# When can the velocity gradient be set to zero?

## Homework Statement

An experimentalist has measured the u-velocity component of a two-dimensional flow

field. It is approximated by

u = (1/3)( xy) (y^2)
It is also known that the v-velocity is zero along the line y=0.

## Homework Equations

∇V=du/dx+dv/dy (partial derivatives)

## The Attempt at a Solution

The solution is found by setting du/dx+dv/dy=0 (partial derivatives), solving the differential equation and then using the boundary conditions at v=0, y=0 to find the constant.
How can we set the velocity gradient equal to zero? Why is it safe to assume that there is no velocity gradient?

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mfb
Mentor
The problem statement should state that the fluid is incompressible.

Chestermiller
Mentor
The velocity gradient is not zero. Do you know the difference between the velocity gradient and the divergence of the velocity? If so, please elaborate.

The problem statement should state that the fluid is incompressible.
Thank you, so if the fluid is incompressible the layers do not have a velocity towards each other? If it was compressible the fluid layers would be able to compress towards each other?

The velocity gradient is not zero. Do you know the difference between the velocity gradient and the divergence of the velocity? If so, please elaborate.
As I understand the velocity gradient is the velocity that the layers of a fluid move in respect to each other. The divergence of the velocity is kind of the total rate of change of the field, since it is a sum of the components partial derivatives.

haruspex
Homework Helper
Gold Member
The divergence of the velocity is kind of the total rate of change of the field,
If the divergence is nonzero there is a net flow in or out of the point. I.e. it is a sink or a source.

Chestermiller
Mentor
If the divergence is nonzero there is a net flow in or out of the point. I.e. it is a sink or a source.
A non-zero divergence also applies to a distributed source or accumulation.

Chestermiller
Mentor
As I understand the velocity gradient is the velocity that the layers of a fluid move in respect to each other. The divergence of the velocity is kind of the total rate of change of the field, since it is a sum of the components partial derivatives.
You have been confusing the gradient of the velocity vector with the divergence of the velocity vector. The divergence of the velocity is not the same as the gradient of the velocity. The gradient is the del operator applied to the velocity vector. The divergence is the del operator dotted with the velocity vector. In post #1, the right hand side of your equation is the divergence of the velocity vector, not the velocity gradient.