Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T06:11:24.391Z Has data issue: false hasContentIssue false
  • English
  • Français

EXACT SOLUTIONS FOR THE POISEUILLE FLOW OF A GENERALIZED MAXWELL FLUID INDUCED BY TIME-DEPENDENT SHEAR STRESS

Part of: Foundations, constitutive equations, rheology

Published online by Cambridge University Press:  21 February 2011

W. AKHTAR ,
CORINA FETECAU  and
A. U. AWAN
W. AKHTAR*
Affiliation:
Institute of Space Technology, P. O. Box 2750, Islamabad, Pakistan (email: [email protected])
CORINA FETECAU
Affiliation:
Department of Theoretical Mechanics, Technical University of Iasi, Iasi 700506, Romania (email: [email protected])
A. U. AWAN
Affiliation:
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Poiseuille flow of a generalized Maxwell fluid is discussed. The velocity field and shear stress corresponding to the flow in an infinite circular cylinder are obtained by means of the Laplace and Hankel transforms. The motion is caused by the infinite cylinder which is under the action of a longitudinal time-dependent shear stress. Both solutions are obtained in the form of infinite series. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases. Finally, the influence of the material and fractional parameters on the fluid motion is brought to light.

Keywords

generalized Maxwell fluidexact solutionstime-dependent shear stress

MSC classification

Secondary: 76A05: Non-Newtonian fluids
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 4 , April 2010 , pp. 416 - 429
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Akhtar, W., Fetecau, C., Tigoiu, V. and Fetecau, C., “Flow of a Maxwell fluid between two side walls induced by a constantly accelerating plate”, Z. Angew. Math. Phys. 60 (2009) 498–510.CrossRefGoogle Scholar
[2]Athar, M., Kamran, M. and Fetecau, C., “Taylor–Couette flow of a generalized second grade fluid due to a constant couple”, Nonlinear Anal. Model. Control 15 (2010) 3–13.CrossRefGoogle Scholar
[3]Bagley, R. L. and Torvik, P. J., “A theoretical basis for the application of fractional calculus to viscoelasticity”, J. Rheol. 27 (1983) 201–210.CrossRefGoogle Scholar
[4]Debnath, L. and Bhatta, D., Integral transforms and their applications, 2nd edn (Chapman and Hall–CRC Press, Boca Raton, FL, 2007).Google Scholar
[5]Fetecau, C., Mahmood, A., Fetecau, C. and Vieru, D., “Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder”, Comput. Math. Appl. 56 (2008) 3096–3108.CrossRefGoogle Scholar
[6]Fetecau, C., Mahmood, A. and 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. Simul. 15 (2010) 3931–3938.CrossRefGoogle Scholar
[7]Fetecau, C., Prasad, S. C. and Rajagopal, K. R., “A note on the flow induced by a constantly accelarating plate in an Oldroyd-B fluid”, Appl. Math. Model. 31 (2007) 647–654.CrossRefGoogle Scholar
[8]Friedrich, C., “Relaxation and retardation functions of the Maxwell model with fractional derivatives”, Rheol. Acta. 30 (1991) 151–158.CrossRefGoogle Scholar
[9]Germant, A., “On fractional differentials”, Philos. Mag. 25 (1938) 540–549.CrossRefGoogle Scholar
[10]Heibig, A. and Palade, L. I., “On the rest state stability of an objective fractional derivative viscoelastic fluid model”, J. Math. Phys. 49 (2008), 043101-22.CrossRefGoogle Scholar
[11]Khan, M., Hyder, A. S., Fetecau, C. and Qi, H., “Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model”, Appl. Math. Comput. 33 (2009) 2526–2533.Google Scholar
[12]Khan, M., Hyder, A. S. and Qi, H., “Exact solutions of starting flows for a fractional Burgers fluid between coaxial cylinders”, Nonlinear Anal. Real World Appl. 10 (2009) 1775–1783.CrossRefGoogle Scholar
[13]Lorenzo, C. F. and Hartley, T. T., “Generalized functions for fractional calculus”, NASA/TP-1999-209424, 1999.Google Scholar
[14]Makris, N., Dargush, D. F. and Constantinou, M. C., “Dynamical analysis of generalized viscoelastic fluids”, J. Engrg. Mech. 119 (1993) 1663–1679.CrossRefGoogle Scholar
[15]Podlubny, I., Fractional differential equations (Academic Press, San Diego, CA, 1999).Google Scholar
[16]Qi, H. and Jin, H., “Unsteady rotating flows of viscoelastic fluid with the fractional Maxwell model between coaxial cylinders”, Acta Mech. Sin. 22 (2006) 301–305.CrossRefGoogle Scholar
[17]Qi, H. and Jin, H., “Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative”, Nonlinear Anal. Real World Appl. 10 (2009) 2700–2708.CrossRefGoogle Scholar
[18]Srivastava, P. N., “Nonsteady helical flow of a visco-elastic liquid”, Arch. Mech. Stos. 18 (1966) 145–150.Google Scholar
[19]Tong, D. and Liu, Y., “Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe”, Internat. J. Engrg. Sci. 43 (2005) 281–289.CrossRefGoogle Scholar
[20]Tong, D., Wang, R. and Yang, H., “Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe”, Sci. China Ser. G 48 (2005) 485–496.CrossRefGoogle Scholar
[21]Wang, S. and Xu, M., “Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus”, Nonlinear Anal. Real World Appl. 10 (2009) 1087–1096.CrossRefGoogle Scholar