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Heat and pressure effect in viscous flow through a pipe

I. General formulation and basic solution

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Summary

The basic relations governing viscous flow of liquids with variable transport properties is considered. A dimensional analysis of the problem is given, and the basic equations of generalizedNewtonian flow through a pipe are derived. A perturbation solution is given for the special case of aNewtonian liquid with pressure and temperature dependent viscosity and expressions for the velocity components, pressure distribution and flow rate are obtained for both isothermal and adiabatic wall conditions. It is shown that pressure and viscous heating effects may result in apparently nonNewtonian behaviour.

Zusammenfassung

Es werden die Grundbeziehungen für das viskose Fließen von Flüssigkeiten mit variablen Transporteigenschaften betrachtet. Das Problem wird dimensionsanalytisch untersucht, und die Grundgleichungen des verallgemeinertennewtonschen Fließens durch ein Rohr werden abgeleitet. Für den Spezialfall dernewtonschen Flüssigkeit mit druck- und temperaturabhängiger Viskosität werden durch Anwendung eines Störungsverfahrens sowohl für isotherme als auch adiabatische Bedingungen an der Wand Ausdrücke für die Geschwindigkeitskomponenten, die Druckverteilung und den Volumenstrom angegeben. Man findet, daß die Druck- und Aufheizungs-Effekte nicht-newtonsches Verhalten vortäuschen können.

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Abbreviations

A, B, C :

material constants

a, a 0,b 0 :

empirical coefficients

a i ,b i ,b ik :

constants in eqs. [76]–[79]

Br:

Brinkman number (eq. [34])

b :

material constant (eq. [62])

C i :

constant (eq. [83])

c :

specific heat of incompressible liquid

C v :

specific heat at constant volume

D :

pipe diameter

D()/Dt :

material derivate

E :

internal energy

E * :

activation energy

\(\dot e,\dot e^{ij} \) :

rate of strain tensor

\(\ddot e\) :

convected derivative of the rate of strain tensor

f,f i :

body force vector

f g :

free volume ratio at glass temperature

f 0 :

free volume ratio at temperatureT 0 and atmospheric pressure

G, G p ,G v :

dimensionless parameter (eq. [48])

\(\tilde g,g^{ij} \) :

the metric tensor

I 2 :

the second invariant of the rate of strain tensor

II :

the second invariant of the deviatoric stress tensor

L :

the pipe length

m :

coefficient of power law liquid (eq. [11])

m 0 :

coefficient of power law at atmospheric pressure and temperatureT 0.

n :

power index in a rheological model (eq. [11])

P :

isotropic pressure

P in :

inlet pressure

P 0 :

inlet pressure obtained by neglecting heat and pressure effects

P ij,\(\tilde P\) :

stress tensor

P′ij,\(\tilde P'\) :

the deviatoric part of the stress tensor

\(\bar P\) :

pressure-to-viscous force ratio (eq.[34])

Pr:

Prandtl number (eq.[34]) dimensionless pressure (P/P 0)

p i (i=0,1⋯):

thei′th approximation for pressure in the perturbation solution

Q :

rate of discharge

Q 0 :

rate of discharge, neglecting heat and pressure effects

q k,\(\overrightarrow q \) :

heat flux vector

R :

pipe radius

\(\tilde R\) :

gas constant

R * :

material constant (eq. [23])

Re:

Reynolds' number (eq. [32])

r :

radial coordinate

s :

power index in rheological models (eq. [12]),s = 1/n

T :

temperature

T 0 :

inlet temperature

T m :

mean temperature (eq. [45])

T g :

glass transition temperature

t :

time

V i,V j ,\(\overrightarrow V \) :

velocity vector

V r ,V z :

velocity components in cylindrical coordinates

V :

specific volume

V 0 :

solid-packed volume

V f :

free volume

z :

axial coordinates

α :

perturbation parameter (eq. [60])

\(\hat \alpha _0 \) :

temperature coefficient at reference condition, subscript 0 (eq. [58])

α f :

expansion coefficient of the free volume

β :

dimensionless parameter (eq. [61])

β 0 :

pressure coefficient at reference condition, subscript 0 (eq. [59])

β f :

compressibility coefficient of free volume

γ :

coefficient of temperature sensitivity (eq. [101])

ε :

the ratioR/L

ζ :

the ratioz/L

η :

the ratioη/η 0

η :

viscosity of generalizedNewtonian liquid

η ap :

apparent viscosity (eq. [41])

η :

apparent viscosity obtained by neglecting heat and pressure effects (eq. [42])

θ :

temperature ratioT/T m (eq. [30])

θ i (i = 0, 1, 2):

i′th approximation of temperature in perturbation solution

λ :

thermal conductivity

µ :

Newtonian viscosity

µ * :

Newtonian viscosity at atmospheric pressure and temperatureT 0

Π :

“internal pressure” in eq. [23]

ρ :

ratio of radii (r/R)

\(\tilde \rho \) :

density

σ i (i=0,1,2⋯):

material functions in eq. [13]

τ :

time ratio (eq. [30])

\(\widetilde\psi \) :

stream function eq. [28])

ψ :

dimensionless stream function (eq.[30])

ψ i (i=0,1,2⋯):

i'th approximation for stream function in perturbation parameter

ω :

volume at absolute zero (eq. [23])

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Authors and Affiliations

  1. Department of Mechanics, Technion — Israel Institute of Technology, Haifa, Israel

    N. Galili, R. Takserman-Krozer & Z. Rigbi

  2. Institute of Theoretical and Experimental Physics, University of Stuttgart, 7000, Stuttgart, West Germany

    R. Takserman-Krozer

Authors
  1. N. Galili
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  2. R. Takserman-Krozer
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  3. Z. Rigbi
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Additional information

This paper is based on the D. Sc. Thesis of the first author supervised by the other authors.

With 12 figures and 3 tables

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Galili, N., Takserman-Krozer, R. & Rigbi, Z. Heat and pressure effect in viscous flow through a pipe. Rheol Acta 14, 550–567 (1975). https://doi.org/10.1007/BF01525306

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