Abstract
The main objective of this work is to demonstrate the agreement between the two-fluid linear Langevin formulation and that described by the extended irreversible thermodynamics (EIT). The two-fluid model, originally proposed by de Gennes, has been widely analyzed by many authors in various flow situations, especially to compare predictions with experimental data of the structure factor in many complex flows. The canonical Langevin equations together with the fluctuation-dissipation theorem ensure consistent thermodynamic behavior for constitutive equations. Therefore, agreement between the EIT formulation and the two-fluid Langevin equations demonstrates the thermodynamic consistency of the EIT formulation. Extension of this analysis to include normal stresses is also considered.
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References
Bautista F, Soltero JFA, Pérez-López JH, Puig JE, Manero O (2000) On the shear banding flow of elongated micellar solutions. J Non-Newtonian Fluid Mech 94:57–66. https://doi.org/10.1016/S0377-0257(00)00128-2
Bautista F, Soltero JFA, Macías ER, Puig JE, Manero O (2002) Irreversible thermodynamics approach and modeling of shear banding flow of wormlike micelles. J Phys Chem B 106:13018–13026. https://doi.org/10.1021/jp0206370
Bautista F, Pérez-López JH, García-Sandoval JP, Puig JE, Manero O (2007) Stability analysis of shear banding flow with the BMP model. J Non-Newtonian Fluid Mech 144:160–169. https://doi.org/10.1016/j.jnnfm.2007.04.001
Bautista F, Muñoz M, Castillo-Tejas J, Pérez-López JH, Puig JE, Manero O (2009) Critical phenomenon analysis of shear banding flow in polymer-like micellar solutions.1. Theoretical approach. J Phys Chem B 113:16101–16109. https://doi.org/10.1021/jp906310k
Bautista F, Fernández V, Macías ER, Pérez-López JH, Escalante JI, Puig JE, Manero O (2012) Experimental evidence of the critical phenomenon of shear banding flow in polymer-like micellar solutions. J Non-Newtonian Fluid Mech 177:89–96. https://doi.org/10.1016/j.jnnfm.2012.03.006
Brochard F, de Gennes P-G(1977) Dynamical scaling of polymers in theta solvents. Macromolecules 10:1157–1161. https://doi.org/10.1021/ma60059a048
Cromer M, Villet MC, Fredrickson GH, Leal LG, Stepanyan R, Bulters MJH (2013) Concentration fluctuations in polymer solution under extensional flow. J Rheol 57:1211–1235. https://doi.org/10.1122/1.4808411
de Gennes PG (1976a) Dynamics of entangled polymer solutions I The Rouse model. Macromolecules 1976(9):587–593. https://doi.org/10.1021/ma60052a011
de Gennes PG (1976b) Dynamics of entangled polymer solutions II Inclusion of hydrodynamic interactions. Macromolecules 9:594–598. https://doi.org/10.1021/ma60052a012
Doi M, Onuki A (1992) Dynamic coupling between stress and composition in polymer solutions and blends. J Phys II France 2:1631–1656. https://doi.org/10.1051/jp2:1992225
Fierro C, Medina-Torres L, Bautista F, Herrera-Valencia EE, Calderas Manero O (2021) The structure factor in flowing worm-like micellar solutions. J Non-Newtonian Fluid Mech 289(17):104469. https://doi.org/10.1016/j.jnnfm.2020.104469
García-Sandoval JP, Bautista F, Puig JE, Manero O (2019) Inhomogeneous flow of wormlike micelles: predictions of the generalized BMP model with normal stresses. Fluids 4:45. https://doi.org/10.3390/fluids4010045
García-Sandoval JP, Bautista F, Puig JE, Manero O (2012) Inhomogeneous flow and shear banding formation in micellar solutions: predictions of the BMP model. J Non-Newtonian Fluid Mech 179-180:43–54. https://doi.org/10.1016/j.jnnfm.2012.05.006
García-Sandoval JP, Bautista F, Puig JE, Manero O (2017) The interface migration in shear banded micellar solutions. Rheol Acta 56:765–778. https://doi.org/10.1007/s00397-017-1031-2
Helfand E, Fredrickson HG (1989) Large fluctuations in polymer solutions under shear. Phys Rev Lett 62:2468–2471. https://doi.org/10.1103/PhysRevLett.62.2468
Ji H, Helfand E (1995) Concentration fluctuations in sheared polymer solutions. Macromolecules 28:3869–3880. https://doi.org/10.1021/ma00115a017
Jou D, Camacho J, Grmela M (1991) On the non-equilibrium thermodynamics of non-Fickian diffusion. Macromolecules 24:3597–3602. https://doi.org/10.1021/ma00012a021
Jou D, Casas-Vázquez J, Lebon G (2010) Extended irreversible thermodynamics. 4th Ed. Berlin: Springer
