Navier–Stokes Equations

In fluid motion under viscous conditions, shear stress is directly proportional to fluid deformation. In incompressible Newtonian fluids, this stress varies linearly with the deformation rate.

Normal stresses depend on pressure and deformation rates in specific directions, defining fluid flow behavior under different pressures.

Shear stresses act tangentially, describing how different fluid layers slide past one another, connecting stress with velocity changes across layers.

Substituting these stress relationships into differential equations of motions forms the Navier-Stokes equations, which balance forces like inertia, pressure, and gravity in a viscous fluid.

The inertia term captures fluid acceleration, showing how a moving fluid resists sudden changes in its speed or direction.

Pressure gradients drive fluid movement from high-pressure to low-pressure areas, while viscous terms represent internal friction within the fluid.

Each directional equation captures internal forces, such as viscosity, and external forces, like gravity, predicting fluid response in various conditions.

The Navier-Stokes equations can be simplified for steady or laminar flow scenarios, allowing simpler analysis in controlled scenarios like boundary layer and Couette flows.