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)
=((ⅈ ∂/∂x+j ∂/∂y+k ∂/∂z) .(ⅈ ∂/∂x+j ∂/∂y+k ∂/∂z))U
∇⋅(∇U) is called the Laplacian ∇²
(∂^2/(∂x^2 )+∂^2/(∂y^2 )+∂^2/(∂z^2 ))U= ∇^2 U=0

 . (U)=(ix+jy+kz) . ((ix+jy+kz) 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)

=((ix+jy+kz) . (ix+jy+kz)) 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

=(2x2+j2y2+k2z2) U\qquad = (\dfrac{\partial^2}{\partial x^2} + j \dfrac{\partial^2}{\partial y^2} + k \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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