Chapter 10: Mechanical properties of fluids

Introduction

Liquids and gases can flow and are therefore called fluids. It is this property that distinguishes liquids and gases from solids in a basic way. Fluids are everywhere around us. Earth has an envelope of air and two-thirds of its surface is covered with water. Water is not only necessary for our existence; every mammalian body consists mostly of water. All the processes occurring in living beings including plants are mediated by fluids. Thus understanding the behaviour and properties of fluids is important.

How are fluids different from solids? What is common in liquids and gases?

Solids have fixed volume and shape, liquids have fixed volume but not fixed shape. Gases on the other hand neither have fixed volume nor fixed shape.

Fluids can be defined as any substance which is capable of flowing.

They don’t have any shape of their own. For example:-water which does not have its own shape but it takes the shape of the container in which it is poured. But when we pour water in a tumbler it takes the shape of the tumbler

Fluids are assumed to be the incompressible (i.e., the density of liquid is not dependent on the variation in pressure and remains constant).

Fluids are also assumed to be non-viscous (i.e., the two liquid surfaces in contact are not pressing any tangential force on each other)

Pressure

Pressure is defined as the physical force exerted on an object. The force applied is perpendicular to surface of objects per unit area.

Mathematically ,  P= F/ A

Unit of pressure is Pascals (Pa).

Since P= F/A   so pressure is inversely proportional to the Area.  If the area would be less, pressure would be more.

For Example:-

Consider a very sharp needle which has a small surface area and consider a pencil whose back is very blunt and has more surface area than the needle.

If we poke a needle in our palm it will hurt as the needle gets pierced inside our skin. Whereas if we poke the blunt side of the pencil into our hand it won’t hurt so much. This is because the area of contact between the palm and the needle is very small therefore the pressure is large.

Whereas the area of contact between the pencil and the palm is more therefore the pressure is less.

Conclusion: Two factors which determine the magnitude of the pressure are:-

• Force – greater the force greater is the pressure and vice-versa.
• Coverage area –greater the area less is the pressure and vice-versa.

Fluid pressure

Normal force exerted by fluid per unit area. This means force is acting perpendicular to the surface of contact.

If a body is submerged in the water, force is exerted by the water perpendicular to the surface of the body. Fluid force exerts itself perpendicularly to any surface in the fluid, no matter the orientation of that surface. Thus, fluid pressure has no intrinsic direction of its own and can be considered as a scalar quantity.

Pressure is a scalar quantity. Because the force here is not a vector quantity but it is the component of force normal to the area.

Dimensional formula for pressure is[ ML^-1T²]. The S.I unit is Pascal (Pa).

Atmospheric pressure: The atmospheric pressure at a point is equal to the weight of the column of air of unit cross-sectional area extending from that point to the top of the atmosphere. Its value is 1.013 ×10⁵ Pa at sea level. Atmospheric pressure drops as altitude increases. It is measured using an instrument called a barometer.

Definition of 1 atm

An atmosphere (atm) is a unit of measurement equal to the average air pressure at sea level at a temperature of 15 degrees Celsius (59 degrees Fahrenheit). One atmosphere is 1,013 millibars, or 760 millimetres (29.92 inches) of mercury. Atmospheric pressure drops as altitude increases

Pascal’s law

Pascal’s law states that if the pressure is applied to uniform fluids that are confined, the fluids will then transmit the same pressure in all directions at the same rate.

Pascal’s law holds good only for uniform fluids.

Let us try to understand this with a suitable example. Consider a vessel of circular shape filled with water which has 4 openings and in the entire openings 4 pistons are attached.

• Apply force on the first piston; this piston will move inward and all other pistons will move outwards.

• This happens because when this piston moves inwards the pressure is exerted on the water. Water transmits this pressure in all the directions.
• The other pistons, except A, moves at the same speed which shows water has exerted pressure in all the directions

Conclusion:-

1. For a uniform fluid in equilibrium, pressure is the same at all points in a horizontal plane. This means there is no net force acting on the fluid; the pressure is the same at all the points.

1. A fluid moves due to the differences in pressure. That means fluid will always move from a point which is at a higher pressure to the point which is at a lower pressure.

