## THERMAL SCIENCE

International Scientific Journal

### NEW METHODS TO PROVIDE EXACT SOLUTIONS FOR SOME UNIDIRECTIONAL MOTIONS OF RATE TYPE FLUIDS

**ABSTRACT**

Based on three immediate consequences of the governing equations corresponding to some unidirectional motions of rate type fluids, new motion problems are tackled for exact solutions. For generality purposes, exact solutions are developed for shear stress boundary value problems of generalized Burgers fluids. Such solutions, for which the shear stress instead of its differential expressions is given on the boundary, are lack in the literature for such fluids. Consequently, the first exact solutions for motions of rate type fluids induced by an infinite plate or a circular cylinder that applies a constant shear f or an oscillating shear f sin(ωt) to the fluid are here presented. In addition, all steady-state solutions can easily be reduced to known solutions for second grade and Newtonian fluids.

**KEYWORDS**

PAPER SUBMITTED: 2013-02-25

PAPER REVISED: 2013-08-08

PAPER ACCEPTED: 2013-09-14

PUBLISHED ONLINE: 2013-09-22

**THERMAL SCIENCE** YEAR

**2016**, VOLUME

**20**, ISSUE

**1**, PAGES [7 - 20]

- Siddiqu, I., Vieru, D., Exact solutions for rotational flow of a fractional Maxwell fluid in a circular cylinder, Thermal Science 16, 2 (2012), pp. 345-355.
- Jamil, M., Khan, N. A., Nazish Shahid, Fractional MHD Oldroyd-B fluid over an oscillating plate, Thermal Science DOI: 10.2298/TSCI110731140J.
- Mahmood, A., On analytical study of fractional Oldroyd-B flow in annular region of two torsionally oscillating cylinders, Thermal Science 16, 2 (2012), pp. 411-421.
- Renardy, M., Inflow boundary condition for steady flow of viscoelastic fluids with differential constitutive laws, Rocky Mountain Journal of Mathematics, 18 (1988), 2, pp. 445-453.
- Renardy, M., An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions, J. Non-Newtonian Fluid Mech. 36 (1990) pp. 419-425.
- Renardy, M., Recent advances in the mathematical theory of steady flow of viscoelastic fluids, J. Non-Newtonian Fluid Mech., 29 (1988) pp. 11-24.
- Rajagopal, K. R., A new development and interpretation of the Navier-Stokes fluid which reveals why the "Stokes assumption" is inapt, Int. J. Non-Linear Mech., 50 (2013) pp. 141-151.
- Waters, N. D., King, M. J., Unsteady flow of an elastico-viscous liquid, Rheol. Acta. 9 (1970), 3, pp. 345-355.
- Fetecau, C., Kannan, K., A note on an unsteady flow of Oldroyd-B fluid, Internat. J. Math. Math. Sci., Volume 2005, (2005). 19, pp. 3185-3194.
- Vieru, D., Fetecau, Corina, Fetecau, C., Unsteady flow of a generalized Oldroyd-B fluid due to an infinite plate subject to a time-dependent shear stress, Can. J. Phys., 88 (2010), 9, pp. 675-687.
- Fetecau, Corina, Imran, M., Fetecau, C., Taylor-Couette flow of an Oldroyd-B fluid in an annulus due to a time-dependent couple, Z. Naturforsch., 66a (2011) pp. 40- 46.
- Jamil, M., Fetecau, C., Imran, M., Unsteady helical flows of Oldroyd-B fluids, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 3, pp. 1378-1386.
- Jamil, M., Fetecau, C., Starting solutions for the motion of a generalized Burgers' fluid between coaxial cylinders, Boundary Value Problems, 14 (2012), pp 1-15.
- Jamil, M., First problem of Stokes for generalized Burgers' fluids, ISRN Mathematical Physics, Volume 2012, Article ID 831063, 17 pages, doi:10.5402/2012/831063.
- Bandelli, R., Rajagopal, K. R., Start-up flows of second grade fluids in domains with one finite dimension, Int. J. Non-Linear Mech., 30 (1995), 6, pp. 817-839.
