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A **Newtonian fluid** is a fluid in which the viscous stresses arising from its flow, at every point, are linearly^{ [1] } correlated to the local strain rate—the rate of change of its deformation over time.^{ [2] }^{ [3] }^{ [4] } That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions.

- Definition
- Incompressible isotropic case
- For anisotropic fluids
- Newtonian law of viscosity
- Fluid model
- Examples
- See also
- References

More precisely, a fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. If the fluid is also isotropic (that is, its mechanical properties are the same along any direction), the viscosity tensor reduces to two real coefficients, describing the fluid's resistance to continuous shear deformation and continuous compression or expansion, respectively.

Newtonian fluids are the simplest mathematical models of fluids that account for viscosity. While no real fluid fits the definition perfectly, many common liquids and gases, such as water and air, can be assumed to be Newtonian for practical calculations under ordinary conditions. However, non-Newtonian fluids are relatively common, and include oobleck (which becomes stiffer when vigorously sheared), or non-drip paint (which becomes thinner when sheared). Other examples include many polymer solutions (which exhibit the Weissenberg effect), molten polymers, many solid suspensions, blood, and most highly viscous fluids.

Newtonian fluids are named after Isaac Newton, who first used the differential equation to postulate the relation between the shear strain rate and shear stress for such fluids.

An element of a flowing liquid or gas will suffer forces from the surrounding fluid, including viscous stress forces that cause it to gradually deform over time. These forces can be mathematically approximated to first order by a viscous stress tensor, which is usually denoted by .

The deformation of that fluid element, relative to some previous state, can be approximated to first order by a strain tensor that changes with time. The time derivative of that tensor is the strain rate tensor, that expresses how the element's deformation is changing with time; and is also the gradient of the velocity vector field at that point, often denoted .

The tensors and can be expressed by 3×3 matrices, relative to any chosen coordinate system. The fluid is said to be Newtonian if these matrices are related by the equation where is a fixed 3×3×3×3 fourth order tensor, that does not depend on the velocity or stress state of the fluid.

For an incompressible and isotropic Newtonian fluid, the viscous stress is related to the strain rate by the simpler equation

where

- is the shear stress ("drag") in the fluid,
- is a scalar constant of proportionality, the
*shear viscosity*of the fluid - is the derivative of the velocity component that is parallel to the direction of shear, relative to displacement in the perpendicular direction.

If the fluid is incompressible and viscosity is constant across the fluid, this equation can be written in terms of an arbitrary coordinate system as

where

- is the th spatial coordinate
- is the fluid's velocity in the direction of axis
- is the th component of the stress acting on the faces of the fluid element perpendicular to axis .

One also defines a total stress tensor , that combines the shear stress with conventional (thermodynamic) pressure . The stress-shear equation then becomes

or written in more compact tensor tensor notation

where is the identity tensor.

More generally, in a non-isotropic Newtonian fluid, the coefficient that relates internal friction stresses to the spatial derivatives of the velocity field is replaced by a nine-element viscous stress tensor .

There is general formula for friction force in a liquid: The vector differential of friction force is equal the viscosity tensor increased on vector product differential of the area vector of adjoining a liquid layers and rotor of velocity:

where – viscosity tensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components – turbulence eddy viscosity.^{ [5] }

The following equation illustrates the relation between shear rate and shear stress:

- ,

where:

*τ*is the shear stress;*μ*is the viscosity, and- is the shear rate.

If viscosity is constant, the fluid is Newtonian.

The power law model is used to display the behavior of Newtonian and non-Newtonian fluids and measures shear stress as a function of strain rate.

The relationship between shear stress, strain rate and the velocity gradient for the power law model are:

- ,

where

- is the absolute value of the strain rate to the (n-1) power;

- is the velocity gradient;

*n*is the power law index.

If

*n*< 1 then the fluid is a pseudoplastic.*n*=1 then the it is Newtonian fluid.*n*> 1 then the fluid is a dilatant.

The relationship between the shear stress and shear rate in a casson fluid model is defined as follows:

where *τ*_{0} is the yield stress and

- ,

where *α* depends on protein composition and *H* is the Hematocrit number.

Water, air, alcohol, glycerol, and thin motor oil are all examples of Newtonian fluids over the range of shear stresses and shear rates encountered in everyday life. Single-phase fluids made up of small molecules are generally (although not exclusively) Newtonian.

