Newton's law of viscosity in 3D, used to derive Navier-Stoke

[cantgetaname]

I'm trying to understand how the Navier-Stokes equations are derived and having trouble understanding how the strain rates are related to shear stresses in three dimensions, what a lot of texts refer to as the 'Stokes relations'.

http://soliton.ae.gatech.edu/labs/windtunl/classes/hispd/hispd06/ns_eqns.html

It's no longer the simple stress=viscosity*velocity gradient of the 1d case, but more complicated than that and none of the books I could get my hands on or the internet could show me how these relations are derived. They are just mentioned there as if they are obvious but I can't see how.

And what does the normal components in viscous shear stress tensor (τ_xx, τ_yy, τ_zz) even mean? How can you have viscous shear stress normal to a surface.

 

[Chestermiller]

I'm trying to understand how the Navier-Stokes equations are derived and having trouble understanding how the strain rates are related to shear stresses in three dimensions, what a lot of texts refer to as the 'Stokes relations'.

http://soliton.ae.gatech.edu/labs/windtunl/classes/hispd/hispd06/ns_eqns.html

It's no longer the simple stress=viscosity*velocity gradient of the 1d case, but more complicated than that and none of the books I could get my hands on or the internet could show me how these relations are derived. They are just mentioned there as if they are obvious but I can't see how.

And what does the normal components in viscous shear stress tensor (τ_xx, τ_yy, τ_zz) even mean? How can you have viscous shear stress normal to a surface.

An outline of the derivation you are looking for is presented in Section 1.2 of Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena.

In terms of your question regarding viscous "shear" stresses normal to a surface, the answer is that these are not shear stresses (at least not in the a coordinate system aligned with the surface). These are normal stresses resulting from the tensile and compressive deformations that are occurring along the coordinate directions. Are you familiar with the 3D version of Hooke's law for solids? There you have tensile normal strains, and you have corresponding tensile normal stresses. This is the analogous thing for a fluid. For example, if you have a rod of a very viscous fluid and you apply tension to the rod, it will experience a rate of deformation in the axial direction. The force you are applying divided by the cross section area of the rod is the normal stress. For an incompressible viscous fluid, the stress is 3 times the viscosity times the rate of extension.

Chet

 

[boneh3ad]

Are you familiar with continuum mechanics? A Newtonian fluid is actually one where the viscous stresses are directly proportional to the rate of strain. If you are familiar with continuum mechanics and tensors, the above equations are easily derived.

 

[cantgetaname]

An outline of the derivation you are looking for is presented in Section 1.2 of Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena.

In terms of your question regarding viscous "shear" stresses normal to a surface, the answer is that these are not shear stresses (at least not in the a coordinate system aligned with the surface). These are normal stresses resulting from the tensile and compressive deformations that are occurring along the coordinate directions. Are you familiar with the 3D version of Hooke's law for solids? There you have tensile normal strains, and you have corresponding tensile normal stresses. This is the analogous thing for a fluid. For example, if you have a rod of a very viscous fluid and you apply tension to the rod, it will experience a rate of deformation in the axial direction. The force you are applying divided by the cross section area of the rod is the normal stress. For an incompressible viscous fluid, the stress is 3 times the viscosity times the rate of extension.

Chet

That book does answer my question (although at present a lot of it goes over my head). Thanks!