Archimedes principle

Archimedes Principle:

• Consider a body partially or fully dipped in a fluid. The fluid exerts a contact force on this body. The resultant of all these contact forces is termed buoyant force or up thrust.
• F=weight of fluid displaced by the body
• This force is termed buoyant force and it acts vertically upwards (opposite to the weight of the body) through the centre of gravity of the displaced fluid. Mathematically,       

F=Vσg

Where V is the volume of displaced liquid and σ is the density of the liquid

• The apparent reduction in weight of body =Up thrust = weight of liquid displaced by the body.

Variation of pressure with depth

Consider a cylindrical object inside a fluid, consider two positions for this object. Fluid is at rest therefore the force along the horizontal direction is zero.

Force along the vertical direction

Consider two positions 1 and 2. Force at position 1 is perpendicular to cross-sectional area   A₁F₁ = P₁

Similarly,  A₂ F₂=P₂ 

Totalthe  force   F_net= F₁+F₂   as F₁ is along negative y axis and is negative and F₂ is along positive y axis and hence positive so we have now,

 F_net= -F₁+F₂ = -P₁A₁+P₂A₂=(P₂-P₁)A    , Taking A₁=A₂=A

This net force will be balanced by the weight of the cylinder. Therefore under equilibrium conditions.

Fnet= mg (weight of cylinder)=ρ Vg( weight of liquid displaced)

Fnet= (P₂-P₁) A=ρg(Ah)   using V= Ah

Which gives  (P₂-P_{1)A =ρghA   ⇒ (P2-P1)=ρgh}

Therefore the difference in the pressure is dependent on the height of the cylinder.

Consider the top of the cylinder exposed to air therefore P₁ = P_a (atmospheric pressure)

Then   P₂= P_a+ρgh

Conclusion:

•  The pressure P₂ , at depth  below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount ρgh. 
• The pressure is independent of the cross sectional or base area or the shape of the container.
• Thus, the pressure P, at depth below the surface of a liquid open to the atmosphere, is greater than atmospheric pressure by an amount ρgh. The excess of pressure, P − Pa, at depth h is called a gauge pressure at that point.

Hydrostatic Paradox

Hydrostatic Paradox means: - hydro = water, static =at rest

Paradox means that something is taking place surprisingly.

• Consider 3 vessels of very different shapes (like thin rectangular shape, triangular and some filter shape) and we have a source from which water enters into these 3 vessels.
• Water enters through the horizontal base which is the base of these 3 vessels. We observe that the level of water in all the 3 vessels is the same irrespective of their different shapes.
• This is because pressure at some point at the base of these 3 vessels is the same.
• The water will rise in all these 3 vessels till the pressure at the top is same as the pressure at the bottom.
• As pressure is dependent only on height therefore in all the 3 vessels the height reached by the water is the same irrespective of difference in their shapes.

This experiment is known as Hydrostatic Paradox.

Applications: Pascal’s law for transmission of fluid pressure

Hydraulic lift:-

Hydraulic lift is a lift which makes use of fluid.  For example: Hydraulic lifts that are used in car service stations to lift the cars.

Principle: -

• Inside a hydraulic lift there are 2 platforms, one has a smaller area and the other one has a larger area. It is a tube-like structure which is filled with uniform fluid.
• There are 2 pistons (P1 and P2) which are attached at both the ends of the tube. Cross-sectional area of piston P1 is A1 and piston P2 is A2.
• If we apply force F1 on P1, pressure gets exerted and according to Pascal’s law pressure gets transmitted in all the directions and same pressure gets exerted on the other end. As a result the Piston P2 moves upwards.

Advantage of using hydraulic lift is that by applying small force on the small area we are able to generate a larger force.

Mathematically ,  F₁= P₁A₁  and F₂= P₂A₂    

Since by Pascal’s law

Hydraulic Brakes

• Hydraulic brakes work on the principle of Pascal’s law.
• According to this law whenever pressure is applied on fluid it travels uniformly in all directions.
• Therefore when we apply force on a small piston, the pressure gets created which is transmitted through the fluid to a larger piston. As a result of this larger force, uniform braking is applied on all four wheels.
• As braking force is generated due to hydraulic pressure, they are known as hydraulic brakes.
• Liquids are used instead of gas as liquids are incompressible.

Effect of gravity on fluid pressure

We can start by stating the relationship between gravity and fluid pressure. We can define both of the terms. We can also write down the formula to find the fluid pressure and see if it is related to gravity. Gravity is a force existing between bodies.

The formula to find the fluid pressure is given by the formula,

P=ρgh

Where ρ is the density of the fluid, g is the acceleration due to gravity and h is the depth of the fluid level.

