Physics: Class 11 (Book 2) — Chapter 9: Mechanical Properties of Fluids

• Shear stress : Fluids offer negligible resistance to shearing forces. A very small shear stress causes continuous deformation (flow). The shearing stress of fluids is approximately 10⁶ \times smaller than that of solids.
• Compressibility : Solids and liquids exhibit low compressibility, meaning their volume changes very little under external pressure. Gases, conversely, are highly compressible and show large variations in density with pressure changes.

The average pressure $P_{av}$ is defined as the normal force acting per unit area:

In a limiting sense, the pressure $P$ at a point is given by:

• Pressure is a scalar quantity because the force in the definition is specifically the component normal to the area, not a vector force.
• The SI unit of pressure is the pascal ( Pa ), where 1 Pa = 1 N m⁻² .
• Another common unit is the atmosphere ( atm ), representing the atmospheric pressure at sea level: 1 atm = 1.013 \times 10⁵ Pa .

The density $\rho$ of a fluid of mass $m$ occupying volume $V$ is defined as:

The SI unit of density is kg m⁻³ . For liquids, density is nearly constant across wide pressure ranges, whereas gas density varies significantly.

The relative density of a substance is the dimensionless ratio of its density to the density of water at 4 °C (which is 1.0 \times 10³ kg m⁻³ ):

According to Pascal's law , the pressure in a fluid at rest is identical at all points that lie at the same horizontal height. Furthermore, any change in pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel.

Consider a fluid element of area $A$ and height $h$ at rest. The pressure difference between two vertical points separated by height $h$ is given by:

If point 1 is shifted to the top surface of the liquid open to the atmosphere, then $P_1$ is replaced by atmospheric pressure $P_a$, and the absolute pressure $P$ at depth $h$ becomes:

The excess pressure $P - P_a = \rho g h$ is defined as the gauge pressure .

The pressure at a certain depth depends solely on the height of the liquid column above it, not on the cross-sectional area or shape of the container. This is demonstrated by the hydrostatic paradox , where connected vessels of different shapes hold liquid up to the same height because the pressure at the base of each vessel is identical.

Hydraulic devices leverage Pascal's law to achieve mechanical advantage. In a hydraulic lift, two pistons of different cross-sectional areas $A_1$ and $A_2$ ($A_1 < A_2$) are connected by an enclosed liquid. Applying a small force $F_1$ on the smaller piston produces a pressure $P = \frac{F_1}{A_1}$. This pressure is transmitted undiminished, resulting in an upward force $F_2$ on the larger piston:

The factor $\frac{A_2}{A_1}$ represents the mechanical advantage of the system, allowing heavy loads to be lifted with minimal effort.

• Streamline flow : A flow is steady if the velocity of each passing fluid particle at any given point remains constant over time. The path taken by a fluid particle in steady flow is called a streamline . No two streamlines can ever cross.
• Turbulent flow : When the fluid velocity exceeds a certain limit called the critical speed , the flow loses its steady character and becomes chaotic, forming whirlpool-like structures called rapids or eddies.

For the steady flow of an incompressible fluid through a pipe of varying cross-section, the mass of fluid entering one end must equal the mass leaving the other end. This leads to the equation of continuity :

Where $A$ is the cross-sectional area and $v$ is the fluid speed. Consequently, fluid speed increases at narrower constrictions and decreases in wider regions.

Along any streamline, the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant:

This principle \implies that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.

The speed of efflux $v_1$ of a liquid from a small hole in an open tank at a depth $h$ below the liquid surface is given by Torricelli's law :

This speed is identical to the speed acquired by a freely falling body dropped from rest through a height $h$.

• Magnus effect : A spinning ball drags air along with its surface. This creates a difference in air speeds above and below the ball, resulting in a pressure difference that exerts a net upward or downward force, causing the ball to deviate from its normal parabolic path.
• Aerofoil (Aircraft Wing) : The wing is shaped so that air flows faster over the top surface than underneath it. According to Bernoulli's principle, this speed difference creates a lower pressure on top and a higher pressure below, generating the dynamic lift necessary to support the aircraft.

When a fluid layer is sandwiched between a fixed plate and a plate moving with velocity $v$, the fluid experiences a velocity gradient. The coefficient of viscosity $\eta$ is defined as the ratio of shearing stress to strain rate:

• The SI unit of viscosity is the poiseuille ( Pl ), equivalent to N s m⁻² or Pa s .
• Viscosity of liquids decreases with increasing temperature, whereas the viscosity of gases increases with temperature.

According to Stokes' law , the viscous drag force $F$ acting on a small spherical body of radius $a$ moving with velocity $v$ through a fluid of viscosity $\eta$ is:

When a sphere falls through a viscous medium, it eventually reaches a constant velocity called the terminal velocity $v_t$, where the downward gravitational force is perfectly balanced by the sum of the upward buoyant force and the viscous drag force:

Where $\rho$ is the density of the sphere and $\sigma$ is the density of the fluid.

Molecules inside a liquid are attracted equally in all directions, whereas molecules at the surface experience a net inward cohesive force. Consequently, surface molecules possess extra potential energy. The work required to increase the surface area of a liquid film by an amount $\Delta A$ is stored as surface energy :

Alternatively, surface tension $S$ is defined as the force per unit length acting in the plane of the interface along a line of unit length:

The angle of contact $\theta$ is the angle between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid. It determines whether a liquid will wet a solid surface or form droplets:

• If adhesive forces exceed cohesive forces, $\theta$ is acute (e.g., water on glass), and the liquid wets the surface.
• If cohesive forces exceed adhesive forces, $\theta$ is obtuse (e.g., mercury on glass), and the liquid forms droplets.

Due to surface tension, the pressure inside a curved liquid interface is always greater than the pressure outside. The excess pressure is given by:

• For a spherical liquid drop (one interface): $$P_i - P_o = \frac{2S}{r}$$
• For a spherical soap bubble (two interfaces): $$P_i - P_o = \frac{4S}{r}$$

When a narrow tube (capillary) is dipped in a liquid, the liquid rises or falls in the tube. The height $h$ of the capillary rise is given by:

Where $a$ is the radius of the capillary tube and $\theta$ is the contact angle. If $\theta$ is acute, the liquid rises ($h > 0$); if $\theta$ is obtuse, the liquid is depressed ($h < 0$).