Rheology and Constitutive Equations. Rheology = Greek verb to flow. Rheology is the study of the flow and deformation of materials.

Transcription

1 Rheology and Constitutive Equations Rheology = Greek verb to flow Rheology is the study of the flow and deformation of materials. The focus of rheology is primarily on the study of fundamental, or constitutive, relations between stress (or force) and deformation in materials, usually fluids. These fundamental relations expressed in terms of constitutive equations which mathematically relate the stress to the strain. A rheologist focuses on the behavior of materials in simple deformations (e.g. simple shear flow, elongational flow) and develops constitutive relations. A physicist (or theorist or material scientist) studies the physical (polymer) dynamics and micromechanics of the material and develops constitutive models. An engineer studies the stresses and structure which is developed in the complex deformations arising in real flow processes. A numerical scientist applies the constitutive models and knowledge of material behavior to simulate flow processes or other flow behavior using numerical methods. Interdisciplinary work and communication is necessary.

2 Rheological Behavior A fluid can be non-newtonian in different ways: 1. Stress is a nonlinear function of the strain or rate-of-strain. 2. Non-zero normal stress differences occur in shear flow due to additional normal viscous stresses which are produced in shear flow. 3. Extensional or elongational viscosity is present in simple uniaxial, biaxial, planar extensional flow. 4. There are time effects or memory effects due to the structure of the fluid. A spectrum of relaxation times is associated with the rates at which structural changes take place. For example, in polymeric liquids, thermally-induced configurational changes of polymer molecules take place via rotations around temporary chemical bonds. This gives the fluid a (fading) memory. A fluid can exhibit one or all of these non-newtonian characteristics. Whether a fluid exhibits a particular non-newtonian characteristic depends on the flow or processing conditions.

3 Rheological Behavior: Categories Two general classifications for many common non-newtonian fluids: 1. Time-independent non-newtonian fluids in which γ is a unique nonlinear function of τ 12 Inelastic non-newtonian or Purely viscous non-newtonian fluids These fluids exhibit the first kind of non-newtonian characteristic given above. 2. Time-dependent non-newtonian fluids in which shear rate γ and shear shear strain γ are related to τ 12 Viscoelastic fluids These fluids usually exhibit all four of the non-newtonian characteristics listed above.

4 Rheological Behavior: Non-Newtonian Fluids Fluids with a shear-rate dependent viscosity are classified as follows: Pseudoplastic, or shear-thinning, fluids: Shear viscosity decreaases as the shear rate increases. Dilatant, or shear-thickening, fluids: Shear viscosity is an increasing function of shear rate. Viscoplastic, or yield value, fluids: Fluids which can sustain an an applied (nonzero) stress without flowing. The stress below which the fluid does not flow is called a yield value or yield stress.

5 Rheological Behavior: Shear-thinning Fluids Shear Viscosity, η [Pa s] Zero-shearrate region Transition region Power law region Upper shear rate region (Often not measureable) η Shear Rate, γ [1/s] Zero-shear rate viscosity: Infinite-shear-rate viscosity: lim ηγ ( ) γ 0 η lim ηγ ( ) γ In many shear thinning fluids, η is very small, e.g. η < O10 ( 2 ), and» η.

6 Power Law Model of Ostwald (1925) and de Waele (1923) ηγ ( ) = κγ n 1 n is the power law exponent (dimensionless) κ is called the consistency index (with units Pa s n, for example) n κ describes the slope of viscosity curve ηγ ( ) in the power law region describes the vertical shift of the power law region n = 1: Newtonian constitutive equation with constant viscosity µ = κ n < 1 : pseudoplastic or shear-thinning (or structure viscous) since the predicted viscosity decreases with increasing shear rate; lim ηγ ( ) γ 0 = and η lim ηγ ( ) = 0 γ n > 1 : dilatant or shear-thickening since the predicted viscosity increases with increasing shear rate; lim ηγ ( ) = 0 and η lim ηγ ( ) γ 0 γ =

7 Power Law Model: Comments 1. The power law model is widely-used and very popular among many engineers. It is simple since it contains only two parameters. Analytical solutions to many problems in simple and defined flow fields can be found. 2. The power law model cannot describe behavior outside the power law region, that is, for small or large values of shear rate. Therefore, its use in CFD programs can lead to large computational errors. If onset of power law region is at low shear rates, then power law model is probably safe to use; If onset of power law region occurs at high shear rates, then power law model is not a good model to use. η η 10-2 γ 10 2 γ Safe Unsafe

