When can the velocity gradient be set to zero?

In summary, an experimentalist has measured the u-velocity component of a two-dimensional flow field and found that 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. The solution to this problem involves setting the velocity gradient equal to zero, solving the differential equation, and using boundary conditions to find the constant. The problem statement should also state that the fluid is incompressible. The velocity gradient and the divergence of the velocity are not the same, and a non-zero divergence can indicate a net flow in or out of a point.
  • #1
ScareCrow271828
22
0

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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  • #2
The problem statement should state that the fluid is incompressible.
 
  • #3
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.
 
  • #4
mfb said:
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?
 
  • #5
Chestermiller said:
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.
 
  • #6
ScareCrow271828 said:
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.
 
  • #7
haruspex said:
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.
 
  • #8
ScareCrow271828 said:
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.
 

Related to When can the velocity gradient be set to zero?

1. Can the velocity gradient ever be zero?

Yes, the velocity gradient can be zero in certain scenarios. This is known as a laminar flow, where the fluid moves in layers without any mixing or turbulence.

2. What factors affect the velocity gradient?

The velocity gradient is affected by the fluid's viscosity, surface tension, and flow rate. These factors determine how easily the fluid can change its velocity as it flows.

3. How does the velocity gradient impact fluid flow?

The velocity gradient is a measure of the change in velocity over a distance. It affects the fluid flow by determining the amount of shear stress, or force, that is applied to the fluid. A higher velocity gradient results in a higher shear stress and can cause turbulence in the flow.

4. Is there a limit to how high or low the velocity gradient can be?

There is no specific limit to the velocity gradient, but it is typically kept within a certain range for practical purposes. If the gradient is too high, it can cause turbulence and affect the accuracy of measurements. If it is too low, it may not provide enough shear stress for certain experiments.

5. How is the velocity gradient calculated?

The velocity gradient is calculated by dividing the change in velocity by the distance over which the change occurs. This can be represented by the formula dV/dx, where dV is the change in velocity and dx is the change in distance. The units of velocity gradient are typically expressed in 1/s or s^-1.

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