Wall shear stress in the Navier-Stokes equation: A commentary

Abstract

[...]the use of Cartesian differential operator ∇=∂∂x+∂∂y+∂∂z and Kronecker's delta δij is valid. Normal stress components (scalar fields, similar to pressure) can be expressed as functions of the second viscosity coefficient λ=−23μ, as hypothesized by Stokes [4,5], convective derivative (∇⋅V) and normal velocity gradient (∂ui∂xi), as: inx-direction:τxx=λ∇⋅V+2μ∂u∂xiny-direction:τyy=λ∇⋅V+2μ∂v∂yinz-direction:τzz=λ∇⋅V+2μ∂w∂z, The shear stress expressions, derived by Navier and Stokes, assumed symmetric tensor, thus equating each two reciprocating shear stress components, which was necessary to maintain the continuum and equilibrium assumptions of the equation, such that:(4) τyx=τxy=μ(∂v∂x+∂u∂y)(5) τxz=τzx=μ(∂u∂z+∂w∂x)(6) τzy=τyz=μ(∂w∂y+∂v∂z) Using Einstein summation convention [6], we write: τij=τji=μ(∂ui∂xj+∂uj∂xi). [...]the direction of action is neutralized in the symmetric assumption, and the shear stress on each plane has become a scalar field [7].