# Velocity gradient in a liquid

When we consider the motion of fluid in terms of many thin layers sliding over each other , we say that layer at a top of a layer forces it to move forward while layer below a layer forces it to move backwards by applying a resistive force $$f_v$$ where $$f_v=\eta A \frac{\text{d}v}{\text{d}h}$$

If velocity gradient is constant then each layer is pushed by a force $$f_v$$ forward and by an equal force $$f_v$$ backwards implying net force on a layer is zero ( with some external force on top layer to keep it moving with constant velocity) and hence every layer moves with constant velocity.

Since each layer has net force zero on it and it moves with constant velocity then why there is a decrease in velocity of layers as we move deeper in the liquid, is this due to some interaction other than viscous forces?

P.S. - I am a high school student who has learned only a basic definition of viscosity.

• If the fluid is situated between two infinite horizontal parallel plates, and the top plate is moving horizontally while the bottom plate is stationary, then at each elevation between the plates, the fluid velocity is constant in the horizontal direction, but it varies in the vertical direction. Why is this a problem for you? Apr 1 at 9:46
• @ChetMiller why/how does it varies in vertical direction , I want to know an explanation for that Apr 1 at 13:03
• If the velocity is zero at the bottom and V at the top, it must be varying in the vertical direction. Apr 1 at 14:30
• @ChetMiller Yes, I know that but I want to know what makes it vary ? Apr 1 at 15:36
• As the first answer indicates, the no-slip boundary condition. Apr 1 at 20:53

## 2 Answers

I think you are unaware of the no-slip boundary condition. In simple words, as you're a high school student,

When a stream of fluid flows over a horizontal surface, the layer of fluid just above the surface comes to rest due to friction which resists the motion of the fluid layer. This is known as the no-slip boundary condition as there is no relative motion between the boundary and the fluid layer right next to the boundary. This layer at rest then applies resistive viscous forces on the top consecutive layers which cause a positive velocity gradient in the upward direction. This force between consecutive fluid layers has the formula that you've correctly mentioned and is the viscous force.

For many general cases, the no-slip boundary condition is valid and even if it's not in some cases, you still know that the external force that initiates the velocity gradient is the friction due to the boundary.

So, you must think just the opposite of what you've written in the question i.e. why does the velocity increase as we go upwards?

Edit: Imagine that there is a stream of fluid with each layer having a constant velocity. Suddenly the flowing fluid encounters a surface at the bottom. The lowermost layer of the fluid will immediately come to rest as it flows over the surface. This layer will pull the layer just above with the force $$f_v$$ where $$f_v=\eta A \frac{\text{d}v}{\text{d}h}$$. This will cause the above layer to decrease in speed and hence the effect of friction of the surface propagates upwards causing the speed of the above layers to reduce. This is what causes the velocity gradient.

The arrows denote magnitude of velocity at that point.

See how on the left, each layer has same velocity. But now that it encounters a boundary, velocity becomes zero at surface and increases upwards. At each point between two layers, the viscous force would be related to velocity gradient at that point.

• This layer at rest then applies resistive viscous forces on the top consecutive layers which cause a positive velocity gradient in the upward direction , how does this causes positive velocity gradient ? that's what my question is? Mar 31 at 18:34
• by positive velocity gradient I mean the velocity increases in upward direction. Mar 31 at 19:27
• I've edited the answer. Ask if you still don't understand. Mar 31 at 19:35

Here is one way to look at this. I think your problem is you are forgetting that this setup has two boundary constraints: 1) the velocity at the top of the "stack" of thin fluid layers is moving with a specific free-stream velocity, and 2) the layer closest to the wall at the bottom of the stack is moving with velocity zero. Try the derivation again including the v = 0 constraint at the wall and see what develops.

• Yes that's my question, the layer closest to wall is moving with velocity zero and topmost layer is moving with specific free stream velocity but why does layers in between move with continuously varying velocities? Mar 31 at 18:33