Skip to main content
SpringerLink
Log in

Free surface instability of a layer of visco-elastic liquid flowing down an inclined plane

  • Published:
Rheologica Acta Aims and scope Submit manuscript

Summary

The present investigation is concerned with the free-surface instability of an incompressible third-order fluid film flowing down an inclined plane under the action of gravity. Applying linear hydrodynamic stability theory, the eigenvalue problems governing the stability of the flow are derived for the general case where the disturbances propagate along an arbitrary direction oblique to the main flow. Long wave solution has been obtained by the method of asymptotic expansion. SmallReynolds number solutions are obtained only for two-dimensional disturbances. Direct numerical method is used to compute the neutral stability curves, damping and amplification curves for various parametric values for the case of second-order fluid with two-dimensional disturbances when wave numbers are not small. The investigation is limited to the study of the free-surface modes.

Zusammenfassung

Die vorliegende Untersuchung behandelt die Instabilität der freien Oberfläche eines Films einer Flüssigkeit dritten Grades, welche auf einer schiefen Ebene unter dem Einfluß der Schwerkraft abwärts fließt. Durch Anwendung der linearen hydrodynamischen Stabilitätstheorie werden die Eigenwertprobleme abgeleitet, welche die Stabilität der Strömung des allgemeinen Falles beherrschen, bei welchem sich die Störungen entlang einer willkürlichen Richtung schräg zur Hauptströmung ausbreiten. Eine langwellige Lösung wurde mittels der Methode der asymptotischen Expansion erhalten. Lösungen mit kleinenReynoldsschen Zahlen werden nur für zweidimensionale Störungen erhalten. Eine direkte numerische Methode wird angewendet, um die Indifferenzkurve sowie Dämpfungs- und Verstärkungskurven für verschiedene Parameterwerte für den Fall einer Flüssigkeit zweiten Grades mit zweidimensionalen Störungen abzuleiten, wenn die Wellenzahlen nicht klein sind. Die Untersuchungen beschränken sich auf die Gegebenheiten einer freien Oberfläche.

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 :

integration constants

\(\overline{\overline A} _n \) :

nthRivlin-Ericksen tensor

\(\bar a\) :

acceleration field

a ξ,a η,a ζ :

acceleration components in (ξ, η, ζ) directions

c :

phase speed of the disturbance (=c r +ic i )

D :

d/dη

D :

\(\frac{d}{{d\bar z}}\) (after eq. [95])

d :

depth of the liquid layer

E :

cotβ/R

F :

Froude number\(\left( { = \bar u_a /\sqrt {gd} } \right)\)

G :

constant defined in eq. [72]

g :

gravitational acceleration

h :

dimensionless displacement of the free surface from its undisturbed position

\(\overline{\overline I} \) :

unit tensor

i :

\(\sqrt { - 1} \)

K 1,K 2,K 3 :

constants defined in eq. [65]

M :

combined third-order viscoelastic parameter

M 1,M 2 :

constants defined in eq. [74]

n :

dimensionless wavenumber

P :

dimensionless primary flow pressure\(\left( { = \bar P/\rho \bar u_a^2 } \right)\)

\(\bar P\) :

primary flow pressure

p :

hydrostatic pressure

p 1 :

dimensionless pressure\({ = p/\rho \bar u_a^2 }\)

p′ :

pressure perturbation

\(\hat p\) :

pressure amplitude function

R :

Reynolds number

R 2 :

µ 2/ρd 2 second-order viscoelastic parameters

R 3 :

µ 3/ρd 2 second-order viscoelastic parameters

R 4 :

µ 4 ū a /ρd 3 third-order viscoelastic parameters

R 5 :

µ 5 ū a /ρd 3 third-order viscoelastic parameters

R 6 :

µ 6 ū a /ρd 3 third-order viscoelastic parameters

S :

dimensionless surface tension parameter\(\left( { = T/\rho \bar u_a^2 d} \right)\)

S ij :

dimensionless primary stress components (i,j = ξ, η ζ)

\(\overline{\overline s} \) :

stress tensor

s ij :

stress components (i,j = 1,2,3 orx,y,z)

T :

surface tension per unit length

t :

time

U :

dimensionless primary flow velocity\(\left( { = \bar u/\bar u_a } \right)\)

\(\bar u\) :

primary flow velocity

\(\bar u_a \) :

average velocity of primary flow

u 1,u 2,u 3 :

dimensionless velocity components along (x, y, z) directions

u′, v′, w′ :

velocity perturbations

\(\hat u,\hat v,\hat w\) :

velocity amplitude functions

\(\bar v\) :

velocity field

v 1,v 2,v 3 :

velocity components along (x, y, z) directions

\(\bar X\) :

body force per unit mass

x,y,z :

rectangularCartesian coordinates

\(\bar z\) :

1 —η

α :

wavenumber alongx direction

β :

angle of inclination of the plane

\(\bar \beta \) :

3 cotβ + S Rα2

γ :

wavenumber alongz direction

δ :

RR 5

ζ :

z/d

η :

y/d

ξ :

x/d

θ :

angle between the direction of wave propagation andx-axis

ε 1 :

viscosity

ε 2,ε 3 :

second-order viscoelastic coefficients

ε 4,ε 5,ε 6 :

third-order viscoelastic coefficients

ρ :

mass density

σ :

RR 3

τ :

dimensionless time\(\left( { = t\bar u_a /d} \right)\)

τ ij :

dimensionless stress components (i,j = ξ, η, ζ)

\(\tilde \tau _{ij} \) :

dimensionless third-order stress components

τ′ ij :

stress perturbations (i,j = ξ, η, ζ)

\(\tilde \tau '_{ij} \) :

third-order stress perturbations

φ :

amplitude function

ψ :

stream function

i, j :

index number

i :

imaginary part

r :

real part

References

  1. Benjamin, T. B. J. Fluid Mech.2, 554 (1957); Corrigendum3, 657 (1958).

    Google Scholar 

  2. Biermann, M. Rheol. Acta7, 138–163 (1968); Corrigendum, p. 295.

    Google Scholar 

  3. Binnie, A. M. J. Fluid Mech.5, 561 (1959).

    Google Scholar 

  4. Castellana, F. S. andC. F. Bonilla, ASME paper 70-HT-32, 1970.

  5. Coleman, B. D., H. Markowitz, andW. Noll, Viscometric Flows of Non-Newtonian Fluids: Theory and Experiment (Berlin-Heidelberg-New York 1966).

  6. Craik, A. D. J. Fluid Mech.33, 33–38 (1968).

    Google Scholar 

  7. Eringen, A. C., Non-linear Theory of Continuous Media (New York 1962).

  8. Giesekus, H. ZAMM42, 32–61 (1962).

    Google Scholar 

  9. Gupta, A. S. J. Fluid Mech.28, 17–28 (1967).

    Google Scholar 

  10. Gupta, A. S. andL. Rai Proc. Camb. Phil. Soc.63, 527–536 (1967),

    Google Scholar 

  11. Gupta, A. S. andL. Rai J. Fluid Mech.33, 87–91 (1968).

    Google Scholar 

  12. Harris, D. L. andW. H. Reid J. Fluid Mech.20, 95 (1964).

    Google Scholar 

  13. Kapitza, P. I. Zh. Eksperim. i. Tear. Fiz.19, 105 (1949).

    Google Scholar 

  14. Kruegar, E. R., A. Gross andR. C. Di Prima J. Fluid Mech.24, 521 (1966).

    Google Scholar 

  15. Lai, W. Phys. Fluids10, 844 (1967).

    Google Scholar 

  16. Lin, C. C., The Theory of Hydrodynamic Stability (Cambridge 1955).

  17. Lin, S. P. Phys. Fluids10, 59 (1967).

    Google Scholar 

  18. Listrov, A. T., Zh. Prikl. Mekh. Tekh. Fiz1965, No. 5.

  19. Markovitz, H. andB. D. Coleman, Adv. in Applied Mech. (New York-London 1964).

  20. Oldroyd, J. G. Proc. Roy. Soc. (London)A200, 523 (1950).

    Google Scholar 

  21. Orchard, S. E. Appl. Sci. Res.A11, 451–464 (1962).

    Google Scholar 

  22. Rivlin, R. S. andJ. L. Ericksen J. Rat. Mech. Anal.4, 681 (1955).

    Google Scholar 

  23. Squire, H. B. Proc. Roy. Soc. (London)A142, 621 (1933).

    Google Scholar 

  24. Ting, L. L. Free Surface Instability of a Layer of Second-Order Liquid Flowing Down an Inclined Plane. Ph. D. Thesis, Rensselaer Polytechnic Inst., Troy, N. Y. 1967.

    Google Scholar 

  25. Vanoene, H. J. Adhesion3, 1–13 (1972).

    Google Scholar 

  26. Whitaker, S. Ind. and Engr. Chem. Funds.3, 132 (1964).

    Google Scholar 

  27. White, J. L. J. App. Polymer Sci.8, 1129 and 2339 (1964).

    Google Scholar 

  28. White, J. L. andA. B. Metzner Amer. Inst. Chem. Eng. J.11, 325 (1965).

    Google Scholar 

  29. Yih, C. S. Quart. Appl. Math.12, 434 (1955).

    Google Scholar 

  30. Yih, C. S. Phys. Fluids6, 321 (1963).

    Google Scholar 

  31. Yih, C. S. Phys. Fluids8, 1257 (1965).

    Google Scholar 

  32. Yih, C. S., Dynamics of Nonhomogeneous Fluids (New York-London 1965).

  33. Bagehi, K. C. Indian J. Mech. Math.3, 108 (1965).

    Google Scholar 

  34. Willson, A. J. Proc. Camb. Phil. Soc.64, 513 (1968).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical Research Department, Scientific Research Staff, Ford Motor Company, 20 000 Rotunda Drive, 48 121, Dearborn, Michigan, USA

    L. L. Ting

Authors
  1. L. L. Ting
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

With 12 figures and 8 tables

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ting, L.L. Free surface instability of a layer of visco-elastic liquid flowing down an inclined plane. Rheol Acta 14, 503–532 (1975). https://doi.org/10.1007/BF01525304

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01525304

Keywords

Search

Navigation