Why Does Maximum Water Velocity Occur at the Surface?

[fonseh]

In the Wiki article on viscous laminar flow in a tube, they do an axial force balance on the shell of fluid between radial locations r and r+Δr. An easier way to analyze the problem is to do an axial force balance on the plug of fluid between r = 0 and arbitrary radial location r:
$$\pi r^2\Delta p+2\pi r L\tau_{rz}(r)=0$$where $\Delta p$ is the pressure drop between the inlet and outlet of the the section of pipe under consideration, L is the axial length of the section of pipe, and $\tau_{rz}(r)$ is the shear stress the radial surface of the plug (i.e., at constant r in the z direction). From this equation, we get that:
$$\tau_{rz}=-\frac{\Delta p}{2L}r$$
This tells us that, for steady flow in a pipe, the shear stress varies linearly with radial distance from the axis.

Where do you get this equation actually ?

 

[Chestermiller]

Where do you get this equation actually ?

It's just a force balance (you remember, like in freshman physics). Please show us what you think the free body diagram looks like.

 

[fonseh]

It's just a force balance (you remember, like in freshman physics). Please show us what you think the free body diagram looks like.

i know that A = pi(r^2) , so dA = 2pi(r)dr , so in $$\pi r^2\Delta p+2\pi r L\tau_{rz}(r)=0$$ , why there is L ? shouldn't it $$= 2\pi r \tau_{rz}(r)=0$$ only ?

 

[Chestermiller]

i know that A = pi(r^2) , so dA = 2pi(r)dr , so in $$\pi r^2\Delta p+2\pi r L\tau_{rz}(r)=0$$ , why there is L ? shouldn't it $$= 2\pi r \tau_{rz}(r)=0$$ only ?

There are pressure forces on the two ends that have to be balanced by the viscous frictional (tangential) force.

 

[Chestermiller]

i know that A = pi(r^2) , so dA = 2pi(r)dr , so in $$\pi r^2\Delta p+2\pi r L\tau_{rz}(r)=0$$ , why there is L ? shouldn't it $$= 2\pi r \tau_{rz}(r)=0$$ only ?

There are pressure forces on the two ends that have to be balanced by the viscous frictional (tangential) force.

 

[fonseh]

There are pressure forces on the two ends that have to be balanced by the viscous frictional (tangential) force.

Can you explain further how to get 2\pi r L\tau_{rz}(r) ??

 

[fonseh]

There are pressure forces on the two ends that have to be balanced by the viscous frictional (tangential) force.

why there is L ?

 

[Chestermiller]