Fluid pressure is the pressure at a point within a fluid arising due to the weight of the fluid. Gravity is the universal force of attraction acting between all matters.

Therefore, according to the formula   P=ρgh  the pressure exerted by a fluid, is directly proportional to the specific gravity at any point and to the height of the fluid above the point.

A fun thing to do: Virtual lab

Below is the link of the simulation of under pressure

Under pressure, this simulation is to understand the concept of pressure.  It will help us understand how pressure varies with depth for different fluids.

What can we do in this simulation?

• We can choose the density of the liquid and can fill the liquid up to the height we want to fill it.
• We can also change the value of gravity and using the meter to measure pressure we can get the value of pressure in terms of Metric unit (Pa) and other units like ‘atm’. We just need to drag the meter and place it to the point where we want to measure the pressure.
• We can also on /off the atmospheric pressure in this simulation to see its effects.

Fluid dynamics

So far we have studied fluids at rest now we will discuss fluid in motion. The study of fluids in motion is known as fluid dynamics. When a water-tap is turned on slowly, the water flow is smooth initially but loses its smoothness when the speed of the outflow is increased. In studying the motion of fluids we focus our attention on what is happening to various fluid particles at a particular point in space at a particular time.

Streamline flow or steady flow

Some streamlines for fluid flow

• The flow of a fluid is said to be steady if at any point, the velocity of each passing fluid particle remains constant within that interval of time.
• Streamline is the path followed by the fluid particle.
• It means that at any particular instant the velocities of all the particles at any point are the same. But the velocity of all the particles won’t be the same across all the points in space.
• Steady flow is termed as ‘Streamline flow’ and ‘Laminar flow’.
• No two streamlines can intersect.
• If two streamlines intersect each other, the particles won’t know which path to follow and what velocity to attain. That is why no two streamlines intersect.

To understand it more clearly let us try to understand this with the help of the example.

• Consider a case when all the particles of fluid passing point A have the same velocity. This means that the first particle will have velocity V1 and second will have velocity V1 and so on. All the particles will have the same velocity V1 at point A.
• At point B, all particles will have velocity V2. Similarly, at point C the velocity of all the particles is V3.

We can see that the velocity is changing from point to point but at one particular point, it is the same.

Equation of Continuity

According to the equation of continuity  A V= constant.

Where A =cross-sectional area and V=velocity with which the fluid flows.

It means that if any liquid is flowing in streamlined flow in a pipe of non-uniform cross-section area, the rate of flow of liquid across any cross-section remains constant.

Consider a fluid flowing through a tube of varying thickness.

Let the cross-sectional area at one end (I) = A1 and the cross-sectional area of the other end (II) = A2.

The velocity and density of the fluid at one end (I) =v1, ρ1respectively, velocity and density of fluid at other end.

Volume covered by the fluid in a small interval of time Δt across left cross-section  =A₁v₁Δt

Volume covered by the fluid in the same time across the right section=A₂v₂Δt

Mass of fluid passes from the left end in time Δt= ρ₁A₁v₁Δt

Mass of fluid passes from the right end in time Δt =ρ₂A₂v₂Δt

If the fluid inside is incompressible which means volume doesn’t change with time and density remains the same then we have ρ₁=ρ₂

So, Finally, we have  A₁v₁ =A₂v₂    Also  Av= constant. 

This is called the equation of continuity.

This equation is termed as the conservation of mass of incompressible fluid.

Turbulent flow

Turbulent flow, a type of fluid (gas or liquid) flow in which the fluid undergoes irregular fluctuations, or mixing, in contrast to laminar flow, in which the fluid moves in smooth paths or layers. In turbulent flow, the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction

This means fluid particles are moving here and there, they are not moving in an organized manner. They all will have different velocities. All the particles are moving here and there randomly.

Bernoulli’s equation

Fluid flow is a complex phenomenon. But we can obtain some useful properties for steady or streamlined flows using the conservation of energy.

Consider a fluid moving in a pipe of varying cross-sectional area. Let the pipe be at varying heights as shown in the figure. We now suppose that an incompressible fluid is flowing through the pipe in a steady flow

Its velocity must change as a consequence of the equation of continuity.   A force is required to produce this acceleration. Which is caused by the fluid surrounding it, the pressure must be different in different regions.

The statement of Bernoulli’s principle

For a streamlined fluid flow, the sum of the pressure (P), the kinetic energy per unit volume  ρ v²/2  and the potential energy per unit volume ρgh remain constant.

