Viscosity

Viscosity is a measure of the resistance of a fluid to deform under shear stress. It is commonly perceived as "thickness", or resistance to pouring. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, water is "thin", having a lower viscosity, while vegetable oil is "thick" having a higher viscosity. All real fluids (except superfluids) have some resistance to shear stress, but an idealized fluid which has no resistance to shear stress is known as an ideal fluid .

Newton's theory

In general, in any flow, layers move at different velocities and the fluid's "thickness" arises from the shear stress between the layers that ultimately opposes any applied force.

Isaac Newton postulated that, for straight, parallel and uniform flow, the shear stress, τ, between layers is proportional to the velocity gradient, ∂u/∂y, in the direction perpendicular to the layers, in other words, the relative motion of the layers.

\tau=\mu \frac{\partial u}{\partial y}

Here, the constant μ is known as the coefficient of viscosity, viscosity, or dynamic viscosity. Many fluids, such as water and most gases, satisfy Newton's criterion and are known as Newtonian fluids. Non-Newtonian fluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity.

The relationship between the shear stress and the velocity gradient can also be obtained by considering two plates closely spaced apart at a distance t. Assuming that the plates are very large, with a large area A, such that edge effects are neglected and that the lower plate is fixed, let a force F be applied to the upper plate. Incidentally, if this force causes the plate to move, the substance is concluded to be a fluid. The velocity of the moving plate and the top , the applied force is proportional to the area and velocity of the plate and inversely proportional to the distance between the plates. Combining these three relations results in the equation F = μ(AU/t). Where mu is the proportionality factor called the absolute viscosity (with units Pa-s or slugs/s-ft). The equation can be expressed in terms of shear stress; ρ = F/A = μ(U/t). U/t is the rate of angular deformation and can be written as an angular velocity, du/dy. Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.

In many situations, we are concerned with the ratio of the viscous force to the inertial force, the latter characterised by the fluid density ρ. This ratio is characterised by the kinematic viscosity, defined as follows:

\nu = \frac {\mu} {\rho}

James Clerk Maxwell called viscosity fugitive elasticity because of the analogy that elastic deformation opposes shear stress in solids, while in viscous fluids, shear stress is opposed by rate of deformation.

Measurement of viscosity

Viscosity is measured with various types of viscometer, typically at 25°C (standard state). For some fluids, it is a constant over a wide range of shear rates. The fluids without a constant viscosity are called Non-Newtonian fluids.

In paint industries, viscosity is commonly measured with a Zahn cup, in which the efflux time is determined and given to customers. The efflux time can also be converted to kinematic viscosities (cSt) through conversion equations.

Also used in paint, a Stormer viscometer uses load-based rotation in order to determine viscosity. It uses units, Krebs units (KU), unique to this viscometer.

Units

Viscosity (dynamic viscosity): ${\mu}$

The SI physical unit of dynamic viscosity (greek symbol: ${\mu}$ ) is the pascal-second (Pa·s), which is identical to 1 kg·m^-1·s^-1. In France there have been some attempts to establish the poiseuille (Pl) as a name for the Pa·s but without international success. Care must be taken in not confusing the poiseuille with the poise named after the same person!

The cgs physical unit for dynamic viscosity is the poise (P) named after Jean Louis Marie Poiseuille. It is more commonly expressed, particularly in ASTM standards, as centipoise (cP). The centipoise is commonly used because water has a viscosity of 1.0020 cP (at 20 °C; the closeness to one is a convenient coincidence).

1 poise = 100 centipoise = 1 g·cm⁻¹·s⁻¹ = 0.1 Pa·s.

1 centipoise = 1 mPa·s.

Kinematic viscosity: $\nu = {\mu} / {\rho}$

Kinematic viscosity (Greek symbol: ${\nu}$ ) has SI units (m²·s^-1). The cgs physical unit for kinematic viscosity is the stokes (abbreviated S or St), named after George Gabriel Stokes . It is sometimes expressed in terms of centistokes (cS or cSt). In U.S. usage, stoke is sometimes used as the singular form.

1 stokes = 100 centistokes = 1 cm²·s⁻¹ = 0.0001 m²·s⁻¹.

Conversion between kinematic and dynamic viscosity, then, is given by $\nu \rho = \mu$ , and so if ν=1 St then

μ=νρ=0.1 kg·m⁻¹s⁻¹·(ρ/(g/cm³))=0.1 poise·(ρ/(g/cm³)). ^*

Molecular origins

The viscosity of a system is determined by how molecules constituting the system interact. There are no simple but correct expressions for the viscosity of a fluid. The simplest exact expressions are the Green-Kubo relations for the linear shear viscosity or the Transient Time Correlation Function expressions derived by Evans and Morriss in 1985. Although these expressions are each exact in order to calculate the viscosity of a dense fluid, using these relations requires the use of molecular dynamics computer simulation.

