Derivation of the Navier–Stokes equations

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids.Distributing the curl: The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids. The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum – a continuous substance rather than discrete particles. Another necessary assumption is that all the fields of interest including pressure, flow velocity, density, and temperature are differentiable, at least weakly. The equations are derived from the basic principles of continuity of mass, momentum, and energy. Sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be applied. This finite volume is denoted by Ω {displaystyle Omega } and its bounding surface ∂ Ω {displaystyle partial Omega } . The control volume can remain fixed in space or can move with the fluid. Changes in properties of a moving fluid can be measured in two different ways. One can measure a given property by either carrying out the measurement on a fixed point in space as particles of the fluid pass by, or by following a parcel of fluid along its streamline. The derivative of a field with respect to a fixed position in space is called the Eulerian derivative while the derivative following a moving parcel is called the advective or material ('Lagrangian' ) derivative. The material derivative is defined as the nonlinear operator: where u {displaystyle mathbf {u} } is the flow velocity. The first term on the right-hand side of the equation is the ordinary Eulerian derivative (i.e. the derivative on a fixed reference frame, representing changes at a point with respect to time) whereas the second term represents changes of a quantity with respect to position (see advection). This 'special' derivative is in fact the ordinary derivative of a function of many variables along a path following the fluid motion; it may be derived through application of the chain rule in which all independent variables are checked for change along the path (i.e. the total derivative). For example, the measurement of changes in wind velocity in the atmosphere can be obtained with the help of an anemometer in a weather station or by observing the movement of a weather balloon. The anemometer in the first case is measuring the velocity of all the moving particles passing through a fixed point in space, whereas in the second case the instrument is measuring changes in velocity as it moves with the flow. The Navier–Stokes equation is a special continuity equation. A continuity equation may be derived from conservation principles of: This is done via the continuity equation, an integral relation stating that the rate of change of some integrated property ϕ {displaystyle phi } defined over a control volume Ω {displaystyle Omega } must be equal to what amount is lost or gained through the boundaries Γ {displaystyle Gamma } of the volume plus what is created or consumed by sources and sinks inside the volume. This is expressed by the following integral continuity equation: