surface tension & viscosity (theory)2009 +1
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The free surface of a liquid actslike a stretched membrane, that
is, the surface of a liquid is in astate of tension as the surface of
an inflated balloon.
Surface Tension
REAL LIQUIDS
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Surface tension is alsoknown as interfacial force,interfacial tension and
surface density.
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It is the property possessedby liquid surfaces wherebythey behave as if they are
covered by a thin elasticmembrane in a state oftension.
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Surface tension is due tointermolecular attraction.
Qualitative definition
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If we draw a straight line on the freesurface of a liquid, this line experiences aforce perpendicular to it but along the
surface.
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B
F
F
A
This force
per unitlength iscalled
surfacetension.
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Thus, it is also numerically equal tothe work to be done to increase thesurface area of a liquid surface by
unity.
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Magnitude-wise and dimensionally, it isequal to the free surface energy which is
defined as the energy per unit area stored
in the surface.
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DefinitionThe surface tension, s, of a liquid
(also called the coefficient ofsurface tension) is defined as the
force per unit length acting in thesurface at right angles to one sideof a line drawn in the surface.
The units of surface tension are Nm-1.
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Cause of surface tension
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B
A
Let us consider a molecule A inside the body
of the liquid as shown in figure.
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B
F A
Inter-molecular forcesin liquids.
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In the volume of aliquid, every moleculesuch as A, as shown in
figure, is surroundedby an equal number ofmolecules on all sides,therefore, no net forceacts on this molecule.
On the otherhand, a molecule on thesurface of the liquid such as B, is
surrounded with a very few molecules onthe vapour side as compared with the
liquid below.
Thus, it experiences a net force F
in the downward direction,
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If one pulls the molecule B
upwards, then one has towork against theintermolecular attraction.
Therefore, molecules on the
free surface of a liquid havepotential energy.B
F A
Inter-molecular forcesin liquids.
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If the moleculeA is shifted to the
position of molecule B, then externalwork has to done in breaking thebonds because at the positionA
molecule forms more bonds with theneighbours as compared to the
position of molecule B.
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B
F
Inter-molecular forcesin liquids.
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B
F A
Inter-molecular forcesin liquids.
Thus, molecules on the surface of a
liquid have more potential energyas compared to the molecules in
the body of the liquid.
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Under the actionof surface tensionforces, the free
surface of a liquid
tends to have theleast area for agiven volume of
liquid
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Since a sphere hasminimum area fora given volume,
therefore,
raindrops andsmall droplets ofmercury are
approximately
spherical in shape
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Surface energySurface energy (also called
free surface energy,) is not thetotal surface energy. It is thetotal surface energy per unitsurface area.
or free surface energy is theamount of work done againstthe force of surface tension, informing the liquid surface ofunit area at a constanttemperature.
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the energy per unit area ofexposed surface.
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Surface tension (s) and surfaceenergy (v)
In figure, a film of liquid, of surfacetension s, is shown stretched across a
horizontal frame PQRS.
Q Q
dx
Force appliedby externalagent
Sliding wirefilm of liquid
fixed frame
R
S
l
PP
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Q Qdx
Force appliedby externalagent
Sliding wirefilm of liquid
fixed frameR
S
l
PP
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The force on the sliding wire PQ of length lis s 2l, since the film has two surfaces. IfPQ is moves to PQ through a distance dxagainst the surface tension, the new area ofsurface A = 2l dx (the film has two sides),and
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Work done W to enlarge surface area
= 2sl dx This equals the increase in thesurface energy v and we then
have 2
2
W l xV
A l x
s d s d
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Q Qdx
Force appliedby external
agent
Sliding wirefilm of liquid
fixed frame
R
S
l
PP
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Thus, s and v are numerically equal;Units and dimensions of surface tension
22[ ]
[ ]
Force MLTMT
Length L
s
SI unit ofs = newton per meter = Nm-1CGS unit ofs = dyne per cm which is morecommonly used.
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Note that The magnitude of a s depends on the
temperature of the liquid and on the mediumon the other side of the free surface.
The surface tension of a liquid decreases byincreasing temperature.
The surface tension of a liquid decreases byadding impurities.
