# Viscosity

The viscosity (μ, from the Latin viscum) is that property of matter, defined as that physical quantity, which is found mainly in the phenomena of transport of a fluid, that describes a fluid’s resistance to flow, or more specifically, is the measure of the resistance of a fluid to gradual deformation by shear stress (τ) or tensile stress.

To quantitatively express the viscosity, consider a liquid at rest contained within a container and suppose to apply a force tangentially to the free surface of the liquid: the superficial layer of the liquid will begin to move with a certain velocity (denoted by $$u$$) transmitting its movement to the underlying layers which will begin to move with gradually decreasing speeds.

Each layer flows with respect to the underlying and overlying one at a certain speed. Between each layer and the adjacent one, friction occurs which interferes with their motion, in a completely analogous way to the sliding friction that is exerted between the surfaces in contact between two bodies. The viscous friction force is exerted on both layers of fluid, in accordance with the action-reaction principle of dynamics. If the two layers slide in the same direction with different speeds, then the frictional force will slow down the faster layer and accelerate the slower one; the internal friction thus tends to uniform the speed value between the various fluid layers.

Fluids resist the relative motion of immersed objects through them as well as to the motion of layers with differing velocities within them. From the microscopic point of view, the viscosity is a property dependent on the amount of internal cohesion forces of the fluid, which are more or less relevant depending on its type and temperature. In particular, in liquids, the viscosity decreases as the temperature increases, whereas in gases it increases (in isochoric conditions, i.e., maintaining the volume of the gas unchanged during the temperature change).

## Kinematic viscosity

Kinematic viscosity describes the behavioral properties of fluids. This physical quantity completes the general framework for the study of the motion field of a moving fluid, as it introduces the fundamental characteristic common to all systems having mass: inertia.

In the definition of dynamic viscosity, we have seen how a fluid subjected to tangential stress prevents the free flow of the various layers of fluid through the braking action of the internal friction to the fluid, absolutely disregarding the density and therefore the mass.

It is clear that this is not sufficient to make a balance in terms of slowing down that cannot be motivated solely by the action of friction, given that the fluid is endowed with mass. So comparing the effect of density on the effect of viscosity, I derive the physical quantity kinematic viscosity:

$\nu=\dfrac{\mu}{\rho}$

which expresses how much motion or non-motion can be transmitted within the fluid. Therefore the fluid brakes the effort not because it is very “viscous dynamically” but because it is very “viscous kinematically.” The SI unit of kinematic viscosity is square meter per second (m2/s).

## Dynamic viscosity

The thermophysical properties of the flow field of fluids are adequately described by the so-called dynamic viscosity. This property provides indications on the intermolecular binding status of the fluid, which also appears to be temperature dependent; indeed:

• in liquids as the temperature increases (and therefore the thermal agitation of the molecules) the dynamic viscosity decreases, as the bonds between the atoms tend to flake leaving the particles freer to “wander”;
• the gaseous, when the temperature increases, have opposite behavior to that of the liquids, that is, the dynamic viscosity tends to increase due to the rise in the probability of collision between the molecules (much more free to move and with higher kinetic energy due to thermal agitation). This leads to greater interactions, and therefore it is as if there were “virtual bonds” that contribute, precisely, to the increase in dynamic viscosity.

Therefore, it is possible to state that the dynamic viscosity describes how the fluid reacts to external action. If we interpose a fluid between two parallel flat plates at a certain distance $$\Delta y$$ and let the upper plate slide with relative motion (holding the lower one), the tangential force applied to it is directly proportional to the relative speed $$\Delta u$$ between the two plates.

From experimental facts, it has been found that the velocity profile varies linearly from the $$u=0$$ value in $$y=0$$ to the $$u=\Delta u$$ value in $$y=\Delta y$$, since each layer of fluid parallel to the moving slab will have interactions both with the one that precedes it and with the one that follows it.

Starting from the first mating layer with the plate and moving solidly with it, this, due to the molecular interactions with the adjacent layer of fluid, will tend to drag it, and the latter on the other hand will tend to brake it and in turn, drag another layer of underlying fluid. This phenomenon is repeated for all the successive layers of the fluid until it reaches zero where the speed of the last layer of fluid will be zero (because the speed of the plate is null). Therefore it is possible to interpret this linear trend as the speed gradient in the direction $$y$$:

$\dfrac{du}{dy}$

The presence of a velocity gradient is determined by the dynamic viscosity of the fluid which shows its reluctance to deform when subjected to a tangential effort. Therefore the dynamic viscosity is the physical quantity of proportionality between cause τ = tangential effort = shearing stress in fluid) and effect (speed gradient):

$\tau=\mu\dfrac{du}{dy}$

From the above considerations, it is possible to imagine, therefore, that the tangential effort is gradually dissipated by the action of friction between the layers of fluid in the direction of the decreasing gradient. The SI unit of dynamic viscosity is the pascal-second (Pa·s), or equivalently kilogram per meter per second (kg·m−1·s−1).

## Viscosity and vortexes

Viscous friction is responsible for the rotation of fluids within a container. If we want to rotate closed water in a bottle, so as to form a vortex, what we do is to rotate the bottle. The rotary motion of the container is transmitted to the fluid thanks to the viscous friction.

When the rotary motion begins, the layers that adhere to the walls of the bottle are the first ones which, under the effect of the frictional force, are dragged and put into rotation. These in turn drag the immediately innermost layers of water with them, which thus start to move and so on, always proceeding inwards, gradually moving away from the walls of the container. The layers of water closest to the rotation axis will start last and those closest to the walls of the bottle have already gained good speed. At the end of the process, however, all fluid layers, regardless of their position, will have reached the same angular velocity.

## References

1. Image-based on https://commons.wikimedia.org/wiki/File:Laminar_shear.svg
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