Velocity Profile Analysis for Force Balance Equation in Fluid Dynamics

[haruspex]

one more question , why the τ is given by - μ / ( du /dr ) ? why y = R -r ? where is r measured from ? where is y measured from ? it's not shown in the diagram

If τ is the shear force per unit area of interface and μ is the viscosity then that equation comes straight from the definition of viscosity.
The author seems to be defining y as R-r, but I don't know why.

 

[haruspex]

i still don't understand why the resultant drag is in opposite direction , could you explain further?

An object rests on a block on a carpet. You yank the carpet. The box slides on the carpet. The carpet's drag on the box is towards you. The object slides on the block. The box's drag on the object is towards you, so the object's drag on the box is away from you. The two drags on the box are in opposite directions.

 

[hotjohn]

An object rests on a block on a carpet. You yank the carpet. The box slides on the carpet. The carpet's drag on the box is towards you. The object slides on the block. The box's drag on the object is towards you, so the object's drag on the box is away from you. The two drags on the box are in opposite directions.

why there is i assume you mean an object rest on a box on a carpet...
but , i can't visualize the situation although i have tried out myself . Based on my observation , the object will moves with the box ?

 

[haruspex]

why there is i assume you mean an object rest on a box on a carpet...
but , i can't visualize the situation although i have tried out myself . Based on my observation , the object will moves with the box ?

It might move with the box, or it might slip. Either way, the box drags the object towards you, so by action and reaction the drag on the box from the object must be away from you.

 

[hotjohn]

It might move with the box, or it might slip. Either way, the box drags the object towards you, so by action and reaction the drag on the box from the object must be away from you.

ok , back to the topic . by taking the example aforementioned as anology , which is box ? which is object ? which is carpet? r ? r+ del (r) ?

 

[haruspex]

ok , back to the topic . by taking the example aforementioned as anology , which is box ? which is object ? which is carpet? r ? r+ del (r) ?

The box is the annulus between r and r+Δr. The carpet is the next smaller annulus (faster moving) and the object is the next larger (slower moving).

 

[hotjohn]

The box is the annulus between r and r+Δr. The carpet is the next smaller annulus (faster moving) and the object is the next larger (slower moving).

all the box , object and carpet are subjected to the same velocity , right ? this is not same as the situation in the water , which at different point , the partcile would have different velocity ?

 

[haruspex]

all the box , object and carpet are subjected to the same velocity , right ? this is not same as the situation in the water , which at different point , the partcile would have different velocity ?

As I framed it, the object slides on the box and the box slides on the carpet, so the three are at different velocities.

 

[Chestermiller]

can someone try to explain why the shear stress force at down and above in diagram 8-11 is opposite in direction ? why in diagram 8-12 , they are in the same direction ?

Drawing it this way really stems from the Cauchy Stress Relationship. This relationship has nothing to do with the type of material that is being deformed. Hot John, unfortunately, until you are taught in your courses the details of how to work with 2nd order tensors like the stress tensor, I'm afraid you're just going to have to accept the fact that you need to draw the specific stress components acting on opposite sides of a body with opposite directions. You had no trouble doing this with the pressure stresses on the ends of the body (which are just normal components of the stress tensor). Mathematically, the normal pressure force is done this way for the same basic reason that the shear stresses are drawn with opposite directions on the other faces of the fluid.