Transcript Sphere Drag Force

Sphere Drag Force
Jace Benoit
Pamela Christian
Amy Stephens
February 11, 2007
Problem Statement
SAE 10 oil at 20°C flows past an 8-cmdiameter sphere. At flow velocities 1, 2,
and 3 m/s, the measured sphere drag
forces are 1.5, 5.3, and 11.2 N,
respectively. Estimate the drag force if the
same sphere is tested at a velocity of 15
m/s in glycerin at 20°C.
Sketch
1 m/s
1.5 N
2 m/s
5.3 N
3 m/s
15 m/s
11.2 N
? in Glycerin
Assumptions
Incompressible Flow
 Steady Flow
 Constant Temperature
 Scaling Law is Valid

Scaling Law
Scaling Laws “convert data from a cheap,
small model to design information for an
expensive, large prototype” (White 294). In this
problem, oil can be used to model information
for glycerin. When scaling laws apply, the
Reynolds number and force coefficient for the
model and prototype (oil and glycerin) are the
same.
Solution
The densities of the SAE 10 oil and glycerin are found in Table
A.3 to be as follows:
ρoil
=
870 kg/m3
ρglycerin =
1260 kg/m3
Solution
Knowing the density (ρ) , length (L or diameter), velocity (V),
and force (F) acting on the sphere, White (5.2) can be
modified to determine the force coefficients (CF) for each
corresponding velocity and force in the oil.
CF = F / ρV2L2
(White 5.2)
Solution
Substituting the known values into the previous equation
yields the following force coefficients for the oil:
CF1 = 1.5 N / (870 kg/m3)*(1 m/s)2*(0.08 m)2
CF1 = 0.2694
CF2 = 5.3 N / (870 kg/m3)*(2 m/s)2*(0.08 m)2
CF2 = 0.2380
CF3 = 11.2 N / (870 kg/m3)*(3 m/s)2*(0.08 m)2
CF3 = 0.2235
Solution
Averaging these three force coefficients gives a single
value to compare to the unknown drag force in the
glycerin. Since the scaling law applies, this averaged
force coefficient can be substituted back into the same
equation with some rearranging to yield the drag force for
glycerin.
FD = CF*ρglycerin*Vglycerin2*L2
FD = (0.2436)*(1260 kg/m3)*(15 m/s)2*(0.08 m)2
FD = 442.0 N
Biomedical Application
Instead of finding a particular force for a given velocity
through experimentation, the force on an object is found
through scaling.
 So instead of testing 20 different velocities to find a
force, we can test 3-4 velocities and develop a curve of
the forces to determine any forces for that fluid.
 This can be useful when designing implants such as
knee replacement or stints. By knowing the velocity and
force an implant can withstand, we can determine the
size of the implant for maximum effectiveness.
 Also using scaling, if an implant is found to have certain
characteristics in a particular fluid, the characteristics of
the implant in another fluid can be predicted.

Reference
White, Frank M. Fluid Mechanics. 5th Ed. McGraw-Hill
Companies, Inc.: New York. 2003.