1998), pp. 44-40.
- Erdogan, M. E. , A note on an unsteady flow of a viscous fluid due to an oscillating plane wall, Internat. J. Non-Linear Mech., 35 (2000), pp. 1-6.
- Rajagopal, K. R., Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid, Acta. Mech., 49 (1983), pp. 281-285.
- Rajagopal, K. R., Bhatnagar, R. K., Exact solutions for some simple flows of an Oldroyd-B fluid, Acta. Mech., 113 (1995), pp. 233-239.
- Hayat, T., Siddiqui, A. M., Asghar, S., Some simple flows of an Oldroyd-B fluid, Internat. J. Engrg. Sci., 39 (2001), pp. 135-147.
- Aksel, N., Fetecau, C., Scholle, M., Starting solutions for some unsteady unidirectional flows of Oldroyd-B fluids, Z. Angew. Math. Phys., 57 (2006) 815-831.
- Khan, M., Asghar, S., Hayat, T., Hall effect on the pipe flow of a Burgers' fluid: An exact solution, Nonlinear Anal.: Real World Appl., 10 (2009), pp. 974-979
- Fetecau, C., Corina Fetecau, Starting solutions for some unsteady unidirectional flows of a Second grade fluid, Internat. J. Engrg. Sci., 43 (2005), pp. 781-789.
- Fetecau, C., Corina Fetecau, Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder, Internat. J. Engrg. Sci., 44 (2006), pp. 788-796.
- Khan, M., Asia Anjum, Qi, H. T., Fetecau, C., On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid , Z. Angew. Math. Phys., 61 (2010), pp. 133-145.
- Khan, M., Asia Anjum, Fetecau, C., Qi, H. T., Exact solutions for some oscillating motions of a fractional Burgers' fluid, Math. Comput. Modelling, 51 (2010), pp. 682-692.
- Mahmood, A., Fetecau, C., Khan, N. A., Jamil, M., Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders, Acta Mech Sin, 26 (2010), pp. 541-550.
- Zheng, L., Zhao, F., Zhang, X., Exact solutions for generalized Maxwell fluid flow due to oscillatory and constantly accelerating plate, Nonlinear Anal.: Real World Appl., 11 (2010), pp. 3744-3751.
- Asia Anjum, Ayub, M., Khan, M., Starting solutions for oscillating motions of an Oldroyd-B fluid over a plane wall, Commun Nonlinear Sci Numer Simulat, DOI: 10.1016/j.cnsns.2011.05.004.
- Bagley, R. L., A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), pp. 201-210.
- Heibig, A., Palade, L. I., On the rest state stability of an objective fractional derivative viscoelastic fluid model, J. Math. Phys., 49 (2008), pp. 043101.
- Song, D. Y., Study of rheological characterization of Fenu-Greek gum with modified Maxwell, J. Chem. Eng., 8 (2008), pp. 85-88.
- Wang, S., Xu, M.: Axial Coutte flow of two kinds of fractional viscoelastic fluids in an annulus. Nonlinear Anal. Real World Appl. 10 (2009) , pp. 1087-1096.
- Fetecau, C., Mahmood, A., Jamil, M., Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress, Commun Nonlinear Sci Numer Simulat, 15 (2010), pp. 3931-3938.
- Tripathi, D., Pandey, S. K., Das, S., Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel, Applied Mathematics and Computation, 215 (2010), pp. 3645-3654.
- Hyat, T., Najam, S., Sjid, M., Ayub, M., Mesloub, S., On exact solutions for oscillatory flows in a generalized Burgers' fluid with slip condition, Z. Naturforsch, 65a (2010), pp. 381-391.
- Hayat, T., Zaib, S., Fetecau, C., Corina Fetecau, Flows in a fractional generalized Burgers' fluid, J. Porus Media, 13 (2010) 725-739.
- Tripathi, D., Peristaltic flow of a fractional second grade fluid through a cylindrical tube, Thermal Science, DOI: 10.2298/TSCI100503061T (2010).
