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The Paradox of Lost Complementary Shear Stress

In engineering fluid mechanics texts it is common to furnish a mechanistic explanation of viscous/turbulent shear stress in a horizontal fluid shear flow in terms of the momentum transport (molecular/turbulent) between two adjacent horizontal fluid layers which have, as a result of the shear in the flow, slightly differing speeds (see Fig. 1). While such an explanation provides a ready insight into the cause of the apparent shear stress on horizontal planes which does exist in such a flow, the same mechanistic explanation when applied to vertical planes leads to the conclusion that no shear stress exists on these planes because the fluid has no velocity and therefore no shear in the vertical direction. Yet Newton's law applied to a rectangular fluid element having horizontal and vertical plane sides leads to the contradictory conclusion that a complementary shear stress must exist on the vertical planes. Herein lies an apparent paradox. The explanation, while not far to seek, is not at all obvious and is rarely to be found in engineering texts, so it is the purpose of this article to make good this general omission. For convenience the explanation below is given in terms of the molecular transport of