In fluid mechanics, shear stresses arise from viscosity, which represents a fluid's internal resistance to deformation. For low-viscosity fluids, like water, these stresses are minimal, simplifying flow analysis by allowing the fluid to be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent, leaving only normal stresses, which act perpendicularly to fluid elements. Notably, pressure — defined as the negative of the normal stress — remains...

Applying the conservation of energy principle or the work-energy theorem to an incompressible, inviscid fluid in laminar, steady, irrotational flow leads to Bernoulli's equation. It states that the sum of the fluid pressure, potential, and kinetic energy per unit volume is constant along a streamline.
All the terms in the equation have the dimension of energy per unit volume. The kinetic energy per unit volume is called the kinetic energy density, and the potential energy per unit volume is...

Thus far, the ideal gas law, PV = nRT, has been applied to a variety of different types of problems, ranging from reaction stoichiometry and empirical and molecular formula problems to determining the density and molar mass of a gas. However, the behavior of a gas is often non-ideal, meaning that the observed relationships between its pressure, volume, and temperature are not accurately described by the gas laws. 

Viscosity measures the resistance a fluid offers to flow and deformation. It results from internal friction between layers of fluid moving relative to one another. Dynamic viscosity, denoted by the Greek letter mu (μ), quantifies the force needed to move one fluid layer over another. For Newtonian fluids like water and air, the relationship between the shearing stress and the rate of shearing strain is linear, meaning their viscosity remains constant regardless of the applied stress.

When a fluid is in constant acceleration, the pressure and buoyant force equations are modified. Suppose a beaker is placed in an elevator accelerating upward with a constant acceleration, a. In the beaker, assume there is a thin cylinder of height h with an infinitesimal cross-sectional area, ΔS.
The motion of the liquid within this infinitesimal cylinder is considered to obtain the pressure difference. Three vertical forces act on this liquid:

The ideal gas law is an approximation that works well at high temperatures and low pressures. The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by considering two factors.
First, the attractive forces between molecules, which are stronger at higher densities and reduce the pressure, are considered by adding to the pressure a term equal to the square of the molar density multiplied by a positive coefficient a. Second, the volume...