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.
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Abbreviations
- e ij :
-
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
- h 1, h2, h3 :
-
metric coefficients
- I 1, I 2, I 3 :
-
invariants
- m :
-
constant
- P :
-
pressure, dynes/cm2
- r :
-
radius, cm
- t :
-
time, sec
- \(\mathop v\limits^ - \) :
-
velocity vector, cm/sec
- v 1, v 2, v 3 :
-
velocities in the x 1 x 2, and x 3 directions, respectively, cm/sec
- v n(t):
-
velocity of the nth node, cm/sec
- x 1, x 2,x 3 :
-
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
- v 0, v l :
-
constant kinematic viscosities, cm2/sec, cm2 secm-2
- ϱ:
-
density, g/cm3
- σij :
-
stress tensor
- Ф:
-
fluid dilation
References
Hughes, W. F. and E. W. Gaylord, Basic Equations of Engineering Science, p. 9 (1964).
Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Second Printing, pp. 101-103 (London-New York 1962).
Skelland, A. H. P., Non-Newtonian Flow and Heat Transfer, p. 4 (London-New York 1966).
Rivlin, R. S., Proc. Ray. Soc. A 193, 261 (London 1948).
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© 1975 Springer-Verlag Berlin Heidelberg
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Cobble, M.H., Smith, P.R., Mulholland, G.P. (1975). Nonlinear motion equations for a Non-Newtonian incompressible fluid in an orthogonal coordinate system. In: Vallet, G., Meskat, W. (eds) Rheological Theories · Measuring Techniques in Rheology Test Methods in Rheology · Fractures Rheological Properties of Materials · Rheo-Optics · Biorheology. Steinkopff, Heidelberg. https://doi.org/10.1007/978-3-662-41458-3_23
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DOI: https://doi.org/10.1007/978-3-662-41458-3_23
Publisher Name: Steinkopff, Heidelberg
Print ISBN: 978-3-7985-0424-0
Online ISBN: 978-3-662-41458-3
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