sphere rolling a inclind plane in a submerged liquid

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Proceedings of the 37th

International & 4th

National Conference on Fluid Mechanics and Fluid Power

FMFP2010

December 16-18, 2010, IIT Madras, Chennai, India

FMFP10 - AM - 01

SPHERE ROLLING DOWN AN INCLINE

SUBMERGED IN A LIQUID

Pravin K. Verekar Department of Mechanical Engineering

Indian Institute of ScienceBangalore, Karnataka, India

Jaywant H. ArakeriDepartment of Mechanical Engineering

Indian Institute of ScienceBangalore, Karnataka, [email protected] [email protected]

ABSTRACTA sphere rolling down an inclined plane submerged

in water is studied experimentally in order to

understand the different forces acting on it. This

forms a class of solid-fluid interaction problems that

include sediment transport, movement of gravel on

ocean floor and river bed due to water currents. The

flow development around the rolling sphere is

elucidated in order to highlight its implications on

the nature of hydrodynamic forces that act on the

sphere. Equation of motion for the sphere is solved

numerically and the experimental data is fitted on

these solutions; the best fit gives the values of theforce coefficients.

Keywords: rolling sphere, inclined plane, flowdevelopment, hydrodynamic forces, force coeffi-

cients.

INTRODUCTIONExperiments are done with a sphere rolling

down an inclined plane submerged in quiescent

water. These experiments are conducted to

determine the hydrodynamic force coefficients.

The experimental setup consists of a glass tank

15 cm wide by 14 cm deep by 61 cm long. At

one end, the glass tank is fixed at the base with

two levelling screws on either side, and at the

other end, the tank rests on a spherical pivot

mounted centrally. The levelling screws allow

the tank to be tilted to the desired angle.

The motivation for this study comes fromthe need to improve understanding of the solid-

fluid interaction problems such as sedimenttransport, movement of gravel on ocean floor

and river bed, etc.

Proceedings of the 37th National & 4th International Conference on Fluid Mechanics and Fluid Power

December 16-18, 2010, IIT Madras, Chennai, India.

FMFP10 - AM - 01



2

EXPERIMENTS

A solid rigid smooth sphere is released from rest on

the inclined bottom glass plane of the tank which is

filled with water. The sphere rolls down the planeunder the influence of gravity. Initial motion of the

sphere is with decreasing acceleration until it attains

terminal velocity. The spheres used for the

experiments along with the inclined plane angles

kept during these experiments are given in Table 1.

The spheres are so chosen because they have good

material homogeneity and are perfectly spherical.

The motion of the sphere is photographed in front

view using Photron FASTCAM PCI R2 model 500

digital camera. Photography for some experimental

runs is done in close view which captures the initial

acceleration portion of the sphere motion while for other runs it is done in far view which captures the

full motion of the sphere till it attains terminal

velocity. The displacement of the sphere in pixels isobtained from its translation in the digital images

and the time increment is known from the framing

rate. The conversion scale for distance is based on

the sphere size in the digital image given by pixels

per mm of the sphere diameter. The paired data of

the displacement and time are used later for the

kinematic analyses of the sphere.

Table 1: Experimental settings Spheredescription

Diameter (cm)

Specific

gravity, U

Inclined plane

angles, T

Ratio of diameter, D to width of tank, W

1 Acrylic sphere No. 1

2.54 1.18 1.8q, 5.7q 0.17

2 Acrylic sphere No. 2 5.08 1.19 1.7q, 2.8q 0.34

3 Pool ball

5.24 1.70 2.9q 0.35

The sphere is released using a pair of tongs

made from aluminium wire of diameter 1.5 mm. In

some experiments a pair of tongs made from steelwire of diameter 1.2 mm is used. To find the angle of

the inclined plane, the difference in height of the

surface of water from the bottom of the tank ismeasured at a separation of 50 cm and 30 cm along

the incline of the bottom of the tank; and then the

average is taken of the two values obtained from the

inverse of sine of the ratio of the difference in height

to separation distance.

