Viscous Flow Around Metal Spheres Terminal Velocity and Acceleration Profile as a Function of Radius

Introduction Characterize viscous flow around a sphere Find dependence of terminal velocity on radius Find dependence of acceleration profile on radius

Introduction Classical problem in fluid dynamics Special flow regime called Stoke’s Flow Viscous forces dominate the flow Often used to determine fluid viscosity

Our Experimental Setup Graduated cylinder filled with glycerin Different sizes of metal balls Digital Camcorder

Experimental Setup 6 different sized steel balls

Experimental Setup

Experimental Setup A video data file of the 6.35mm balls

Theoretical Models Terminal Velocity

Anticipated terminal velocity v. radius.

Theoretical Models Acceleration Profile

Theoretical Models

Theoretical Models

Theoretical Models

Theoretical Models

Theoretical Models Navier Stokes Analysis Non-dimensionalizing the Eqns

Theoretical Models For Stokes Flow Re<<1 So the Equations simplify to

Theoretical Models Navier Stokes Analysis

Theoretical Models Analytical Soln for the Sphere

Theoretical Models The Analytical Expression for Drag Force F matches Dimensional Analysis Stoke’s Law

Results & Analysis Used video from camcorder to find experimental speeds Calculated theoretical speeds using model Compared: Experimental Theoretical Predicted Scaling Rate from Dimensional Analysis (V ~ r^2)

Results & Analysis

Results & Analysis Error sources Viscosity is a function of temperature!

Results & Analysis Error Sources (cont.) Bubbles effectively reduce viscosity when they’re in a ball’s path Bubbles effectively increase buoyancy when they’re piggybacking on a ball Sidewall effects (disruption of flow lines) Instrument resolution (time and distance)

Results & Analysis Velocity Profile Analysis Terminal velocity reached for smallest ball in 0.007 seconds, faster than camera. Reached for largest ball in 0.303 second, but times and distances involved were still too fast:

Results & Analysis

Results & Analysis

Conclusion Experimental terminal velocity matches with dimensional analysis and theoretical model significant errors due to temperature and other effects Acceleration profile cannot be measured with current equipment resolution is too low relative to phenomena to be observed