## Kerala Plus One Physics Notes Chapter 10 Mechanical Properties of Fluids

Introduction

Something that can flow is a fluid. So gases and liquids are fluids. Between liquids and gases, it is the gases that flow more easily. And among liquids water flows more easily than honey and syrup. Fluids have no rigidity but they have bulk modulus. A liquid has a free surface and cannot be compressed much. Gases have no free surface, they fill the container completely even at very low pressures and can be compressed easily. We shall study more about this here.

Pressure

- In fluids, the molecules are free to move. When they move, they collide with the walls of the container and exert a force on the walls. The force is acting normal perpendicular to the surface.
- The force acting per unit area normal to the surface is called pressure.

- Its unit is N/m
^{2}or Pascal (Pa). Another common unit of pressure is atmosphere. - Dimensional formula of pressure is [ML
^{-1}T^{-2}].

Note:

The pressure exerted by the atmosphere at sea level is called the atmospheric pressure.

1 atm = 1.0313 ×10^{6} Pa = 760 torr

Measurement of Fluid Pressure:

The figure below shows a pressure measuring device placed in a fluid-filled vessel. It consists of a small piston of area Δ A resting on a spring.

The spring can measure the force acting on the cylinder. The force exerted by the fluid on the inward surface of the piston is balanced by the outward spring force. This spring force can be measured and thus we find out the pressure exerted by the fluid (same as spring force).

Example: Some phenomenon that can be explained using pressure are:

- A sharp knife (lesser edge area) cuts better than a blunt knife (more edge area).
- Nails have pointed ends. Even for small forces the pressure applied by it on the surface is large and hence it can penetrate (go inside) easily.
- Camels find it easier to walk on sand than humans because their foot area is larger and thus less pressure on the sand.

Pascal’s Law

- The pressure in a fluid at rest is same at all points if we ignore gravity.
- If the pressure in a liquid is changed at a particular point, the change is transmitted to the entire liquid without being diminished in magnitude.
- The pressure exerted at any point on an enclosed liquid is transmitted equally in all directions.

Hydraulic life:

Construction: It is a simple application of Pascal’s law. There are two cylinders

C_{1} and C_{2} of cross-sectional areas A_{1} and connected, by a pipe. Pistons are fitted in both cylinders. The load to be lifted is kept on the platform of the piston with more area

(A_{2}> A_{1}). The cylinders and pipe contain a liquid.

Working:

A force is applied on the smaller piston. Then pressure exerted on the liquid is,

According to Pascal’s law, the same pressure ‘P’ is transmitted to the larger piston.

∴ Force on larger piston is

Hence by making the ratio A_{2}/A_{1}, large, very heavy loads like cars can be lifted by using a small force.

Note:

Here work done (F.s) by force F_{1} i$ same as that done by F_{2}. The piston P_{1} moves down by a larger distance than P_{2} Thus the product F.s, hence work done is same for both.

Pressure exerted by a liquid column:

Consider the vessel given in the figure filled with a liquid of density, ρ. The weight of the liquid column exerts a pressure on the bottom of the vessel.

Variation of pressure with depth:

As shown in the figure consider a liquid at rest in a container. Assume that liquid pressure is same at all points which are at same depth. Let P_{1} and P_{2} be the pressures at some top point 1 and bottom point 2 respectively.

Consider forces in vertical direction:

Force due to liquid pressure at top F_{1} = P_{1} A_{1} downwards

Force due to liquid pressure at bottom F_{2} = P_{1},A_{2} upwards

Weight of liquid cylinder downwards, W = mg = Volume × ρ × g= Ahpg (ρ= density of liquid) As liquid is in equilibruim, net force downward = net force upward

f_{1 +} w = f_{2 }f_{2}-f_{1 =} w

P_{2}A- P_{1}A = Ahρg

P_{2} – P_{1} = hρg

If point 1 is considered at the surface of the liquid then P, can be replaced by atmospheric pressure P_{g} and P_{2} by P in the above equation.

Then, P -P_{a} = hρg

P = P_{a} + hρg

Therefore, the pressure at depth h below the liquid surface is greater than the atmospheric pressure by an amount hρg.