FLOW DEVELOPMENT AROUND THE

SPHERE

The flow development around the sphere is

explained here. As the sphere begins to roll, theReynolds Re based on instantaneous velocity

(hereafter called instantaneous Reynolds

number) are in the creeping viscous flow

regime. For creeping flow or Stokes flow

regime, Re is less than 1; this translates for the

experiments carried out as: (i) for acrylic

sphere No. 1, the instantaneous velocity V of

the sphere less than 0.004 cm/s, and (ii) for

acrylic sphere No. 2 and pool ball as thevelocity less than 0.002 cm/s. What is noticed

is that, the Stokes flow is momentary and

appears to have no consequence on the later flow development. Creeping viscous flows are

explained in Langlois, 1964.

The starting flow over the rolling sphere

can be treated as irrotational; here the fluid

particles glide over the surface of the sphere

and the velocity field has a potential which is

completely defined by knowing the

instantaneous normal velocity of the surface of

the sphere; this means that the irrotational

motion is entirely without memory (Lighthill,

1986). Cox & Cooker (2000) have found that

for an irrotational flow past a sphere touching atangent plane, the flow velocity is singular at

the point of contact and the flow speeds around

the point of contact are large. Also it is noted

that the influence of the tangent plane is small

one radius away and the flow there is very

similar to that for an isolated sphere.

The starting potential flow over the sphereis brief as the no-slip condition takes hold at

the solid-fluid interface. The fluid particles

next to the surface adhere to it and thesucceeding adjacent fluid layers are sheared

until the outer edge of the boundary layer. The boundary layer is a highly strained flow and is

the region where all the vorticity is confined.

This thin viscous layer is pressed upon by the

outer potential flow, and its growth is governed

by the momentum diffusivity of the fluid and



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the convective velocity of the outer flow. The fluid

particles in the boundary layer in overcoming the

viscous friction lose kinetic energy; hence this

retarded layer of flow cannot follow the curvature of

the sphere at the rear side as the fluid particles withsmaller kinetic energy cannot overcome the positive

pressure gradient present in this region and the

boundary layer separates; initially forming a

separation bubble at lower Re, and later in time

shedding lumps of vorticity at higher instantaneous

Re (see Schlitchting, 1968). Boundary conditions no

longer govern the separated flow and as Lighthill

(1986) puts it “all of the memory in a fluid flow lies

in its vorticity; which, once generated is subject to

convection and diffusion.” Stuart (1963) points out

that “at large times, the boundary layer flow

becomes quasi-steady, in the sense that it behaveslike a steady flow of boundary-layer theory with

instantaneous outer flow speed.”

Since the flow velocities are higher around the bottom hemisphere near the point of contact than

those around the top hemisphere, there is a

differential shedding of the vortex sheet (boundary

layer) that rolls up at the rear of the sphere. The

circulating eddies detaching from the lower region

are convected with higher velocity; they are pushed

upwards by the outer potential flow and are pressed

sideways by the slower rotating eddies above them;

rotating fluid elements acquire an upward andtransverse velocity components. This streamwise

spiral vorticity cannot be captured properly in 2-D

flow visualisation.

Adding complexity to the flow features are two

more phenomena peculiar to the rolling sphere: (i) a

primary flow in the boundary layer which is

opposing the potential flow in the top hemisphere

and which is along the potential flow in the lower

hemisphere; and (ii) a secondary flow along the

surface, spiralling away from the poles towards the

equator (Howarth, 1951). Three dimensional

boundary layers are discussed in Moore (1956).Visualizations of the wake formation behind a

rolling sphere in steady uniform flow have been

done by Stewart et al. (2008) using fluorescein dye.

The Reynolds numbers are varied from 75 to 350.

They have defined a parameter “the rotation rate of

the sphere, D” which is the ratio of thetangential velocity on the surface of the sphere

with respect to the centre of the moving sphere

to the translation velocity of the sphere in

ground reference frame. The case D=1represents sphere rolling down an inclined

plane. Their visualizations show a steady wake

mode for low Re with transition to unsteady

wake mode occurring at around Re=100. In thesteady wake mode, they observe that the

opposing motion of the top surface of the

sphere to the outer flow creates a zone of

recirculating fluid behind the sphere and the

dye escapes this recirculation zone via a single

tail along the centreline of the body. In the

unsteady wake mode, they observe that the

shedding from the top of the sphere takes the

form of hairpin vortices similar to that for an

isolated sphere.

The flow features that evolve in time

around the sphere determine the nature of the

hydrodynamic forces that act on it.

