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So the velocity field will have positive divergence everywhere. Similarly, if the gas is cooled, it will contract. There will be more room for gas particles in any volume, so the external pressure of the fluid will cause a net flow of gas volume inward through any closed surface. Therefore, the velocity field has negative divergence everywhere.
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.
The volume rate of flow of liquid through a source or sink (with the flow through a sink given a negative sign) is equal to the divergence of the velocity field at the pipe mouth, so adding up (integrating) the divergence of the liquid throughout the volume enclosed by S equals the volume rate of flux through S. This is the divergence theorem. [2]
At the same time, the divergence can also be obtained from Navier–Stokes equation by taking its divergence. In particular, for incompressible flow, where ∇ ⋅ u = 0 {\displaystyle \nabla \cdot \mathbf {u} =0} , with body forces given by − ∇ U {\displaystyle -\nabla U} , the Lamb vector divergence reduces to
The pressure value that is attempted to compute, is such that when plugged into momentum equations a divergence-free velocity field results. The mass imbalance is often also used for control of the outer loop. The name of this class of methods stems from the fact that the correction of the velocity field is computed through the pressure-field.
If the fluid is incompressible (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation: [3] =, which means that the divergence of the velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero, hence a flow of water through a ...
The second vector calculus identity above states that the divergence of the curl of a vector field is zero. Since the (incompressible) mass continuity equation specifies the divergence of flow velocity being zero, we can replace the flow velocity with the curl of some vector ψ so that mass continuity is always satisfied: