1. Sign of the Shear Stress for Flow in a Tube
Concentric layers of fluid for flow in
a tube. Layers in the center move faster
and exert a frictional force on the adjacent
outer layer. The outer layers exert and equal
and opposite force on the adjacent inner layer.
τrz is the shear stress between adjacent layers of fluid in
laminar flow through a tube as well as at the wall. In
determining the sign of the shear stress, you must first
decide what is the surface of interest. Is it the force due to
the stress acting on the wall or a layer of fluid? The unit
normal n to that surface indicates the orientation of this
surface as the positive or negative coordinate direction. If
for example you are looking at the force on an
inner layer by an outer layer of fluid, the normal will be in
the positive r-direction. The shear stress will be negative if
the flow is in the positive z-direction(because the slower layer
causes a negative force on the layer of interest; n >0; F <0) or
positive if the flow is in the negative z-direction(n >0; F >0).
Exercise: Think about the sign of τrz if you are interested in the force
on an outer layer by an inner layer of fluid.