What does 'dv and dx' mean in f = eta X A (dv/dx)?

  • Thread starter Indranil
  • Start date
  • #1
177
11

Homework Statement


What does 'dv and dx' mean in f = eta A (dv/dx) in coefficient of viscosity?

2. Homework Equations

What does 'dv and dx' mean in f = eta A (dv/dx) in coefficient of viscosity?

The Attempt at a Solution


As I know v = velocity and x = distance. then what does 'dv' and 'dx' express here. Kindly explain.
 

Answers and Replies

  • #2
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
11,447
4,466
Like any derivative, it means the rate at which v varies with respect to x. Imagine running at top speed on asphalt and then you hit a muddy field that gets muddier the farther in you get. Even though you are pumping your muscles as hard as you can, the deeper you get, the lower your speed. The rate at which your speed decreases with distance is dv/dx, in this example a negative number.
 
  • #3
Buzz Bloom
Gold Member
2,477
452
Hi Indranil:

I am not sure I understand what is puzzling you. I will tell you what it means to me, and I hope that will be helpful.

dv/dx is the rate of change of velocity as position changes.

You can also write dv/dx = dv/dt / dx/dt.
Do you understand that? Since dx/dt = v, and dv/dt = a (acceleration) you can say that the meaning of dv/dt is the value you get when you divide the acceleration by the velocity.

I hope this helps.

Regards,
Buzz
 
  • #4
fresh_42
Mentor
Insights Author
2021 Award
16,447
15,501
Here's what Wikipedia has to say about it: https://de.wikipedia.org/wiki/Viskosität (They used ##y## instead of ##x##.)
800px-Definition_Viskositaet.png



In the experiment, it can be shown that, ideally, the force ##F## necessary to move the top plate is proportional to the area ##A##, the speed difference ##\Delta v## and antiproportional to the spacing of the plates ##\Delta y##, that is:
##F \sim A## and ##F \sim \Delta v## and ##F \sim \dfrac{1}{\Delta y}##

This gives the equation ##F = \eta A {\dfrac{\Delta v}{\Delta y}}## and as continuous, infinitesimal form ##F = \eta A {\dfrac{d v}{d y}}##.
 

Attachments

  • 800px-Definition_Viskositaet.png
    800px-Definition_Viskositaet.png
    92.9 KB · Views: 2,112
  • Like
Likes Chestermiller and Delta2
  • #6
21,948
4,996
I agree with @fresh_42 . This equation is typically used to introduce Newton's law of viscosity, based on the most simple example. In this example, you have a fluid contained between 2 large parallel plates, with the lower plate (y=0) stationary and the upper plate y=h) moving at velocity V in the positive x-direction. The fluid x-velocity in the region between the plates is linear, and given by:$$v_x=V\frac{y}{h}$$. The force in the positive x-direction required to cause the fluid to shear with this velocity profile is given by Newton's law of viscosity as:
$$F=\eta A\frac{dv_x}{dy}=\eta A\frac{V}{h}$$where ##\eta## is the fluid viscosity and A is the plate area. The derivative of the x-velocity with respect to y is referred to as the "shear rate."

The example given by Buzz Bloom, rather than involving shear of the fluid, is referred to as an elongational deformation. In this case, dv/dx is called the "rate of elongation", and the tensile force involves 3 times the viscosity.
 
  • Like
Likes Indranil and Delta2
  • #7
Delta v\Delta X = Viscosity Gradient

Change in Velocity/Change in Depth = VELOCITY GRADIANT = Uniformly CHANGE

dv/dx = Small CHANGE IN velocity / small change in depth Non Uniform
 

Related Threads on What does 'dv and dx' mean in f = eta X A (dv/dx)?

  • Last Post
Replies
7
Views
14K
Replies
1
Views
774
Replies
7
Views
1K
  • Last Post
Replies
2
Views
7K
  • Last Post
Replies
7
Views
5K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
5
Views
795
  • Last Post
Replies
4
Views
1K
Replies
14
Views
1K
Top