Fluid mechanics is concerned with the behavior of liquids and gases. Density \(\rho = m/V\), measured in kg/m³, is the most fundamental property. Pascal's law states that pressure applied to a confined fluid is transmitted equally in all directions throughout the fluid, a principle that underlies hydraulic systems. Archimedes' principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid displaced.
Two cornerstone equations govern the flow of fluids. The continuity equation for incompressible flow, \(A_1 v_1 = A_2 v_2\), shows that the product of cross-sectional area and velocity is constant along a streamline. Bernoulli's equation expresses energy conservation along a streamline for steady, incompressible, inviscid flow: \(P + \tfrac{1}{2}\rho v^2 + \rho g h =\) constant. Together, these equations explain many practical phenomena, from the lift on an aircraft wing to the flow through a Venturi meter.
The Reynolds number \(Re = \rho v D / \mu\) is a dimensionless quantity that predicts whether flow is laminar or turbulent. For pipe flow, the transition typically occurs at a critical Reynolds number of about 2300. Laminar flow is characterized by smooth, orderly layers, while turbulent flow involves chaotic fluctuations and vigorous mixing. Viscosity \(\mu\) is the fluid property that quantifies internal resistance to flow, with the shear stress related to the velocity gradient by \(\tau = \mu (dv/dy)\).