The viscosity is an intensive property of a fluid that measures its internal resistance to motion or deformation. It plays an important role in areas such as aerodynamics and reservoir engineering since it determines the nature of the flow of a given fluid, such as air, water, and oil.

A tangible everyday example of the comparison between two fluids’ viscosities is the one between honey and milk. Intuitively, honey is more viscous than milk. This can be seen in an experiment such as the one in the picture below, where the viscous fluid (right) is more difficult to flow than the less viscous fluid (left).

The most basic ideas of the mathematics of fluid mechanics — including its structure and formulations — emerged between the late seventeenth century and the first half of the eighteenth century. More advanced and involved concepts such as turbulence, discontinuities, and viscosity were introduced in the nineteenth and twentieth centuries.

The field of fluid dynamics started being scientifically defined with Newton’s Principia Mathematica in 1687, analyzing for the first time the dynamics of fluids. Newton treated fluids such as air as a particle agglomerate. It was only with Leonhard Euler that the differential and continuum form of fluid dynamics was developed1

${}^{1}$.

It was before Newton, however, that many important questions began to appear. Christiaan Huygens was interested in studying the effects of bodies inside fluids since he was a student of ballistics, and therefore studied how air resistance worked. The problem of determining the dynamics of a body in relative motion — with a fluid surrounding it — is represented through the problem of resistance and was in many aspects intrinsically related to the study of viscosity.

The mathematical models of fluid dynamics are mainly based on the mass conservation, momentum balance, and energy conservation, together with the constitutive relations of the fluid. When coupled, the conservation principles of energy, momentum, and mass can form the Navier-Stokes equation, which is used to describe the motion of many viscous fluids.

The mass conservation can be stated as follows:

∂ρ∂t+∇⋅(ρu⃗ )=0(1)

$$\begin{array}{}\text{(1)}& \frac{\mathrm{\partial}\rho}{\mathrm{\partial}t}+\mathrm{\nabla}\cdot \left(\rho \overrightarrow{u}\right)=0\end{array}$$

where ρ

$\rho $is the mass density field as a function of space and time, and u⃗

$\overrightarrow{u}$is the velocity field also as a function of space and time. This is known as the strong formulation of the mass conservation law, or the equation of continuity.

The momentum conservation can be written as2

${}^{2}$:

∂ρu⃗ ∂t+∇⋅(ρu⃗ ⊗u⃗ )+∇p(ρ)=μ∇2u⃗ +(λ+μ)∇(∇⋅u⃗ )+f⃗ (2)

$$\begin{array}{}\text{(2)}& \frac{\mathrm{\partial}\rho \overrightarrow{u}}{\mathrm{\partial}t}+\mathrm{\nabla}\cdot \left(\rho \overrightarrow{u}\otimes \overrightarrow{u}\right)+\mathrm{\nabla}p(\rho )=\mu {\mathrm{\nabla}}^{2}\overrightarrow{u}+\left(\lambda +\mu \right)\mathrm{\nabla}\left(\mathrm{\nabla}\cdot \overrightarrow{u}\right)+\overrightarrow{f}\end{array}$$

where μ

$\mu $is the shear viscosity coefficient, λ

$\lambda $is the second coefficient of viscosity, f⃗

$\overrightarrow{f}$are the body forces such as gravity, where f⃗ +ρ⋅g⃗

$\overrightarrow{f}+\rho \cdot \overrightarrow{g}$, and ⊗

$\otimes $is the outer product operator, so that u⃗ ⊗u⃗ =u⃗ ×u⃗ T

$\overrightarrow{u}\otimes \overrightarrow{u}=\overrightarrow{u}\times {\overrightarrow{u}}^{T}$.

These equations form the Navier-Stokes system. It’s common to approximate λ

$\lambda $as η−23μ

$\eta -\frac{2}{3}\mu $, where η

$\eta $is the bulk viscosity.

