Laplacian vector field

In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:

Laplace's equation

From the vector calculus identity it follows that
that is, that the field v satisfies Laplace's equation.
However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a Laplacian vector field.

Cauchy-Riemann equations

A Laplacian vector field in the plane satisfies the Cauchy–Riemann equations: it is holomorphic.

Potential of Laplacian field

Suppose the curl of is zero, it follows that can be expressed as the gradient of a scalar potential which we define as :
since it is always true that.
Other forms of can be expressed as
When the field is incompressible, then
And substituting equation 1 into the equation above yields
Therefore, the potential of a Laplacian field satisfies Laplace's equation.

Potential flow theory

The Laplacian vector field has an impactful application in fluid dynamics. Consider the Laplacian vector field to be the velocity potential which is both irrotational and incompressible.
Irrotational flow is a flow where the vorticity,, is zero, and since, it follows that the condition is satisfied by defining a quantity called the velocity potential, such that, since always holds true.
Irrotational flow is also called potential flow.
If the fluid is incompressible, then conservation of mass requires that
And substituting the previous equation into the above equation yields which satisfies the Laplace equation.
In planar flow, the stream function can be defined with the following equations for incompressible planar flow in the xy-plane:
When we also take into consideration, we are looking at the Cauchy-Reimann equations.
These equations imply several characteristics of an incompressible planar potential flow. The lines of constant velocity potential are perpendicular to the streamlines everywhere.