Summary
Equations describing the laminar, gravitationally accelerated flow of power-law liquid films flowing inside of a vertical pipe have been studied mathematically. The flow depends on the values of the flow behavior index, the initial film thickness, the magnitude of the initial uniform velocity, and the distance traveled by the fluid along the vertical wall. Values for the film thickness, boundary-layer thickness as well as entrance length are obtained numerically. The numerical solutions show the magnitude of deviations from the flow of Newtonian fluids that can be expected with power-law fluid characteristics.
Zusammenfassung
Es werden die Gleichungen, welche eine laminare schwerkraftbeschleunigte Filmströmung von Ostwald-deWaele-Flüssigkeiten längs der Innenseite eines senkrechten Rohrs beschreiben, mathematisch analysiert. Die Strömung hängt vom Fließindex, der anfänglichen Filmdicke, der Größe der gleichförmigen Anfangsgeschwindigkeit und der Laufstrecke ab. Die Werte von Filmdicke und Eintrittslänge werden numerisch bestimmt. Diese Lösungen zeigen die Größenordnung der Abweichungen von der Strömung newtonscher Flüssigkeiten, die bei Ostwald-deWaele-Flüssigkeiten erwartet werden können.
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Abbreviations
- G :
-
Dimensionless parameter, (3U 0)n m/(gh n+10 )
- g :
-
Gravitational acceleration, ft/sec2
- h :
-
Liquid film thickness, ft;h 0 initial value;\(\bar h = h/h_0\)
- k :
-
Fluid consistency index, lb f sec nft−2
- m :
-
k/ϱ, lb f lb -1 m secnft
- n:
-
Flow behavior index, dimensionless
- R :
-
Radius of the considered curved surface, ft
- r :
-
Radial distance in cylindrical coordinates, ft
- U 0 :
-
Initial velocity, ft/sec
- U s :
-
Free stream velocity, ft/sec
- V r :
-
Velocity component inr-direction in cylindrical coordinates, ft/sec
- V z :
-
Velocity component inz-direction in cylindrical coordinates, ft/sec
- z :
-
Coordinate axis, distance from original in direction of flow, ft;\(\bar z = z/h_0\), dimensionless
- z * :
-
Entrance length, ft;\(\bar z^ * = z^ * /h_0\), dimensionless
- δ :
-
Boundary-layer thickness, ft;\(\bar \delta = \delta /h_0\);δ * =δ at end of entrance region;\(\bar \delta ^ * = \delta ^ * /h_0\)
- η :
-
Defined by Eq. [8], dimensionless;η * =η when\(\bar z = \bar z^ *\)
- ξ :
-
Dimensionless parameter, (R −r)/δ
- ϱ :
-
Fluid density, lb m /ft3
- τ rz :
-
Shear stress inz-direction on surface normal tor, lb f /ft2
References
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Ibid, p. 81.
Bruley, D. F. A.I.Ch.E. J.11, 945 (1965).
Chien, S. F. J. Appl. Mech.33, 222 (1966).
Hassan, N. A. J. Appl. Mech.34, 535 (1968).
Haugen, R. J. Appl. Mech.35, 631 (1968).
Yang, T. M. T. andD. W. Yarbrough J. Appl. Mech.40, 290 (1973).
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Yang, T.M.T., Yarbrough, D.W. Laminar flow of non-Newtonian liquid films inside a vertical pipe. Rheol Acta 19, 432–436 (1980). https://doi.org/10.1007/BF01524016
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DOI: https://doi.org/10.1007/BF01524016