The Peculiar Mathematical Structure of Non-Newtonian Flows

1. The Peculiar Mathematical Structure of Non-Newtonian Flows Meng Xu Department of Mathematics University of Wyoming

2. Motivation and Goal • Motivation • Fluids exhibiting nonlinear constitutive laws are encountered in industrial applications (e.g. slurries, ceramics, toothpaste, gels, mineral oils, suspensions). • Modeling the dynamics of such fluids in processes (e.g. pipelining, extrusion, emulsification), is achallenging task with mathematicaldifficulty. • Goal of the talk • derive the mathematical formula governing Non-Newtonian flows • study of Change of type in delayed die swell

3. Some examples of Non-Newtonian flows Aside from air and water, most fluids in physical and industrial processes are “Non-Newtonian“,(no simple relationship between the stress and the rate of strain)

4. Dumbbell model • a couple of massless beads connected by a spring • Convector vector: • Forces: 1.spring force 2.friction force (Stokes drag) 3.stochastic force Dumbbell model of a polymer molecule

5. the spring exerts a spring force • The friction force is exerted by the surrounding fluid, and is given by a Stokes drag: • is the solvent viscosity, is the radius of the bead, is the velocity of the bead, and is the velocity of the surrounding fluid. We shall write for • The stochastic force is due to Brownian motion, we denote it on each bead by and Assumptions: • Positive direction is defined from to • Each macroscopic volume element contains a large number of polymer molecules • The distribution of their connector vectors can be described by a probability density

6. The force balance law Linear approximation: The result is the equation: Methods of stochastic differential equations can be used to convert the above stochastic differential equation to a Fokker-Planck equation.

7. Goal: derive the Fokker-Planck equation and the upper convected model (UCM) equation Ito’s formula (Brownian Configuration fields) Stochastic differential equation of force balance ---spring constant ---Boltzmann’s coefficient ---temperature ---friction coefficient ---unit tensor

8. Put stochastic differential equation into the Ito’s formula Taking the average on both sides, we get This is the Fokker-Planck Equation under the dumbbell model.

9. The number of connectors with orientation that intersect a plane with normal vector is proportional to , where is the number density of polymer molecules. Stress tensor: Under linear force law: UCM

10. Change of type in viscoelastic flows Delayed die swell: A viscoelastic fluid emerging from a nozzle forms a jet whose diameter is substantially larger than that of the nozzle. Usually this swelling of the jet occurs immediately when the fluid leaves the nozzle, but in some cases it can be delayed.

11. The equations governing viscoelastic flows are: This is a quasi-linear system of ten equations, which we can put in the form

12. The following analysis is based on a one-dimensional approximation where all quantities are regarded as homogeneous across the jet. Conservation of mass: Momentum balance law: is the coordinate along the axis of the jet, the cross-sectional area, and the velocity of the fluid, denote the force in the jet and the density. The force is given by , where is the stress tensor • Assumptions: • the constitutive law for is that of the UCM fluid • and are approximately diagonal (axisymmetric and homogeneous across the jet)

13. Constitutive law: Set , The characteristic speeds of the system are the roots of the equation (twice) This system of equations is non-strictly hyperbolic

14. Further study plan • Change of type of flow models in applied field, such as geological flows • Stochastic analysis and control theory • Fluid dynamics courses in mathematics and geology departments

15. References: • Mathematical Analysis of Viscoelastic Flows---M. Renardy • Rheol. Acta 8 (1968), 411---H. Giesekus • Perry’s Chemical Engineers’ Handbook---R.H. Perry • Dynamics of Polymeric Liquids---R.B. Bird