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Fluid Dynamics and the Navier-Stokes Equations


Contents

Navier-Stokes Equations
Steady Flow
Lift for Aeroplane and Helicopter
High and Low Pressure Cells
Spiral Flow
Hydrostatics
Formation of Spherical Body
Magnetohydrodynamics (MHD) and the Formation of Jet(s)

Navier-Stokes Equations

Fluid dynamics deals with the motion of liquids and gases, which when studied macroscopically, appear to be continuous in structure. All the variables are considered to be continuous functions of the spatial coordinates and time. The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the flow of fluids. They model weather, the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena. The Navier-Stokes equations for irrotational flow, i.e., for x u = = 0, are:

where u = velocity vector field, = thermodynamic internal energy, p = pressure, T = temperature, = density,
= viscosity, KH = heat conduction coefficient, F = external force per unit mass, , and .

The Navier-Stokes equations are time-dependent and consist of a continuity equation for conservation of mass, three conservation of momentum equations and a conservation of energy equation. There are four independent variables in the equation - the x, y, and z spatial coordinates, and the time t; six dependent variables - the pressure p, density , temperature T, and three components of the velocity vector u. Together with the equation of state such as the ideal gas law - p V = n R T, the six equations are just enough to determine the six dependent variables. In general, all of the dependent variables are functions of all four independent variables. Usually, the Navier-Stokes equations are too complicated to be solved in a closed form. However, in some special cases the equations can be simplified and may admit analytical solutions (see "Differential Equation" for a very brief introduction).

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