I do not think there is a philosophical problem. It is only a problem of point of view. The flow is slowed down by the friction, therefore the stress is in in the opposite direction of the flow speed.

On the other hand, theft plate is dragged by the friction, therefore the force is in the flow direction.

thanks for the text. They are a bit lengthy. I searched through 'find tab' to get the relevant text on shear stress, but it could not be found. I think I should carefully study it to get the answer to my question. Anyway, I would be pleased, if you could possibly elucidate in a few sentences, your own idea about the answer.

I don't think that your question is well suited. The problem is that the "stress" is a tensor, therefore is associated to two directions, that is:

ei tauij ej

so that when you analyse tau12|wall= du/dy|wall that is the scalar function associated to two directions, the direction x of the flow and the direction y normal to the plane of the wall.

I do not think there is a philosophical problem. It is only a problem of point of view. The flow is slowed down by the friction, therefore the stress is in in the opposite direction of the flow speed.

On the other hand, theft plate is dragged by the friction, therefore the force is in the flow direction.

I have my own idea illustrated and described here:

http://www.saebamirahmadi.com/2018/01/an-upgraded-aerodynamic-model-and.html

It may take two directions

As shown in your figure, the stress of fluid is in in the opposite direction of the flow speed.

thank you all. It might be concluded that, it may take two directions. This is what I concluded through online search, some texts, and also by the comments here. No certainty in what direction it has. It could be a scientific conundrum.

How it can be both directions exactly I do not understand. I've recommended a clear answer before that is correct.

1. define your coordinate system

2. define to what body you apply your stress (either fluid or solid)

3. get a single, clear and simple definition of the direction of the stress applied to that body (either fluid or solid).

Consider you are rubbing palm of your hands against the other, both in movement. The friction you get from rear view is against the direction of the friction from front view and also for upper view and lower view this is in. The palms move in symmetry and are of no discrimination in any direction. This is a contradiction, and is the quandary.

A drawing with definitions of coordinate systems and arrows of motion, shear (velocity derivatives) and shear stresses would be more helpful for the discussion, as the one you posted in your question.

In the figure Wall shear stress 1.png all is defined well (velocity direction, velocity derivative of the fluid) with one exception - to which body the $\tau$ applies. is it a $\tau_w$ from the word "wall" that applies to the by the fluid on the solid or is it $\tau_w$ that is a "wall effect on the fluid" that arises from solid, and it applies to the fluid? of course there are two equal shear stresses, one apply to fluid and one to surface and they're in two opposite directions, in respect to the coordinate system that is attached to that body. If the body is moving in any direction in the world, it's irrelevant as the fluid velocity is measured relative to the body in that coordinate system. At least that's what I understand from that figure.

Thank you all for your participation in the discussion. It turns out that the answer is to be figured out.

The answer by G. Collona is right. It is true for a "no slip" flow. However, for "slip" flow, the shear stresses applied to the fluid and to the plate are not the same.

here the wall shear stress is in the reverse direction.

http://www.thermopedia.com/content/5018/101BLFig1.gif

https://www.google.com/search?q=What+is+the+direction+of+wall+shear+stress+at+boundary-layer+over+flat+plate%3F&rlz=1C1GCEA_enIR778IR778&source=lnms&tbm=isch&sa=X&ved=0ahUKEwjTm6STp-DYAhVoQZoKHSUnCOMQ_AUICigB#imgrc=65JWnQtq599QuM:

Immagine that the wall is not fixed but can slip (like tapis roulant). The direction of the slip is a consequence of the force (stress X surface) applied on that direction.

By looking at NS equations you will find shear strss terms having negative sign and always opposing the fluid motion.

In coordinates fixed in the plate, the stress component on the plate is in the direction of the flow or the x direction. Physically, the force needed to keep the plate in place is in the direction opposite to the flow which balances the the integrated stress component over the plate in the positive x direction.

Here, I have illustrated my own resolution.

Thank you all. I am looking forward to know about your comments and probably objections over my resolution.

I think you are not considering the issue in the right way. First, we use to associate the stress to the velocity gradient when we introduce the Newtonian linear relation. But you question could extend to a more general framework, for example use your hand to apply a stress over a solid plane (or viceversa).

Then, the concept of limit you want to apply is somehow debatable. The derivative du/dy evaluated at the limit y=0 can be done only by doing the limit from the inner fluid zone. But to define the existence of a derivative, we should ensure that the limit exists and is the same from both side.

I still think that the answer is much more simple if you consider separately the region of fluide that exerts a stress on the wall and the wall that exerts a friction on the fluid layers.

The force on the plate is obtained by taking the dot product of the stress tensor at a point on the plate from the fluid side with the exterior normal to the plate at the same point, which is a vector. The component of the force on the plate in the x direction, as depicted in the original picture, is independent of the pressure, and when the fluid is Newtonian it becomes equal to the viscosity coefficient times the slope of the velocity at the wall, which is positive.

I guess I am now in possession of enough material, to decide how to reconcile the dispute. Thank you all.

Looking at the fluid exclusively, the balance of forces implies that the driving force (pressure gradient) is negated by wall drag. So from the point of reference of the fluid, the shear stress tensor is acting opposite to the main flow direction: with a negative x-component (and zero y-component).

Newton's third law implies that the wall feels a stress of opposite direction and equal magnitude, thus with positive x-component.

First draw a free body diagram:

https://en.wikipedia.org/wiki/Free_body_diagram

The wall is one body, and the fluid the other. Then apply all the conventions from your 1st semester mechanics class, and you'll have the correct answer.

@Nils Beck

Hey dude, you’ve disregarded the whole discussion. You should be recommended to the Nobel Prize committee. But nevermind, forget it ( I am kidding). Thank you to participate in this discussion and add your answer. Sincerely, Saeb.

I think, the stress is in in the opposite direction of the flow speed.

Not a direct answer to your point, but when gas is removed at the walls the velocity, hence momentum of the remaining gas reduces which in turn causes an increase in pressure. If the wall flow is great enough the pressure at the exit of the conduit will be greater than at the entrance. This can be counter-intuitive if unaware of the effect.

If it thinks that the plate moves itself in sense of right to left side the force of friction in the cap border generated about the plate place gives to a pulling power in the same direction but in the opposite direction to the movement of the plate, or that the force of friction in the plate is from left side to right