Surface Tension & Viscosity (Theory)2009 +1

8/2/2019 Surface Tension & Viscosity (Theory)2009 +1

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1Sunday, March 18,2012

1B.M.Sharma Academy of Physics


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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

2

Surface tension is alsoknown as interfacial force,interfacial tension and

surface density.

Sunday, March 18,2012



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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.

3

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.

4

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.

5

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

7

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.

8

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,




9/909

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





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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.

10

B

F





11/9011

B

F A


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.

14

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


fixed frameR

S

l

PP

16

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

17

Q Qdx

Force appliedby external

agent


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.

21

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.

22

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.

(b)Sunday, March 18,2012



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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 :

31

Due to internal pressure,p2

Due to external pressure,p1 Due to surface tension of the liquid, s




32/90

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).

32

p R1( )

p R2( )

s (2 )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

33

p R1( )

p R2( )

s (2 )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

35

p R1( )

p R2( )

s [2(2 )]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.

36

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

convex upward.Sunday, March 18,2012 39B.M.Sharma Academy of Physics


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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




44/90

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.

45

TT

qq

q q

R

h





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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

46

TT

qq

q q

R

h





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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.

47

TT

qq

q q

R

h


2R

R

Meniscus of watersurface in aglass tube.




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48

TT

qq

q q

R

h


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

49

TT

qq

q q

R

h


1

2g R h




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50Sunday, March 18,2012 50B.M.SDharma Academy Of

Physics


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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).


Physics


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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.

52

A = areaof surface v

F

l


52B.M.SDharma Academy OfPhysics


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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.

53

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


Physics


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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.


Physics


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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.


Physics


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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.


Physics

U it d di i f


58/90

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


Physics

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.

59Sunday, March 18,2012 59B.M.SDharma Academy OfPhysics


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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.


C ffi i t f D i i it


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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.


Poiseuilles formula


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Poiseuille s formulaThe streamlines for steady flow in a

circular pipe are shown in figure.

62

Steady flow



Everywhere they are parallel to the


63/90

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.

63

Steady flow




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The surfaces of equal velocity are the

surfaces of concentric cylinders.

64

Steady flow



An expression for the volume of


65/90

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.



67/90

liquid flow

l

P = atmospheric pressure

P + p p

r


We have V = k x ry (p/l)z


68/90

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].


V = k x ry (p/l)z


69/90

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



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.


Stokes law


71/90

Stokes lawThe streamlines for a fluid flowing

slowly past a stationary solid sphereare shown in figure.

Viscous fluid



72/90

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.

73

Viscous fluid



Viscous forces are thereby brought


74/90

Viscous forces are thereby broughtinto play and constitute theresistance experienced by the movingsphere.

74

Viscous fluid



If we make the posible assumption


75/90

(i) the viscosity h of the fluid,

(ii) the velocity v and radius r ofthe sphere,

75

If we make the posible assumptionthat the viscous retarding force Fdepends on



then an expression can be derived


76/90

then an expression can be derivedfor F by the method ofdimensions. Thus

F = k hx vy rz

76

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



[MLT-2] = [ML-1 T-1]x [LT-1] y [L]z


77/90

[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


A detailed treatment first done by


78/90

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).


TERMINAL VELOCITY


79/90

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.



80/90

Fallingsphere

F (viscous drag)

U (fluid upthrust)

Viscous liquid

W (weight of sphere)


The resultant downward force is


81/90

Fallingsphere

F (viscous drag)

U (fluid upthrust)

Viscous liquid

W (weight of sphere)

81

(W U F) and causes the sphere toaccelerate until its velocity becomes

constant, W U F = 0



The sphere then continues to fall with a


82/90

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

82

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



Hence


83/90

Hence

3 34 4 6 03 3

tr g r g rv s h 22 ( )

9t

r gv

s h


Example of terminal velocity


84/90

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.



85/90

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.


3 Falling down with the help of a parachute


86/90

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.


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


87/90

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


Fluid Resistance


88/90

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.


Capillaries in series


89/90

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.


Capillaries in parallel


90/90

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

Surface Tension & Viscosity (Theory)2009 +1

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