8 Cross Model η η = [ 1+ ( λγ ) m ] 1 η m is the Cross law exponent, which is related to the power law exponent, n, via m = 1 n (dimensionless); λ is a time constant (units of time, e.g. seconds), describing the transition region in the viscosity curve; 1 Specifically, γ = -- is the shear rate at which the fluid changes from the constant shear rate behavior to the power law λ behavior; is the zero-shear-rate viscosity, which is assumed finite (units of viscosity, e.g. Pa s); η is the infinite-shear-rate viscosity, which is assumed finite (units of viscosity, e.g. Pa s). m = 0: Newtonian behavior ( n = 1) m > 0: Shear-thinning behavior ( n < 1) m < 0: Shear-thickening behavior ( n > 1)

9 Cross Model assuming shear thinning behavior For low shear rates ( γ «1): lim γ 0 η η η 1 = lim γ ( λγ ) m = 1 since ( λγ ) m 0 as γ 0 when m > 0; Therefore, η as γ 0 For high shear rates ( γ» 1): η lim [ η] = lim η γ γ 1 + ( λγ ) m = η since ( λγ ) m as γ 0 when m > 0; That is, η η as γ, For intermediate shear rates: ( λγ ) m ( λγ ) m = λ n 1 γ n 1 so that η η ( η )λ n 1 γ n 1 or η η κγ n 1 where κ = ( η )λ n 1 ; Now, for η» η, this reduces to η κγ n 1, which describes the power law region.

10 Carreau-Yasuda Model η η = [ 1+ ( λγ ) a ] ( n 1) a η a is the parmeter describing the transition region between the zero-shear-rate viscosity region and the power law region (dimensionless); n is the power law exponent, which desribes the slope of ( η η ) ( η ) is the power law region (dimensionless); λ is a time constant, describing the transition region in the viscosity curve (units of time, e.g. seconds); 1 Specifically, γ = -- is the shear rate at which the fluid changes from the constant shear rate behavior to the power law λ behavior; Comments: 1. If a = 2, then this is called the Bird-Carreau model (1968) or simply the Carreau model. 2. For many shear thinning fluids, a 2.

11 Yield Value Fluids A fluid is said to have a yield value, or a yield stress, if it can sustain an applied (nonzero) stress without flowing. The yield value, or yield stress, is the stress below which there is no relative flow. Let τ 0 denote the yield stress. Then: τ ij = ηγ ( )γ ij when τ 2 2 τ 0 γ ij = 0 when τ 2 2 < τ 0 where ηγ ( ) is the apparent viscosity of the material beyond the yield point ηγ ( ) = τ ηˆ ( γ ) γ ηˆ ( γ ) is the constitutive equation for the fluid after the yield stress is reached Comments: 1.For yield value fluids, the viscosity approaches infinity at small shear rates, i.e. lim ηγ ( ) = γ 0 2.In order to solve yield-value flow problems with the velocity-based FEM, we must usually modify the yield value constitutive model slightly due to the condition γ ij = 0

12 Bingham Model The fluid behaves like a Newtonian fluid after the yield stress has been reached, i.e. ηˆ ( γ ) = µ = constant Therefore, the viscosity law is ηγ ( ) µ τ 0 = γ and the stress after yield is τ = µγ + τ 0 for τ 2 2 τ 0

13 Bingham Model: Computational Implementation We approximate the model at low shear rates to avoid η. ηγ ( ) µ τ 0 = γ for γ γ c ηγ ( ) µ τ 0 = ( 2 γ γ γ c ) c for γ < γ c where γ c is the critical shear rate below which viscosity is linear in shear rate. Flexibility in choosing value of γ c, but it should be chosen so that it produces no significant difference in the results of the flow problem being solved. An appropriate value of γ c depends on the particular flow problem, the flow parameters or conditions, and the material parameters. Comments: This approximation allows the fluid to undergo deformation below the yield stress, but the magnitude of the motion can be made as small as desired by decreasing γ c. As γ 0, the approximation approaches the true Bingham model.

14 Herschel-Bulkley Model The fluid behaves like a power law fluid after the yield stress has been reached, i.e. ηˆ ( γ ) = κγ n 1 Therefore, the viscosity law is ηγ ( ) = κγ n 1 + τ γ and the stress after yield is τ = κγ n + τ 0 for τ 2 2 τ 0

15 Herschel-Bulkley Model: Computational Implementation We approximate the model at low shear rates to avoid η. ηγ ( ) κγ n 1 τ 0 = γ for γ γ c n 1 τ 0 ηγ ( ) = κγ c γ c for γ < γ c where γ c is the critical shear rate below which viscosity is constant. Flexibility in choosing value of γ c, but it should be chosen so that it produces no significant difference in the results of the flow problem being solved. An appropriate value of γ c depends on the particular flow problem, the flow parameters or conditions, and the material parameters.