AMS 599Special Topics in Applied

MathematicsLecture 5

James Glimm

Department of Applied Mathematics and Statistics,

Stony Brook University

Brookhaven National Laboratory

A Chemical Processing Problem

• Separation of spent nuclear fuel rods• Plan to bury them ran into political

opposition• New plan is to greatly reduce the volume• Separation of different fuel components• Re-use remaining fuel and only bury a

small fraction of spent fuel rod• Separation has many stages and steps;

we consider only one.

A Fluid Dynamics Problem

• Couette flow• Inner and outer cylinder• Inner rotates, outer is stationary• As rotation speed increases, a series of

transitions occur in the flow pattern– Laminar– Vortex rings (Taylor vortices)– Wavy vortices– Wavy, wiggly vortices– Turbulent flow

Couette Flow

• A widely studied flow as it serves as a test for theories of the onset of turbulence

• We are not concerned with this and are nowhere near the transition point. High rotation rate well past turbulent transition point

• We study two phase Couette flow– Oil based fluid– Water based fluid

Contactor

• The device we study is called a contactor• Two immiscible fluids in a high speed Couette

flow• Produces fine scale droplets and bubbles of

heavy and light fluid• Produces a large surface area• Chemical processing occurs at the surface

between the two phases• Optimal processing has a large surface area

Overall goal

• Simulate of small region of the contactor flow• Determine the surface area and the flow

properties of the two phases in a vicinity of the interface

• Thus estimate a possible chemical reaction rate as limited by surface area and diffusion to the surface

• Verify and validate• Use results to calibrate a macroscopic model

Macroscopic Model

• Simulations of the full contactor do not attempt to describe the surface between the fluids

• Surface area is assumed and diffusion to surface, giving an effective reaction rate

• Effective reaction rate is a key unknown (parameter) in the model.

• With good parameters, macroscopic model can succeed

• We aim to supply this parameter by simulation• The parameter will also be determined

experimentally so there will be a cross check (validation)

The Team/Collaborators

• Stony Brook University– James Glimm– Xaiolin Li– Xiangmin Jiao– Hyunkyung Lim– Shuqiang Wang– Navamita Ray– Yijie Zhou– Bryan Clark

• Oak Ridge National Laboratory– Valmor de Almeida

• Manhattan Community College– Brett Sims

Contactor

Photographs of 2 phase flow droplets 10 to 100 microns

1 mm x 1 mm field of view

Simulation Region

Droplet distribution as measuredby image processing

Simulation Equations

• Two phase incompressible Navier-Stokes equation– Two fluids, oleac (o) and acqueous (w)– Different densities– Immiscible, so that at each point, time instant, there is

only one phase present– Boundary between the fluids is called the interface

• Three phase incompressible Navier-Stokes equation– Extra phase is air (a)

Incompressible Navier-Stokes Equation (3D)

( )

0

dynamic viscosity

/ kinematic viscosity

density; pressure

velocity

t v v v P v

v

P

v

Total time derivatives

( ) particle streamline

( ) ( ) / velocity

Lagrangian time derivative

= derivative along streamline

Now consider Eulerian velocity ( , ).

On streamline, ( ( ), )

x t

v t dx t dt

D

Dt

vt x

v v x t

v v x t t

Dv

Dt

acceleration of fluid particle

v vv

t x

Euler’s EquationForces = 0

inertial force

Pressure = force per unit area

Force due to pressure =

other forces 0

S V

Dv

Dt

Pds Pdx

DvP

Dt

Conservation form of equationsConservation of mass

0

Conservation of momentum

other forces

v

t x

v vv

t t tv

v v P vx x

v v vP

t x

Momentum flux

( ) 0; flux of U

flux of momentum

stress tensor

Now include viscous forces. They are added to

'

' viscous stress tensor

ik ik i k ik i k

ik ik ik

ik

UF U F

tv v P

P v v v v

P

Viscous Stress Tensor

' depends on velocity gradients, not velocity itself

' is rotation invariant; assume ' linear as a function of velocity gradients

Theorem (group theory)

2'

3

C

i k i iik ik

k i i i

v v v v

x x x x

orollary: Incompressible Navier-Stokes eq. constant density

v v vP v

t x

Taylor Couette Vortices

Two Phase NS Equationsimmiscible, Incompressible

• Derive NS equations for variable density• Assume density is constant in each phase with a jump

across the interface• Compute derivatives of all discontinuous functions using

the laws of distribution derivatives– I.e. multiply by a smooth test function and integrate formally by

parts• Leads to jump relations at the interface

– Away from the interface, use normal (constant density) NS eq.– At interface use jump relations

• New force term at interface– Surface tension causes a jump discontinuity in the pressure

proportional to the surface curvature. Proportionality constant is called surface tension