FORCES ACTING ON THE SPHERE

The sphere rolls under the influence of the

component of the resultant of the weight force

minus the buoyancy force along the plane. Thisdriving force can be expressed as

sin s f Vol g U U T where

s U is the density

of the sphere, f

U is the density of the fluid,

Vol is the volume of the sphere, g is the

acceleration due to gravity and T is the angle

of the incline.

When the driving force accelerates the

sphere in still water, it needs to also acceleratethe surrounding mass of water. This rate of

change of momentum of the surrounding mass

of fluid appears as a resisting force; this isaccounted by considering an increase in the

mass of the sphere by adding a separate mass,

what is called added-mass (also called virtual mass), to the mass of the sphere. The added-

mass is taken as a factor Ca times the mass of

the displaced fluid. The added-mass force is



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given by f

dV Ca Vol

dt U where Ca is the added-mass

coefficient and V is the velocity of the sphere.The other hydrodynamic resisting force to the

motion of the sphere, the drag force, originates from

the tangential skin friction ( skin drag ) and thedifference in the quasi-steady normal pressure at fore

and aft of the sphere ( form drag ). At the initial times

when the boundary layer has not separated, the

relative value of the skin drag to the total drag is

high; while on separation of the boundary layer, it isthe relative value of the form drag to the total drag

that is higher; the higher form drag corresponds tothe low pressures at the rear of the sphere when the

separation occurs. The drag force is given by

1

2 f Cd A V V U where Cd is the drag coefficient

and A is the projected area of the sphere

perpendicular to the main flow.

Also opposing the motion is the force of friction at the point of contact. When the sphere is in

pure rolling, this force is given by2

I dV

R dt where I

is the mass moment of inertia of the sphere about the

diameter and R is the radius of the sphere.

THEORY

The equation of motion of the sphere is given below.

2

2

1sin

2 s s f f f

dV dV I dV Vol gVol Ca Vol Cd V A

dt dt R dt U U U T U U

(1)

The above equation is solved for the solid sphere

where the ratio s U

U is written as U . The following

equation is arrived at.

23

1 .4 1 s in4

d V V C a g C d

d t D U U T

(2)

During the early part of the motion of the

sphere, when the flow over the sphere is potential,the dominant hydrodynamic resistance comes from

the inertia of the added-mass. During the later part of

the motion, when the boundary layer has developed

and then separated, the hydrodynamic drag is the

main opposing force.

Similar inclined plane experiments have

been done previously by Carty (1957), Garde

& Sethuram (1969), and Jan & Chen (1997).

The values of Cd measured by Garde &Sethuram are higher than Carty’s values, while

those measured by Jan & Chen are

intermediate. The first two studies are not

concerned with finding Ca, while Jan & Chen

determine Ca as 2.0 using insufficient data.

Chhabra & Ferreira (1999) have solved the

Eq. (1) analytically using Jan & Chen’s drag

and added-mass coefficient data. They use a

single expression for Cd–Re relation, which

gives the best fit to the Cd–Re equations of Jan

& Chen (1997). TheCd–Re

relation used by

them is given below.5321.906

Cd 0.861 0.1 Re < 10 Re

(3)

Equations for the potential flow past a

sphere touching a tangent plane have been

solved numerically by Cox & Cooker (2000).

They find the added-mass coefficient Ca as

0.621.

EXPERIMENTAL OBSERVATIONS AND

ANALYSES

Collateral experiments on unidirectional,

uniform, unsteady flow past the unconstrained

acrylic sphere No. 2 resting on a horizontal

plane in the unsteady water tunnel facility in

the laboratory have shown that the sphere rolls

on the glass surface without slipping. A sphere

rolls without sliding when the rolling friction is

less than the sliding friction. As explained in

Starzhinskii (1982), pure rolling occurs when

the ratio of the coefficient f r of rolling friction

to the radius R of the sphere is less than

coefficient f s of sliding friction; this statementis independent of the magnitude of the resultant

driving force when this force passes through

the centroid of the homogeneous sphere.

Practically the coefficient f r depends solely on

the materials of the rolling body and of the

inclined plane; and if the materials of the



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rolling body and of the inclined plane are sufficiently

hard, the force of the rolling friction is very small(Strelkov, 1978). Usually the ratio f r /R is

considerably smaller than the coefficient f s of sliding

friction (Starzhinskii, 1982). Verification that thesphere is in pure rolling is important for the validity

of Eq. (1).

