How Does Shear Stress Direction Relate to Fluid Flow on an Incline?

[RagincajunLA]

Hey guys, I'm trying to learn fluid mechanics while at an internship and i just have a quick question. Let's say i have a fluid flowing down and incline plane with a quadratic velocity profile. Which direction does the shear stress (tau) act?
I know the formula for shear stress is tau = mu*(du/dy) where u is the velocity profile and mu is the viscosity. In the problem I am working on, the derivative of the velocity profile (du/dy) points down the incline, does that mean shear stress also points down the incline?

the reason i ask such a basic question is that the hint on the problem says the shear stress must counteract the parallel weight of the fluid. the weight points down the slope so this is saying the shear stress points up the slope, but the formula above is saying it also points down the slope

 

[W R-P]

I think your diagram is similar to fig. 6.6 http://www.creatis.insa-lyon.fr/~dsarrut/bib/others/phys/www.mas.ncl.ac.uk/%257Esbrooks/book/nish.mit.edu/2006/Textbook/Nodes/chap06/node12.html

The layer of fluid DIRECTLY in contact with the surface is at rest( due to boundary layer conditions).This layer will in turn 'pull back' or 'oppose' the motion,of layer of fluid directly above it,similar to friction.These shear forces act between the successive layers of the fluid. So if we view the fluid in it's entirety,the shear force acting on it is up the slope since it moves downslope.
I hope this is clear enough.

 

[FourierFaux]

The boundary conditions for a fluid traveling down the slope can be described as a fluid will be at rest at the interface between the fluid and the boundary. (Incidentally if the boundary is moving, the fluid is still at rest with respect to the boundary). I believe this is correct (If I'm wrong; correct me)

Here is an interesting video of a lava flow (although you could do this same thing with some warm maple syrup with black pepper liberally sprinkled onto it's surface):



Watch carefully in the first five seconds how the top surface of the lava moves with respect to the rest of the 'fluid'. You can almost see the effects of the boundary conditions in the first five seconds of this video.

 

[boneh3ad]

Is is up the plane. You forgot the minus sign in your formula.

 

[W R-P]

If you really want to go into lava flow, http://www.itg.cam.ac.uk/people/heh/Paper221.pdf
The top layer solidifies due to cooling,creating a tube like surface for the rest of the lava to flow,that's why the top layer(dark) moves slower than the rest(orange).

 

[FourierFaux]

Interesting paper, yes. And interesting idea from you, W R-P.

*In most materials (all?) the viscosity of a fluid varies with Temperature. In the system of lava flow, the differences in temperature within the system are great enough to allow phase transitions between states.

It looks like s/he used order of magnitude calculations... very highly approximated and simplified. (Very complex system, I can't say I blame them) If one wanted to attempt to simulate this kind of a system... it would be interesting. Would it be the kind of thing that would produce chaotic results (i.e. results whose final behavior can be radically different under varying initial conditions).

It seems that one would need several things to properly simulate this system...

1) A proper model for phase transitions within the lava.
2) A proper model for viscosity, effusion and surface tension as a function of Temperature.
3) A method to account for the different chemical compositions of the lava (I doubt very seriously that it's a homogeneous material)

This is not a complete list, of course. To attack the problem this way seems like it would be very frustrating, and you'd still only be producing approximations for such systems anyways. Does the ground behave like putty when a 3000 degree multi-ton mass flows over it? It seems even outlining the boundary conditions would be hair-grayingly complex.

Of course, this could fun...

 

[FourierFaux]

*edit: idea came from author of paper, not W R-P originally.

 

[FourierFaux]

http://www.chust.org/lava-flows/lava-flow-simulation.pdf

/\____ Here's a comparison of methods using cellular automata algorithms to attack this problem.