Navier stokes equeations, shear term

In summary, the conversation discusses the derivation of the Navier-Stokes equations and specifically focuses on the term d/dx(tau_xx), which is a shear stress in the x-direction that is acting on a surface with an x-normal. The participants of the conversation have different understandings of this term and its physical meaning. Some suggest that it is related to the viscosity of the fluid, while others mention the theory of Reynolds stress. Ultimately, there is confusion and a need for further explanation and clarification.
  • #1
navalstudent
6
0
Hey!, I was repeating for myself a course I had from a earlier year, fluid mechanics. I looked at the derivation of the navier stokes equations, and there is one term that does not give meaning to me.


Take a look at the x-momentum equation here:
http://www.grc.nasa.gov/WWW/K-12/airplane/nseqs.html

The term I don't get is the d/dx(tau_xx). I mean what does this term mean physically? It is a shear stress in the x-direction that is acting on the the surface wit an x-normal? Tau_xy and tau_xz is easy to understand from ordinary mechanics, but not tau_xx. I tought only the term -d/dx(P) would give a normal stress in the x-direction.

So can someone please explain to me how we can have a shear stress in the x-direction acting on the x surface(y-z-plane).
 
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  • #2
That is an odd way to write the equations; I thought the diagonal components of the stress tensor (tau_xx, tau_yy, tau_zz) end up in the pressure term.
 
  • #3
Andy Resnick said:
That is an odd way to write the equations; I thought the diagonal components of the stress tensor (tau_xx, tau_yy, tau_zz) end up in the pressure term.

Hello Andy Resnick.

That is what I also tought at first, but the fact is that these terms are not the classical preassure terms. And I still have not been able to find a physical explanation for them.

Could one say that since the fluid has viscosity, we will have a "glue-effect", so that the normal strains contains preassure terms, and since the fluid is sticky, the viscosity will "drag" the fluid forward?(if we have a velocity gradient in the normal-direction). Note: I have not talked about shear-stresses here.
 
  • #4
here's what I dug up:

keywords: Turbulence modeling, dynamic viscocity (aka absolute viscocity), reynold's stress

wiki said:
It should also be noted that the theory of the Reynolds stress is quite analogous to the kinetic theory of gases, and indeed the stress tensor in a fluid at a point may be seen to be the ensemble average of the stress due to the thermal velocities of molecules at a given point in a fluid. Thus, by analogy, the Reynolds stress is sometimes thought of as consisting of an isotropic pressure part, termed the turbulent pressure, and an off-diagonal part which may be thought of as an effective turbulent viscosity.

http://en.wikipedia.org/wiki/Reynolds_stress

Dynamic Viscocity:
http://www.engineeringtoolbox.com/dynamic-absolute-kinematic-viscosity-d_412.html
 
  • #5
I tried parsing my go-to reference for this stuff (Non-Linear field theories of Mechanics, Handbuch der Physics vol III/3) and was promptly confused.

They do write down a general constitutive relation for fluids T + p1, and there is no restriction on the Cauchy stress tensor, but by the time they get to Korteweg's theory, I got lost in a maze of tensor representation theorems.

I wish the GRC site gave a little more information, instead of just tossing out a formula.
 

1. What are Navier-Stokes equations?

Navier-Stokes equations are a set of partial differential equations that describe the motion of a fluid in space. These equations were developed by French mathematician Claude-Louis Navier and Irish mathematician George Gabriel Stokes in the 19th century.

2. What is the shear term in Navier-Stokes equations?

The shear term in Navier-Stokes equations represents the rate of deformation of a fluid due to forces acting on it. It is a measure of the amount of shear stress, or force per unit area, that is applied to the fluid.

3. Why is the shear term important in Navier-Stokes equations?

The shear term is important because it accounts for the internal friction of a fluid, which is essential for accurately predicting the behavior of fluids in real-world applications. It plays a crucial role in understanding fluid dynamics and is used in various fields such as aerodynamics, hydrodynamics, and weather forecasting.

4. How is the shear term calculated in Navier-Stokes equations?

The shear term is calculated by taking the gradient of the velocity field, which represents the change in velocity over a given distance. This gradient is then multiplied by the viscosity of the fluid, which is a measure of its resistance to deformation.

5. Can the shear term be neglected in Navier-Stokes equations?

No, the shear term cannot be neglected in Navier-Stokes equations. While it may be small in some situations, it is still a crucial component of the equations and neglecting it can lead to inaccurate predictions of fluid behavior. In certain cases, such as low Reynolds number flows, the shear term may be negligible but it is always important to consider it for a more accurate understanding of fluid dynamics.

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