mb.pejvak |
September 26, 2013 23:19 |
The concept in discussion is what is called a local or a non-local property of a field. That means that, if the property in discussion is local, its value can be defined in each point of a field and its value only depends on the values of other properties/variables in the point of study or observation. In other words, everything is locally determined by the values in each point.
A non-local property of a field is that where its value in each point in the field depends on the values of properties or variables all over the space where the field is defined. Due to the fact that for an incompressible fluid, the pressure field satisfies a Laplace equation with a source term (Poissons equation), it is possible to formally solve the pressure field as a function of the source term by means of the Biot-Savart law (really it is a Green's function). The result is an integral over all the definition space of the pressure field and the value of the pressure at each point depends on values of other properties/variables all over this definition space. Therefore, pressure is a non-local variable and so is velocity since it depends on the gradient of pressure, the gradient of this volume integral over the definition space. The vorticity equation has not a direct dependence on pressure and, therefore, vorticity is a local variable.
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