# Shear stress direction and the velocity gradient

#### Rahulx084

yes, understood , this is for steady flow right?

#### Chestermiller

Mentor
yes, understood , this is for steady flow right?
Yes. So now what do you get if you divide the equation by $2\pi r \delta r \delta z$ and then take the limit as $\delta r$ and $\delta z$ approach zero?

#### Rahulx084

Sir , I think there should not be $\delta_{z}$ in the third term of force balance in which we have pressure

#### Rahulx084

$\frac 1 r$ $\frac {dr\tau_{rz}} {dr}$ + $\frac {dp} {dz}$ , this is what I'm getting sir .

#### Chestermiller

Mentor
$\frac 1 r$ $\frac {dr\tau_{rz}} {dr}$ + $\frac {dp} {dz}$ , this is what I'm getting sir .
Excellent. So now you have some actual experience at applying the Cauchy Stress Relationship to a problem. Does that help answering your questions?

#### Rahulx084

sir but how to get the term in #19 ? and sir one more doubt If we have a cubical element how we are gonna write the term #19 ? Is it going to remain the same??

#### Chestermiller

Mentor
sir but how to get the term in #19 ? and sir one more doubt If we have a cubical element how we are gonna write the term #19 ? Is it going to remain the same??
The term in #19 follows from the equations for the stress tensor components for a viscous Newtonian fluid. Are you familiar with these equations?

When you have a cubical element (or rectangular parallelepiped), you use the stress tensor in component form for Cartesian coordinates.

#### Rahulx084

I think I'm not familiar with that. Actually sir this isn't in my course but I'm learning it because it seems so interesting plus your great explanations . Can you provide me with only the results of cubical element or if there is any source where I can find it so that I can look after , or maybe you tell me sir.

#### Chestermiller

Mentor
I think I'm not familiar with that. Actually sir this isn't in my course but I'm learning it because it seems so interesting plus your great explanations . Can you provide me with only the results of cubical element or if there is any source where I can find it so that I can look after , or maybe you tell me sir.
See page 29 of http://web.mit.edu/2.25/www/pdf/viscous_flow_eqn.pdf

#### Rahulx084

can you just give me the result for stress tensor of cubical and parallelopiped one ? It would be so nice of you . Thanks

#### Chestermiller

Mentor
can you just give me the result for stress tensor of cubical and parallelopiped one ? It would be so nice of you . Thanks
See Eqns. 43 of that same reference.

"Shear stress direction and the velocity gradient"

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