P+ ρv²/2 + ρgh= constant

Assumptions in Bernoulli’s equation:

1. Fluid flows through a pipe of varying width.
2. Pipe is located at changing heights.
3. Fluid is incompressible.
4. Flow is laminar.
5. No energy is lost due to friction: applicable only to non-viscous fluids.

When a fluid is at rest i.e. its velocity is zero everywhere, Bernoulli’s equation becomes P1 + ρgh1 = P2 + ρgh2; (P1 − P2) = ρg (h2 − h1)

Speed of Efflux - Torricelli's law

The word efflux means flow outward. Torricelli discovered that the speed of efflux from an open tank is given by a formula identical to that of a freely falling body.

Consider a tank containing a liquid of density ρ with a small hole in its side at a height y1 from the bottom (see figure given below). The air above the liquid, whose surface is at height y2, is at pressure P. From the equation of continuity, we have   A1 v1= A2v2 

If the cross-sectional area of tank A2 is much larger than that of the hole (A2 >>A1), then we may take the fluid to be approximately at rest at the top, i.e. v2 = 0. Now applying the Bernoulli equation at points 1 and 2 and noting that at the whole P1 = Pa, the atmospheric pressure.

From Bernoulli’s equation we have

Simplifying we get

   

By taking y2-y1=h 

So finally we have

Since P1=Pa and P2=P  

Then,

Case 1: When the tank is open from the top P = Pa, then

This is the speed of a freely falling body. And the above equation is known as Torricelli’s law

Case 2: When the Tank is not open to the atmosphere but P>>Pa.

Therefore 2gh is ignored as it is very large, hence v1= √2P/ρ.

The velocity with which the fluid will come out of the container is determined by the pressure at the free surface of the fluid alone.

Venturimeter

Venturimeter is a device to measure the flow of incompressible liquid.

It consists of a tube with a broad diameter having a larger cross-sectional area but there is a small constriction in the middle.

It is attached to a U-tube manometer. One end of the manometer is connected to the constriction and the other end is connected to the broader end of the Venturimeter.

The U-tube is filled with a fluid whose density is ρ.

A1= cross-sectional area at the broader end, v1 = velocity of the fluid.

A2=cross-sectional area at the constriction, v2= velocity of the fluid.

The principle behind this meter has many applications. Filter pumps or aspirators, Bunsen burners, atomizers and sprayers used for perfumes or to spray insecticides work on the same principle.

Blood flow and heart attack

Bernoulli’s principle helps in explaining blood flow in arteries.

• When the artery gets constricted due to the accumulation of plaque on its inner walls. In order to drive the blood through this constriction, a greater demand is placed on the activity of the heart.
• The speed of the flow of the blood in this region is raised which lowers the pressure inside and the artery may collapse due to the external pressure.
• As the blood rushes through the opening, the internal pressure once again drops due to the same reasons leading to a repeat collapse. This may result in a heart attack.

Dynamics lift

Dynamic lift is the force that acts on a body, such as an airplane wing, a hydrofoil or a spinning ball, by virtue of its motion through a fluid. In many games such as cricket, tennis, baseball, or golf, we notice that a spinning ball deviates from its parabolic trajectory as it moves through air.

Dynamic lift on airplane wings:-

Dynamic lift is most popularly observed in airplanes.

Whenever an airplane is flying in the air, due to its motion through the fluid here fluid is air in the atmosphere. Due to its motion through this fluid, there is a normal force that acts on the body in the vertically upward direction. This force is known as Dynamic lift.

Consider an airplane whose body is streamlined. Below the wings of the airplane, there is air that exerts an upward force on the wings. As a result, airplane experiences a dynamic lift.

Magnus Effect

Dynamic lift by virtue of spinning is known as the Magnus effect.

Magnus effect is a special name given to dynamic lift by virtue of spinning.   Example:-Spinning of a ball.

Case1:-When the ball is not spinning.

• The ball moves in the air, it does not spin, the velocity of the ball above and below the ball is the same.
• As a result there is no pressure difference. (ΔP= 0). Therefore there is no dynamic lift.

Case2:- When the ball is moving in the air as well as spinning.

• When the ball spins it drags the air above it therefore the velocity above the ball is more as compared to the velocity below the ball.
• As a result, there is a pressure difference; the pressure is more below the ball. Because of the pressure difference, there is an upward force which is the dynamic lift.

