# Divergence of velocity vector

by Dasa . 15 May 2023

## Definition:

Divergence calculates a scalar from a vector.

It reduces the Tensor value by one. For example, the divergence of a velocity vector (Tensor of value 1) will be a scalar (Tensor of value 0).

When we compute the divergence of a velocity vector in a finite volume, it gives a scalar value that is the mass flux. Here the scalar value is mass flux per unit of time.

Consider a velocity vector of

$U = u\space i + v\space j + w\space k$

The Nabla is equal to

$\nabla = (\dfrac{\partial}{\partial x} i + \dfrac{\partial}{\partial y} j + \dfrac{\partial}{\partial z} k)$

Then the divergent of U is

$\nabla . U = div\space U = (\dfrac{\partial}{\partial x} i + \dfrac{\partial}{\partial y} j + \dfrac{\partial}{\partial z} k) . U$
$\qquad = (\dfrac{\partial}{\partial x} i + \dfrac{\partial}{\partial y} j + \dfrac{\partial}{\partial z} k) . (u\space i + v\space j + w\space k)$
$\qquad = (\dfrac{\partial}{\partial x} u + \dfrac{\partial}{\partial y} v + \dfrac{\partial}{\partial z} w)$

If we dot a Nabla with a vector we get a vector that is called divergence.

## Divergence theorem:

It states that the accumulation of a vector in a finite volume is equal to the sum of that vector over the surfaces.
Consider an infinitesimal volume having sides as dx, dy, and dz as below. And a velocity vector vj enters the surface dS (=dxdz) and leaves at the opposite surface located at a distance of dy. Then change is the finite volume is the integral over the surfaces

$= vi\space dx\space dz - (vi+\dfrac{\partial}{\partial y} \space vj\space j\space dy) \space dx \space dz$

$= -\dfrac{\partial}{\partial y} \space vj \space j \space dy \space dx \space dz$

$= \dfrac{\partial}{\partial y} \space v \space dV$

When we sum the other two faces, we get

$= (\dfrac{\partial}{\partial x} \space u + \dfrac{\partial}{\partial y} \space v + \dfrac{\partial}{\partial z} \space w) \space dV$

$= \nabla \space . \space U dV$

Note: Here, we have yet to prove that the quantity calculated is independent of the coordinate system.
When a vector field has zero divergences, it is said to be “Solenoidal”.

The divergence theorem states that for any closed surfaces, the net total flux through the surfaces must be zero.