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Something has recently begun to trouble me, specifically, the presence of the viscous shear term - ie $\mu \nabla^2 \mathbf{u}$ - in the Navier-Stokes equations.

I'm somewhat troubled by the presence of this term because I'm not sure that shear forces can cause translational motion of the center of mass of a continuum body. If shear forces are, in fact, unable to cause translational motion, then they contribute nothing to the acceleration of a fluid element's center of mass, right?

Perhaps I should be asking a more fundamental question - does $\frac{D\mathbf{u}}{Dt}$ in the Navier-Stokes equations represent the accleration of the center of mass of a fluid element? If so, it seems like my objection is well-founded. If not, what are we determining the acceleration of, exactly?

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  • $\begingroup$ How would one define the center of mass of an infinitesimal volume? $\endgroup$
    – Kyle Kanos
    Commented Nov 13 at 22:32
  • $\begingroup$ Presumably the same way one would define the center of mass of a body of non-"infinitesimal" size. I suppose what you're saying is that in the limit of very small dimensions, there is no distinction between what we might term the "center of mass" of the fluid element and any other point within the fluid element? $\endgroup$
    – Zachary Candelaria
    Commented Nov 13 at 22:36
  • $\begingroup$ What kind of force wouldn't affect the motion? $\endgroup$
    – D. Halsey
    Commented Nov 13 at 23:25
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    $\begingroup$ Please see this answer to Are two forces required to cause shearing?. (Quick summary: A single shear load definitely drives translation.) Does this resolve the issue? $\endgroup$
    – Chemomechanics
    Commented Nov 13 at 23:25
  • $\begingroup$ @Chemomechanics it does indeed! Thanks for sharing $\endgroup$
    – Zachary Candelaria
    Commented Nov 14 at 1:18

2 Answers 2

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An unbalanced shear load on an element causes both translation and rotation; by the parallel-axis theorem, the load can be replaced by a force acting on the center of the element plus a moment around that center:

(Please ignore the "✖" annotation; I made the animation to contrast with an equilibrium state in the context of elasticity:

Here, fully four loads are required to prevent both translation and rotation. The important thing to note in the context of the current question is that the balancing doesn't come automatically.)

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A constant/homogeneous shear stress can't, but a gradient of a shear stress can. That's why the viscous force term has a $\nabla^2$, not a $\nabla$.

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