# Velocity of water across depth

## Homework Statement

Here's the velocity profile of the water across the depth . I dont understand why The max velocity occur at the top surface[/B]

## The Attempt at a Solution

Is it because the friction on the top surface of water only comes from the below , whereas for the water below the top surafce , it has higher friction from the moving on the top , and moving water beneath it .... Is my concept correct ? [/B]

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BvU
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Hi.
What's holding back the water ? The viscous friction with something that doesn't move, namely the bottom. So all transport of horizontal momentum takes place in a vertical direction - as you can see from the variation in velocity going from 0 at the bottom to max at the surface.

Chestermiller
Mentor

## Homework Statement

Here's the velocity profile of the water across the depth . I dont understand why The max velocity occur at the top surface[/B]

## The Attempt at a Solution

Is it because the friction on the top surface of water only comes from the below , whereas for the water below the top surafce , it has higher friction from the moving on the top , and moving water beneath it .... Is my concept correct ? [/B]
The friction at the top surface (i.e., the shear stress) is zero. There is friction at the bottom surface because of the non-slip boundary condition.

• fonseh
The friction at the top surface (i.e., the shear stress) is zero. There is friction at the bottom surface because of the non-slip boundary condition.
Do you mean as the depth increases , the shear stress increases, causing the velocity to decreases parabolically from top surface to the bottom >

Chestermiller
Mentor
Do you mean as the depth increases , the shear stress increases, causing the velocity to decreases parabolically from top surface to the bottom >
Yes

• fonseh
Yes
can you explain further why the shear stress will increase with depth ?

Chestermiller
Mentor
can you explain further why the shear stress will increase with depth ?
You do understand that the shear stress is zero at the upper surface, correct? And you do understand that the fluid does not slip at the wall, correct.

• fonseh
y
You do understand that the shear stress is zero at the upper surface, correct? And you do understand that the fluid does not slip at the wall, correct.
yes , i understand both .... Why the shear stress will incresae down the depth ?

Chestermiller
Mentor
y

yes , i understand both .... Why the shear stress will incresae down the depth ?
Since the fluid at the wall is stationary (the velocity at the wall is zero), and you have a volumetric flow rate, the velocity has to be higher away from the wall. This means that the velocity gradient at the wall must be positive. From Newton's law of viscosity, that means that there is a shear stress at the wall. So you have shear stress at the wall and no shear stress at the interface with the air. So the shear stress must be increasing with distance from the wall.

• fonseh
Since the fluid at the wall is stationary (the velocity at the wall is zero), and you have a volumetric flow rate, the velocity has to be higher away from the wall. This means that the velocity gradient at the wall must be positive. From Newton's law of viscosity, that means that there is a shear stress at the wall. So you have shear stress at the wall and no shear stress at the interface with the air. So the shear stress must be increasing with distance from the wall.
why ? ?

Chestermiller
Mentor
why ? ?
If the velocity were zero everywhere, there would be no flow.

If the velocity were zero everywhere, there would be no flow.
why it has to increase parabolically ? Why the shape of graph cant be others ? I mean the graph has still positive gradient ...

Chestermiller
Mentor
why it has to increase parabolically ? Why the shape of graph cant be others ? I mean the graph has still positive gradient ...
To obtain that result, you need to actually solve the flow problem involving the differential force balance and Newton's law of viscosity.

• fonseh
To obtain that result, you need to actually solve the flow problem involving the differential force balance and Newton's law of viscosity.
I still doesnt get it . Can you show the problem involving the differential force balance and Newton's law of viscosity???

Chestermiller
Mentor
I still doesnt get it . Can you show the problem involving the differential force balance and Newton's law of viscosity???
It's too extensive to show here. Are you familiar with the derivation of the Hagen-Poiseuille law for laminar viscous flow in a pipe.

• fonseh
It's too extensive to show here. Are you familiar with the derivation of the Hagen-Poiseuille law for laminar viscous flow in a pipe.
Can you provide some link and explain briefly here ?

Chestermiller
Mentor
Can you provide some link and explain briefly here ?

• fonseh
BvU
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2019 Award
Bird Stewart and Lightfoot are even better friends. Whole derivation in 2.2 in my edition.

• fonseh and Chestermiller
Bird Stewart and Lightfoot are even better friends. Whole derivation in 2.2 in my edition.
What do you mean ?

Since the fluid at the wall is stationary (the velocity at the wall is zero), and you have a volumetric flow rate, the velocity has to be higher away from the wall. This means that the velocity gradient at the wall must be positive. From Newton's law of viscosity, that means that there is a shear stress at the wall. So you have shear stress at the wall and no shear stress at the interface with the air. So the shear stress must be increasing with distance from the wall.
Do you mean the shear stress increase to max from the wall to the center ? Which means the shear stresss is max at the center at particular depth ?

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Chestermiller
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Chestermiller
Mentor
Do you mean the shear stress increase to max from the wall to the center ? Which means the shear stresss is max at the center at particular depth ?
No. It is zero at the top and maximum at the wall. It decreases linearly with distance from the wall.

No. It is zero at the top and maximum at the wall. It decreases linearly with distance from the wall.
can you show the graph of shear stress ? I'm very confused now ... Do you mean the shear stress increases from zero at the top down to the bottom ?

The red part means for a particular depth , the shear stress decreases from the wall to the center ?