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# Navier-Stokes model with viscous strength

In the laminar mode interactions among molecules generate friction between layers of water that slide with respect to each other. This friction triggers the shear stress, which is traditionally presumed to be linearly proportional to the velocity gradient. The proportionality coefficient characterizes the viscosity of water. Remarkably, the standard Navier-Stokes model surmises that materials never fail – the transition to turbulence can only be triggered by some kinematic instability of the flow. This premise is probably the reason why the Navier-Stokes theory fails to explain the so-called subcritical transition to turbulence with the help of the linear instability analysis. When linear instability analysis fails, nonlinear instability analysis can be resorted to, but, despite the occasional uses of this approach, it is intrinsically biased to require finite flow perturbations which do not necessarily exist.

In the present work we relax the traditional restriction on the perfectly intact material and introduce the parameter of *fluid viscous strength*, which enforces the breakdown of internal friction. We develop a generalized Navier-Stokes constitutive model which unites two modes of the Newtonian flow: inviscid ideal and linearly viscous. We use the new model to analyze the Couette flow between two parallel plates to find that the lateral infinitesimal perturbations can destabilize the laminar flow. Furthermore, we use the results of the recent experiments on the onset of turbulence in pipe flow to calibrate the viscous strength of water. Specifically, we find that the maximal shear stress that water can sustain in the laminar flow is about one Pascal. We note also that the introduction of the fluid strength suppresses pathological stress singularities typical of the traditional Navier-Stokes theory and uncovers new prospects in the explanation of the remarkable phenomenon of the delay of the transition to turbulence due to an addition of a small amount of long polymer molecules to water.

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## Comments

## Navier-Stokes model with viscous strength

This is a very interesting paper. I

reached a similar conclusion in a recent book entitled “Thermodynamic Limit to

the Existence of Inanimate and Living Systems”, in which I

considered the thermodynamic

implications of the principles of thermodynamics and of the fact that

the free energy that a finite system can store isothermally must be

finite. In Ch. 11

of that book I applied that theory to predict the conditions (and, in

particular the Reynolds number) at which the laminar fluid flow of a fluid ceases

to be physically admissible and, thus, must become turbulent. I also

calculated the maximum shear strength of water at 20°C, as can be obtained from

some recent experimental results on plane Couette flow taken from the

literature. The result that I obtained is τmax= 0.114 Pa, which is not

terribly far from the result of about 1 Pa that you obtained from experimental data

on the transition to turbulence in pipe flow.

It seems that our insights on the

problem are akin in many respects, and I think that that it may be useful to

discuss them a little more. The field seems ripe for a quite important

revolution of the kind that occurred some decades ago concerning the limit

strength of elastic structures and their post elastic behaviour.

## Re: Navier-Stokes model with viscous strength

Dear Andreas,

Thanks. I find our conclusions very reasonable . Your number perfectly fits mine because experiments on the transition to turbulence are difficult to do. I would be happy with a viscous strength of water in the range from 0.1 to 100 Pa.

Can you put your chapter here on iMechanica?

Best,

Kosta

## Re: Navier-Stokes model with viscous strength

Hi Kosta,

I asked the publisher but he expressed some reservations about posting the whole Ch.11 in addition to the part of it which is already available from the book site.

Anyway, the calculations to determine the limit shear strength from a plane Couette flow experiment are quite simple. For this experiment, τmax is given by

τmax= η v*/d

where v* is the velocity of the moving plate with respect to the stationary one, d is the distance between the two plates and η is the fluid viscosity.

For water at 20°C, we have that η=10-3[N sec/m2]. If we accept the results reported in:

Tillmark N., Alfredson P.H.: Experiments in plane Couette flow.

J. Fluid Mech.235, 89-102 (1992),we have that the velocity at which the flow starts to become turbulent is v*=72 10-3 [m sec-1], while d=5 [mm]. Then, the above formula gives the value τmax = 14.1 10-3 [Pa], which is reported in Ch.11 of the book. (The value τmax= 0.114 Pa that is reported in my post above is not correctly quoted!)

Regards,

Andreas

## Re: Navier-Stokes

I see. Well, the experiments on the transition to turbulence are difficult. So, probably, there is a range of two-three orders of magnitude where the strength can be found. I used experiments by Avila et al which were very recent - 2011. Which experiments should be trusted is always a dilemma