Note:

The excess pressure, P-P_{s }at depth h is called gauge pressure

Hydrostatic paradox:

The pressure exerted by a liquid column depends only on the height of the liquid column and not on the shape of the containing vessel.

The pressure at the bottom is the same, as vertical height ‘h’ is the same.

Mercury Barometer :

Torricelli used this device to measure atmospheric pressure.

A glass tube open at one end and having a length of one meter is filled with mercury. The open end is temporarily closed by a thumb and the tube is inverted in a cup of mercury. The thumb is removed and the mercury level in the tube falls down a little and comes to rest at a height of 76 cm above the mercury level in the dish.

The pressure at the upper end of A of the column is vacuum and thus P_{A} = 0

Consider a point C on the surface of the mercury in the dish. The pressure at C is equal to the atmospheric pressure. As B and C are at same horizontal level, pressure at B and C are same.

P_{B} = P_{c} = atmospheric pressure, P_{g }Suppose the point B is at a depth h below A If p is the density of mercury,

P_{B} = P_{A} + hρg

P_{a} = 0 + hρg

P_{a}= hρg

For mercury barometer h = 76 cm = 0.76 m,

ρ=13.6 x 10^{3} kgm^{-3}, g = 9.8 ms^{-2 }Atmospheric pressure, P_{a}= 0.76 × 13.6 × 10^{3 }× 9.8

= 1.01 × 10^{5} Pa

Manometer:

It is a device to used to measure the pressure in a closed vessel containing a gas. It consists of a U-tube having some liquid. One end of the tube is open to atmosphere and the other end is connected to the vessel.

The pressure of the gas is equal to the pressure at A = pressure at B

= P_{c} + hρg

= P_{a} + hρg

where P_{a} is the atmospheric pressure, h = BC = difference in levels of the liquid in the two arms and p is the density of the liquid.

Streamline Flow:

Assume that all the particles reaching the point ‘A’ moves along the same path and have same velocity. Also assume that all the particles reaching B will have same velocity at B although different from that of point A .Also assume if one particle passing though A has gone through B, then all the particles passing through A goes through B. Such a flow of fluid is called a streamline flow.

In streamline flow or steady flow the velocity of fluid particles reaching a particular point is the same at all time. Thus, each particle follows the same path as taken by a previous particle passing through that point.

Equation of Continuity

Consider a non-viscous (no internal friction) and in compressible (density constant during flow) liquid flowing steadily between the sections A and B of a pipe of varying cross-section .Let the area of cross section at A be A_{1}, and that at B be A_{2} Let the speed of fluid be v_{1} at A and v_{2} at B.

As, m = volume x density

= area of cross section × length × density Mass of fluid that flows through A in time

At, m_{1}= A_{1}v_{1}p_{1} Δt

Mass of fluid that flows through B in time

At, m_{2} = A_{2}v_{2}p_{2}, Δt

By conservation of mass, m, = m_{2 }A_{1}v_{1}p_{1} Δt = A_{2}v_{2}p_{2}, Δt and

p_{1} = p_{2} (in compressible fluid)

A_{1}v_{1 }= A_{2}v_{2}

The product of the area of cross-section and the speed remains the same at all points of a tube of flow. This is called the equation of continuity and expresses the law of conservation of mass in fluid dynamics.

Note:

Equation of continuity also tells us that v α1/a. This explains why speed of water emerging from a PVC pipe increases when we press its outlet with our fingers and hence decreases its area of cross-section.

Bernoulli’s Principle

In words.

As we move along a streamline, Pressure energy + Kinetic energy + Potential energy, per unit volume = Constant. This is Bernoulli’s theorem.

Velocity of efflux: Torricelli’s law:

Torricelli’s law may be stated as the velocity of efflux through a hole at a depth ‘h’ will be equal to the velocity gained by a freely falling body when it travels a distance ‘h’.

ie., v= fegh

Venturimeter:

It is a device used to measure the rate of flow of a liquid through a pipe. It is an application of Bernoulli’s princpile.

Dynamic lift:

Dynamic lift is the force that acts on a body due to its motion through a fluid (air). (This dynamic lift can be partly explained on the basis of Bernoulli’s principle

CAQ

Question 1.

