Key Concepts of Fundamental Navier-Stokes Equations to Know for Mathematical Fluid Dynamics

The Fundamental Navier-Stokes Equations are key to understanding fluid behavior. They encompass mass conservation, momentum, and energy principles, forming a framework for analyzing both incompressible and compressible flows, essential for predicting real-world fluid dynamics in various applications.

1. Continuity equation

   • Represents the principle of mass conservation in fluid dynamics.
   • States that the rate of change of mass within a control volume is equal to the net mass flow into the volume.
   • For incompressible flows, simplifies to the divergence of velocity being zero.
   • Essential for linking velocity fields to density changes in compressible flows.

2. Momentum equation

   • Describes the motion of fluid particles and the forces acting on them.
   • Incorporates body forces (like gravity) and surface forces (like pressure and viscous forces).
   • Fundamental for predicting flow patterns and behavior under various conditions.
   • Forms the basis for the Navier-Stokes equations, which govern fluid motion.

3. Energy equation

   • Accounts for the conservation of energy within a fluid system.
   • Includes terms for internal energy, kinetic energy, and potential energy.
   • Essential for understanding thermal effects and energy transfer in fluid flows.
   • Can be used to analyze heat transfer and thermodynamic processes in fluids.

4. Incompressible Navier-Stokes equations

   • Simplified form of the Navier-Stokes equations for fluids with constant density.
   • Assumes that the divergence of the velocity field is zero.
   • Commonly used in engineering applications where density changes are negligible.
   • Provides a framework for analyzing laminar and turbulent flows.

5. Compressible Navier-Stokes equations

   • General form of the Navier-Stokes equations applicable to compressible flows.
   • Accounts for variations in density, pressure, and temperature within the fluid.
   • Important for high-speed flows, such as those encountered in aerodynamics.
   • Requires additional equations of state to relate pressure, density, and temperature.

6. Vorticity equation

   • Describes the rotation of fluid elements and the circulation within a flow field.
   • Provides insights into the formation of vortices and turbulence.
   • Can be derived from the momentum equation and is useful for analyzing complex flows.
   • Helps in understanding the effects of viscosity and boundary layers.

7. Stream function formulation

   • A mathematical approach to simplify the analysis of two-dimensional incompressible flows.
   • The stream function is a scalar function whose contours represent streamlines.
   • Eliminates the need to solve the continuity equation explicitly.
   • Useful for visualizing flow patterns and calculating flow rates.

8. Reynolds number and its significance

   • A dimensionless quantity that characterizes the flow regime (laminar or turbulent).
   • Defined as the ratio of inertial forces to viscous forces in a fluid.
   • Critical for predicting flow behavior and transition points between laminar and turbulent flow.
   • Influences design and analysis in various engineering applications.

9. Boundary conditions for Navier-Stokes equations

   • Essential for solving the Navier-Stokes equations in practical applications.
   • Types include no-slip conditions, free-slip conditions, and inflow/outflow conditions.
   • Determine how the fluid interacts with solid boundaries and other fluids.
   • Influence the stability and uniqueness of the solution to the equations.

10. Non-dimensionalization of Navier-Stokes equations

    • A technique to simplify the equations by removing units and scaling variables.
    • Helps in identifying dominant forces and simplifying complex problems.
    • Facilitates comparison of different flow scenarios and experimental results.
    • Aids in the development of similarity solutions and scaling laws in fluid dynamics.