Abstract
Given a general velocity field consistent with the stagnation point flow, can the viscoelastic stresses arising in the flow of an upper convected Maxwell fluid found by solving the constitutive equation also satisfy the momentum equation? Consideration is given to the study of the stress tensor arising in the steady flow of an upper convected Maxwell (UCM) fluid with a velocity field consistent with the stagnation point flow. By the method of characteristics, exact solutions to the partial differential equations arising in the approximating model of the viscoelastic stresses in the flow of an upper convected Maxwell (UCM) fluid are obtained for the three components of the stress tensor, for reasonably general velocity fields. We are able to account for the effects of variable boundary data at the inflow by considering the viscoelastic stresses over two spatial variables. Furthermore, we assume a relatively general velocity field. As a special case, some results present in the recent literature are obtained; it is known that these special case solutions do not satisfy the momentum equation. In the general case we consider, we find that the general solution will not satisfy the momentum equation except in a limited restricted case. We discuss how this shortcoming might be rectified by use of a more general velocity field.
Similar content being viewed by others
References
Renardy M (2006) A comment on smoothness of viscoelastic stresses. J Non-Newton Fluid Mech 138:204–205
Oliveira MSN, Pinho FT, Poole RJ, Oliveira PJ, Alves MA (2009) Purely-elastic flow asymmetries in flow-focusing devices. J Non-Newton Fluid Mech 160:31–39
Becherer P, Morozov AN, van Saarloos W (2008) Scaling of singular structures in extensional flow of dilute polymer solutions. J Non-Newton Fluid Mech 153:183–190
Becherer P, van Saarloos W, Morozov AN (2009) Stress singularities and the formation of birefringent strands in stagnation flows of dilute polymer solutions. J Non-Newton Fluid Mech 157:126–132
Van Gorder RA, Vajravelu K, Akyildiz FT (2009) Viscoelastic stresses in the stagnation flow of a dilute polymer solution. J Non-Newton Fluid Mech 161:94–100
Phan-Thien N (1983) Plane and axi-symmetric stagnation flow of a Maxwellian fluid. Rheol Acta 22:127–130
Arratia PE, Thomas CC, Diorio J, Gollub JP (2006) Elastic instabilities of polymer solutions in cross-channel flow. Phys Rev Lett 96:144502
Poole RJ, Alves MA, Oliveira PJ (2007) Purely elastic flow asymmetries. Phys Rev Lett 99:164503
Lagnado RR, Phan-Thien N, Leal LG (1985) The stability of two-dimensional linear flows of an Oldroyd-type fluid. J Non-Newton Fluid Mech 18:25–59
Astarita G (1979) Objective and generally applicable criteria for flow classification. J Non-Newton Fluid Mech 6:69
Mompean G, Thompson RL, Mendes PRS (2003) A general transformation procedure for differential viscoelastic models. J Non-Newton Fluid Mech 111:151
Thompson RL, Mendes PRS, Naccache MF (1999) A new constitutive equation and its performance in contraction flows. J Non-Newton Fluid Mech 86:375
Acknowledgements
The author thanks the referees for comments which have led to improvement in the presentation of the results. R.A.V. was supported in part by an NSF research fellowship.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Van Gorder, R.A. Do general viscoelastic stresses for the flow of an upper convected Maxwell fluid satisfy the momentum equation?. Meccanica 47, 1977–1985 (2012). https://doi.org/10.1007/s11012-012-9568-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-012-9568-8