Summary
A new framework for viewing steady extensional flow is presented: Steady orthogonal stagnation flow is an ideal which one should strive to reach in an actual experiment. The kinematics and the stress boundary conditions are developed for stagnation flow in general and in two special cases: the impingement of two circular streams and of two planar sheets. Contemplating this ideal clarifies the advantages and disadvantages of current experiments, thereby pointing the way towards new experiments; a number are suggested. Axisymmetric (with\(\dot \varepsilon _x \) > 0) and planar stagnation flow within a lubricated die look particularly promising. Some preliminary experimental results are given for uniaxial extension of a polyacrylamide solution in a water lubricated die.
Zusammenfassung
Es werden neue Leitlinien zur Beurteilung von stationären Dehnströmungs-Experimenten vorgeschlagen: Die stationäre orthogonale Staupunktströmung wird als ein Ideal herausgestellt, das man in Experimenten zu erreichen versuchen sollte. Die Kinematik und die Randbedingung für das Spannungsfeld werden für allgemeine Staupunktströmungen hergeleitet und auf zwei Sonderfälle angewendet: die axialsymmetrische und die ebene Staupunktströmung. Der Vergleich dieser idealen Strömung mit wirklichen Experimenten verdeutlicht deren jeweilige Vor- und Nachteile und weist so auf neue Experimente hin. Mehrere neue Experimente werden vorgeschlagen. Die axialsymmetrisch nach außen gerichtete und die ebene Staupunktströmung in Düsen mit geschmierten Wandungen erscheinen besonders erfolgversprechend. Einige vorläufige experimentelle Ergebnisse für die einachsige Dehnströmung einer Polyacrylamidlösung in einer mit Wasser geschmierten Düse werden vorgestellt.
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Abbreviations
- a —:
-
ratio of extension rates\({{\dot \varepsilon _z } \mathord{\left/ {\vphantom {{\dot \varepsilon _z } {\dot \varepsilon _y }}} \right. \kern-\nulldelimiterspace} {\dot \varepsilon _y }}\)
- A m 2 :
-
cross-sectional area of fluid stream
- F N:
-
force on cross-section of stream
- g m/s2 :
-
gravitational acceleration
- h m:
-
thickness of planar stream, figure 3
- l m:
-
length of planar stream, figure 3
- n —:
-
normal vector to stream surface
- p Pa:
-
pressure, eqs. [19], [20]
- r m:
-
coordinate
- R m:
-
radius of prescribed circular cross-section at positionx = ±x 0, eq. [5]
- Δt s:
-
residence time, eq. [10]
- t —:
-
tangential vector on stream surface, eqs. [25], [26]
- T Pa:
-
stress tensor
- v m/s:
-
velocity
- \(\dot V\) m3/s:
-
volume flow rate
- x,y,z m:
-
principal coordinates
- \(\dot \gamma \) s−1 :
-
rate of strain tensor, eq. [3]
- \(\dot \varepsilon _i \) s−1 :
-
principal rate of extension
- η Pa s:
-
shear viscosity
- τ Pa:
-
extra stress tensor
- n :
-
normal to stream surface
- t :
-
tangential to stream surface
- 0:
-
reference position in flow field, eq. [4]
- x, y, z :
-
direction of coordinates
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Winter, H.H., Macosko, C.W. & Bennett, K.E. Orthogonal stagnation flow, a framework for steady extensional flow experiments. Rheol Acta 18, 323–334 (1979). https://doi.org/10.1007/BF01515825
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DOI: https://doi.org/10.1007/BF01515825