Viscous Flow around Metal Spheres

Math 6514, Industrial Mathematics
Katharina Baamann, Cornelius Ejimofor, Alan Michaels, Alec Muller

In this project, we analyzed the dynamic viscosity of chemical-grade glycerin by measuring the terminal velocity of metallic spheres of varying diameters dropped in a graduated cylinder. A mathematical analysis using non-dimensionalization of Stokes fluid flow was then compared to the data to determine the accuracy of the approximation and receive an ultimate estimation for the viscosity of the glycerin.

The experimental setup consisted of a 1000 mL plastic graduated cylinder filled to approximately 1100 mL (measurements with a ruler show the added demarcation) with glycerin. Metal ballbearings were dropped from the surface of the fluid, and a camera was used to record the sphere's descent. Frame-by-frame analysis of the video footage yielded rough estimates of the ball's location within 0.03 seconds accuracy for statistically consistent results. Trials of 6 different sized ballbearings (counting BB's and birdshot) and 50-75 drops per type of ball were used to come up with the average velocities. An emperical assumption was used to determine when the balls reached terminal velocity near the top of the tube, and the analysis agreed.

After collecting the data, the curves were compared to the theoretical projections and the non-dimensionalizations.

First are the Powerpoint presentation slides .

Here is the Matlab *.m file with simple data plots.

A couple of sample Movie files to see how data was collected.

Here are the Excel Data Files from which the results were extracted.

Here are the Pictures collected of the setup.

Otherwise, here is the full report as a Word document and as html:

FINAL REPORT

Georgia Institute of Technology

School of Mathematics

MATH 6514A: Industrial Mathematics

Fall 2002

Professor

John McCuan

Katharina Baamann

Cornelius Ejimofor

Alan Michaels

Alec Muller

December 7, 2002

The purpose of our project was to model the fluid flow over a sphere in a viscous fluid, and to design and carry out an experiment to test this model. This is a classical problem in fluid dynamics, and the experimental solution is useful for measuring viscosity in many industrial applications.

Our system consisted of metal balls falling through glycerin under the force of gravity. The first goal was to determine the dependence of a ball’s terminal velocity on its radius. Our second goal, if possible, was to determine how the acceleration of a ball to terminal velocity from rest depended on the radius.

We began with a basic balance of the physical forces exerted on the ball when the terminal velocity had been attained. Terminal velocity suggests zero net acceleration and therefore no net force acting on the ball. The derivation below demonstrates a non-dimensionalization used to determine that the terminal velocity is directly proportional to the square of the balls’ radii.

The second analysis that would provide interesting results is that of the transient behavior of the ball’s motion when initially introduced to the glycerin. The goal here is to find a reasonable estimate for the distance a ball will travel before it reaches terminal velocity, and to determine whether or not it will be practical to measure the acceleration profile using the equipment we have.

Similarly we get that velocity x’ is:

From these expressions we observe that the displacement profile of the ball decays exponentially to a linear transient with time. Correspondingly the velocity profile decays exponentially to a constant value (-b/a). This value can be compared to experimental values of terminal velocity.

We know from Stokes’ Analysis that the Drag Force on the sphere is given by:

F_Drag = 6 p m a U

F_Drag = Drag Force (N)

m = Kinematic Viscosity (m²/s)

a = radius of the sphere (m)

U = Velocity of Sphere (m/s) (Constant Terminal Velocity)

If we analysis the sphere as a free body we observe that it is acted on by only three forces. These are its weight, the buoyancy force and the drag on the body.

Applying Newton’s Law in the direction of falling (the vertical direction) we get

SF_y = 0

F_Buoyancy + F_Drag - W = 0

F_Buoyancy + F_Drag = W

V_sphere * r_fluid * g + F_Drag = M *g

F_{Drag =} M *g - V_sphere * r_fluid * g

Using Stokes’ Law

6 p m a U = M *g - V_sphere * r_fluid * g

U = (M *g - V_sphere * r_fluid * g ) / (6 p m a)

= 1/(6 p m a) *( M - V_sphere * r_fluid) g

If we know the density of the sphere and it is a true sphere we can further simplify to get

M = V_sphere * r_sphere

U = 1/(6 p m a) (V_sphere * r_sphere - 4/3 p *a³ * r_fluid) g

= 1/(6 p m a) (V_sphere * r_sphere - 4/3 p *a³ * r_fluid) g

= 1/(6 p m a) * V_sphere * (r_sphere - r_fluid) g

= 1/(6 p m a) * (4/3 p *a³ )* (r_sphere - r_fluid) g

Where:

F_Buoyancy = Buoyancy Force on Sphere (N)