Discuss some applications of Dynamic lift

Answer:

- Fig (1) shows the streamlines formed in concentric circles when a ball spins about an axis perpendicular to its horizontal motion.
- The streamlines due to translatory motion of the bail is in fig (2). Here the air rushes backward to fill in the empty space left by the ball.
- Fig (3) shows a ball which is moving and spinning at the same time. The ball is moving forward and the air is moving backward relative to it. So the velocity of air above the bail relative to it is larger and below it is smaller. This difference in velocities of air causes a pressure difference between the lower and upper face and there is a net upward force on the ball. This dynmaic lift due to spinning is called Magnus effect.

Viscosity:

Viscosity is liquid friction. When liquid layer moves over another liquid layer, there is a

force of friction between the liquid layers, op-posing the motion of layers.

The coeffecient of viscosity q for fluid is defined as the ratio of shearing stress to the strain rate.

The SI unit of viscosity is poiseiulle or Nsm^{-2 }Note:

The viscosity of liquids decreases with temperature while it increases in the case of gases.

Stoke’s law:

According to Stoke’s law, the backward viscous force acting on a small spherical body of radius r moving with uniform velocity v through fluid of viscosity q is given by F= 6m\rv

Reynolds Number:

The turbulence in a fluid is determined by a dimension parameter called Reynolds number.

Reynolds number

where ρ is the density of a fluid, v is the speed of flow of the liquid, ‘d’ is the dimension of pipe, q is the viscosity of the liquid. R_{e} is the dimensionless number.

The flow is streamline when the R_{e} is below 1000, unsteady when R_{e} is in between 1000 and 2000 and turbulent when R_{e} > 2000.

Surface Tension:

Surface tension (σ) is the property due to which the free surface of a liquid at rest behaves like an elastic stretched membrane tending to contract so as to occupy minimum surface area.

Thus it is measured as the force acting per unit length of an imaginary line drawn on the liquid surface, the direction of force being perpendicular to this line and tangential to the liquid surface.

Surface Energy:

The extra energy possesed by the molecules of surface film of unit area compared to the molecules in the interior is called surface energy. It is equal to the work done in increasing the area of the surface film by unit amount.

Surface energy and surface tension:

Consider a the frame ABCD in which AB is movable. Let us dip this frame in soap solution. The wire AB is pulled inwards due to surface tension of the film formed, with a force:

Here instead of ‘l’ we see ‘2l’ because the soap film has two free surfaces.

Let AB be moved out through distance x’ to a new position A B’

Work done = Force x distance = 2σ×l × x

Increase in surface area of film = 2lx

Thus numberically the surface energy of liquid is equal to its surface tension.

Drops and Bubbles:

Due to surface tension, the liquid surface always tends to have the minimum surface area. For a given volume, a sphere has a minimum surface area. Hence, small drops and bubbles of a liquid assume spherical shape.On the other hand, for bigger drops the effect of gravity predominates over surface tension and the drop gets flattened.

Consider a spherical drop of radius R. Let σ be the surface tension of the liquid. Due to its spherical shape, there is an excess pressure p inside the drop over that on outside.

The radius of spherical drop increases from R to R+dR under excess pressure ρ.

Note:

In the case of spherical bubble, it has air both inside and outside. Hence it has two surfaces inner and outer. When the radius of the bubble increases, the area of both surfaces will increases by 8πRdR.

Hence the total increase in S.A = 16πRdR

So excess pressure,

Capillary Rise:

When a clean capillary tube is dipped in a liquid, which wets it, the liquid immediately rises in the tube. This is called capillary rise.

Consider a capillary tube of radius r dipped vertically in a liquid of density p and surface tension σ. The meniscus of the liquid inside the tube is concave. Thereby the liquid rises through the tube to some height. Let ‘R’ be the radius of the meniscus, 0 be the angle of contact and h the height of the liquid column in the tube with respect to the level of liquid outside.

Here the hydrostatic pressure exerted by the liquid column becomes equal to the excess pressure ρ.

Therefore at equilibrium we have p = hρg

Note:

Hence liquid rises more in a narrower tube than in a wider tube.

## Leave a Reply