im having difficulty understanding the criteria of signs settled down by BSL transport phenomena in the derivation of Stokes law in chapter 2

at the page 59 it takes the molecular momentum-flux tensor negative when integrating the normal force on the solid of the sphere

but at page 60 when the book makes the integration of the tangential force they take the molecular -flux tensor as positive why?

in the first case the fluid has higher $r$ value than the sphere so the tensor is negative but in the second case the fluid also has higher $r$ value but here the tensor is positive wtf, BSL just does not explain this at all

  • $\begingroup$ Consider to spell out acronyms. $\endgroup$ – Qmechanic Jun 27 at 8:03

The stress vector exerted by the fluid on the surface of the sphere at r = R, according to their notation, is given by $(-p-\tau_{rr})\mathbf{i_r}-\tau_{r\theta}\mathbf{i_{\theta}}$. So there is no problem with what they do on page 59 with respect to the normal component of the force.

But, on page 60, they realize that the tangential force is going to be oriented in the $-\theta$ direction, so they dot the stress vector with a unit vector in this direction, to obtain $[(-p-\tau_{rr})\mathbf{i_r}-\tau_{r\theta}\mathbf{i_{\theta}}]\centerdot (-\mathbf{i_{\theta}})=+\tau_{r\theta}$

In my judgment, they play it a little fast and loose with the mathematics, but what they did is ultimately correct.

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  • $\begingroup$ And why they did not dot product the stress vector at r=R With the -i r unit vector isn't supposed that the normal force is pointing in the negative radial direction? Or it is actually included in the sign convention? $\endgroup$ – Esteban Soto Montijo Jul 1 at 23:20
  • $\begingroup$ They did dot the stress tensor with -ir to get the stress vector exerted by the fluid on the surface (as I have written it). $\endgroup$ – Chet Miller Jul 2 at 0:37
  • $\begingroup$ If you dot that vector with -ir I get +p since Tau rr at R is 0 $\endgroup$ – Esteban Soto Montijo Jul 2 at 0:41
  • $\begingroup$ I didn’t say you dot the stress vector. I said you dot the stress tensor. The stress vector then already automatically has the correct direction $\endgroup$ – Chet Miller Jul 2 at 1:54
  • $\begingroup$ No, it's $P(\mathbf{-i_r})$. So the pressure of the fluid is acting in the minus r direction on the surface. You should be aware that the tensor sign convention used in this book (with positive tensor components representing compression) is the opposite of that used in most mechanics developments. The version of the Cauchy stress relationship for such a framework is: The stress vector exerted by region A on region B at the interface between them is equal to the stress tensor dotted with a unit normal drawn from A to B. $\endgroup$ – Chet Miller Jul 2 at 19:29

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