- Liu, Y., Zheng, L., Unsteady MHD Couette flow of a generalized Oldroyd-B fluid with fractional derivative, Comput. Math. Appl., 61 (2011), pp. 443-450.
- Zheng, L., Liu, Y., Zhang, X., Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear. Anal.: Real World Appl., DOI: 10.1016/j.nonrwa.2011.02.016.
- Fetecau, C., Corina Fetecau, Jamil, M., Mahmood, A., Flow of fractional Maxwell fluid between coaxial cylinders, Arch. App. Mech., 81 (2011), pp. 1153-1163.
- Jamil, M., Rauf, A., Zafar, A. A., Khan, N. A., New exact analytical solutions for Stokes' first problem of Maxwell fluid with fractional derivative approach, Comput. Math. Appl., 62 (2011), pp. 1013-1023.
- Jamil, M., Zafar, A. A., Khan, N. A., Translational flows of an Oldroyd-B fluid with fractional derivatives, Comput. Math. Appl., 62 (2011), pp. 1540-1553.
- Siddique, I., Vieru, D., Exact solutions for rotational flow of a fractional Maxwell fluid in a circular cylinder, Thermal Science, (2011), DOI: 10.2298/TSCI 101228072S.
- Tripathi, D., Gupta, P. K., Das, S., Influence of slip condition on peristaltic tansport of a viscoelastic fluid with model, Thermal Science, 15 (2011), pp. 501-515.
- Tripathi, D., Peristaltic transport of fractional Maxwell fluids in uniform tubes: Application of an endoscope, Computers and Mathematics with Applications, 62 (2011), pp. 1116-1126.
- Tripathi, D., Peristaltic transport of a viscoelastic fluid in a channel, Acta Astronautica, 68 (2011)1379-1385.
- Tripathi, D., Pandey, S. K., Das, S., Peristaltic transport of a generalized Burgers' fluid: Application to the movement of chyme in small intestine, Acta Astronautica, 69 (2011), pp. 30-38.
- Tripathi, D., Numerical study on peristaltic flow of generalized Burgers' fluids in uniform tubes in presence of an endoscope, International Journal for Numerical Methods in Biomedical Engineering, doi: 10.1002/cnm.1442.
- Beg, O. A., Sim, L., Zueco, J., Bhargava, R., Numerical study of magnetohydrodynamic viscous plasma flow in rotating porous media with Hall currents and inclined magnetic field influence, Communications in Nonlinear Science and Numerical Simulation , 15 (2010), pp. 345-359.
- Pandey, S. K., Tripathi, D., Influence of magnetic field on the peristaltic flow of a viscous fluid through a finite-length cylindrical tube, Applied Bionics and Biomechanics, 7 (2010), pp. 169-176.
- Pandey, S. K., Tripathi, D., Peristaltic flow characteristics of Maxwell and Magnetohydrodynamic fluids in finite channels, Journal of Biological Systems, 18 (2010), pp. 621-647.
- Podlubny, I., Fractional Differential Equations, Academic press, San Diego, (1999).
- Mainardi, F., Fractional calculus and waves in viscoelasticity: An introduction to mathematical models, Imperial College Press, London (2010).
- Mathai, A. M., Saxena, R. K., Haubold, H. J., The H-functions: Theory and Applications, Springer, New York (2010).
- Debnath, L., Bhatta, D., Integral Transforms and Their Applications (Second Edition), Chapman & Hall/CRC, (2007).
- Tan, W. C., Xu, M. Y., Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model, Acta Mech. Sin., 18 (2002), pp. 342-349.
- Qi., H. T., Xu, M., Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B model, Acta Mech. Sin. 23 (2007), pp. 463-469.
- Tan., W. C., Xu, M. Y., The impulsive motion of flat plate in a generalized second grade fluid, Mech. Res. Comm., 29 (2002), pp. 3-9.