Viscosity

Most of the fluids are not ideal ones and offer some resistance to motion. This resistance to fluid motion is like internal friction analogous to friction when a solid moves on a surface. It is called viscosity. This force exists when there is relative motion between layers of the liquid.

Suppose we consider a fluid like oil enclosed between two glass plates as shown. The bottom plate is fixed while the top plate is moved with a constant velocity v relative to the fixed plate. If oil is replaced by honey, a greater force is required to move the plate with the same velocity. Hence we say that honey is more viscous than oil.

Stoke’s law

The force that retards a sphere moving through a viscous fluid is direct ∝ to the velocity and the radius of the sphere, and the viscosity of the fluid.

Mathematically: - F =6πηrv where

Let retarding force F ∝ v where v =velocity of the sphere

F ∝ r where r=radius of the sphere

F ∝ η where η=coefficient of viscosity   and 6π = constant

Stokes law is applicable only to the laminar flow of liquids. It is not applicable to turbulent law.

Example: - Falling raindrops

Consider a single raindrop, when the raindrop is falling it is passing through air. The air has some viscosity; there will be some force that will try to stop the motion of the raindrop.

Initially, the raindrop accelerates but after some time it falls with constant velocity.

As the velocity increases the retarding force also increases. There will be vicious force Fv and bind force Fb acting in the upward direction. There will also be Fg gravitational force acting downwards.

After some time Fg = Fr (Fv+Fb)

Net Force is 0. If force is zero as a result acceleration also becomes zero.

Terminal Velocity

Terminal velocity is the maximum velocity of a body moving through a viscous fluid.

It is attained when the force of resistance of the medium is equal and opposite to the force of gravity.

As the velocity is increasing the retarding force will also increase and a stage will come when the force of gravity becomes equal to the resistance force.

After that point velocity won’t increase and this velocity is known as terminal velocity.  It is denoted by ‘vt’.Where t=terminal.

Liquid Surfaces

Certain properties of free surfaces:-

Whenever liquids are poured in any container they take the shape of that container in which they are poured and they acquire a free surface.

Consider a case if we pour water inside the glass it takes the shape of the glass with a free surface at the top.

Top surface of the glass is a free surface. Water is not in contact with anything else, it is in contact with the air only. This is known as free surfaces. Liquids have free surfaces. As liquids don’t have a fixed shape they have only fixed volume.

Free surfaces have additional energy as compared to inner surfaces of the liquid called surface energy.

Surface Energy

Surface energy is the excess energy exhibited by the liquid molecules on the surface compared to those inside the liquid. This means liquid molecules at the surface have greater energy as compared to molecules inside it.

Suppose there is a tumbler and when we pour water in the tumbler, it takes the shape of the tumbler. It acquires a free surface.

Case 1: When molecules are inside the liquid:-

Suppose there is a molecule inside the water, there will be several other molecules that will attract that molecule in all directions.

As a result, this attraction will bind all the molecules together. This results in negative potential energy of the molecule as it binds the molecule.

To separate this molecule a huge amount of energy is required to overcome potential energy. Some external energy is required to move this molecule and it should be greater than the potential energy.

Therefore a large amount of energy is required by the molecules which are inside the liquid.

Case2: When the molecules are at the surface:-

When the molecule is at the surface, half of it will be inside and half of it is exposed to the atmosphere.

The lower half of the molecule, it will be attracted by the other molecules inside the liquid. But the upper half is free. The negative potential energy is only because of the lower half.

But the magnitude is half as compared to the potential energy of the molecule which is fully inside the liquid. So the molecule has some excess energy, because of this additional energy which the molecules have at the surface different phenomena happen like surface energy, and surface tension.

Liquids always tend to have the least surface area when left to themselves.

As more surface area will require more energy as a result liquids tend to have less surface area.

Surface Tension

Surface tension is the property of the liquid surface which arises due to the fact that surface molecules have extra energy.

Surface tension is the surface energy per unit area of the liquid surface. It can be also defined as Force per unit length on the liquid surface

Surface tension(S) =Surface Energy/area

At any interface (it is a line that separates two different mediums) the surface tension always acts in the equal and opposite directions and it is always perpendicular to the line at the interface.

A fluid will stick to a solid surface if the surface energy between fluid and solid is smaller than the sum of energies between solid-air and fluid-air.

This means Ssf (solid-fluid) < Sfa (fluid air) + Ssa (Solid air)

Stretching a film (a) A film in equilibrium ;(b) the film stretched an extra distance.