F_Drag = Total Drag Force on Sphere (N)

W = Weight of Sphere (N)

V_sphere = Volume of Sphere (m³)

r_fluid = Fluid Density (kg/m³ )

g = Acceleration due to Gravity (m/s²)

Hence we see that the terminal velocity depends on the radius of the sphere, the density of the sphere, a universal constant, and some fluid properties. Unfortunately it turns out that the fluid properties are functions of temperature, a strong function in the case of viscosity. Neglecting that variation we should expect that for fixed sphere or different sphere of the same material and size the velocity should be constant. Also comparing different sphere of the same material but different sizes the velocity should scale with the square of velocity. The expression for U here agrees with the terminal velocity calculated from the transient analysis in the previous section.

In fluid dynamics the flow field in a general fluid can be described by the Navier-Stokes equations. For incompressible viscous flow these equations can be given as the following vector equations.

Conservation of Mass Equation

Conservation of Momentum

The above equation relates the various energy terms in the flow in the manner:

Inertial Forces (F=ma) = Body Forces + Pressure Forces + Viscous Forces

For our analysis the only body force is the weight of the fluid. We can neglect that term in comparison to the magnitude of the other terms present. This simplifies the momentum equation to:

We can expand the total derivative as:

We can now non-dimensionalize our equation by selecting a set of characteristic scales.

v = v*/U, x = x*/L, p = p*/(mU/L), t=t*/(L/U)

The term rUL/m is a well known non-dimensional parameter, Reynolds Number, Re. This term represents the ratio of the inertial to the viscous terms in the flow. For our problem the viscous terms, due to the viscosity in our fluid, glycerin, are going to be much greater than the inertia forces due to gravity. Since Re <<1 then we can drop the first two term in the Navier-Stokes momentum equation. We get a much simplified differential equation. These equations are Stokes’ Flow equations.

We should observe that by dropping those terms we loose our observability of transient effects. We are implying since the term multiplying them, Reynolds Number, was very small the transient effect would have very little effect on the problem solution. We have essential reduced the problem to a steady state one. In regimes where this is not true we cannot use the Stokes’ Flow analysis.

Solution for the Sphere

We can solve this equation around a sphere as follows.

We find that the solution of the velocity field is related to that for a special function called the stream function Y. The stream function is defined such that the continuity equation is always satisfied when using it, thus simplifying the computations. The stream function can be shown to satisfy the biharmonic function ⁴Y = 0. The velocities can be obtained from the stream function as:

Solving further using either the separation of variables approach or a similarity solution we can find solutions the stream function (and hence the velocities) as follows.

Using these we find

The shear stress is given by the product of the viscosity and the divergence of velocity.

We can find the total drag force by integrating the effect of the shear and pressure forces around the body.

Hence we see that 2/3 of the total drag force can be accounted for by the effect of the shear stress on the sphere. The remaining 1/3 of the force is created by the pressure field around the body. We also observe that we get the same expression for drag force as we got from our dimensional analysis as we expected.

We tested our models using a large glycerin-filled graduated cylinder, and six series of metal balls. The balls ranged in size from 1.59mm to 12.70mm, and we performed 55 trials for each size. We used a digital camcorder to record the drops, and played back the video frame by frame to obtain distance vs. time data. Time was recorded in 1/100ths of a second, while distance was recorded in ml. Measuring the cylinder with a ruler gave us a conversion factor from ml to meters.

Our equipment had definite limits on its precision. The graduated cylinder was marked in increments of 10ml, which corresponded to a resolution of 5.18mm. The video took 30 frames per second, giving us a resolution of 0.033 seconds. In reality, however, our data were not quite this accurate in either domain. Blurriness and jumpiness in the video increased both of these error sources by about a factor of two. To avoid even more blurriness, we zoomed in on the range between 100 and 900-950 ml.

While dropping the balls, we noticed that each ball tended to entrap a tiny (millimeter-sized) bubble of air in its wake, and that these bubbles would separate at random distances down the cylinder. Since the glycerin is very viscous, it takes several minutes for a bubble to float to the top, so there were dozens of bubbles in solution by the end of each 55 ball run.

After 6 runs of the different sized balls, we did two abbreviated runs with one of the medium sized balls to see how practical it would be to measure the transient response. We had difficulty doing this, because the marks on the cylinder stopped almost 10cm below the rim of the cylinder. The transient response required us to drop the balls from the surface (our model assumed zero velocity at zero height), but we could only reach 5cm below the rim of the cylinder. We added a scale of our own, but even with this scale we were limited to the same levels of precision we had earlier in the experiment.

