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In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible.^[1] If the field is denoted as v, then it is described by the following differential equations:

{\begin{aligned}\nabla \times \mathbf {v} &=\mathbf {0} ,\\\nabla \cdot \mathbf {v} &=0.\end{aligned}}

Laplace's equation

From the vector calculus identity $\nabla ^{2}\mathbf {v} \equiv \nabla (\nabla \cdot \mathbf {v} )-\nabla \times (\nabla \times \mathbf {v} )$ it follows that

\nabla ^{2}\mathbf {v} =\mathbf {0}

that is, that the field v satisfies Laplace's equation.^[2]

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 ${\bf {v}}=(xy,yz,zx)$ 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 $\mathbf {u}$ is zero, it follows that (when the domain of definition is simply connected) $\mathbf {u}$ can be expressed as the gradient of a scalar potential (see irrotational field) which we define as $\phi$ :

\mathbf {u} =\nabla \phi \qquad \qquad (1)

since it is always true that $\nabla \times \nabla \phi =0$ .^[3]

Other forms of $\mathbf {u} =\nabla \phi$ can be expressed as

$u_{i}={\frac {\partial \phi }{\partial x_{1}}}\quad ;\quad u={\frac {\partial \phi }{\partial x}},v={\frac {\partial \phi }{\partial y}},w={\frac {\partial \phi }{\partial z}}$ .^[3]

When the field is incompressible, then

$\nabla \cdot u=0\quad {\textrm {or}}\quad {\frac {\partial u_{j}}{\partial x_{j}}}=0\quad {\textrm {or}}\quad {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0$ .^[3]

And substituting equation 1 into the equation above yields

\nabla ^{2}\phi =0.

^[3]

Therefore, the potential of a Laplacian field satisfies Laplace's equation.^[3]

Potential flow theory

The Laplacian vector field has an impactful application in fluid dynamics. Consider the Laplacian vector field to be the velocity potential $\phi$ which is both irrotational and incompressible.

Irrotational flow is a flow where the vorticity, $\omega$ , is zero, and since $\omega =\nabla \times u$ , it follows that the condition $\omega =0$ is satisfied by defining a quantity called the velocity potential $\phi$ , such that $u=\nabla \phi$ , since $\nabla \times \nabla \phi =0$ always holds true.^[3]

Irrotational flow is also called potential flow.^[3]

If the fluid is incompressible, then conservation of mass requires that

${\textstyle \nabla \cdot u=0\quad {\textrm {or}}\quad {\frac {\partial u_{j}}{\partial x_{j}}}=0\quad {\textrm {or}}\quad {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0}$ .^[4]

And substituting the previous equation into the above equation yields $\nabla ^{2}\phi =0$ which satisfies the Laplace equation.^[4]

In planar flow, the stream function $\psi$ can be defined with the following equations for incompressible planar flow in the xy-plane:

$u={\frac {\partial \psi }{\partial y}}\quad {\textrm {and}}\quad v=-{\frac {\partial \psi }{\partial x}}$ .^[3]

When we also take into consideration $u={\frac {\partial \phi }{\partial x}}\quad {\textrm {and}}\quad v={\frac {\partial \phi }{\partial y}}$ , we are looking at the Cauchy-Reimann equations.^[3]

These equations imply several characteristics of an incompressible planar potential flow. The lines of constant velocity potential are perpendicular to the streamlines (lines of constant $\psi$ ) everywhere.^[4]

References

^ Arfken, George B; Weber, Hans J; Harris, Frank E (2013). "Vector Analysis". Mathematical Methods for Physicists: A Comprehensive Guide (7th ed.). Waltham, MA: Elsevier Inc. pp. 154–5. ISBN 978-0-12-384654-9.
^ Claycomb, J. R. (2018). "Vector Calculus". Mathematical Methods for Physics: Using MATLAB and Maple. Dulles, VA: Mercury Learning and Information. p. 199. ISBN 978-1-68392-098-4.
^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ Brennen, Christopher E (2004). "Incompressible, Inviscid, Irrotational Flow". Internet Book on Fluid Dynamics. Retrieved December 9, 2024.
^ ^a ^b ^c Techet, Alexandra (2005). "Hydrodynamics (13.012): 2005reading4". MIT OpenCourseWare. Retrieved December 9, 2024.
^ Abreu-Blaya, R; Bory-Reyes, J; Moreno-Garcia, T; Peña‐Peña, D (May 10, 2008). "Laplacian decomposition of vector fields on fractal surfaces". Mathematical Methods in the Applied Sciences. 31 (7): 849–857. doi:10.1002/mma.952 – via Wiley Online Library.
^ Choi, Hon Fai; Blemker, Silvia S (October 25, 2013). Sampaolesi, Maurilio (ed.). "Skeletal muscle fascicle arrangements can be reconstructed using a Laplacian vector field simulation". PLOS ONE. 8 (10): e77576. doi:10.1371/journal.pone.0077576.