# What is the cause of normal stress on a fluid element?

#### benny_91

In the derivation of Navier Stokes equation there is a term for normal stress acting on the fluid element. While the cause of normal stress is the static pressure which is already present in the equation doesn't this mean that the same force on the fluid element is repeated twice with different names?

Last edited:
Related Aerospace and Astronautics Engineering News on Phys.org

#### DrDu

Pressure is the isotropic part of the stress tensor by definition. It is not the cause of stress.

• Orodruin

#### benny_91

we know that stress is a result of resistance to externally applied force. For example shear stress is a result of resistance to viscous force acting between fluid layers. On the same lines could you please tell me the cause of normal stress on fluid elements?

#### DrDu

we know that stress is a result of resistance to externally applied force. For example shear stress is a result of resistance to viscous force acting between fluid layers. On the same lines could you please tell me the cause of normal stress on fluid elements?
Hm, this may partly be a question of wording, but shear stress is the force per surface acting parallel to a given surface. In the case of liquids I would consider it to be rather the cause of shearing than the result of something. In the case of pressure, pressure is the cause of volume change but admittedly, you may look at it just the other way round. There is a difference between static and dynamic pressure which is due to volume viscosity, the latter being somewhat hard to grasp. My favourite example is the following:
Consider nitrogen dioxide in the production which you, as an aircraft engineer you, will be heavily involved. It is also used as a rocket propellant.
At room temperature it has an interesting property, namely it is partially dimerised:
$\mathrm{2NO_2\leftrightharpoons N_2O_4}$
By the law of mass action, you can change this equilibrium changing the volume available to the gas. Upon compression, more N2O4 will be formed and on decompression more NO2. However, equilibrium is not established instantaneously but takes some time. That means that if you compress this gas rapidly, its pressure will be higher as when you compress it slowly, despite both components behaving nearly like ideal gases. That's an example of dynamic pressure and volume viscosity. It will be present for all real gases as the gas molecules attract due to van der Waals interactions and the establishment of a new distance distribution upon compression takes some time, too.

#### Chestermiller

Mentor
In a Newtonian fluid, for which the Navier Stokes equations apply, the normal stresses involve a combination of a contribution from pressure and a contribution from viscous stresses. Even though, as novices, we learn that viscous shear stresses are caused by velocity gradients normal to the velocity vector, when this is properly (tensorially) generalized to 3 dimensions, we find that the viscous contribution of normal stress is determined by the gradient of the velocity component in the direction of that velocity component. The pressure contribution and the viscous contribution are each expressed as separate terms in the Navier Stokes equations. Even in pure shear flow between parallel plates, if we rotate the coordinate axes, we find that there are viscous normal components of stress in the new coordinate directions. For more details on this, see Chapter 1 of Transport Phenomena by Bird, Stewart, and Lightfoot.

• DrDu

#### DrDu

Namely in the main axis frame of the stress tensor, there are only normal stresses.

#### Chestermiller

Mentor
Namely in the main axis frame of the stress tensor, there are only normal stresses.
And, for a fluid that is deforming, these principal normal stresses include viscous contributions.

#### Chestermiller

Mentor
Namely in the main axis frame of the stress tensor, there are only normal stresses.
The rotated axes I described in post #5 are not necessarily principal directions of stress, and, for these axes, there are still normal stress components in the coordinate directions.

#### benny_91

So you mean to say that normal stresses in moving fluid are caused by viscocity as well as static pressure. Did I get it right?

#### Chestermiller

Mentor
So you mean to say that normal stresses in moving fluid are caused by viscocity as well as static pressure. Did I get it right?
Yes

#### benny_91

So does this mean that part of the static pressure force acting on the fluid element causes it to accelerate while the other part deforms it volumetrically thereby producing normal stress in the fluid element under consideration?

#### Chestermiller

Mentor
So does this mean that part of the static pressure force acting on the fluid element causes it to accelerate while the other part deforms it volumetrically thereby producing normal stress in the fluid element under consideration?
No. The static pressure force contributes to acceleration and deforming, as does the viscous force, since both of these add up to the total force (neglecting gravity). And volumetric deformation is only one of the infinite range of deformations that a fluid can experience. All these deformations contribute to the viscous portion of the stress tensor. Furthermore, incompressible fluids do not experience volumetric deformation at all. Are you aware of the relationship between the velocity gradients, the pressure, and the components of the stress tensor.

#### benny_91

No I am not aware of the relationship. Please could you direct me to the proper book or online material where i can read and understand it?

#### Chestermiller

Mentor
No I am not aware of the relationship. Please could you direct me to the proper book or online material where i can read and understand it?
Transport Phenomena, Bird, Stewart, and Lightfoot

#### Chestermiller

Mentor
$$\sigma_{i,j}=-p\delta_{i,j}+\eta\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right)$$
where $\sigma_{i,j}$ is a component of the stress tensor, p is the pressure, $\delta_{i,j}$ is the isotropic identity tensor, $\eta$ is the fluid viscosity, $v_i$ is a component of the velocity vector, and $x_i$ is a Cartesian spatial coordinate. This is the equation for the components of the stress tensor in terms of the pressure and the velocity gradients, for a Newtonian fluid. Note the separate contributions of the pressure and the viscous term to the overall stress.

Last edited: