Laplacian vector field

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:

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

Laplace's equation

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

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

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

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

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

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

${\displaystyle u_{i}={\frac {\partial \phi }{\partial x_{1}}}\quad$ ;\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

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

And substituting equation 1 into the equation above yields

${\displaystyle \nabla ^{2}\phi =0.}$ ^[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${\displaystyle \phi }$which is both irrotational and incompressible.

Irrotational flow is a flow where the vorticity, ${\displaystyle \omega }$, is zero, and since ${\displaystyle \omega =\nabla \times u}$, it follows that the condition ${\displaystyle \omega =0}$ is satisfied by defining a quantity called the velocity potential ${\displaystyle \phi }$, such that ${\displaystyle u=\nabla \phi }$, since ${\displaystyle \nabla \times \nabla \phi =0}$ 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 ${\displaystyle \nabla ^{2}\phi =0}$ which satisfies the Laplace equation. ^[4]

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

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

When we also take into consideration ${\displaystyle u={\frac {\partial \phi }{\partial x}}\quad {\textrm {and}}\quad v={\frac {\partial \phi }{\partial y}}}$, 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 ${\displaystyle \psi }$) everywhere. ^[4]

Further reading

The Laplacian vector field theory is being used in research in mathematics and medicine. Math researchers study the boundary values for Laplacian vector fields and investigate an innovative approach where they assume the surface is fractal and then must utilize methods for calculating a well-defined integration over the boundary. ^[5] Medical researchers proposed a method to obtain high resolution in vivo measurements of fascicle arrangements in skeletal muscle, where the Laplacian vector field behavior reflects observed characteristics of fascicle trajectories. ^[6]

See also

• Potential flow
• Harmonic function

References

1. 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.
2. 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.
3. Brennen, Christopher E (2004). "Incompressible, Inviscid, Irrotational Flow". Internet Book on Fluid Dynamics. Retrieved December 9, 2024.
4. Techet, Alexandra (2005). "Hydrodynamics (13.012): 2005reading4". MIT OpenCourseWare. Retrieved December 9, 2024.
5. 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.
6. 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 .