Why does water stick to glass but Mercury doesn’t?

In the case of water and glass, water sticks to glass because the surface energy of water and glass is less than the surface energy between water and air and between glass and air. S.E (w-g) < S.E (w-a) +S.E (g-a)

In the case of mercury, Surface Energy between mercury and glass S.E (m-g), Surface energy between mercury and air S.E (m-a), and Surface Energy between air and glass S.E (a-g). E (m-g)> S.E (m-a) +S.E (a-g)

Angle of Contact

The angle of contact is the angle at which a liquid interface meets a solid surface. It is denoted by θ. It is different at the interfaces of different pairs of liquids and solids.

For example - A droplet of water on a lotus leaf. The droplet of water (Liquid) is in contact with the solid surface which is a leaf.

This liquid surface makes some angle with the solid surface. This angle is known as the angle of contact.

Water forms a spherical shape on the lotus leaf but it splits on the table.

Significance of Angle of Contact

Angle of contact determines whether a liquid will spread on the surface of a solid or it will form droplets on it.

• If the Angle of contact is obtuse: then droplets will be formed.
• If the Angle of contact is acute: then the water will spread.

Case1: When the droplet is formed

Consider we have a solid surface, a droplet of water which is liquid and air.

The solid-liquid interface is denoted by Ssl, the solid air interface is denoted by Ssa and the liquid-air interface is denoted by Sla.

The angle which Ssl makes with Sla. It is greater than 90 degrees. Therefore a droplet is formed.

Case 2: When water just spreads

The angle at which liquid forms with a solid surface is less than 90 degrees.

Drops and Bubbles

Why are water and bubbles drop?

Whenever liquid is left to itself it tends to acquire the least possible surface area so that it has the least surface energy so it has the most stability.

Therefore for more stability, they acquire the shape of a sphere, as the sphere has the least possible area.

Spherical Shape

Distinction between Drop, Cavity and Bubble

1. Drop: - Drop is a spherical structure filled with water.
2. There is only one interface in the drop.
3. The interface separates water and air.

Water droplets

Cavity: -Cavity is a spherical shape filled with air. In the surroundings there is water and in the middle, there is a cavity filled with air. There is only one interface that separates air and water.

Example: - bubble inside the aquarium.

Bubble: - In a bubble there are two interfaces. One is air and water and another is water and air. Inside a bubble there is air and there is air outside. But it consists of a thin film of water.

Pressure inside a drop and a cavity

Pressure inside a drop is greater than the pressure outside.

Suppose there is a spherical drop of water of radius ‘r’ which is in equilibrium.

Consider there is an increase in radius which is Δr.

Therefore Extra Surface energy = Surface Tension(S) x area

Sla x 4π(r+Δr) 2 – Slax4πr2

After calculating   Extra Surface energy=8πr Δr Sla

At Equilibrium, Extra Surface energy = Energy gain due to the pressure difference

8πr Δr Sla = (Pi - Po) 4πr2xΔr

Where Pi= Pressure inside the drop and Po = Pressure outside the drop.

After calculation Pi - Po = 2 Sla/r

Pressure inside a Bubble

• Pressure inside a bubble is greater than the pressure outside.
• As bubble has 2 interfaces, Pi-Po=2Sla/r x 2
• Therefore, Pi-Po=4Sla/r

Conclusion: - In general, for a liquid-gas interface, the convex side has a higher pressure than the concave side.

Capillary Rise

In Latin the word Capilla means hair. Due to the pressure difference across a curved liquid-air interface, the water rises up in a narrow tube in spite of gravity.

Consider a vertical capillary tube of circular cross-section (radius a) inserted into an open vessel of water.

The contact angle between water and glass is acute. Thus the surface of the water in the capillary is concave. As a result, there is a pressure difference between the two sides of the top surface. This is given by

(Pi – Po) = (2S/r) = 2S/ (a sec θ) = (2S/a) cos θ     (i)

Thus the pressure of the water inside the tube, just at the meniscus (air-water interface) is less than the atmospheric pressure.

Consider the two points A and B. They must be at the same pressure,

P0 + h ρ g = Pi = PA      (ii)

Where ρ is the density of water, and h is called the capillary

h ρ g = (Pi – P0) = (2S cos θ)/a     (By using equations (i) and (ii))

Therefore the capillary rise is due to surface tension. It is larger, for a smaller radius.