Gases

Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. The kinetic theory of gases allows accurate prediction of the behaviour of gaseous viscosity, in particular that, within the regime where the theory is applicable:

Viscosity is independent of pressure; and
Viscosity increases as temperature increases.

Liquids

In liquids, the additional forces between molecules become important. This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial. Thus, in liquids:

Viscosity is independent of pressure (except at very high pressure); and
Viscosity tends to fall as temperature increases (for example, water viscosity goes from 1.79 cP to 0.28 cP in the temperature range from 0°C to 100°C); see temperature dependence of liquid viscosity for more details.

The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases.

Viscosity of some common materials

Some dynamic viscosities of Newtonian fluids are listed below:

Gases (at 0 °C):

	viscosity (Pa·s)
hydrogen	8.4 × 10^-6
air	17.4 × 10^-6
xenon	21.2 × 10^-6

Liquids (at 25 °C):

	viscosity (Pa·s)
ethanol	^a 1.074 × 10^-3
acetone	^a 0.306 × 10^-3
methanol	^a 0.544 × 10^-3
propanol	^a 1.945 × 10^-3
benzene	^a 0.604 × 10^-3
water	^a 0.890 × 10^-3
nitrobenzene	^a 1.863 × 10^-3
mercury	^a 1.526 × 10^-3
sulfuric acid	^a 24.2 × 10^-3
glycerol	^a 934 × 10^-3
olive oil	81 × 10^-3
castor oil	0.985
molten polymers	10³
pitch	2.3 × 10⁸
glass	10⁴⁰

^a Data from CRC Handbook of Chemistry and Physics, 73^rd edition, 1992-1993.

Fluids with variable compositions, such as honey, can have a wide range of viscosities.

A more complete table can be found here

Can solids have a viscosity?

Amorphous solids, such as glass, may be considered to have viscosity, on the basis that all solids flow, to some small extent, in response to shear stress. This has led some to the view that solids are simply liquids with a very high viscosity, typically greater than 10¹² Pa·s. This position is often adopted by supporters of the widely held misconception that glass flow can be observed in old buildings.

However, others argue that solids are, in general, elastic for small stresses while fluids are not. Even if solids flow at higher stresses, they are characterized by their low-stress behavior. Viscosity may be an appropriate characteristic for solids in a plastic regime. The situation becomes somewhat confused as the term viscosity is sometimes used for solid materials, for example Maxwell materials, to describe the relationship between stress and the rate of change of strain, rather than rate of shear.

These distinctions may be largely resolved by considering the constitutive equations of the material in question, which take into account both its viscous and elastic behaviors. Materials for which both their viscosity and their elasticity are important in a particular range of deformation and deformation rate are called viscoelastic. In geology, earth materials that exhibit viscous deformation at least three times greater than their elastic deformation are sometimes called rheids.

One example of solids flowing which has been observed since 1930 is the Pitch drop experiment.

Bulk viscosity

The trace of the stress tensor is often identified with the negative of one third of the thermodynamic pressure, which only depends upon the equilibrium state potentials like temperature and density. However, in general, the trace of the stress tensor is the sum of thermodynamic pressure contribution plus another contribution which is proportional to the divergence of the velocity field. This constant of proportionality is called the bulk viscosity.

Eddy viscosity

In the study of turbulence in fluids, a common practical strategy for calculation is to ignore the small-scale vortices (or eddies) in the motion and to calculate a large-scale motion with an eddy viscosity that characterizes the transport and dissipation of energy in the smaller-scale flow. Values of eddy viscosity used in modeling ocean circulation may be from 5x10⁴ to 10⁶ Pa·s depending upon the resolution of the numerical grid.

Fluidity

The reciprocal of viscosity is fluidity, usually symbolised by φ (=1/μ) or F (=1/η), depending on the convention used, measured in reciprocal poise (cm·s·g^-1), sometimes called the rhe. Fluidity is seldom used in engineering practice.

The concept of fluidity can be used to determine the viscosity of an ideal solution. For two components (a and b), the fluidity of a solution of a and b is:

F ≈ + [χ(b)F(b)

which is only slightly simpler than the equivalent equation in terms of viscosity:

η ≈ 1/+χ(b)/η(b)

Where χ = mole fraction of a or b and η = the viscosity of pure a or b

The linear viscous stress tensor

(see Hooke's law and strain tensor for an analogous development for linearly elastic materials.)

