1. Pressure in Fluids

Fluids, both liquids and gases, exert pressure, which is defined as force per unit area ($\text{P} = \frac{F}{A}$). In a fluid at rest, pressure is exerted equally in all directions. The pressure in a fluid increases with depth due to the weight of the fluid above. This relationship is described by the hydrostatic pressure formula: $\text{P} = \text{P}_0 + \rho gh$, where $\text{P}_0$ is the pressure at the surface, $\rho$ is the fluid density, $g$ is the acceleration due to gravity, and $h$ is the depth. This principle explains why the pressure at the bottom of a swimming pool is greater than at the surface.

2. Buoyancy and Archimedes' Principle

Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object. Archimedes' Principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. If the buoyant force is greater than the object's weight, it floats; if it's less, it sinks; and if they are equal, it remains suspended. This principle explains why ships made of dense materials like steel can float; their overall shape displaces a volume of water whose weight equals the ship's weight.

3. Fluid Dynamics

Fluid dynamics studies how fluids move. An ideal fluid is assumed to be incompressible (constant density) and non-viscous (no internal friction). The Bernoulli's principle is a key concept in fluid dynamics, stating that for an ideal fluid in steady flow, the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline: $\text{P} + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$. This principle explains phenomena like the lift on an airplane wing and the operation of atomizers. Real fluids, however, exhibit viscosity, which affects their flow.

4. Viscosity and Stokes' Law

Viscosity is a measure of a fluid's resistance to flow, essentially internal friction. It quantifies the force required to move one layer of fluid past another. For laminar flow of a spherical object through a viscous fluid, Stokes' Law describes the viscous drag force as $F_d = 6\pi \eta rv$, where $\eta$ is the coefficient of viscosity, $r$ is the radius of the sphere, and $v$ is its velocity. Fluids with high viscosity, like honey, flow slowly, while those with low viscosity, like water, flow easily. Viscosity is temperature-dependent; for liquids, it generally decreases with increasing temperature.

5. Surface Tension and Capillarity

Surface tension is the tendency of liquid surfaces to shrink into the minimum surface area possible. It arises from the cohesive forces between liquid molecules, causing the surface molecules to be attracted inwards more strongly than outwards. This phenomenon allows insects to walk on water and causes water to form droplets. Capillarity is the ability of a liquid to flow in narrow spaces without the assistance of, and even in opposition to, external forces like gravity. This is due to the combined effects of surface tension and adhesive forces between the liquid and the walls of the narrow tube.

6. Additional: Continuity Equation

The Continuity Equation is a statement of conservation of mass for fluid flow. For an incompressible fluid flowing through a pipe, it states that the product of the cross-sectional area ($A$) and the fluid velocity ($v$) is constant along the flow path: $A_1 v_1 = A_2 v_2$. This means that if the area of the pipe decreases, the velocity of the fluid must increase to maintain the same flow rate. This principle is evident in how water flows faster from a hose when the nozzle is narrowed, a common observation in households.