Viscosity: Definition, Unit & Formula

The category of fluids encompasses many different substances that can be distinguished from each other in numerous ways, including chemical composition, polarity, density and so on. Another property of fluids is a quantity known as viscosity.

Suppose you have a cup of water and a cup of syrup. When you pour the liquids from these cups, you notice a distinct difference in how each liquid flows. The water pours out quickly and easily while the syrup pours more slowly. This difference is due to a difference in their viscosities.

Viscosity is a measure of a fluid's resistance to flow. It can also be thought of as a measure of a fluid's thickness or its resistance to objects passing through it. The greater the resistance to flow, the higher the viscosity, so in the previous example, the syrup has a higher viscosity than water.

Viscosity is caused by internal friction between the molecules in a fluid. Think of a flowing fluid as consisting of layers moving in relation to each other. These layers rub against each other, and the greater the friction, the slower the flow (or the more force required to achieve flow). This friction is caused by intermolecular forces which impacts the internal resistance of a fluid

Many factors can affect a substance's viscosity; among these is temperature. Recall that temperature is a measure of the average kinetic energy per molecule in a substance. A higher average kinetic energy – temperature increase – per molecule results in faster-moving molecules and the resulting viscosity decreases. If you warm syrup up in a microwave, for example, you might notice that it flows more easily.

The viscosity of liquids is different from gas. When dealing with this less contained state of matter, a higher temperature actually causes them to "thicken," and their viscosity increases with temperature. This is because for gasses at low temperatures, the molecules rarely collide or interact with each other, while at higher temperatures there are many more collisions. As a result, the gasses' resistance to flow increases. Looking at the viscosity of gasses can be very useful when dealing with atmospheric science and weather predictions; the atmosphere can be described by flow rate of various layers with viscous forces.

The shape of the molecules in a fluid can also affect the viscosity. Rounder molecules can roll past each other more easily than molecules with branches and less uniform shapes. (Imagine pouring a bucket of marbles out versus pouring a bunch of jacks.)

Two factors that relate to the mathematical formulation of viscosity are shear stress and shear rate. In order to understand the formal definition of viscosity, it is first important to understand the definitions of these quantities.

Consider the method of approximating fluid flow as layers of fluid flowing past each other. If we think of a flowing fluid like this, the shear stress is the force pushing one layer across another divided by the area of the layers. More formally, this can be stated as the ratio of the force F applied with the cross-sectional area A of the material that is parallel to the applied force.

Shear stress is often denoted with the Greek letter tau τ, and hence the corresponding mathematical expression is:

\(\tau = \frac{F}{A}\)

Shear rate is essentially the rate at which the fluid layers are moving past each other. More formally it is defined as follows:

\(\dot{\gamma}=\frac{\Delta v}{x}\)

Where Δ‌v‌ is the difference in velocity between two layers, and ‌x‌ is the layer separation.

The notation of γ with the dot is because γ is the shear, and a first derivative (the rate of change) of a variable is often denoted with a dot above the associated variable. Using calculus, the continuous shear rate would be given as ‌dv/dx‌ instead and is also referred to as the velocity gradient.

Viscosity comes in a few different types. There is dynamic viscosity, also called absolute viscosity, which is usually the viscosity referred to when simply saying "viscosity." But there is also kinematic viscosity, which has a slightly different mathematical formulation.

Dynamic or absolute viscosity is the ratio of shear stress to shear rate, as shown in the following equation:

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

A common formulation of this relationship is called Newton's equation and is written as follows:

\(F = \mu A \frac{u}{y}\)

Where μ is the coefficient of viscosity, A is area of the surface, u is speed, and y is the separation of the two surfaces trapping the liquid.

Kinematic viscosity is defined as the absolute viscosity divided by mass density:

\(v = \frac{\mu}{\rho}\)

where ⍴ is the density of the liquid.

Consider two fluids that might have the same dynamic viscosity, but different mass densities. These two fluids will pour out of a container at different rates under the influence of gravity because an equal quantity of each will have different gravitational forces acting on them (proportional to their masses). The kinematic viscosity takes this into account by dividing by the mass density, and hence can be thought of as a measure of resistance to flow under the influence of gravity alone.

Using the SI units of viscosity, shear stress was in N/m² and shear rate was in (m/s)/m = 1/s, so dynamic viscosity has units of Ns/m² (newton-seconds per meter squared) or Pa s (pascal-seconds). However, the most common unit of viscosity is the dyne-second per square centimeter (dyne s/cm²) where 1 dyne = 10^-5N. One dyne-second per square centimeter is called a poise after French physiologist Jean Poiseuille. One pascal-second is equal to 10 poise.

