a) The property of a liquid due to which its free surface tries to have minimum surface area and behaves as it were under tension some what like that of a stretched elastic membrane is called surface tension.
b) A small drop of liquid or a soap bubble is spherical in shape because due to surface tension the liquid surface tries to have minimum surface area and for a given volume sphere has minimum surface area.
c) Surface tension is defined as the force per unit length in the place of the liquid surface, acting at right angles on either side of an imaginary line drawn in that surface.
d) The work, required to be done in increasing the free surface area of a liquid by unity at constant temperature, is defined as surface tension.
e) The surface tension is due to cohesion between the molecules of a liquid, hence it is a molecular phenomenon.
h) It is a scalar quantity with dimensions are
Surface tension and spring constant have same units and dimensions.
i) Surface tension depends only on the nature of liquid and is independent of the area of surface or length of line considered.
Note:
a) The force required to lift a thin wire from surface of water is F = T × 2L.
b) The force required to lift a circular plate from surface of water is F = T× 2Πr.
c) The force required to lift a ring from surface of water is F = T (2Πr1+ 2Πr2).
Where r1 = inner radius; r2 = outer radius.
d) The force required to lift a rectangular plate from surface of water is F = 2T(L+b).
Factors Influencing surface tension:
a) Temperature:
Surface tension of a liquid decreases with rise in temperature. But for copper and melted cadmium surface tension temperature.
b) Impurities:
i) The dust particles or lubricating materials on the liquid surface decrease its surface tension.
ii) If the impurity is completely soluble then on mixing it in the liquid, its surface tension increases. For example on dissolving salt in water, its surface tension increases.
iii) If the impurity is partially soluble in a liquid, then its surface tension decreases. For example on mixing detergent in water its surface tension decreases.
Surface Energy (E)
Due to property of surface tension, free surface of a liquid is always under tension and tends to have minimum surface area. If the area of liquid surface is increased, work will be done very much similar to that done in stretching a rubber sheet. This work is stored as potential energy in the surface and the amount of this energy per unit area of the surface under isothermal condition is called surface energy.
a) The additional potential energy of molecules per unit area of surface layer of liquid is defined as surface energy.
b) Surface energy is numerically equal to the work done per unit increase in surface area of free liquid surface. 
c) Surface energy is equal to surface tension.
d) If the amount of heat present in the liquid surface per unit area is H, then the total surface energy is E = T + H.
iii) Total surface area decreases
iv) Surface energy decreases.
v) Temperature of liquid increases.
vi) Energy is released.
h) Splitting of a bigger drop into smaller drops:
When a drop of radius R splits into n smaller drops, then
iii) Total surface area increases and surface energy increases.
iv) Temperature of liquid decreases.
v) Energy is required
Excess pressure:
Due to the property of surface tension, a drop or bubble tries to contract and so compresses the matter enclosed. This in turn increases the internal pressure which prevents further contraction and equilibrium is achieved.
a) So in equilibrium, the pressure inside a bubble or drop is greater than out
The excess pressure in case of a bubble in air is
c) Excess pressure is inversely proportional to the radius of the bubble or drop i.e., pressure inside a smaller bubble or drop is higher than inside large bubble or drop. This is why, when two bubbles of different size are put in communication with each other, the air will rush from smaller to larger bubble, so that smaller will shrink while larger will expand.
Angle of contact:
When the free surface of a liquid comes in contact of a solid, it becomes curved near the place of contact.
a) "The angle between the tangent to the liquid surface at the point of contact and the solid surface, inside the liquid, is called the angle of contact".
b) Angle of contact is different for different pairs of solids and liquids.
c) For mercury and glass the angle of contact is 135º.
d) For ordinary water and glass the angle of contact is nearly 8º.
e) For pure water and clean glass plate angle of contact is 0º.
f) Angle of contact increases on increasing the temperature.
g) Angle of contact decreases on adding soluble impurity to a liquid.
h) Angle of contact does not depend on the inclination of the tube.
Capillarity:
a) The phenomenon of rise or depression of liquids in a capillary tube is known as capillarity.
b) The capillary rise 'h' is given by the formula 
Where r = radius of capillary tube
h = rise or fall of liquid
g = acceleration due to gravity
d = density of liquid
θ = Angle of contact 
The capillary depends on the nature of liquid and solid both i.e., T, d, θ and r.
i) If θ = 90º; meniscus is plane, h = 0 so no capillarity.
ii) If θ < 90º; meniscus is concave towards air, i.e., the liquid will rise in the capillary tube as incase of water in a glass tube.
iii) If θ > 90º; meniscus is convex, i.e., the liquid will descend in the capillary tube as in case of mercury in a glass tube.
iv) When the capillary tube is tilted from the vertical by an angle α, then the vertical height 'h' of the liquid column remains the same, of course the length of liquid in capillary tube increases
1.Viscosity:
The property of the fluid due to which it opposes the relative motion between its different layers is called viscosity.
