CE 213 - Fluid Mechanics Capillary Effect, Compressibility and...

CE 213 - Fluid Mechanics

Capillary Effect, Compressibility and Pascal’s Law

Bachu Anilkumar

Department of Civil and Environmental EngineeringIIT Patna

Anil (IITP) | CE 213 - Fluid Mechanics | Lecture - 3 | August 7, 2019

1Learning Objectives

Surface tension and few special casesCapillary effectCompressibilityPascal’s law


2Surface Tension

Surface tension, σ =FL

(1)

Points to rememberThat surface tension should occur only whenthere is a surface of separation between twofluids

Surface tension of the liquid is not theproperty of the liquid itself, it is a binaryproperty of the liquid and the gas.

Interfacial tension


3Surface TensionCase 1: Fluid on one Side and air on another Side of a thin fluid film

pi and po = Pressure on its convex and concave sides respectively

2σr2dθ2 sin(dθ1

2) + 2σr1dθ1 sin(

dθ2

2) = (pi − po)r1r2dθ1dθ2 (2)

pi − po = ∆p = σ

(1r1

+1r2

)(3)


4Surface TensionCase 2: Air on both sides of a thin liquid film

2x2σr2dθ2 sin(dθ1

2) + 2x2σr1dθ1 sin(

dθ2

2) = (pi − po)r1r2dθ1dθ2 (4)

pi − po = ∆p = 2xσ(

1r1

+1r2

)(5)


5Surface TensionSpecial Cases

Case i: Spherical Liquid Drop

The radius of curvature is same in all directionsr1 = r2 = r i.e. the radius of the dropInteracting with air from only one side

∆p = σ

(1r

+1r

)=

2σr. (6)

Case ii: Spherical Bubble

The radius of curvature is same in all directionsr1 = r2 = rInteracting with air from both sides (inside the bubble and outside the bubble).

∆p = 2σ(

1r

+1r

)=

4σr. (7)


6Surface TensionSpecial Cases

Case iii: Cylindrical Liquid Jet

The radius of curvature is r is one direction and∞ in another directionr1 = r and r2 =∞

∆p = σ

(1r

+1∞

)=σ

r. (8)


7Capillary Effect

The tendency of a fluid to be raised or suppressed in a narrow tube, or capillary tube

Examples

Rise of kerosene through a cotton wick inserted into the reservoir of a kerosene lampRise of water to the top of tall trees


8Capillary EffectContact angles

Interface Contact angleMercury–glass 140Water–glass 0Water–paraffin 107Water–silver 90Organic liquids (most)–glass 0Ethyl alcohol–glass 0Kerosene–glass 26


9Capillary Effect

Capillary effect can be explainedmicroscopically by considering cohesiveforces (the forces between likemolecules, such as water and water)and adhesive forces (the forces betweenunlike molecules, such as water andglass).The relative magnitudes of these forcesdetermine whether a liquid wets a solidsurface or not.


10Capillary Effect

W = ~Fsurface (9)

W = mg = ρVg = ρg(πR2h) (10)

W = ~Fsurface → ρg(πR2h) = 2πRσscosφ

Capillary rise,h =2σs

ρgRcosφ (11)


11Capillary Effect


12Compressibility

Volume of a fluid changes with a changein its temperature or pressure.Expansion - Heated up or depressurizedContraction - Cooled down orpressurizedFluids act like elastic solids with respectto pressure

Bulk modulus of elasticity,E = lim∆v→0

(−∆P

∆vv

)


13Compressibility

Bulk modulus of elasticity,E = lim∆v→0

(−∆P

∆vv

)= −v

dpdv

For a given mass of body, ρv = m (constant)

dρρ

+dvv

= 0→ −dvv

=dρρ

E = −vdpdv

= ρdpdρ

(12)

Interpretation:The fluids for which the k is high, it requires very high pressure to change a definiteamount of density of fluid.For a given change in pressure, the change in volume will be very small, if the valueof k is very high


14Compressibility

Ewater = 2x105 KN/m2

Eair = 101 KN/m2

Air is 104 times compressible than water at atmospheric conditions.

All incompressible fluids exhibit incompressible flow in nature ?

All compressible fluids exhibit compressible flow in nature ?

When to consider the compressibility of fluid/ What is the criteria to decide whether thefluid is compressible or incompressible - Based on Mach number of the flow


15Compressibility

Criteria: Considering whether the change in pressure brought about by the fluid motioncauses large change in volume or density

Bernoulli equation:

O(∆p) = O(ρv2

2)

From Eq. (12),

E = ρdpdρ→ dρ

ρ=

dpE,

dρρ

=ρv2

2E→ dρ

ρ=

12

v2

Eρ

,

W.K.T. the velocity of sound, a =√

E/ρ

∴dρρ

=12

v2

a2 →12

Ma2

Ma is the ratio of velocity of flow to the acousticvelocity in the flowing medium


16Compressibility

Considering a maximum relative change in density of 5% a criterion of an incompressibleflow,

dρρ< 0.05→ 1

2Ma2 < 0.05→ Ma < 0.33

CriterionMa < 0.33, incompressible flow else compressible flow

For air at STP conditions, a = 335m/s

A Mach number of 0.33 corresponds to a velocity of 110 m/s.

∴ Flow of air up to a velocity of 100 m/s under standard condition can be considered asincompressible flow.


17Compressibility

FLUID STATICS


18Fluid Statics

A fluid element, in isolation from its surroundings, will experience two types of forces:

Body forcesForces act throughout the body of fluid and distributed over entire massCaused by external factors: gravitational, electro magnetic, or electro static fields

Surface forcesForces exerted its surroundings through direct contact at the surface.Two components:

Normal force - component along the normal to the areaShear force - along the plane of the area.


19Fluid StaticsNormal Stresses in a Stationary Fluid

Assumption: Fluid is at restNo shear stresses and tensile stressesonly normal forces - compressive in nature

Equations of static equilibrium

ΣFx = σx

(∆y∆z

2

)− σn∆Acosα = 0.

ΣFy = σy

(∆z∆x

2

)− σn∆Acosβ = 0.

ΣFz = σz

(∆x∆y

2

)−σn∆Acosγ−ρg

6(∆x∆y∆z) = 0.

ΣFx ,ΣFy ,ΣFz - net forces acting on fluid elementcosα, cosβ, cosγ - direction cosines of the normal to the inclined plane


20Fluid StaticsNormal Stresses in a Stationary Fluid

∆Acosα = ∆y∆z2

∆Acosβ = ∆x∆y2

∆Acosγ = ∆y∆x2

Upon equating and substituting,

σx = σy = σz = σn (13)


21Pascal’s Law

Pascal’s lawNormal stress at any point in a fluid atrest are

Directed towards the point from alldirectionsEqual magnitude

σx = σy = σz = −p (14)

where p is a scalar quantity, defined ashydrostatic/fluid static/thermodynamicpressure

THANK YOU !!

Date post:	08-Mar-2021
Category:	Documents
Upload:	others
View:	5 times
Download:	0 times

CE 213 - Fluid Mechanics Capillary Effect, Compressibility and...

Documents