While the viscous stresses are generated by physical phenomena that depend strongly on the nature of the medium, the viscous stress tensor is only a description the local momentary forces between adjacent parcels of the material, and not a property of the material.
22.
In the Newtonian fluid model, the relationship is by definition a linear map, described by a viscosity tensor that, multiplied by the strain rate tensor ( which is the gradient of the flow's velocity ), gives the viscous stress tensor.
23.
Where } } is the material derivative operator, also denoted by in capital notation as } }, is the flow velocity, is the local fluid density, is the local pressure, is the viscous stress tensor and represents the sum of the external body forces.
24.
At this outer boundary distant from the airfoil, the velocity and pressure are well represented by the velocity and pressure associated with a uniform flow plus a vortex, and viscous stress is negligible, so that the only force that must be integrated over the outer boundary is the pressure.
25.
However, when the tangential stress is sufficiently vigorous compared to ? / D, the surface can be deformed into a steady tapering shape, which allows the continuous and smooth acceleration of the liquid under the combined actions of the pressure drop ?P and the tangential viscous stress ?s on the liquid surface.
26.
In particular, the tensor equation "'c? "'} } relating elastic stresses to strains is entirely similar to the equation "'?? & # x307; "'} } relating the viscous stress tensor and the strain rate tensor in flows of dynamical stresses ( related to the " rate " of deformation ).
27.
This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent stress \ left (-\ rho \ overline { u _ i ^ \ prime u _ j ^ \ prime } \ right ) owing to the fluctuating velocity field, generally referred to as the Reynolds stress.
28.
Either way, the strain rate tensor expresses the rate at which the mean velocity changes in the medium as one moves away from the point except for the changes due to rotation of the medium about as a rigid body, which do not change the relative distances of the particles and only contribute to the rotational part of the viscous stress via the rotation of the individual particles themselves . ( These changes comprise the vorticity of the flow, which is the curl ( rotational ) of the velocity; which is also the antisymmetric part of the velocity gradient .)