Equation (2) is solved for the velocity V of the

sphere using MATLAB differential equation solver

ode45 (refer Shampine et al., 2003). Cd is taken as

given in Eq. (3) and Ca is treated as a parameter; the

values of Ca are chosen as 0.5, 0.621, 1.0, 1.5, and

2.0. The value 0.5 comes from the potential flow

solution for an isolated sphere; the value 0.621 is

taken from the potential flow solution for a sphere

touching a tangent plane; the value 2.0 is as given by

Jan & Chen (1997) for a rolling sphere on theincline. The values of U , T and D come from the

experimental settings, which are given in Table 1.

The MATLAB code plots following graphs: distance

X travelled by the sphere versus time t , velocity V of

the sphere vs time t , and velocity V of the sphere vs

dimensionless distance X/D. Experimental data

points are superposed on these graphs. Velocity for

the experimental data is obtained by fitting cubic

spline for distance-time points and then

differentiating the spline. Typical graphs for an

experimental run for acrylic sphere No. 1 (diameter

D=2.54cm) and inclination T = 5.7q are shown inFig. 1 to Fig. 3, and for acrylic sphere No. 2

(diameter D=5.08cm) and inclination T = 2.8q are

shown in Fig. 6 to Fig. 8.

Figure 1 gives the distance travelled by theacrylic sphere No. 1 plotted against time for the

entire motion of the sphere till it reaches terminal

velocity. The coloured curves are obtained from the

numerical solutions to the differential equation for

different values of Ca; and it is found that the

experimental points fall close to the green curve (Ca

=0.621) for the entire motion of the sphere. This is

also seen in Fig. 2 and Fig. 3. The sphere attainsterminal velocity at about time t=2.5s and distance

X/D=6. Taking the green curve as the best fit for the

experimental points, the various forces, namely, the

added-mass force, the drag force and the rollingfriction, are calculated as the sphere rolls down the

Fig. 1. Distance vs time diagram

Fig. 2. Velocity vs time diagram

Fig. 3. Velocity vs X/D diagram



6

incline; and the same are normalised by the driving

force. The normalised forces are shown in Fig. 4 andFig. 5. It is seen from Fig. 5 that the dominant

influence of the added mass force at starting times is

for a distance X/D up to about 0.3.

Fig. 4. Normalised force vs time diagram

Fig. 5. Normalised force vs X/D diagram

For acrylic sphere No. 2, Fig. 6 gives plot of

distance travelled vs time, Fig. 7 gives plot of

velocity vs time, and Fig. 8 gives plot of velocity vs

X/D. It is seen from these plots that the experimental points at initial times fall close to the green curve

(Ca =0.621) but later when nearing terminal velocity

(at time t=3.5s and X/D=3) they deviate and fall

below the green line. Calculations show that the Cd at the terminal velocity for the experimental points is

Fig. 6. Distance vs time diagram


Fig. 8. Velocity vs X/D diagram



7


Fig. 10. Normalised force vs time diagram

Fig. 11. Normalised force vs X/D diagram

1.2 times higher than that given by the green

line. Taking Ca as 0.621 and Cd as 1.2 timesthat given by Eq.(3), the differential equation

of the motion of the sphere is solved. This

solution and the experimental points are shownin Fig. 9. The solution fits the experimental

points adequately, except at the region where

the acceleration ends and the terminal velocitystarts. Taking this solution as the fit, the

different normalised forces are calculated and

shown in Fig. 10 and Fig. 11. From Fig. 11, itis seen that the dominant influence of the added

mass force at starting times is for a distance

X/D up to about 0.3.

Terminal velocity V t is found from the

experimental data. Reynolds number Ret is

calculated at terminal velocity. Coefficient of

drag Cd expt is calculated by equating drag force

to the driving force, which it balances when the

sphere has attained terminal velocity. Cd lit. is

the drag coefficient obtained from Eq.(3),

which is given by Chhabra & Ferreira (1999).

Results are presented in Table 2.

Table 2: Experimental resultsSphere

description Dcm

D/W T V t cm/s

Ret Cd expt Cd lit.