In general, when considering incompressible flow, where the fluid is not compressed or expanded rapidly, η≈0

$\eta \approx 0$and λ≈−23μ

$\lambda \approx -\frac{2}{3}\mu $, which is called the Stokes hypothesis. In this case, λ

$\lambda $is commonly disconsidered from the equation. Also, the bulk viscosity coefficient is hard to measure and to find values for in literature. Also, the divergence of the velocity, ∇⋅u⃗

$\mathrm{\nabla}\cdot \overrightarrow{u}$, is usually so small that the entire term which includes the bulk viscosity is commonly neglected. In the case of incompressible or isochoric flow, assuming ρ=1

$\rho =1$, the divergence of the velocity completely vanishes, and the Navier-Stokes system can be simplified to:

∇⋅u⃗ =0(3)

$$\begin{array}{}\text{(3)}& \mathrm{\nabla}\cdot \overrightarrow{u}=0\end{array}$$

∂u⃗ ∂t+∇⋅(u⃗ ⊗u⃗ )+∇p=μ∇2u⃗ +f⃗ (4)

$$\begin{array}{}\text{(4)}& \frac{\mathrm{\partial}\overrightarrow{u}}{\mathrm{\partial}t}+\mathrm{\nabla}\cdot \left(\overrightarrow{u}\otimes \overrightarrow{u}\right)+\mathrm{\nabla}p=\mu {\mathrm{\nabla}}^{2}\overrightarrow{u}+\overrightarrow{f}\end{array}$$

This system is known as the incompressible Navier-Stokes system.

As can be seen from the above equations, the viscosity plays an important role in the determination of the dynamics of a fluid. Viscosity is a property of the fluid that determines the internal resistance of the fluid to motion. It can be qualitatively verified that a fluid with higher viscosity is more resistant to motion by comparing oil to water, for example, given that oil is a more viscous fluid than water.

From the incompressible Navier-Stokes equations, it follows that viscosity is the diffusion coefficient, associated with the Laplacian of the velocity.

There are several types of viscosity. The most commonly used in the field of fluid dynamics is the shear — or dynamic viscosity — represented here as μ

$\mu $. However, when dealing with shock waves and other phenomena that include high and rapid compression of the fluid, the bulk viscosity η

$\eta $can’t be neglected, being related to important concepts such as the sound attenuation.

The kinematic viscosity ν=μρ

$\nu =\frac{\mu}{\rho}$appears frequently in fluid mechanics and heat transfer and is related to flows under the gravity force.

One common expression that uses kinematic viscosity is the Reynolds number, which relates the momentum to the viscous forces of a fluid. The Reynolds number is defined as Re=uLν

$Re=\frac{uL}{\nu}$, where L

$L$and u

$u$are the characteristic length and velocity. In a pipe flow, for example, L

$L$would be the diameter of the pipe, while u

$u$would be the average velocity of the fluid. The Reynolds number is commonly used to measure whether a flow is turbulent or laminar. For example, under practical conditions, the flow of water inside a pipe is laminar if Re≤2300

$Re\le 2300$, transitional if 2300≤Re≤4000

$2300\le Re\le 4000$, and turbulent if Re≥4000

$Re\ge 4000$3

${}^{3}$.

Another type of viscosity is the apparent viscosity, which is defined when dealing with non-Newtonian flows.

The shear or dynamic viscosity is commonly obtained with a velocity profile experiment. One of the most important examples of a velocity profile is the one in a boundary layer as is shown in the picture below, adapted from 1

${}^{1}$, where the section of a body such as a wing is represented in a current flow:

The arrows represent the velocity field. The velocity profile near the fixed object approaches zero, and at the exact boundary, the velocity is zero. This is known as the no-slip condition, which states that the fluid sticks to the surface of a body. The boundary layer is the region where viscosity effects are most significant in body-fluid interactions.

The velocity profile experiment is usually made with two parallel and sufficiently large plates with a non-zero relative tangential velocity, zero normal velocity, and a fluid between the two plates. If for simplicity the experiment is made such that one plate is fixed while the other moves tangentially with a tangential velocity u=V

$u=V$, then a velocity profile would appear in the fluid. The fluid in contact with the fixed plate would have zero velocity, while the fluid in contact with the moving plate would have the same velocity as it, namely V

$V$. The picture below illustrates the boundary layer experiment described:

If the fluid is Newtonian, where the rate of deformation is linearly proportional to the shear stress, the shear viscosity can be obtained in an experiment of one-dimensional shear flow, such as the velocity profile experiment, expressed as follows3

${}^{3}$:

μ=τdudy(5)

$$\begin{array}{}\text{(5)}& \mu =\frac{\tau}{\frac{du}{dy}}\end{array}$$

where τ

$\tau $is the shear stress acting on the fluid layer in contact with the moving plate, and dudy

$\frac{du}{dy}$is the variation of the velocity in relation to the height.