Jou D, Casas Vázquez J, Criado-Sancho M (2011) Thermodynamics of fluids under flow. 2nd Ed. Berlin: Springer
Lai J, Fuller GG (1994) Structure and dynamics of concentration fluctuations in a polymer blend solution under shear flow. J Polym Sci B Polym Phys 32:2461–2474. https://doi.org/10.1002/polb.1994.090321503
Manero O, Bautista F, Soltero JFA, Puig JE (2002) Dynamics of worm-like micelles: the Cox-merz rule. J Non-Newtonian Fluid Mech 106(1):1–15. https://doi.org/10.1016/S0377-0257(02)00082-4
Manero O, Perez-López JH, Puig JE, Bautista F (2007) A thermodynamic approach to rheology of complex fluids: the generalized BMP model. J Non-Newtonian Fluid Mech 146:22–29. https://doi.org/10.1016/j.jnnfm.2007.02.012
Milner ST (1993) Dynamical theory of concentration fluctuations in polymer solutions under shear. Phys Rev E 48:3674–3691. https://doi.org/10.1103/PhysRevE.48.3674
Onuki A (1989) Elastic effects in the phase transition of polymer solutions under shear flow. Phys Rev Lett 62:2472–2475. https://doi.org/10.1103/PhysRevLett.62.2472
Onuki A (1990) Dynamic equations of polymers with deformations in semidilute regions. J Phys Soc Jpn 59:3423–3426. https://doi.org/10.1143/JPSJ.59.3423
Onuki A (1997) Phase transitions of fluids in shear flow. J Phys Condens Matter 9:6119–6157. https://doi.org/10.1088/0953-8984/9/29/001
Öttinger HC, Grmela M (1997) Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys Rev E 56:6633–6655
Saito S, Takenaka M, Toyoda N, HashimotoT (2001) Structure factor of semidilute polymer solution under continuous shear flow: numerical analysis of a two-fluid model and comparison with experiments. Macromolecules 34:6461–6473. https://doi.org/10.1021/ma0021390
Acknowledgements
We acknowledge the financial support from project IN 100620 from DGAPA-UNAM and the scholarship CONACYT CVU 896737.
Glossary
\( \underset{\_}{\underset{\_}{B}} \)Strain tensor
\( \underset{\_}{\underset{\_}{L}} \)Rate of deformation tensor
\( \underset{\_}{\underset{\_}{D}} \)Symmetric part of the rate of strain tensor \( \underset{\_}{\underset{\_}{L}} \)
\( \underset{\_}{J} \)Mass flux
\( \nabla {\underset{\_}{J}}^s \)Is the symmetric part of the tensor \( \nabla \underset{\_}{J} \)
φ ≡ η−1 Is the fluidity (inverse of the viscosity)
\( {\varphi}_s\equiv {\eta}_s^{-1} \)Solvent fluidity
φo Fluidity at zero shear rate
φ∞ Fluidity at high shear rates
G0 Elastic modulus
β0 Phenomenological coupling coefficient
β1 Coupling coefficient of the mass flux
β2 Phenomenological coefficient
\( {\underset{\_}{\underset{\_}{\sigma}}}_p \)Stress tensor
\( {\underset{\_}{\underset{\_}{\overset{\nabla }{\sigma}}}}_p \)Upper-convected derivative of the stress tensor
λ Relaxation time related to structure building
λmShear-dependent relaxation time
k Kinetic constant
μ Chemical potential
μNeqNon-equilibrium chemical potential
ϕ Dispersed phase concentration
\( \underset{\_}{v} \)Velocity vector
ρ Density
p Hydrostatic pressure
π Osmotic pressure
A Structural parameter
F Free energy
χ Osmotic susceptibility
ζ Friction coefficient
kb Boltzmann constant
T Temperature
Dc Cooperative diffusion coefficient
Kos Osmotic modulus
DG Diffusion coefficient of the micellar network
Θ Thermal noise
δ Denotes property fluctuation
δ(t − t′) Dirac delta function
z \( =\nabla \nabla :\underset{\_}{\underset{\_}{\mathit{\mathsf{\sigma}}}} \)
\( S\left(\underset{\_}{q}\right) \)Structure factor
\( \underset{\_}{q} \)Scattering vector
\( \hat{q} \)Unit scattering vector
ω Frequency
\( \left\langle \underset{\_}{r}\ \underset{\_}{r}\right\rangle \)Configuration tensor
\( {\xi}_{ve}^2 \)Viscoelastic correlation length
\( \underset{\_}{\underset{\_}{\mathcal{I}}} \)Correlation matrix
\( {\mathcal{L}}_{ij} \)Onsager kinetic coefficients
\( {\mathcal{I}}_q \)Steady-state structure factor
N1 First normal stress difference
N2 Second normal stress difference
ψ Normal stress coefficient
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Fierro, C., Bautista, F., García-Sandoval, J.P. et al. Compatibility of the generalized BMP model and the two-fluid Langevin formulations. Rheol Acta 60, 751–761 (2021). https://doi.org/10.1007/s00397-021-01290-4
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DOI: https://doi.org/10.1007/s00397-021-01290-4