Airy wave theory: Difference between revisions

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'''Palierne equation''' connects the [[dynamic modulus]] of [[emulsion]]s with the dynamic modulus of the two [[Phase (matter)|phases]], size of the droplets and the interphase [[surface tension]]. The equation can also be used for [[Suspension (chemistry)|suspensions]] of [[viscoelastic]] solid particles in viscoelastic fluids.<ref name=Pal>{{cite journal |doi=10.1002/pen.21065 |last=Pal |first=Rajinder |title=A New Viscoelastic Model for Emulsions and Suspensions|journal=Polymer Engineering and Science |volume=48 |year=2008 |issue=7 |month=July |pages=1250..1253}}</ref> The equation is named after French [[rheology|rheologist]] [[Jean-François Palierne]] who has proposed the equation in 1991.<ref>{{cite journal |doi=10.1007/BF01331356 |last=Palierne |first=Jean-François |title=Linear Rheology of viscoelastic emulsions with interfacial tension|journal=Rheologica Acta |volume=29 |issue=3 |year=1991 |pages=204}}</ref>
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For the dilute emulsions Palierne equation looks like:
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:<math>G^*=G^*_m(1+5\phi H^*)</math>
where <math>G^*</math> is the dynamic modulus of the emulsion, <math>G^*_m</math> is the dynamic modulus of the continuous phase (matrix), <math>\phi</math> is the volume fraction of the disperse phase and the <math>H^*</math> is given as
:<math>H^*=\frac{(G^*_d-G^*_m)(19G^*_d+16G^*_m)+(4\sigma/R)(5G^*_d+2G^*_m)}{(2G^*_d+3G^*_m)(19G^*_d+16G^*_m)+(40\sigma/R)(G^*_d+G^*_m)}</math>
where <math>G^*_d</math> is the dynamic modulus of the disperse phase, <math>\sigma</math> is the surface tension between the phases and <math>R</math> is the [[radius]] of the droplets.<ref name=Pal/>
 
For the suspension of solid particles the value of <math>H^*</math> is given as<ref name=Pal/>
:<math>H^*=\frac{G^*_d-G^*_m}{2G^*_d+3G^*_m}</math>
 
The Palierne equation is usually extended for the finite volume concentrations of the  disperse phase <math>\phi</math> as:<ref name=Pal/>
:<math>G^*=G^*_m\frac{1+3\phi H^*}{1 - 2 \phi H^*}</math>
 
==References==
{{reflist}}
 
[[Category:Non-Newtonian fluids]]
[[Category:Colloidal chemistry]]
[[Category:Composite materials]]

Revision as of 14:06, 27 February 2014

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