Navier–Stokes Equations

In viscous fluid motion, shear stress is directly proportional to the rate of fluid deformation. For incompressible Newtonian fluids, this relationship is linear, meaning stress increases uniformly with deformation rate. Shear stresses act tangentially across fluid layers, describing how adjacent layers slide past one another and connecting tangential stress to velocity differences between layers.

Normal stresses depend on pressure and deformation rates in specific directions, determining how fluid flows under varying pressures. Shear stresses, conversely, act tangentially across fluid layers, explaining how adjacent layers slide relative to one another. Together, normal and shear stresses define the internal forces within a fluid that influence its movement and behavior.

The inertia term captures fluid acceleration, showing how a moving fluid resists sudden changes in speed or direction and demonstrates momentum conservation. The pressure gradient term drives fluid movement by creating a net force from high-pressure to low-pressure areas. Together, these terms balance forces with viscous effects to predict fluid response under various conditions.

The Navier-Stokes equations are formed by substituting stress relationships into differential equations of motion. These stress relationships include normal stresses dependent on pressure and deformation rates, and shear stresses representing internal friction. The resulting equations balance inertial, pressure, and gravitational forces in a viscous fluid, enabling prediction of fluid behavior under various conditions.

Viscous terms in the Navier-Stokes equations represent internal friction arising from molecular interactions within the fluid. Depending on velocity gradients present, these viscous forces act to slow down fluid movement. Each directional equation captures viscous stress as an internal force alongside external forces like gravity, predicting how fluids resist motion.

The Navier-Stokes equations can be simplified for steady or laminar flow scenarios, where flow is smooth and orderly. These simplifications enable more straightforward analysis in controlled scenarios like steady laminar flow between parallel plates and boundary layer flows near surfaces, making complex fluid problems more manageable.

Each directional Navier-Stokes equation captures both internal forces, such as viscous stress and pressure gradients, and external forces like gravity. These equations allow prediction of fluid motion across various conditions by accounting for how gravity influences acceleration and flow patterns in viscous fluids alongside pressure and inertial effects.