Pure stress modes for linear viscoelastic flows with variable coefficients

  • Debanjana Mitra
  • Mythily Ramaswamy
  • Michael RenardyEmail author


We consider the equations of a linear Maxwell fluid with spatially varying coefficients. Pure stress modes are solutions with zero velocity but nonzero stresses. We derive equations to characterize such solutions. In two dimensions, we find that under generic hypotheses only certain “trivial” solutions exist. In three dimensions, on the other hand, there exist nontrivial solutions. To get them, we derive a system of partial differential equations whose type (elliptic or hyperbolic) depends on the sign of the Gauss curvature of level surfaces of the relaxation time.


Linear viscoelastic flow Maxwell fluid Stress modes 

Mathematics Subject Classification

Primary 76A10 



  1. 1.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. 17, 35–92 (1964)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Boldrini, J.L., Doubova, A., Fernández-Cara, E., González-Burgos, M.: Some controllability results for linear viscoelastic fluids. SIAM J. Control Optim. 50, 900–924 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Carleman, T.: Sur un problème dunicité pur les systèmes déquations aux dérivées partielles à deux variables indépendantes. Ark. Mat. Astr. Fys. 26B(17), 1–9 (1939)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chowdhury, S., Mitra, D., Ramaswamy, M., Renardy, M.: Approximate controllability results for linear viscoelastic flows. J. Math. Fluid. Math. 19, 529–549 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    do Carmo, Manfredo P.: Differential geometry of curves and surfaces. PrenticeHall Inc, Englewood Cliffs (1976)zbMATHGoogle Scholar
  6. 6.
    Doering, C.R., Eckhardt, B., Schumacher, J.: Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers. J. Non-Newton. Fluid Mech. 135, 92–96 (2006)CrossRefGoogle Scholar
  7. 7.
    Doubova, A., Fernández-Cara, E., González-Burgos, M.: Controllability results for linear viscoelastic fluids of the Maxwell and Jeffreys kinds. C. R. Acad. Sci. Paris Sr. I Math. 331, 537–542 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Doubova, A., Fernández-Cara, E.: On the control of viscoelastic Jeffreys fluids. Syst. Control Lett. 61, 573–579 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 19, 2nd edn, p. xxii+749. American Mathematical Society, Providence (2010)Google Scholar
  10. 10.
    Kupferman, R.: On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation. J. Non-Newton. Fluid Mech. 127, 169–190 (2005)CrossRefGoogle Scholar
  11. 11.
    Rogers, R.C., Renardy, M.: An Introduction to Partial Differential Equations. Texts in Applied Mathematics, 13, 2nd edn. Springer, New York (2004)Google Scholar
  12. 12.
    Renardy, M.: Are viscoelastic flows under control or out of control? Syst. Control Lett. 54, 1183–1193 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Renardy, M.: Stressmodes in linear stability of viscoelastic flows. J. Non-Newton. Fluid Mech. 159, 137–140 (2009)CrossRefGoogle Scholar
  14. 14.
    Wloka, J.T., Rowley, B., Lawruk, B.: Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Debanjana Mitra
    • 1
  • Mythily Ramaswamy
    • 2
  • Michael Renardy
    • 3
    Email author
  1. 1.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia
  2. 2.T.I.F.R Centre for Applicable MathematicsBangaloreIndia
  3. 3.Department of MathematicsVirginia TechBlacksburgUSA

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