Explaining and modelling the rheology of polymeric fluids with the kinetic theory Dmitry Shogin University of Stavanger The National IOR Centre of Norway IOR Norway 2016 Workshop April 25, 2016
Overview Introduction The FENE Dumbbell Model Analytical results Steady shear flow Steady shearfree flows Experimental results
Polymer solutions Polymer solutions behave as non-newtonian fluids: The viscous stress tensor τ ij describes the transfer of momentum in the fluid; knowledge of this tensor is required to construct and solve the equations of fluid dynamics. For Newtonian fluids, e. g. water and oil, τ ij is proportional to the velocity gradients: ( vi τ ij = η + v ) j, x j x i where the viscosity η is a constant. Using this relation leads to the well-known Navier-Stokes equations of fluid dynamics. For polymers it is essentially different.
Viscosity Polymer solutions demonstrate shear-thinning, dvs. viscosity drops as shear rate grows.
Normal stress Polymer solutions contract along the streamlines; Non-zero stress in a perpendicular direction is produced; In a flow between two parallel plates, a force is needed to hold the plates in place: For Newtonian fluids, the normal stress is identically zero.
Rod-climbing
The challenge Many (very useful) correlations exist for the non-newtonian viscosity. Why do we need models based on the Kinetic theory? Non-Newtonian viscosity is not enough to describe a flow other than a simple shearing one. A physical model would predict the behaviour of the polymer solutions in different kinds of flows. Understanding of the underlying mechanisms would help to predict and explain the effects of temperature, polymer concentration, salinity of the solvent and other physical and chemical factors.
Polymer molecules What makes polymers so special? Polymers are macromolecules. They are constructed from one or several types of simple structural units (monomers) and may have an extremely high molecular weight (up to 10 5 10 7 g/mol). A huge amount of possible configurations. The molecules continuously change their configurations due to the thermal motion and to the complex intramolecular interactions.
The FENE Dumbbell Model
The Dumbbell A polymer molecule is represented by two spherical beads connected by an elastic spring. The polymer solution is assumed to be sufficiently dilute, and the solvent is an incompressible Newtonian fluid.
Molecular dynamics Forces acting on the beads: Hydrodynamic drag (of Stokes type, with a coefficient ζ) Brownian forces The connector force from the spring The inertial forces (mass acceleration) External forces The forces allow to calculate the configurational distribution function ψ andtheviscousstresstensorτ.
Stress tensor The contributions of the forces to the momentum transfer can be calculated: or τ = η s γ n QF (c) + }{{}}{{}} nktδ {{} solvent spring motion τ = η s γ + nζ 4 QQ (1). Some notations: γ = { v} + { v} T is the rate-of-strain tensor; Angle brackets denote the configuration-space average; δ is the unit tensor.
Connector force Hookean spring can be extended without limits: does not yield non-newtonian fluid dynamics. A FENE (Finitely Elongated Nonlinear Elastic) spring is a working alternative: F (c) = H Q 1 (Q/Q 0 ) 2. Nearly Hookean (with coefficient H ) at small extensions...... but strongly nonlinear at larger extensions; Finally, cannot be stretched beyond the length Q 0.
Distribution function Known issues The configurational distribution function, ψ, is required to calculate the averages in τ ij ; In equilibrium (no flow) the exact expression for ψ is known; Out of equilibrium, ψ can be found from a (partial differential) diffusion equation; The diffusion equation must be solved at each point of the flow - complicates the CFD procedures. Isthereawaytogoaroundthisproblem?
Peterlin s closure The FENE-P assumption QQ QQ 1 Q 2 /Q0 2 1 Q 2 /Q0 2 + εq2 0δ The ε-containing term is introduced to improve the approximation. Here ε is a constant determined from the requirement that the trace of the above is true at equilibrium. The closure allows to avoid determining ψ at each point of the flow by excluding the averages using mathematical transformations.
The constitutive equations (FENE-P) Zτ p + λ H τ p(1) λ H {τ p (1 εb)nktδ} DlnZ Dt Z =1+ 3 ( 1 εb tr(τ ) p). b 3nkT = (1 εb)nktλ H γ A two-parameter model b = HQ0 2 /kt: a dimensionless parameter characterizing the degree of nonlinearity of the spring; λ H = ζ/4h : a time parameter reflecting some typical timescale for configurational changes.
Analytical results
Steady shear flow
Steady shear flow The constitutive equation reduces to algebraic form; The viscous stress tensor has only two independent components; Exact analytical expressions for these components can be obtained; Two material functions, the non-newtonian viscosity and the First normal stress coefficient, can be extracted: τ yx = τ xy = η( γ) γ τ xx τ yy = Ψ 1 ( γ) γ 2.
Viscosity 1/2
Viscosity 2/2
Normal stress 1/2
Normal stress 2/2
Analytical expansions 1. At very low shear rates: 1 η η s + nktbλ H b +5 Ψ 1 nktbλ 2 (b +2) H (b +5) 2 2. At high shear rates: ( ) λ 1/3 H η η s nktb 2(b +2) 2 ( γ) 2/3 ) 2λ 2 1/3 Ψ 1 nktb( H ( γ) 4/3 b +2
Steady shearfree flows The velocity field is given by v x = 1 2 ɛx, v y = 1 2 ɛy, v z = ɛz; ɛ is the elongation rate; ɛ >0: elongational flow; ɛ <0: biaxial stretching.
Steady shearfree flows The viscous stress tensor is diagonal: τ =diag(τ xx,τ xx,τ zz ); Characterized by the normal stress difference τ zz τ xx = η( ɛ) ɛ; η( ɛ) is called the elongational (Trouton, extensional) viscosity; For Newtonian fluids, η =3η and is constant.
Elongational flow
Biaxial stretching
Analytical expansions 1. At very low elongation rates: 2. At high elongation rates: η 3η s +3nkTbλ H 1 b +5 For ɛ + : For ɛ : η 3η s +2nkTbλ H η 3η s + 1 2 nktbλ H
Experimental results
Equipment Measurements are performed using the rotational rheometer at IRIS laboratory Output includes viscosity and normal force as functions of the shear rate.
Polymer samples Flopaam 5115 SH polymer: Commercial polymer Terpolymer Mm 15 10 6 g/mol Polydispersity unknown At different concentrations In salt (NaCl) water brines with different salinity
Results
Results
Results
Results
Results
Results
Results
Objectives Determine the actual values of b and λ H for the polymer solutions of interest; Estimate the accuracy of the FENE-dumbbell model; Check the theoretically predicted dependence of the material functions on the polymer concentration and temperature; Investigate the model parameters b and λ H as functions of the salinity of the brine; Ultimately beabletosimulatetheflowsofpolymers dissolved in salt water brines in channels with different geometries.
Acknowledgements The Research Council of Norway ConocoPhillips Skandinavia AS BP Norge AS Det Norske Oljeselskap AS Eni Norge AS Maersk Oil Norway AS DONG Energy A/S, Denmark Statoil Petroleum AS ENGIE E&P NORGE AS Lundin Norway AS Halliburton AS Schlumberger Norge AS Wintershall Norge AS
Thank you for attention!