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The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.[1]

Viscosity can be conceptualized as quantifying the internal frictional force that arises between adjacent layers of fluid that are in relative motion. For instance, when a fluid is forced through a tube, it flows more quickly near the tube's axis than near its walls. In such a case, experiments show that some stress (such as a pressure difference between the two ends of the tube) is needed to sustain the flow through the tube. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion: the strength of this force is proportional to the viscosity.

A fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at very low temperatures in superfluids. Otherwise, the second law of thermodynamics requires all fluids to have positive viscosity;[2][3] such fluids are technically said to be viscous or viscid. A fluid with a high viscosity, such as pitch, may appear to be a solid.

## Etymology

The word "viscosity" is derived from the Latin viscum ("mistletoe"). Viscum also referred to a viscous glue derived from mistletoe berries.[4]

## Definition

### Simple definition

Illustration of a planar Couette flow. Since the shearing flow is opposed by friction between adjacent layers of fluid (which are in relative motion), a force is required to sustain the motion of the upper plate. The relative strength of this force is a measure of the fluid's viscosity.
In a general parallel flow, the shear stress is proportional to the gradient of the velocity.

In materials science and engineering, one is often interested in understanding the forces, or stresses, involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the rate of change of the deformation over time. These are called viscous stresses. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the distance the fluid has been sheared; rather, they depend on how quickly the shearing occurs.

Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow.

In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed ${\displaystyle u}$ (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from ${\displaystyle 0}$ at the bottom to ${\displaystyle u}$ at the top.[5] Each layer of fluid moves faster than the one just below it, and friction between them gives rise to a force resisting their relative motion. In particular, the fluid applies on the top plate a force in the direction opposite to its motion, and an equal but opposite force on the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed.

In many fluids, the flow velocity is observed to vary linearly from zero at the bottom to ${\displaystyle u}$ at the top. Moreover, the magnitude ${\displaystyle F}$ of the force acting on the top plate is found to be proportional to the speed ${\displaystyle u}$ and the area ${\displaystyle A}$ of each plate, and inversely proportional to their separation

A fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at very low temperatures in superfluids. Otherwise, the second law of thermodynamics requires all fluids to have positive viscosity;[2][3] such fluids are technically said to be viscous or viscid. A fluid with a high viscosity, such as pitch, may appear to be a solid.

The word "viscosity" is derived from the Latin viscum ("mistletoe"). Viscum also referred to a viscous glue derived from mistletoe berries.[4]

## Definition

### Simple definition

Illustration of a planar Couette flow. Since the shearing flow is opposed by friction between adjacent layers of fluid (which are in relative motion), a force is required to sustain the motion of the upper plate. The relative strength of this force is a measure of the fluid's viscosity.
In a general parallel flow, the shear stress is proportional to the gradient of the velocity.

In materials science and engineering, one is often interested in understanding the forces, or stresses, involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the rate of change of the deformation over time. These are called viscous stresses. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the distance the fluid has been sheared; rather, they depend on how quickly the shearing occurs.

Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow.

In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed ${\displaystyle u}$ (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from ${\displaystyle 0}$ at the bottom to ${\displaystyle u}$materials science and engineering, one is often interested in understanding the forces, or stresses, involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the rate of change of the deformation over time. These are called viscous stresses. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the distance the fluid has been sheared; rather, they depend on how quickly the shearing occurs.

Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow.

In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed ${\displaystyle u}$ (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from ${\displaystyle 0}$ at the bottom to ${\displaystyle u}Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow.$

In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed ${\displaystyle u}$ (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from ${\displaystyle 0}$ at the bottom to ${\displaystyle u}$ at the top.[5] Each layer of fluid moves faster than the one just below it, and friction between them gives rise to a force resisting their relative motion. In particular, the fluid applies on the top plate a force in the direction opposite to its motion, and an equal but opposite force on the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed.

In many fluids, the flow velocity is observed to vary linearly from zero at the bottom to ${\displaystyle u}$ at the top. Moreover, the magnitude ${\displaystyle F}$ of the force acting on the top plate is found to be proportional to the speed ${\displaystyle u}$ and the area ${\displaystyle A}$ of each plate, and inversely proportional to their separation ${\displaystyle y}$:

The proportionality factor ${\displaystyle \mu }$ is the viscosity of the fluid, with units of ${\displaystyle {\text{Pa}}\cdot {\text{s}}}$ (pascal-second). The ratio ${\displaystyle u/y}$ is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction perpendicular to the plates (see illustrations to the right). If the velocity does not vary linearly with ${\displaystyle y}$, then the appropriate generalization is

${\displaystyle \tau =\mu {\frac {\partial u}{\partial y}},}$${\displaystyle \tau =F/A}$, and ${\displaystyle \partial u/\partial y}$ is the local shear velocity. This expression is referred to as Newton's law of viscosity. In shearing flows with planar symmetry, it is what defines ${\displaystyle \mu }$. It is a special case of the general definition of viscosity (see below), which can be expressed in coordinate-free form.

Use of the Greek letter mu (${\displaystyle \mu }$) for the viscosity is common among mechanical and chemical engineers, as well as physicists.[6]Greek letter mu (${\displaystyle \mu }$) for the viscosity is common among mechanical and chemical engineers, as well as physicists.[6][7][8] However, the Greek letter eta (${\displaystyle \eta }$) is also used by chemists, physicists, and the IUPAC.[9] The viscosity ${\displaystyle \mu }$ is sometimes also referred to as the shear viscosity. However, at least one author discourages the use of this terminology, noting that ${\displaystyle \mu }$ can appear in nonshearing flows in addition to shearing flows.[10]

In very general terms, the viscous stresses in a fluid are defined as those resulting from the relative velocity of different fluid particles. As such, the viscous stresses must depend on spatial gradients of the flow velocity. If the velocity gradients are small, then to a first approximation the viscous stresses depend only on the first derivatives of the velocity.[11] (For Newtonian fluids, this is also a linear dependence.) In Cartesian coordinates, the general relationship can then be written as

${\displaystyle \mu _{ijk\ell }}$