Module 15 : Vector fields, Gradient, Divergence and Curl
Lecture 44 : Gradient Divergence and Curl [Section 44.1]
This is called the continuity equation of a compressible fluid flow without sinks or sources. The fluid flow is said to be steady , if is independent of time. In that case and hence the equation of flow is
If is also a constant, i.e., the fluid has uniform density (incompressible), we have the equation to be This is also the necessary condition for the incompressibility of the fluid flow.
44.1.4
Visualizing Divergence:
We saw in the previous example that if we treat a vector-field as the velocity-field of a steady flow of an incompressible fluid flow, then at a point means that the flow has no source or sink. We say fluid flow has source at a point if at that point and has a sink at a point if at that point. Thus, if we represent as a vector (arrow), then at a point where there is a sink, there are more arrows going in that point than the number of arrows that going out of it. At a source point the opposite happens, i.e., there are more arrows going out than coming in. Or, we can say that the flow is ‘diverging' at that point. One can also treat as a force field. Then as an arrow indicates the acceleration of a point See an interactive visualization at the end of the section.
44.1.5
Note:
Note that in examples 44.1.2 and 44.1.3, we represented physical quantities in terms of vectors, which of course depend upon coordinate systems. For example, our definition of divergence depended upon the vector representation
of the vector field Does that mean that physical phenomenon depend upon the choice of coordinates? One can show that this is not so. In fact, all that quantities like dot-product, cross product, divergence are independent of the choice of coordinates.
is independent of time. In that case 
and hence the equation of flow is 
is also a constant, i.e., the fluid has uniform density (incompressible), we have the equation to be 
This is also the necessary condition for the incompressibility of the fluid flow. 
as the velocity-field of a steady flow of an incompressible fluid flow, then 
at a point means that the flow has no source or sink. We say fluid flow has source at a point 
if 
at that point and has a sink at a point 
if 
at that point. Thus, if we represent 
as a vector (arrow), then at a point 
where there is a sink, there are more arrows going in that point than the number of arrows that going out of it. At a source point the opposite happens, i.e., there are more arrows going out than coming in. Or, we can say that the flow is ‘diverging' at that point. One can also treat 
as a force field. Then 
as an arrow indicates the acceleration of a point 
See an interactive visualization at the end of the section. 
Does that mean that physical phenomenon depend upon the choice of coordinates? One can show that this is not so. In fact, all that quantities like dot-product, cross product, divergence are independent of the choice of coordinates. 
