Skip to main content
SpringerLink
Log in

Turbulent flow of mildly elastic fluids through rotating straight circular tubes

  • Published:
Applied Scientific Research Aims and scope Submit manuscript

Abstract

The turbulent flow of mildly elastic drag reducing fluids through a straight tube rotating around an axis perpendicular to its own is analysed using boundary layer approximations. The momentum integral approach is used and the governing equations have been solved numerically using the Runge-Kutta-Merson method. The influence of the Deborah number on the velocity distribution and the boundary layer thickness has been exemplified through the analysis.

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 :

radius of the rotating tube

A, B :

constant in equation (12)

C :

pressure gradient in z-direction, −∂p*/∂z as given by equation (21)

D :

characteristic angular velocity component in equation (38)

D 0 :

dimensionless value of D as given by equation (48)

D 1 :

angular velocity component defined by equation (63)

D 3 :

coefficient in equation (63)

De:

Deborah number (\(\left( {\frac{{\theta _1 u^{*2} }}{v}} \right)\))

f :

friction factor for turbulent flow in stationary straight tubes

f rt :

friction factor for rotating tubes given by equation (61)

f 1, f 2, f 3, f 4, f 5 :

functions defined by equations (52)–(56)

F(x):

arbitrary function of x defined by equation (11)

p :

pressure appearing in equation (4)

p*:

pressure in rotating system defined by equation (4)

r :

distance in the radial direction

Re:

Reynolds number defined by equation (28)

u :

radial velocity component

u*:

friction velocity (\(\left( { = \sqrt {\frac{{\tau _w }}{\rho }} } \right)\))

U :

velocity component in the x-direction

U 1 :

velocity scale near pipe wall as given by equation (40)

v :

axial velocity component

V :

velocity component in the y-direction

w :

velocity component in the z-direction

w 0 :

dimensionless value of w 1 given by equation (49)

w 1 :

axial velocity component at the edge of the boundary layer

w m :

value of the W at the centre of the tube

W :

velocity component in the z-direction

x, y, z :

three directions of the co-ordinate system

α, β :

functions of De for dilute drag reducing fluids, respectively appearing in equation (36)

α*:

integral defined by equation (59)

δ :

boundary layer thickness

δ :

dimensionless boundary layer thickness defined by equation (47)

δ 1 :

dimensionless boundary layer thickness defined by equation (62)

δ 2 :

coefficient in equation (62)

θ :

angular co-ordinate

θ 1 :

fluid relaxation time

μ :

viscosity of a drag reducing fluid/Newtonian fluid

ν :

kinematic viscosity

ξ :

radial distance from the wall (=a−r)

ρ :

density of the fluid

τ w :

wall shear stress

τ rθ , τ rz :

shear stress components defined by equations (34) and (35) respectively

ψ :

shear function defined in equation (10)

Ω:

angular velocity

References

  1. J.L. Lumley, in: W.R. Sears (ed.) Annual Review of Fluid Mechanics, Annual Review Inc., Palo Alto, Cal., 1 (1969) 367.

    Google Scholar 

  2. G.F. Gadd, Encyclopedia of Polymer Science & Tech. 15 (1971).

  3. R. Darby, Naval Research Laboratory, Memo Report 2446 (1972).

  4. J.W. Hoyt, Trans. ASME, J. Basic Engg. 94 (1972) 258.

    Google Scholar 

  5. M.T. Landahl, Proc. 18th Intern. Theor. and Appl. Mechanics, Moscow, ed. E. Becker and G.K. Mikhailov, p. 177 (1973).

  6. J.L. Lumley, J. Polym. Sci., Macromolecular Reviews, 7 (1973) 263.

    Google Scholar 

  7. M.C. Fisher and R.L. Ash, NASA TMX-2894 (1974).

  8. J.A. Palyvos, Report No. 741, Thermodynamics and Transport Phenomena Laboratory, National Technical University, Athens, 147, Greece (1974).

  9. P.S. Virk, AIChE 21 (1975) 625.

    Article  Google Scholar 

  10. R.H.J. Sellin, J.W. Hoyt and O. Scrivener, Part I: Basic aspects, J. Hydraulic Res. 20 (1982) 29.

    Google Scholar 

  11. R.H.J. Sellin, J.W. Hoyt, J. Pollert and O. Scrivener, Part II: Present applications and future proposals, J. Hydraulic Res. 20 (1982) 235.

    Google Scholar 

  12. A.V. Shenoy, Colloid Polym. Sci. 262 (1984) 319.

    Article  Google Scholar 

  13. S.H. Barua, Quart. J. Mech. Appl. Math. 14 (1963) 61.

    MathSciNet  Google Scholar 

  14. G.S. Benton, J. Fluid Mech. 23 (1956) 123.

    MATH  MathSciNet  Google Scholar 

  15. G.S. Benton and D. Boyer, J. Fluid Mech. 26 (1966) 69.

    ADS  Google Scholar 

  16. R.W. Gunn, B. Mena K. Walters, and ZAMP 25 (1974) 591.

    Google Scholar 

  17. H.M. Ito and K. Nanbu, J. Basic Engg., Trans. ASME, 383 (1971).

  18. I. Mori and W. Nakayama, Int. J. Heat Mass Transfer 11 (1968) 1027.

    Google Scholar 

  19. K. Nanbu, Memoirs of Institute of High Speed Mechanicis, Tohoku University, Japan, 25, No. 262 (1970).

    Google Scholar 

  20. D.T. Jones, J.R. Trevors and K. Walters, ZAMP 18 (1967) 774.

    Google Scholar 

  21. R.A. Mashelkar and G.V. Devarajan, Trans. Instn. Chem. Engrs. 54 (1976) 100.

    Google Scholar 

  22. R.A. Mashelkar and G.V. Devarajan, Trans. Instn. Chem. Engrs. 54 (1976) 108.

    Google Scholar 

  23. R.A. Mashelkar and G.V. Devarajan, Trans. Instr. Chem. Engrs. 55 (1977) 29.

    Google Scholar 

  24. A.V. Shenoy, V.R. Ranade and J.J. Ulbrecht, Chemical Engg. Communications 5 (1980) 269.

    Google Scholar 

  25. A.V. Shenoy and R.A. Mashelkar, Int. Eng. Chem. Proc. Des. Dev. 22 (1983) 165.

    Google Scholar 

  26. D.W. Dodge and A.B. Metzner, AIChE J. 5 (1959) 189.

    Article  Google Scholar 

  27. G.V. Devarajan, Ph.D. Thesis, University of Salford, U.K. (1976).

Download references

Author information

Authors and Affiliations

  1. Chemical Engineering Division, National Chemical Laboratory, 411008, Pune, India

    A. V. Shenoy

Authors
  1. A. V. Shenoy
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

NCL Communication No. 3354.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shenoy, A.V. Turbulent flow of mildly elastic fluids through rotating straight circular tubes. Appl. Sci. Res. 43, 39–54 (1986). https://doi.org/10.1007/BF00385727

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00385727

Keywords

Search

Navigation