The Navier–Stokes equations are a set of partial differential equations that describe the motion of viscous fluid substances. They are fundamental to fluid dynamics, forming the core mathematical framework used to model how fluids, such as water, air, or oil, move under the influence of forces like pressure gradients, gravity, and viscosity. Solving these complex equations allows scientists and engineers to predict flow patterns in diverse applications, including the design of aircraft wings, the study of ocean currents, and the simulation of atmospheric weather systems. Because the equations account for momentum transfer within the fluid, they are critical tools in aerospace engineering, civil infrastructure design, and advanced scientific research requiring an understanding of fluid mechanical behavior.