Viscosity is a molecular phenomena which arises between molecules of different layers of the fluid. It can also be thought as fluid friction or internal friction.
2. Viscous Force:
When a fluid sets into motion there exist internal tangential forces between different layers of fluid which try to destroy the relative motion between layers. These tangential forces are called viscous forces.
3. Newton's law of viscous force:
"Co-efficient of viscosity is a measure of viscosity and is defined as the tangential force required per unit area to maintain unit velocity gradient between the layers".
Note: i) η can also be called as dynamic viscosity.
ii) It depends only on the nature of the fluid and is independent of area and velocity gradient.
iii) S.I. unit: Newton - second / meter2 (or) (poiseullie)
C.G.S unit: dyne - second / cm2 (or) (Poise)
1 Poise= 10-1 Nm2s
iv The dimensional formula of η is ML-1 T -1
v Viscosity of liquids is much greater than that of gas.
iv The viscosity of liquids decreases with increase in temperature.
The viscosity of gases increases with increase in temperature.
vii The viscosity of liquids increases with increase in density. The viscosity of liquids decreases with increase in density.
viii The viscosity of liquids (except water) increases with increase in pressure.
The viscosity of gases is practically independent of pressure.
5. Poisiseuille's Equation:
V → volume of liquid flowing per second
P → pressure difference across the tube
r → radius of the tube
l → length of the tube
η → coefficient of viscosity of liquid.
6. Stoke's Law:
Whenever a body moves through a fluid, the body experiences a viscous dray (force) opposing its motion. According to stoke it depends directly on
i. Viscosity of the fluids
ii. Size (radius) of the body
iii. Velocity of the body.
If a sphere of radius 'r' moves with velocity 'v' through fluid of viscosity η, then
7. Terminal velocity:
When spherical metallic ball falls into a tank of fluid, it attains constant velocity after it moves through a short distance. This constant velocity is called Terminal velocity.
Note: i. Terminal velocity is independent of height from where it is falling.
ii. Terminal velocity depends on the difference of densities of the body and fluid
if e > σ, the terminal velocity is in downward direction.
if e < σ , the terminal velocity is in upward direction (bubble rise). where e is density of the body, σ density of the fluid.
8. Streamline flow and turbulent flow:
Streamline flow : The flow of a liquid, in which every particle follows the same path as the preceding particle follows, is defined as streamline flow.
Turbulent flow : The irregular flow of a liquid is known as turbulent flow, in which the particle does not follow the same path as the preceding particle fol lows, is defined as turbulent flow.
9. Critical velocity:
The maximum velocity of a liquid necessary to maintain its streamline flow, is defined as critical velocity, below which its flow remains streamline and above which it becomes turbulent, is defined as critical velocity.
10. Reynold numbers:
The ratio of the inertial force to viscous force for a liquid is defined as Reynolds number.
Note: i) NR is a dimentionless constant.
ii) If NR < 2000, the flow is steady.
iii) If NR > 3000, the flow is non - steady.
iv) If 2000 < NR < 3000, the flow changes from steady to turbulent or vice - versa called as transition state.
11. Non viscous and incompressible fluid:
Non viscous fluid:
The fluid whose viscosity is zero, is known as non-viscous fluid.
Incompressible fluid:
The fluids which can not be compressed is known as incompressible fluids. When an incompressible fluid is pressed then it does not undergo any change in its volume or density.
12. Bernoullis Theorem:
When an incompressible, irrotational and non viscous fluid flows steadily through a tube of Non-uniform cross-section.
13. Important points on equation of continuity: When an incompressible and non viscous fluid flows steadily through a tube of non-uniform cross section, the product of area of cross section and velocity of flow is same at every point in the tube.
Av = constant or A1 v1 = A2 v2
This is the equation of continuity and represents the conservation of mass in case of moving fluids.
i) Velocity of flow increases as the corss sectional area decreases and vice-versa.
ii) The number of streamlines starting from A is same as that ending on B. At end
A, streamlines are widen and velocity of flow of liquid is less. At end B, streamlines are crowded and velocity of flow is more.
crowded streamlines indicate region of high velocity while widely spaced streamlines indicates low velocity region.
iii) The cross sectional area available to flowing water goes on increasing as we approach the bottom of a canal or river. So velocity of flow decreases. This explains why the "deep water runs slow"
iv) As the velocity of flow increases from A to B there must be some accelerating force indicating that pressure at A is greater than at B.
'' Hence pressure is greater at a point where the velocity is small and vice Versa"
14. Differences between viscosity and solid friction:
i) Force of viscosity depends upon the area of layers. The force of friction does not depend on the area of surfaces in contact.
ii) Force of viscosity depends upon the relative velocity of two layers. Force of friction does not depend on relative velocity of two bodies in contact.