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Application of surface tensionPlane and curved surfaces (cavities, dropsand bubbles)
Examples of plane and curved surfaces(cavities, drops and bubbles) are shown infigure.
A = Plane surface
B = Cavities
Drop Bubble
A
B
B
B
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The bubbles, like soap bubbles, are likeblown-up balloons, air inside and airoutside with a thin liquid film in between
them.
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Bubble
B
B
B
This thin film naturally has two freesurfaces, one inside and the other outside.
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Cavities generally have air inside and liquidoutside. They have one interface or freesurface.
Drops generally have water or some otherliquid inside and air or some gas outsidethem. They too have one exposed surface.
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B = Cavities
Drop
B
B
B
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Pressure on the two sides of a planeliquid surface
Vapour side
Liquid side
S
S
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Pressure difference between the twosides of curved liquid surfaces
A molecule lying on the surface of a liquid is
attracted by other molecules on the surfacein all directions.
If the surface is plane, then the molecule isattracted equally in all directions. Hence the
resultant force on the molecule due tosurface tension is zero.
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Pressure on the two sides of a curvedliquid surface
Figure shows why surface tension forces (s)do not let away patterns remain inequilibrium on liquid surfaces. Crests (C)
have net downward surface tension forcesand troughs (T) have net upward surface
tension forces.
S S
C
S S
T
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If the surface is convex, then a resultantcomponent of all the forces of attraction
acting on every molecule acts normal to thesurface and is directed inwards.
(b)
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Similarly, if the surface is concave, thenevery molecule experiences a resultant
force due to surface tension acting normally
outwards.
(c)
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(c)2
Obviously, for the equilibrium of acurved surface, there must be a
difference of pressure between its two
sides so that the excess pressure forcemay balance the resultant force due to
surface tension.
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On the concave face of a curvedsurface there is always an excesspressure over the convex face of
the surface.
The magnitude of excess pressurecan be obtained by studying the
formation of air and soap bubbles.
Excess Pressure
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p R1( )
p R2( )
s (2 )R
The cross-section of an airbubble of radius .R
Excess Pressure in an Air BubbleThe figure shows one-half cross-section of anair bubble formed inside liquid. It is in
equilibrium under the action of three forces :
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Due to internal pressure,p2
Due to external pressure,p1 Due to surface tension of the liquid, s
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If R is the radius of the air bubble, then theforces due to external and internal pressures
arep1(R2) andp2(R2), respectively.Since the surface tension acts around thecircumference of the bubble, therefore, the
force of surface tension is s (2R).
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p R1( )
p R2( )
s (2 )R
The cross-section of an airbubble of radius .R
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Thus, from the condition of equilibrium,
2 2
2 1( ) ( ) (2 ) p R p R R s or,
2 12p pRs
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p R1( )
p R2( )
s (2 )R
The cross-section of an airbubble of radius .R
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Excess Pressure in Soap BubbleA soap bubble forms two liquid surfaces incontact with air, one inside the bubble and the
other outside the bubble. The figure shows one-half cross-section of the soap bubble. Byconsidering its equilibrium, we get,
2 2
2 1( ) ( ) [2(2 )] p R p R R s
or,2 1
4p pRs
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p R1( )
p R2( )
s [2(2 )]R
The cross-section of an airbubble of radius .R
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If a narrow glass tube openat both ends is pushed inwater, water rises in the tube
to a height above its surface.
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Capillarity
The narrower the tube, the greateris the height to which water rises.
This phenomenon is known as capillarity.Sunday, March 18,2012
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Capillary action with water. Narrower the tubegreater the height.
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The free surface(meniscus) of theliquid which rises inthe capillary tube is
concave upward.Sunday, March 18,2012 37B.M.Sharma Academy of Physics
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When the same capillarytubes are placed in mercury,the liquid is depressed belowthe outside level.
The depression increase asthe diameter of the capillarytube decreases.
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Capillary action with mercury. Narrower the
tube greater the depression in the tube.The meniscus of liquid
which falls in thecapillary tube is
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Adhesion
is the attractive forcebetween the molecules of solidsand liquids or between themolecules of two different liquids.
Cohesion is the attractive forceamong the molecules of the sameliquid.
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If the adhesive forces arestronger than the cohesive forces
then the liquids wets the solidsurface, as water wets thesurface of glass.