In physics, the **Navier–Stokes equations**, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

The **vorticity equation** of fluid dynamics describes evolution of the vorticity **ω** of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid . The equation is:

**Shear stress**, often denoted by **τ**, is the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section of the material. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

A **power-law fluid**, or the **Ostwald–de Waele relationship**, is a type of generalized Newtonian fluid for which the shear stress, *τ*, is given by

A **Bingham plastic** is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is named after Eugene C. Bingham who proposed its mathematical form.

**Hemorheology**, also spelled **haemorheology**, or **blood rheology**, is the study of flow properties of blood and its elements of plasma and cells. Proper tissue perfusion can occur only when blood's rheological properties are within certain levels. Alterations of these properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit and mechanical properties of red blood cells. Red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. Because of that, blood behaves as a non-Newtonian fluid. As such, the viscosity of blood varies with shear rate. Blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. Therefore, blood is a shear-thinning fluid. Contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. Blood viscosity also increases with increases in red cell aggregability.

In fluid dynamics, the **Reynolds stress** is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum.

In seismology, **S-waves**, **secondary waves**, or **shear waves** are a type of elastic wave and are one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves.

**Shear rate** is the rate at which a progressive shearing deformation is applied to some material.

**Stokes flow**, also named **creeping flow** or **creeping motion**, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.

**Fluid mechanics** is the branch of physics concerned with the mechanics of fluids and the forces on them. It has applications in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, oceanography, meteorology, astrophysics, and biology.

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

**Volume viscosity** is a material property relevant for characterizing fluid flow. Common symbols are or . It has dimensions, and the corresponding SI unit is the pascal-second (Pa·s).

The **Herschel–Bulkley fluid** is a generalized model of a non-Newtonian fluid, in which the strain experienced by the fluid is related to the stress in a complicated, non-linear way. Three parameters characterize this relationship: the consistency *k*, the flow index *n*, and the yield shear stress . The consistency is a simple constant of proportionality, while the flow index measures the degree to which the fluid is shear-thinning or shear-thickening. Ordinary paint is one example of a shear-thinning fluid, while oobleck provides one realization of a shear-thickening fluid. Finally, the yield stress quantifies the amount of stress that the fluid may experience before it yields and begins to flow.

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.

The **Reynolds number** is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow. These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation. The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar to turbulent flow, and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full size version. The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behaviour on a larger scale, such as in local or global air or water movement and thereby the associated meteorological and climatological effects.

The **Oldroyd-B model** is a constitutive model used to describe the flow of viscoelastic fluids. This model can be regarded as an extension of the Upper Convected Maxwell model and is equivalent to a fluid filled with elastic bead and spring dumbbells. The model is named after its creator James G. Oldroyd.

The **viscous stress tensor** is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point.

In continuum mechanics, the **strain-rate tensor** or **rate-of-strain tensor** is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the gradient of the flow velocity. In fluid mechanics it also can be described as the **velocity gradient**, a measure of how the velocity of a fluid changes between different points within the fluid. Though the term can refer to the differences in velocity between layers of flow in a pipe, it is often used to mean the gradient of a flow's velocity with respect to its coordinates. The concept has implications in a variety of areas of physics and engineering, including magnetohydrodynamics, mining and water treatment.

In fluid mechanics, **Helmholtz minimum dissipation theorem** states that *the steady Stokes flow motion of an incompressible fluid has the smallest rate of dissipation than any other incompressible motion with the same velocity on the boundary*. The theorem also has been studied by Diederik Korteweg in 1883 and by Lord Rayleigh in 1913.

- ↑ Panton, Ronald L. (2013).
*Incompressible Flow*(Fourth ed.). Hoboken: John Wiley & Sons. p. 114. ISBN 978-1-118-01343-4. - ↑ Batchelor, G. K. (2000) [1967].
*An Introduction to Fluid Dynamics*. Cambridge Mathematical Library series, Cambridge University Press. ISBN 978-0-521-66396-0. - ↑ Kundu, P.; Cohen, I.
*Fluid Mechanics*. p. (page needed). - ↑ Kirby, B. J. (2010).
*Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices*. Cambridge University Press. ISBN 978-0-521-11903-0. - ↑ Volobuev, A. N. (2012).
*Basis of Nonsymmetrical Hydromechanics*. New York: Nova Science Publishers, Inc. ISBN 978-1-61942-696-2.

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