Advertisement

Nonlinear motion equations for a Non-Newtonian incompressible fluid in an orthogonal coordinate system

  • M. H. Cobble
  • P. R. Smith
  • G. P. Mulholland

Abstract

Much interest has developed in recent years in the flow of non-Newtonian fluids and many excellent articles and books have appeared upon this subject (1, 2, 3, 4). However, most of the solutions which have been published have been for steady flows. In this paper the nonlinear equations of motion in general orthogonal coordinates are developed for the non-steady flow of an incompressible non-Newtonian fluid. A solution is presented for the non-steady Hagen-Poiseuille flow through a circular pipe.

Nomenclature

eij

strain rate tensor

\(\mathop F\limits^ - \)

body force density, dynes/cm3

F1, F2, F3

components of body force density, dynes/cm3

g

acceleration of gravity

H

function of time

h1, h2, h3

metric coefficients

I1, I2, I3

invariants

m

constant

P

pressure, dynes/cm2

r

radius, cm

t

time, sec

\(\mathop v\limits^ - \)

velocity vector, cm/sec

v1, v2, v3

velocities in the x 1 x 2, and x 3 directions, respectively, cm/sec

vn(t)

velocity of the nth node, cm/sec

x1, x2,x3

coordinate directions

z

coordinate, cm

δ

unit tensor

δij

Kronecker delta

Δij

2eij

nabla

εijk

alternating unit tensor

η

non-Newtonian viscosity, dynes/cm2

η0, η1

constant viscosities, dynes sec/cm2, dynes secm/cm2

θ

angle, radians

v0, vl

constant kinematic viscosities, cm2/sec, cm2 secm-2

ϱ

density, g/cm3

σij

stress tensor

Ф

fluid dilation

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1).
    Hughes, W. F. and E. W. Gaylord, Basic Equations of Engineering Science, p. 9 (1964).Google Scholar
  2. 2).
    Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Second Printing, pp. 101-103 (London-New York 1962).Google Scholar
  3. 3).
    Skelland, A. H. P., Non-Newtonian Flow and Heat Transfer, p. 4 (London-New York 1966).Google Scholar
  4. 4).
    Rivlin, R. S., Proc. Ray. Soc. A 193, 261 (London 1948).ADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1975

Authors and Affiliations

  • M. H. Cobble
    • 1
  • P. R. Smith
    • 1
  • G. P. Mulholland
    • 1
  1. 1.Mechanical Engineering DepartmentNew Mexico State UniversityLas CrucesUSA

Personalised recommendations