If the cohesive forces arestronger than the adhesive forcesthan the liquids does not wet thesolid surface, as mercury does not
wet the surface of glass.
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The wetting and non-wetting action
of a liquid can also be explained interms of the angle of contact.
If a tangent is drawn on themeniscus of the liquid at the line ofcontact between the liquid and the
surface,
then the angle of contact is definedas the angle between the tangentand the solid surface measured
through the liquid.
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If the angle of contact isacute i.e., q < 90, then theliquid wets the surface.
If the angle of contact isobtuse, i.e.,q > 90, then theliquid does not wet the
surface.
q
Liquid
Solid
q90(b) Non-Wetting Liquid
q
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The figure shows a magnified cross-section of acapillary tube of radius R. Since the angle ofcontact q is acute, therefore, water tends tomaximize its area of contact with the glasssurface, thus it rises in the capillary tube.
Determination of Capillary Rise
TT
qq
q q
R
h
Rise of liquid in acapillary tube.
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In terms of forces, one can imagine
that the vertical component of the forceof surface tension pulls the liquid up inthe tube to a height such that this forceis able to balance the weight of theliquid in the tube.
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TT
qq
q q
R
h
Rise of liquid in acapillary tube.
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Since the force of surface tension acts
along the circumference of the freesurface of the liquid, therefore, it isgiven by
(2 )T Rs
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TT
qq
q q
R
h
Rise of liquid in acapillary tube.
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The weight of water inside the tubeis given ( ')W V V g
where V = volume of the cylinder of
radius R and height h,
and V is the volume of water lyingbelow the meniscus and above thecylinder of height h.
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TT
qq
q q
R
h
Rise of liquid in acapillary tube.
2R
R
Meniscus of watersurface in aglass tube.
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TT
qq
q q
R
h
Rise of liquid in acapillary tube.
2R
R
Meniscus of watersurface in aglass tube.
i.e., 2V R hand 2 3
2
' ( ) 3V R R R or,31
'3
V RSince v is negligible with respect to V,therefore,
thus, W= (R2h)g
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By using the condition ofequilibrium, we get, cosT W
or,2
(2 )cos ( )R R h gs q or,
2cos
gRhs qThus
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TT
qq
q q
R
h
Rise of liquid in acapillary tube.
1
2g R h
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VISCOSITY
Viscosity is internal friction offered
by a fluid itself against its own flow.
If adjacent layers of a material aredisplaced laterally over each other,
the material is called under shear.This is what happens in case of allactual fluids (as against ideal fluids).
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Coefficient of viscosity (h)When a fluid is flowing as in figure,the top layer has the maximumvelocity while he bottom most layermay be sticking with the floor andhaving zero velocity.
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A = areaof surface v
F
l
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Hence as we go down from top-layerto the bottom layer, velocity
decreases. This is called velocitygradient which means change invelocity per unit distance.
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A = areaof surface v
F
l
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Coefficient of viscosity (h) is definedas the tangential force Frequired perunit area A to maintain unit velocitygradient perpendicular to thedirection of flow.
Tangential stress
Velocity gradienth /
/
F A
vh /
/
F A
dv dh
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Those liquids for whom h does notdepend upon velocity gradient (e.g.,water and glass) are calledNewtonian fluids.
For some liquids h decreases astangential stress increases e.g.,
paints, glues and liquid cements.These liquids are called thixotropic.
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Effect of temperature on viscosity
Experiments show that h of a liquiddecreases sharply with increase in itstemperature and becomes zero at its boilingtemperature. On the other hand, h of gasesincrease with temperature.
Viscosity versus sliding friction
Viscosity is the opposition to the flow of onelayer of a liquid over the other layer of theliquid, while sliding friction is theopposition to the slipping of one solidsurface over another solid surface.
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Point of similarity
1. Both oppose relative motion.
2. Both are due to molecular forces.
Point of dissimilarity
1. Total viscous force depends upon the
areas involved whereas total frictionalforce in independent of areas.
2. h depends upon velocity gradient exceptin case of Newtonian liquids, whereas mdoes not depend on velocity.
