Key Points

Fluids are substances that can flow, such as liquids and gases. Pressure (P) is defined as the normal force (F) acting per unit area (A), calculated as $P = \frac{F}{A}$. Its SI unit is the Pascal (Pa).

Pascal's law states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

Based on Pascal's law, hydraulic lifts multiply force. The relationship between input force $F_1$ on area $A_1$ and output force $F_2$ on area $A_2$ is $F_2 = F_1 \frac{A_2}{A_1}$.

The pressure in a fluid at rest increases with depth. The pressure P at a depth h is given by $P = P_a + \rho g h$, where $P_a$ is atmospheric pressure and $\rho$ is the fluid density.

Absolute pressure is the total pressure at a point, while gauge pressure is the difference between absolute pressure and atmospheric pressure: $P_g = P - P_a = \rho g h$.

In steady or streamline flow, the velocity of fluid at any point is constant over time. For an incompressible fluid, the equation of continuity states $A_1 v_1 = A_2 v_2$, where A is the cross-sectional area and v is the fluid velocity.

For a non-viscous, incompressible fluid in steady flow, the sum of pressure (P), kinetic energy per unit volume ($\frac{1}{2}\rho v^2$), and potential energy per unit volume ($\rho g h$) is constant along a streamline: $P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$.

The speed of efflux (v) of a fluid from a small hole at a depth h below the surface in an open container is given by $v = \sqrt{2gh}$, which is the same speed as a freely falling object from height h.

Viscosity is the internal friction of a fluid that resists flow. The coefficient of viscosity ($\eta$) is the ratio of shearing stress to the strain rate, given by $\eta = \frac{F/A}{v/l}$.

Stokes' law describes the viscous drag force (F) on a small sphere of radius 'a' moving with velocity 'v' through a fluid of viscosity $\eta$. The formula is $F = 6\pi\eta a v$.

An object falling through a fluid reaches a constant terminal velocity ($v_t$) when the gravitational force is balanced by the buoyant force and viscous drag. It is calculated as $v_t = \frac{2a^2(\rho - \sigma)g}{9\eta}$, where $\rho$ is the object's density and $\sigma$ is the fluid's density.

Surface tension (S) is the force per unit length acting in the plane of the liquid surface. It is also equal to the surface energy per unit area, arising from the cohesive forces between liquid molecules.

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 wets a solid surface.

Due to surface tension, the pressure inside a spherical liquid drop is greater than the pressure outside. This excess pressure is given by $\Delta P = P_i - P_o = \frac{2S}{r}$, where r is the radius of the drop.

A soap bubble has two surfaces (inner and outer), so its excess pressure is twice that of a liquid drop of the same radius: $\Delta P = P_i - P_o = \frac{4S}{r}$.

Capillarity is the tendency of a liquid to rise or fall in a narrow tube. The height (h) of the liquid is given by $h = \frac{2S \cos\theta}{a \rho g}$, where 'a' is the radius of the tube and $\theta$ is the angle of contact.