In regions not close to solids, it is common to neglect viscous forces, since inertial and pressure forces dominate in those regions. These regions are called inviscid flow regions. When in inviscid regions, it is common to adopt the Euler system, which models the motion of an inviscid fluid:

∂ρ∂t+∇⋅(ρu⃗ )=0(6)

$$\begin{array}{}\text{(6)}& \frac{\mathrm{\partial}\rho}{\mathrm{\partial}t}+\mathrm{\nabla}\cdot \left(\rho \overrightarrow{u}\right)=0\end{array}$$

∂ρu⃗ ∂t+∇⋅(ρu⃗ ⊗u⃗ )+∇p(ρ)=f⃗ (7)

$$\begin{array}{}\text{(7)}& \frac{\mathrm{\partial}\rho \overrightarrow{u}}{\mathrm{\partial}t}+\mathrm{\nabla}\cdot \left(\rho \overrightarrow{u}\otimes \overrightarrow{u}\right)+\mathrm{\nabla}p(\rho )=\overrightarrow{f}\end{array}$$

These equations are directly derived from the compressible Navier-Stokes system by removing the viscosity-dependent terms. Solving this system might present difficulties due to the lack of a diffusion term, which causes singularities and shock waves to appear in the solution. When dealing with numerical solutions, the formulation must be carefully adjusted in order to avoid instability and non-physical solutions due to the lack of diffusivity. This can, for example, be circumvented by the introduction of artificial diffusivity terms.

The Navier-Stokes equations shown earlier are derived considering a linear relation between the shear stress and the velocity gradient in the momentum balance equation. The mathematics of non-Newtonian fluids can become extremely involved. In such cases, the viscosity will be a function of the velocity. The field of rheology studies the flow of non-Newtonian fluids, and many models for viscosity can be found in the literature4

${}^{4}$.

In the case of non-Newtonian flow, the ratio τdudy

$\frac{\tau}{\frac{du}{dy}}$is called the apparent viscosity, and might not be constant. The relation of the shear stress with the rate of deformation classifies the type of fluid. For example, a fluid for which the apparent viscosity increases with the rate of deformation is called a dilatant fluid, while a fluid for which the apparent viscosity decreases with the rate of deformation is called a pseudoplastic fluid.

The SimScale public projects library provides a number of computational experiments which explore the nature of viscous flows.

Beginning with the no-slip condition, the picture below shows the geometry of a relatively long pipe which will be used in the simulation of a fluid flow:

The velocity profile inside the pipe is shown below. It can be noticed that the velocity on the boundary is zero.

This project can be found on the following link: Pipeflow

Compressible air flow is explored in the following project, which analyzes an airfoil, demonstrating the capability of SimScale when dealing with compressible flows.

A common non-Newtonian fluid is blood. SimScale also features projects simulating blood flow. The following project simulates the blood flow inside an artery bifurcation. It compares three blood vessel cases: one healthy, one moderately blocked, and one severely blocked.

Non-Newtonian fluid simulation

If there is one thing we love in Germany, it is beer. This is why we could not resist and simulated pouring ourselves one liter of beer in a Maßkrug. In the simulation below you can see the velocity of the beer which is why you’ll only see a layer moving up the Krug.

However, we can ensure you that there is one liter of beer in the glass.

1

${}^{1}$Calero, J. S., “The Genesis of Fluid Mechanics, 1640-1780”, 2008

2

${}^{2}$Feireisl, E., Karper, T. G., Pokorny, M., “Mathematical Theory of Compressible Viscous Fluids”, 2016

3

${}^{3}$Çengel, Y. A., “Fluid Mechanics: Fundamentals and Applications”, 2014

4

${}^{4}$Irgens, F., “Rheology and Non-Newtonian Fluids”, 2014