Skip to main content
SpringerLink
Log in

Flow of a generalized second grade fluid between heated plates

  • Contributed Papers
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

We examine the fully developed flow of a generalized fluid of second grade between heated parallel plates, due to a pressure gradient along the plate. The constant coefficient of shear viscosity of a fluid of second grade is replaced by a shear dependent viscosity with an exponentm. If the normal stress coefficients are set equal to zero, this model reduces to the standard power-law model. We obtain the solution for the case when the temperature changes only in the direction normal to the plates for the two most commonly used viscosity models, i.e. (i) when the viscosity does not depend on temperature, and (ii) when the viscosity is an exponentially decaying function of temperature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

A 1,A 2 :

Kinematical tensor

b :

Body force

C :

Dimensionless parameter related to the pressure gradient

h :

Separation between the plates

L :

Velocity gradient

m :

Power-law index

M :

Constant appearing in the Reynolds viscosity model

p :

Pressure field

\(\hat p,\hat \bar p\) :

Modified pressure field

q :

Heat flux vector

r :

Radiant heating

T :

Cauchy's stress tensor

l :

Unit tensor

v :

Velocity vector

V :

Characteristic velocity

x :

Axis along the plate

y :

Axis perpendicular to the plate

α1, α2 :

Normal stress coefficient

ε:

Specific internal energy

Γ:

Dimensionless parameter related to the viscous dissipation

ϕ:

Conservative body force field

η:

Specific entropy

κ:

Thermal conductivity

μ:

Coefficient of viscosity

μ0 :

Reference viscosity

Π:

Second invariant of the stretching tensor

θ:

Temperature

θ1 :

Temperature of the lower plate

θ2 :

Temperature of the upper plate

ϱ:

Density

ψ:

Specific Helmholtz free energy

div:

Divergence

grad:

Gradient

tr:

Trace

References

  1. Tsai, C. Y., Novack, M., Roffe, G.: Rheological and heat transfer characteristics of flowing coal-water mixtures. Final Report, DOE/MC/23255-2763 (1988).

  2. Ekmann, J. M., Wildman, D. J., Chen, J. L. S.: Laminar flow studies of highly loaded suspensions in horizontal pipes. Second International Symp. Slurry Flows, ASME FED38, pp. 85–92 (1986).

    Google Scholar 

  3. Schowalter, W. R.: Mechanics of non-Newtonian fluids. New York: Academic Press 1978.

    Google Scholar 

  4. Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Handbuch der Physik (Flügge, S. ed.) III/3. Berlin Heidelberg New York: Springer 1965.

    Google Scholar 

  5. Rivlin, R. S., Ericksen, J. L.: Stress deformation, relations for isotropic materials. J. Rat. Mech. Anal.4, 323–425 (1955).

    Google Scholar 

  6. Man, C. S., Shields, D. H., Kjartanson, B., Sun, Q. K.: Creep of ice as a fluid of complexity 2: the pressuremeter problem. Proc. 10th CANCAM,1, pp. A347-A348 (1985).

    Google Scholar 

  7. Man, C. S., Sun, Q. K.: On the significance of normal stress effects in the flow of glaciers. J. Glaciology33, 268–273 (1987).

    Google Scholar 

  8. Szeri, A. Z., Rajagopal, K. R.: Flow of a non-Newtonian fluid between heated parallel plates. Int. J. Non-Linear Mech.20, 91–101 (1985).

    Google Scholar 

  9. Rajagopal, K. R.: Thermodynamics and stability of non-Newtonian fluids. Ph. D. Thesis, University of Minnesota, USA (1978).

    Google Scholar 

  10. Rajagopal, K. R.: Boundedness and uniqueness of fluids of the differential type. Acta Ciencia India18, (1982).

  11. Dunn, J. E.: On the free energy and stability of nonlinear fluids. J. Rheology26, 43–68 (1982).

    Google Scholar 

  12. Dunn, J. E., Rajagopal, K. R.: A critical review and analysis of fluid of the differential type (to appear).

  13. Dunn, J. E., Fosdick, R. L.: Thermodynamics, stability, and, boundedness of fluids of complexity 2 and fluids of second grade. Arch. Rat. Mech. Anal.56, 191–252 (1974).

    Google Scholar 

  14. Straughan, B.: Energy stability in the Benard problem for a fluid of second grade. J. Appl. Math. Phys. (ZAMP)34, 502–508 (1983).

    Google Scholar 

  15. Rajagopal, K. R., Na, T. Y.: Natural convection flow of a non-Newtonian fluid between two vertical flat plates. Acta Mech.54, 239–244 (1985).

    Google Scholar 

  16. Massoudi, M., Christie, I.: Natural convection flow of a non-Newtonian fluid between two concentric vertical cylinders. Acta Mech.82, 11–19 (1990).

    Google Scholar 

  17. Gupta, G., Massoudi, M.: Heat transfer and flow of a modified second grade fluid. In: Fundamentals of heat transfer in non-Newtonian fluids. ASME HTD174, 7–12 (1991).

    Google Scholar 

  18. Winter, H. H.: Viscous dissipation in shear flows of molten polymers. Adv. Heat Transfer13, 205–267 (1977).

    Google Scholar 

  19. Hutter, K.: Theoretical glaciology. Dordrecht: Reidel 1983.

    Google Scholar 

  20. Mansutti, D., Rajagopal, K. R.: Flow of a shear-thinning fluid between intersecting planes. Int. J. Non-Linear Mech.26, 769–775 (1991).

    Google Scholar 

Download references

Author information

Author notes

  1. G. Gupta

    Present address: Department of Mechanical Engineering, University of Pittsburgh, 15261, Pittsburgh, PA, USA

  2. M. Massoudi

    Present address: U.S. Department of Energy, Pittsburgh Energy Technology Center, P.O. Box 10940, 15236, Pittsburgh, PA, USA

Authors and Affiliations

    Authors
    1. G. Gupta
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. M. Massoudi
      View author publications

      You can also search for this author in PubMed Google Scholar

    Rights and permissions

    Reprints and permissions

    About this article

    Cite this article

    Gupta, G., Massoudi, M. Flow of a generalized second grade fluid between heated plates. Acta Mechanica 99, 21–33 (1993). https://doi.org/10.1007/BF01177232

    Download citation

    • Received:

    • Revised:

    • Issue Date:

    • DOI: https://doi.org/10.1007/BF01177232

    Keywords

    Search

    Navigation