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The Navier–Stokes equations form a vector continuity equation describing the conservation of linear momentum. If the fluid is incompressible (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation: [ 3 ] ∇ ⋅ u = 0 , {\displaystyle \nabla \cdot \mathbf {u} =0,} which means that the ...
Therefore, the continuity equation for an incompressible fluid reduces further to: = This relationship, =, identifies that the divergence of the flow velocity vector is equal to zero (), which means that for an incompressible fluid the flow velocity field is a solenoidal vector field or a divergence-free vector field.
Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid): = + = + where is a vector representing external forces. Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged, [ b ] to yield:
In the analysis of a flow, it is often desirable to reduce the number of equations and/or the number of variables. The incompressible Navier–Stokes equation with mass continuity (four equations in four unknowns) can be reduced to a single equation with a single dependent variable in 2D, or one vector equation in 3D.
Using the Maxwell equations, one can see that the electromagnetic stress–energy tensor (defined above) satisfies the following differential equation, relating it to the electromagnetic tensor and the current four-vector , + = or , + =, which expresses the conservation of linear momentum and energy by electromagnetic interactions.
Since d 2 = 0, the 3-form J satisfies the conservation of current (continuity equation): = = The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time ...
The Cauchy momentum equation is a vector partial differential equation put ... (for which the continuity equation holds ... Finally in convective form the equations ...
The differential forms of these equations require that there is always an open neighbourhood around the point to which they are applied, otherwise the vector fields and H are not differentiable. In other words, the medium must be continuous[no need to be continuous][This paragraph need to be revised, the wrong concept of "continuous" need to be ...