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U it d di i f
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Units and dimensions ofh/
/
F A Fl
v vA
h 2 1 11 2
[ ][ ][ ]
[ ][ ]
MLT LML T
LT L
SI unit ofh = pascal second i.e. Pa sCGS unit ofh = poise1 Pa s = 1 Ns m-2
1 poise = 1 dyne s cm-2
1 deca poise = 10 poise
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There is yet another unit of h, calledpoiseuille which is identical with the SI unitof viscosity (the pascal second),.
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Newtonian and non-newtonian fluids
If a fluid such that its coefficient ofviscosity remains same whatever are thespeeds of its various layers which areflowing in contact with each other, it iscalled a newtonian fluid. If it is not so, it is
called a non-newtonian fluid.
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Coefficient of Kinematic viscositySymbol:v.
It is the ratio of coefficient of viscosity (h)to the fluid density ().It is used in modifying the equations of
motion of a perfect fluid to include theterms due to a real fluid.
The units of kinematic viscosity are metersquare per second.
At room temperature, water has a kinematicviscosity of 10-6 m2 s-1.
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Coefficient of Dynamic viscosityThe ordinary viscosity coefficient (h) isoften called coefficient of dynamic viscosityto distinguish it from kinematic viscosity toavoid confusion.
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Poiseuilles formula
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Poiseuille s formulaThe streamlines for steady flow in a
circular pipe are shown in figure.
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Steady flow
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Everywhere they are parallel to the
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Everywhere they are parallel to theaxis of the pipe and representvelocities varying from zero at the
wall of the pipe to a maximum at itsaxis.
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Steady flow
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The surfaces of equal velocity are the
surfaces of concentric cylinders.
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Steady flow
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An expression for the volume of
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An expression for the volume ofliquid passing per second, V, througha pipe when the flow is steady, can
be obtained by the method ofdimensions.
Steady flow
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It is reasonable to assume that Vdepends on
(i) the coefficient ofviscosity h of the liquid.(ii) the radius r of thepipe and
(iii) the pressure gradientp/l causing the flow,where p is the
pressure differencebetween the ends ofthe pipe and l is itslength.
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liquid flow
l
P = atmospheric pressure
P + p p
r
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We have V = k x ry (p/l)z
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We have V = kh ry (p/l)Where x, y and z are the indices to be foundand k is a dimensionless constant.
The dimensions
of V are [L3T-1],
ofh [ML-1 T-1],of r [L],
of p [MLT-2/L2], i.e., [ML-1 T-2]
(since pressure = force/area),
and of l [L]. Hence the dimensionsof p/lare [ML-2 T-2].
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V = k x ry (p/l)z
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Equating dimensions,
[L3
T-1
]= [ML-1
T-1
]X
[L]y
[ml-2
T-2
]z
Equating indices of M, L and T on both sides,
0 = x + z
3 = -x = y 2z
-1 = -x 2z
Solving, we get x = -1, y = 4 and z = 1.Hence
4
8prV
l
h69
V = khx ry (p/l)z
Poiseuilles formula
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Limitations of Poiseuilles formula
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Limitations of Poiseuille s formula(i) The flow is capillary tube must be
streamlined.(ii) The pressure difference (p) across ends
of capillary must be constant.
(iii) The layers of liquid in contact withwalls of capillary are assumed to be atrest and the velocity of liquid layersgoes on increasing towards the axis.
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Stokes law
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Stokes lawThe streamlines for a fluid flowing
slowly past a stationary solid sphereare shown in figure.
Viscous fluid
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When the sphere moves slowly ratherthan the fluid, the pattern is similar
but the streamlines then flow theapparent motion of the fluid particlesas seen by someone on the movingsphere.
Viscous fluid
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In this latter case, it is known thatthe layer of fluid in contact with thesphere moves with it, thus creating avelocity gradient between this layerand other layers of fluid.
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Viscous fluid
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Viscous forces are thereby brought
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Viscous forces are thereby broughtinto play and constitute theresistance experienced by the movingsphere.
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Viscous fluid
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If we make the posible assumption
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(i) the viscosity h of the fluid,
(ii) the velocity v and radius r ofthe sphere,
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If we make the posible assumptionthat the viscous retarding force Fdepends on
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then an expression can be derived
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then an expression can be derivedfor F by the method ofdimensions. Thus
F = k hx vy rz
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Where x, y and z are the indices to thefound and k is a dimensionless constant.
The dimensional equation is
[MLT-2] = [ML-1 T-1]x [LT-1] y [L]z
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[MLT-2] = [ML-1 T-1]x [LT-1] y [L]z
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[MLT ] = [ML T ] [LT ] y [L]
Equating indices of M, L and T on both
sides. 1 = x1 = -x + y = z
-2 = -x y
Solving, we get x = 1, y = 1 and z = 1Hence F = k h v r
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A detailed treatment first done by
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A detailed treatment, first done byStrokes, gives k = 6 and so
F = 6hvrThis expression, called Stokes
law,only holds for steady motion in a
fluid of infinite extent (otherwise the
walls and bottom of the vessel affectthe resisting force).
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TERMINAL VELOCITY
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Now consider the sphere fallingvertically under gravity in a viscousfluid. Three forces act on it.
(i) its weight W,acting downwards;
(ii) the upthrust Udue to weight offluid displaced,acting upwards;and
(iii) the viscous dragF, actingupwards.
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Fallingsphere
F (viscous drag)
U (fluid upthrust)
Viscous liquid
W (weight of sphere)
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The resultant downward force is
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Fallingsphere
F (viscous drag)
U (fluid upthrust)
Viscous liquid
W (weight of sphere)
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(W U F) and causes the sphere toaccelerate until its velocity becomes
constant, W U F = 0
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The sphere then continues to fall with a
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The sphere then continues to fall with aconstant velocity, known as its terminalvelocity, of say vt.
Now 34
3W r g Where is the density
of the sphere
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and34
3
U r g sWhere s is the density of the fluid. Also, ifsteady conditions still hold when velocity vtis reached then by Stokes law
6 tF rvh
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Hence
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Hence
3 34 4 6 03 3
tr g r g rv s h 22 ( )
9t
r gv
s h
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Example of terminal velocity
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Example of terminal velocity
1. Formation of clouds
When water-vapours present in theatmosphere condense, small droplets areformed. The weight of these droplets in airis very small. Therefore they attain the
terminal velocity very soon due to viscosityof air. Because the value of their terminalvelocity is very small, they appear to float inthe sky as clouds.
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2. Relative velocities of rain drops
As terminal velocity is proportional to thesquare of radius i.e., vT r2, therefore smallrain drops fall with small velocities andlarge drops fall with large velocities.
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3 Falling down with the help of a parachute
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3. Falling down with the help of a parachute
When a person jumps down from a flying
aeroplane, his parachute is closed, henceinitially his velocity increases very rapidlywhile the viscosity of air tries to reduce hisvelocity.
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When the parachute opens, the viscosity ofair exerts greater viscous force in upwarddirection (since viscous force is directlyproportional to surface area); due to which
the velocity of person begins to decreaseand finally he attains the terminal velocityand reaches safe on ground.
Fluid Resistance
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Fluid Resistance
The Poiseuilles formula may be expressed
as4
3
4/ sec
8 8 /
pr p pV m
l l r R
h h Where is called the fluid resistance.
48 lR
rh
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Fluid Resistance
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Now, capillary is a tube with narrow orifice.
Thus, application of Poiseuilles formulashows that the resistance offered by theliquid for its flow in capillary is directlyproportional to the length of the capillary
and inversely proportional to the fourthpower of its radius.
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Capillaries in series
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p
If two capillaries are connected in series,then
(i) Volume of liquid flowing per secondthrough each capillary is same i.e., V1 =V2 = V.
(ii) The net pressure difference acrosswhole system is equal to the sum ofpressure difference across separatecapillaries i.e., p = p1 + p2.
(iii) Net fluid resistance is the sum of fluidresistances of separate capillaries i.e. R= R1 + R2.
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Capillaries in parallel
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p p
If two capillaries are connected in parallel,then
(i) pressure difference across each capillaryremains the same i.e.. p1 = p2 = p
(ii) The total volume flowing per second is
the sum of a volume of liquid flowingthrough separate capillaries i.e. V = V1+ V2
(iii) The net fluid resistance R is given by
1 2
1 1 1
R R R