See Appendix A for our distance vs. time data in ml and 1/100ths of a second. We ignored the minutes except when a ball crossed over into a new minute in the course of traveling from the top marker to the bottom one.

To get this data, we replayed the videos of the experiments and recorded the position and time for the balls at the top (~900 ml) and bottom (~100ml) of the cylinder. We did a preliminary analysis for each ball size to make sure it had reached terminal velocity by the point where we recorded the top position and time measurements.

To analyze our data, we converted it into meters and seconds in Excel. We found that our values for terminal velocity would start off relatively constant, and then ramp up and become more sporadic (see figure 1 below).

Figure 1. Terminal Velocity for the 6.35mm ball

We hypothesized that this is due to the air bubbles dropped by the balls. The first balls to drop will experience very little error in speed due to air bubbles, because the path in front of them will be clear. As more balls are dropped and more air bubbles clutter the fluid, however, it becomes increasingly likely that a ball will encounter one or more bubbles on its way down. The bubble effectively lowers the fluid viscosity, which will decreases the ball’s transit time and increases its speed. By this hypothesis, the greater the likelihood a ball has of hitting a bubble, the greater the amount of noise there should be in the velocity data.

This trend follows perfectly, as you can see in Appendix B, which shows the velocities vs. trial number for each of the 6 different diameters. The two smallest balls have very little ‘bubble noise’ because they have small surface areas and they rarely encounter them. The next three smallest have little noise in the beginning (when there are few bubbles in the solution to hit), but more toward the end, where they hit many. The largest two (which were done right after the other diameters, so there were already bubbles in solution) have huge surface areas and hit bubbles constantly throughout the run.

To get theoretical values for terminal velocity, we used the drag formula: D = 6p m rv, where m is the viscosity, r is the radius, and v is the velocity. Then we equated this force to the buoyant and gravitational forces and solved for v.

Now we can plot the velocities to see how they agree with our dimensional analysis and our theoretical values. Since our drag force depended on r, and our body forces depended on r³, we expect our terminal velocity to increase with r². Plotting terminal velocity vs. r² for each of the six balls gives us the following plot.

Figure 2. Terminal Velocity vs. Radius for the 6 different sized balls

As one can readily see, the relationship is fairly linear but our experimental velocities are almost all too high. What could cause this discrepancy? One possible source of error is the fact that the viscosity of glycerin varies tremendously with temperature (0.280Pa*s@40C, 1.420Pa*s@20C, a 5-fold difference). We didn’t realize this while doing the experiment, so we didn’t even record the temperature, much less try to keep it constant. Our theoretical values used a room temperature of 23 C, but a 2 degree increase would have lowered the viscosity (and thus increased the terminal velocity) by about 20%. Since we did some trials on different days, we could have easily had that much variation from one run to the next.

An additional source of error (although much smaller) is the effect of the air bubbles that piggyback on the balls as they fall. An air bubble artificially increases the buoyant force of the glycerin, which makes the ball take longer to reach the bottom:

Here, we plugged in values for our errors to determine whether or not it was practical to measure the velocity profile as a function of radius. We start with the worst-case scenario, that of the smallest sphere (it should reach terminal velocity the quickest).

Since the ball reaches terminal velocity during approximately 0.23 video frames, the terminal velocity would be impossible to measure using the present equipment. The other end of the spectrum would be the quarter-inch ball which intuitively reaches terminal velocity at a later time after being dropped.

The Reynolds numbers for the various spheres were computed. The values were found to range from 0.02 for the smallest balls to 5.9 for the largest. These values are will below the transition value of 2300 for pipe flow indicating that the flow is strictly laminar as we have assumed. These low values of Reynolds number, either on the order of or much less than one support our presumption that the inertial forces are relatively small and Stokes’ flow exists.

In summary, we were able to confirm that our experimental measurements of terminal velocity agreed with our dimensional analysis and our theoretical predictions, although our errors were fairly large, which we expect mostly come from the temperature variations of the glycerin viscosity. The physical problem is essentially non-variant with time, the velocity transient decaying too quickly to measure or materially affect the motion. We also determined that it would not be practical to measure the acceleration profile of the balls with the equipment we had, because our measurement error outweighed the phenomena we were trying to measure for every single size – just barely for the largest one, and by many orders of magnitude for the smallest.

Some suggestions to be looked at in any further study of this problem should be mentioned. We would suggest that some means be identified of getting high frequency recording of the transient stage of the motion. This could either be using a high speed camera or a light strobe/multiply exposed film arrangement. An automated and repeatable system should be developed to drop the spheres. The effect of the sidewall (size of the ball relative to tube size) and the variations due to using different fluids should also be investigated.