The SI unit of kinematic viscosity is simply m², though a more common unit in the CGS system is the square centimeter per second, which is called a stoke (St) after Irish physicist George Stokes.

Units for viscosity are often measured in centipoise (1/100th of a poise).

Most liquids have viscosities between 1 and 1,000 mPa s while gasses have low viscosity, usually between 1-10 μPa s. The viscosity of water is about 1.0020 mPa s while the viscosity of blood is between 3 and 4 mPa s (lending new meaning to the saying that blood is thicker than water!)

Cooking oils have viscosities between around 25 to 100 mPa s, while motor oil and machine oils have viscosities on the order of a few hundred mPa s.

The air you breathe has a viscosity of about 18 μPa s.

Molten glass is one of the most viscous fluids there is with a high viscosity approaching infinity as it solidifies. At its melting point, the viscosity of glass is about 10 Pa s, while this increases by a factor of 100 at its working point and by a factor of more than 10¹¹ at its annealing point.

A Newtonion fluid is one in which the shear stress is linearly related to the shear rate. In a such a fluid, the viscosity for that fluid is a constant value. (In a non-Newtonian fluid, viscosity ends up being a dynamic function of another variable, such as time or force.)

Not surprisingly, Newtonion fluids are easier to work with and model. Conveniently, many common fluids are Newtonion to a good approximation. Some behavior that non-Newtonian fluids might exhibit include fluids in which the viscosity changes with shear rate, and fluids that become less or more viscous when shaken, agitated or disturbed.

Water and air are examples of Newtonion fluids. Examples of non-Newtonian fluids are non-drip paint, some polymer solutions and even blood. One grade school favorite non-Newtonian fluid is oobleck – a mixture of cornstarch and water that acts almost solid when worked with quickly, and then melts when left alone.

Viscosity can be measured in several different ways. These include using instruments such as a viscometer, or any number of DIY experiments.

Viscometers are best used on Newtonian fluids and tend to work via one of two ways. Either a small object moves through a stationary fluid, or the fluid flows past a stationary object. By measuring the associated drag, the viscosity can be determined. Capillary viscometers work by determining the time required for a certain volume of fluid to flow through a capillary tube of a certain length. Falling ball viscometers measure the time it takes for a ball to fall through a sample under the influence of gravity.

To measure viscosity of non-Newtonian fluids, a rheometer is often used. Rheology is the name of a branch of physics that studies the flow of fluids and soft solids and observes how they deform. A rheometer allows for more variables to be determined when measuring viscosity since non-Newtonian fluids do not have constant viscosity values. The two main types of rheometers are shear rheometers (which control the applied shear stress) and extensional rheometers (which operate based on applied external shear stress).

DIY Viscosity Measurement

The following describes how you can measure the viscosity of a fluid at home using a few simple materials. In order to apply this method, however, you will first need Stokes' law. Stokes' law relates the drag force F on a small sphere moving through a viscous fluid to the viscosity, radius of the sphere r and terminal velocity of the sphere v, via:

\(F = 6 \pi \mu r v\)

Now that you have this law, you can create your own falling ball viscometer.

1. Calculate the Density of the Fluid

Calculate the density of the fluid by weighing a known volume of the fluid and dividing its mass by the volume.

2. Calculate the Density of the Ball

Calculate the density of the ball by first measuring its diameter and using the formula V = 4/3πr³ to calculate its volume. Then weigh the ball and divide the mass by the volume.

3. Measure Terminal Velocity

Measure the terminal velocity of the ball as it falls through the fluid in the graduated cylinder. In a thick fluid, the marble will reach a constant speed fairly quickly. Time how long it takes for the ball to pass between two marked points on the graduated cylinder, and then divide that distance by the time to determine the velocity.

4. Calculate Viscosity Using Stokes' Law

The viscosity of the fluid can be found using Stokes' law and solving for viscosity:

\(\mu = \frac{F}{6 \pi r v}\)

Where F in this case is the drag force. In order to determine the drag force, you must write the net force equation and solve for it. The net force equation when the ball is at terminal velocity is:

\(F_{net} = F_b + F – F_g = 0\)

Where ‌_F_b‌ is buoyant force and ‌F_g_‌ is the gravitational force. Solving for F and plugging in expressions, you get:

\(F = F_g – F_b = \rho_b V_b g – \rho_f V_b g = \frac{4}{3} \pi r^3 (\rho_b – \rho_f)\)

Where ‌_V_b‌ is the volume of the ball, ‌ρ_b‌ is the density of the ball and ‌ρ_f _‌is the density of the fluid.

Hence the formula for viscosity becomes:

\(\mu = \frac{2 r^2 g (p_b – p_f)}{9v}\)

Simply plug in your measured values for radius of the ball, the density of the ball and of the fluid, and the terminal velocity to compute the final result.