.

exp

lit

t

Cd

Cd

Acrylic sph. 1 2.54 0.17 1.8q 3.9 990 1.23 1.19 1.0

Acrylic sph. 1 2.54 0.17 5.7q 7.5 1900 1.06 1.03 1.0

Acrylic sph. 2 5.08 0.34 1.7q 5.6 2800 1.19 0.98 1.2

Acrylic sph. 2 5.08 0.34 2.8q 7.4 3800 1.13 0.95 1.2

Pool ball 5.24 0.35 2.9q 14.9 7800 1.09 0.95 1.2

DISCUSSION AND CONCLUSIONS

It is confirmed experimentally that the Ca for a

sphere rolling down an inclined plane is closeto 0.621 and it holds good at D/W ratio of 0.17

and 0.35. The equation (Eq. (3)) for Cd for the

rolling sphere given by Chhabra and Ferreira

(1999) is valid for D/W ratio 0.17 but not for 0.35. At D/W =0.35, the Cd is higher by 1.2

times that given by the Eq. (3). It may be

mentioned that the Cd-Re relation given byEq.(3) is for a rolling sphere down an incline

and cannot be used for a flow past a stationary



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sphere touching a tangent plane since the wake

profile in the two cases is likely to be different. Italso cannot be used when the rolling of a sphere on a

horizontal plane is induced by a flow of fluid taking

place around it. So the Cd-Re relation for a rollingsphere is problem specific. No such restriction

applies for Ca except that the sphere is touching a

tangent plane.From Fig. 5 and Fig. 11, it is seen that the

dominant influence of the added mass force at

starting times is for a distance X/D up to about 0.3.

NOMENCLATURE

T angle of incline

U specific gravity

U f density of fluid

U s density of sphere

A projected area of sphere

Ca added-mass coefficient

Cd drag coefficient

Cd expt drag coefficient from experiment calculated

at terminal velocity

Cd lit. drag coefficient from eq. given by Chhabra

& Ferreira calculated at terminal velocity

D diameter of sphere f r coefficient of rolling friction

f s coefficient of sliding friction

g acceleration due to gravity

I mass moment of inertia of sphere about its

diameter

R radius of sphere

Re instantaneous Reynolds number

Ret Reynolds number at terminal velocity

t time

V instantaneous velocity of sphere

V t terminal velocity of sphere

Vol volume of sphere

W width of tank

X instantaneous distance travelled by sphere

REFERENCES

Chhabra, R. P., Ferreira, J. M., 1999. An

analytical study of the motion of a sphererolling down a smooth inclined plane in an

incompressible Newtonian fluid, Powder

Technology, 104 pp. 130–138.

Cox, S. J., Cooker, M. J., 2000. Potential flow

past a sphere touching a tangent plane, Journalof Engineering Mathematics, 38 pp. 335–370.

Jan, C., Chen, J., 1997. Movements of a sphere

rolling down an inclined plane, Journal of

Hydraulic Research, 35 pp. 689–706.

Langlois, W. E., 1964. Slow Viscous Flow.Macmillan, New York.

Lighthill, J., 1986. An Informal Introduction toTheoretical Fluid Mechanics. Clarendon Press,

Oxford.

Moore, F. K., 1956. Three-dimensional

Boundary Layer Theory, Advances in Applied

Mechanics, 4 (2) 159-228.

Schlichting, H., 1968. Boundary Layer Theory,

sixth ed. McGraw-Hill, New York, pp. 24–43.

Shampine, L. F., Gladwell, I., Thompson, S.,

2003. Solving ODEs with MATLAB,Cambridge University Press, New York, pp. 1–

131.

Starzhinskii, V. M., 1982. An Advanced

Course of Theoretical Mechanics, Mir Publi-

shers, Moscow, pp. 88–89.

Stewart, B. E., Leweke, T., Hourigan, K.,

Thompson, M. C., 2008. Wake formation

behind a rolling sphere, Physics of Fluids, 20,

071704, 1–4.

Strelkov, S. P., 1978. Mechanics, Mir

Publishers, Moscow, pp. 259–269.



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Stuart, J. T., 1963. Unsteady Boundary Layers. In:

Rosenhead, L., (Ed.), Laminar Boundary Layers.Clarendon Press, Oxford, pp. 349–408.

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