- Erdogan, M. E., On unsteady motion of a second grade fluid over a plane wall, Int. J. Non-Linear Mech., 38 (2003), 7, pp. 1045-1051.
- Yao, Y., Liu, Y., Some unsteady flows of second grade fluid over a plane wall, Nonlinear Anal. Real World Appl., 11 (2010), 5, pp. 4302-4311.
- Vieru, D., Fetecau, C., Sohail, A., Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate, Z. Angew. Math. Phys., 62 (2011), 1, pp. 161-172.
- Fetecau, C., Fetecau, Corina, Rana, M., General solutions for the unsteady flow of second-grade fluids over an infinite plate that applies arbitrary shear to the fluid, Z. Naturforsch., 66a (2011) 753-759.
- Rajagopal, K. R., Srinivasa, A.R., A thermodynamic framework for rate type fluid models, J. Non-Newtonian Fluid Mech.,, 88 (2000) 207-227.
- Muralikrishnan, J., Rajagopal, K. R., Review of the uses and modeling of bitumen from ancient to modern times, Appl. Mech. Rev., 56 (2003), 2, 149-214.
- Chopra, P. N., High temperature transient in olivine rocks, Tectonophysics, 279 (1977), 1-4, 93- 111.
- Jackson, I., Laboratory measurements of seismic wave dispersion and attenuation: Recent progress. In Earth's Deep Interior: Mineral Physics and Tomography from the Atomic to the Global Scale, Geophys. Monogr. Ser., vol. 117, edited by S. Karato et al., 2000, pp. 265-289, AGU, Washington, D.C., doi:10.1029/GM117p0265, 2000.
- Fetecau, C., Hayat, T., Fetecau, Corina, Steady-state solutions for some simple flows of generalized Burgers fluids, Int. J. Non-Linear Mech., 41 (2006), 8, 880-887.
- Tong, D., Shan, L., Exact solutions for generalized Burgers' fluid in an annular pipe, Meccanica 44 (2009), 4, 427-431.
- Fetecau, Corina, Hayat, T., Khan, M., Fetecau, C., A note on longitudinal oscillations of a generalized Burgers fluid in cylindrical domains, J. Non-Newtonian Fluid Mech., 165 (2010) 350- 361.
- Tong, D., Starting solutions for oscillating motions of a generalized Burgers' fluid in cylindrical domains, Acta Mech., 214 (2010), 3-4, 395-407.
- Kuros, A., Cours d'algebre superieure, Edition Mir Moscow, 1973.
- Christov, I. C., Christov, C. I., Comment on "On a class of exact solutions of the equations of motion of a second grade fluid" by C. Fetecau and J. Zierep (Acta Mech. 150, 135-138, 2001) Acta Mech., 215 (2010), 1-4, 25-28.
- Vieru, D., Fetecau, Corina, Rana, Mehwish, Starting solutions for the flow of second grade fluids in a rectangular channel due to an oscillating shear stress, AIP Conference Proceedings Vol. 1450, pp. 45-54 (2012), DOI:10.1063/1.4724116.
- Fetecau, C., Vieru, D., Fetecau, Corina, Effect of the side walls on the motion of a viscous fluid induced by an infinite plate that applies an oscillating shear stress to the fluid, Cent. Eur. J. Phys., 9 (2011), 3, 816-824.
- Jamil, M., Khan, N. A., Helical flows of fractionalized Burgers fluids, AIP Advances, 2, (2012), 15 pages: DOI: 10.1063/1.3694982.
- Rajagopal, K. R., Bhatnagar, R. K., Exact solutions for some simple flows of an Oldroyd-B fluid, Acta Mech. 113 (1995), 1-4, 233-239.
- Debnath, L., Bhatta, D., Integral Transforms and their Applications (2nd edn), Chapman & Hall/CRC, Boca Raton, 2007.
- Tolstov, G. P., Serii Fourier, Editura Tehnica, 1955 (translation from the Russian language).
- Stokes, G. G., On the effect of the rotation of cylinders and spheres about their own axes in increasing the logarithmic decrement of the arc of vibration, (Mathematical and Philosophical Papers 5) Cambridge: Cambridge University Press, England, 1886.
- Rajagopal, K. R., Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid, Acta Mech., 49 (1983), 3-4, 281-285.