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Laplacian of velocity potential

by Dasa . 26 May 2023

We know that the gradient of a scalar (velocity potential) gives a vector (Direction of the steepest ascent of the velocity).

Also, the divergence of a vector (Velocity vector) gives a scalar (mass flux).

The divergence of gradient of velocity in a finite volume will give the mass flux. The divergence of gradient is called Laplacian.

If the mass flux is conservative then the Laplacian of a velocity is zero, since the mass flow into the finite volume minus the mass flow out of the finite volume must be zero.

∇ . (∇U)=(i∂∂x+j∂∂y+k∂∂z) . ((i∂∂x+j∂∂y+k∂∂z) U)\nabla \space . \space (\nabla U) = (i \dfrac{\partial}{\partial x} + j \dfrac{\partial}{\partial y} + k \dfrac{\partial}{\partial z})\space . \space ((i \dfrac{\partial}{\partial x} + j \dfrac{\partial}{\partial y} + k \dfrac{\partial}{\partial z}) \space U)

=((i∂∂x+j∂∂y+k∂∂z) . (i∂∂x+j∂∂y+k∂∂z)) U\qquad= ((i \dfrac{\partial}{\partial x} + j \dfrac{\partial}{\partial y} + k \dfrac{\partial}{\partial z})\space . \space (i \dfrac{\partial}{\partial x} + j \dfrac{\partial}{\partial y} + k \dfrac{\partial}{\partial z})) \space U

=(∂2∂x2+∂2∂y2+∂2∂z2) U\qquad = (\dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2} + \dfrac{\partial^2}{\partial z^2})\space U

=∇2U\qquad = \nabla^2 U

∇2\nabla^2 is called the Laplacian operator.

If the mass flux is conservative

∇2U=0\nabla^2 U = 0

Please read this page for Gradient and Divergence.


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