AMS 599 Special Topics in Applied Mathematics Lecture 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 measured by 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)

Total time derivatives

Euler’s Equation

Conservation form of equations

Momentum flux

Viscous Stress Tensor

Taylor Couette Vortices

Two Phase NS Equations immiscible, 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