Viscous forces in a fluid are a function of the rate at which the fluid velocity is changing over distance. The velocity at any point $\mathbf{r}$ is specified by the velocity field $\mathbf{v}(\mathbf{r})$ . The velocity at a small distance $d\mathbf{r}$ from point $\mathbf{r}$ may be written as a Taylor series:

\mathbf{v}(\mathbf{r}+d\mathbf{r}) = \mathbf{v}(\mathbf{r})+\frac{d\mathbf{v}}{d\mathbf{r}}d\mathbf{r}+\ldots

where $d\mathbf{v}/d\mathbf{r}$ is shorthand for the dyadic product of the del operator and the velocity:

\frac{d\mathbf{v}}{d\mathbf{r}} = \begin{bmatrix}

\frac{\partial v_x}{\partial x}&\frac{\partial v_x}{\partial y}&\frac{\partial v_x}{\partial z}\\ \frac{\partial v_y}{\partial x}&\frac{\partial v_y}{\partial y}&\frac{\partial v_y}{\partial z}\\ \frac{\partial v_z}{\partial x}&\frac{\partial v_z}{\partial y}&\frac{\partial v_z}{\partial z} \end{bmatrix}

This is just the Jacobian of the velocity field. Viscous forces are the result of relative motion between elements of the fluid, and so are expressible as a function of the velocity field. In other words, the forces at $\mathbf{r}$ are a function of $\mathbf{v}(\mathbf{r})$ and all derivatives of $\mathbf{v}(\mathbf{r})$ at that point. In the case of linear viscosity, the viscous force will be a function of the Jacobian tensor alone. For almost all practical situations, the linear approximation is sufficient.

If we represent x, y, and z by indices 1, 2, and 3 respectively, the i,j component of the Jacobian may be written as $\partial_i v_j$ where $\partial_i$ is shorthand for $\partial /\partial x_i$ . Note that when the first and higher derivative terms are zero, the velocity of all fluid elements is parallel, and there are no viscous forces.

Any matrix may be written as the sum of an antisymmetric matrix and a symmetric matrix, and this breakdown is independent of coordinate system, and so has physical significance. The velocity field may be approximated as:

v_i(\mathbf{r}+d\mathbf{r}) = v_i(\mathbf{r})+\frac{1}{2}\left(\partial_i v_j-\partial_j v_i\right)dr_i + \frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)dr_i

where Einstein notation is now being used in which repeated indices in a product are implicitly summed. The second term on the left is the asymmetric part of the first derivative term, and it represents a rigid rotation of the fluid about $\mathbf{r}$ with angular velocity $\omega$ where:

\omega=\mathbf{\nabla}\times \mathbf{v}=\frac{1}{2}\begin{bmatrix}

\partial_2 v_3-\partial_3 v_2\\ \partial_3 v_1-\partial_1 v_3\\ \partial_1 v_2-\partial_2 v_1 \end{bmatrix}

For such a rigid rotation, there is no change in the relative positions of the fluid elements, and so there is no viscous force associated with this term. The remaining symmetric term is responsible for the viscous forces in the fluid. Assuming the fluid is isotropic (i.e. its properties are the same in all directions), then the most general way that the symmetric term (the rate-of-strain tensor) can be broken down in a coordinate-independent (and therefore physically real) way is as the sum of a constant tensor (the rate-of-expansion tensor) and a traceless symmetric tensor (the rate-of-shear tensor):

\frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right) = \frac{1}{3}\partial_k v_k \delta_{ij}+\left(

\frac{1}{2}\left(\partial_i v_j+\partial_j v_i\right)-\frac{1}{3}\partial_k v_k \delta_{ij}\right)

where $\delta_{ij}$ is the unit tensor. The most general linear relationship between the stress tensor $\mathbf{\sigma}$ and the rate-of-strain tensor is then a linear combination of these two tensors :

\sigma_{visc;ij} = \zeta\partial_k v_k \delta_{ij}+

\eta\left(\partial_i v_j+\partial_j v_i-\frac{2}{3}\partial_k v_k \delta_{ij}\right)

where $\zeta$ is the coefficient of bulk viscosity (or "second viscosity") and $\eta$ is the coefficient of (shear) viscosity.

The forces in the fluid are due to the velocities of the individual molecules. The velocity of a molecule may be thought of as the sum of the fluid velocity and the thermal velocity. The viscous stress tensor described above gives the force due to the fluid velocity only. The force on an area element in the fluid due to the thermal velocities of the molecules is just the hydrostatic pressure. This pressure term ( $p\delta_{ij}$ ) must be added to the viscous stress tensor to obtain the total stress tensor for the fluid.

\sigma_{ij} = p\delta_{ij}+\sigma_{visc;ij}\,

The infinitesimal force $dF_i$ on an infinitesimal area $dA_i$ is then given by the usual relationship:

dF_i=\sigma_{ij}dA_j\,

Etymology

The word "viscosity" derives from the Latin word "" for mistletoe. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds.

External links

Gas Dynamics Toolbox Calculate coefficient of viscosity for mixtures of gases using VHS model

Viscosity Page A table of items sorted by viscosity in centipoise (cP)

Physical Characteristics of Water A table of water viscosity as a function of temperature

References