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The formula to calculate average shear stress τ or force per unit area is: [1] =, where F is the force applied and A is the cross-sectional area.. The area involved corresponds to the material face parallel to the applied force vector, i.e., with surface normal vector perpendicular to the force.
For a Newtonian fluid wall, shear stress (τ w) can be related to shear rate by = ˙ where μ is the dynamic viscosity of the fluid. For non-Newtonian fluids, there are different constitutive laws depending on the fluid, which relates the stress tensor to the shear rate tensor.
The wall shear stress has the SI unit of pascals (Pa). ... However, since the advent of the calculator, ease of calculation is no longer a major issue, ...
It is defined as the ratio between the local shear stress and the local flow kinetic energy density: [1] [2] = where f is the local Fanning friction factor (dimensionless); τ is the local shear stress (units of pascals (Pa) = kg/m 2, or pounds per square foot (psf) = lbm/ft 2);
The logarithmic law of the wall is a self similar solution for the mean velocity parallel to the wall, and is valid for flows at high Reynolds numbers — in an overlap region with approximately constant shear stress and far enough from the wall for (direct) viscous effects to be negligible: [3]
Shear velocity is linked to the Darcy friction factor by equating wall shear stress, giving: = where f D is the friction factor. [1] Shear velocity can also be defined in terms of the local velocity and shear stress fields (as opposed to whole-channel values, as given above).
This is only the average stress, actual stress distribution is not uniform. In real world applications, this equation only gives an approximation and the maximum shear stress would be higher. Stress is not often equally distributed across a part so the shear strength would need to be higher to account for the estimate. [2]
In the hydrodynamic entrance region, the wall shear stress, , is highest at the pipe inlet, where the boundary layer thickness is the smallest. Shear stress decreases along the flow direction. [ 6 ] That is why the pressure drop is highest in the entrance region of a pipe, which increases the average friction factor for the whole pipe.