# Fluid Mechanics~ Wall shear stress

In summary, the conversation is about determining the wall shear stress and finding an expression for it using Newton's Law of Viscosity and solving for y in a given equation. The final result for the wall shear stress is (μUπ)/(2δ).

## Homework Statement

My main problem here is that I do not understand what they are asking. What is the wall shear stress? Do they mean the stress at the "floor" (or whatever you want to call it)?

If so, I am assuming I use Newton's Law of Viscosity $\tau=\mu\frac{du}{dy}$ since this is 1-dimensional flow.

Would the wall shear stress $\tau_w$ be given by

$$\tau_w=\int_{y=0}^\delta \tau(y)\, dy$$

and then the problem reduces to finding an expression for $\tau(y)$

Or am I way off here? thanks

I just found http://www.cfd-online.com/Wiki/Wall_shear_stress" [Broken]. So I guess my interpretation was wrong.

So, according to this definition, I should have:

$$\tau_w=\mu*\frac{d}{dy}[U\sin(\frac{\pi y}{2\delta})]|_{y=0}$$Is that all?

Last edited by a moderator:
I just found http://www.cfd-online.com/Wiki/Wall_shear_stress" [Broken]. So I guess my interpretation was wrong.

So, according to this definition, I should have:

$$\tau_w=\mu*\frac{d}{dy}[U\sin(\frac{\pi y}{2\delta})]|_{y=0}$$Is that all?

Not that anyone will respond to this (since no one ever looks in this forum), but I am assuming that if the above is correct, than part (b) is as simple as solving

$$\frac{1}{2}\tau_w=\mu*\frac{d}{dy}[U\sin(\frac{\pi y}{2\delta})]$$

for y.

Sound good? Good

So I have:

$$\frac{\tau_w}{2}=\mu U\cos(\frac{\pi y}{2\delta})*\frac{\pi}{2\delta}$$

$$\Rightarrow \frac{\tau_w\delta}{\mu U\pi}=\cos(\frac{\pi y}{2\delta})$$

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71 views, no responses. Yesssss I am just going to keep chatting it up with myself.

Maybe I can set a new record?

Has there ever been a thread locked where there was only one speaker?

I need help. (The kind PF cannot offer)

τw = μ×$\frac{d}{dy}$[Usin($\frac{\pi y}{2δ}$)]$|y=0$

so that, τw = 180$\pi$μ

and τ = μ×$\frac{d}{dy}$[Usin($\frac{\pi y}{2δ}$)] when τ = τw/2

thus, 90$\pi$μ = μ×$\frac{d}{dy}$[Usin($\frac{\pi y}{2δ}$)]

$\frac{1}{2}$ = cos($\frac{\pi y}{2δ}$)

$\frac{\pi}{3}$ = $\frac{\pi y}{2δ}$

y = 0.02 m

I just found This. So I guess my interpretation was wrong.

So, according to this definition, I should have:

$$\tau_w=\mu*[U\sin(\frac{\pi y}{2\delta})]|_{y=0}$$

Is that all?

Yes. If you actually apply the above formula, you get

τw=(μUπ)/(2δ)

Is that what you got?

## What is wall shear stress?

Wall shear stress is a measure of the frictional force acting on a fluid as it moves along a solid boundary, such as a wall. It is caused by the velocity gradient between the fluid and the wall, and is an important factor in understanding the behavior of fluids in motion.

## How is wall shear stress calculated?

Wall shear stress is calculated by multiplying the dynamic viscosity of the fluid by the velocity gradient at the wall. It is typically measured in units of force per area, such as N/m^2 or Pa.

## What factors affect wall shear stress?

The magnitude of wall shear stress is influenced by several factors, including the density and viscosity of the fluid, the velocity of the fluid, and the roughness of the wall surface. Changes in any of these factors can alter the wall shear stress and affect the behavior of the fluid.

## Why is wall shear stress important?

Wall shear stress is an important concept in fluid mechanics because it has a significant impact on the movement and behavior of fluids. It is essential for understanding the forces that act on objects in a fluid, and it is also used in the design and analysis of various engineering systems, such as pipelines and channels.

## How can wall shear stress be controlled or manipulated?

There are several ways to control or manipulate wall shear stress, depending on the specific application. For example, in pipes, the surface roughness of the inner wall can be altered to change the amount of wall shear stress. In open channels, the flow rate and channel geometry can be adjusted to control the wall shear stress. Additionally, the use of additives or lubricants in the fluid can also affect the wall shear stress.