# Newtonian fluid

Last updated

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] 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.

## Contents

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.

Understanding whether a fluid is Newtonian or not is important in certain industrial processing industries including food processing and pharmaceutical manufacturing. In these industries, the nature of the fluid being processed, and whether or not its viscosity changes when exposed to force, can affect product attributes such as texture, taste, and appearance. [4]

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.

## Definition

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 ${\displaystyle \tau }$.

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 ${\displaystyle v}$ at that point, often denoted ${\displaystyle \nabla v}$.

The tensors ${\displaystyle \tau }$ and ${\displaystyle \nabla v}$ 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 ${\displaystyle \mathbf {\tau } =\mathbf {\mu } (\nabla v)}$ where ${\displaystyle \mu }$ is a fixed 3×3×3×3 fourth order tensor, that does not depend on the velocity or stress state of the fluid.

### Incompressible isotropic case

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

${\displaystyle \tau =\mu {\frac {du}{dy}}}$

where

${\displaystyle \tau }$ is the shear stress ("drag") in the fluid,
${\displaystyle \mu }$ is a scalar constant of proportionality, the shear viscosity of the fluid
${\displaystyle {\frac {du}{dy}}}$ 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

${\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)}$

where

${\displaystyle x_{j}}$ is the ${\displaystyle j}$th spatial coordinate
${\displaystyle v_{i}}$ is the fluid's velocity in the direction of axis ${\displaystyle i}$
${\displaystyle \tau _{ij}}$ is the ${\displaystyle j}$th component of the stress acting on the faces of the fluid element perpendicular to axis ${\displaystyle i}$.

One also defines a total stress tensor ${\displaystyle \mathbf {\sigma } }$, that combines the shear stress with conventional (thermodynamic) pressure ${\displaystyle p}$. The stress-shear equation then becomes

${\displaystyle \mathbf {\sigma } _{ij}=-p\delta _{ij}+\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)}$

or written in more compact tensor notation

${\displaystyle \mathbf {\sigma } =-p\mathbf {I} +\mu \left(\nabla \mathbf {v} +\nabla \mathbf {v} ^{T}\right)}$

where ${\displaystyle \mathbf {I} }$ is the identity tensor.

### For anisotropic fluids

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

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:

${\displaystyle {d}\mathbf {F} {=}\mu _{ij}\,\mathbf {dS} \times \mathrm {rot} \,\mathbf {u} }$

where ${\displaystyle \mu _{ij}}$ – viscosity tensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components – turbulence eddy viscosity. [5]

### Newtonian law of viscosity

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

${\displaystyle \tau =\mu {du \over dy}}$,

where:

• τ is the shear stress;
• μ is the viscosity, and
• ${\textstyle {\frac {du}{dy}}}$is the shear rate.

If viscosity is constant, the fluid is Newtonian.

#### Power law model

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:

${\displaystyle \tau =-m\left\vert {\dot {\gamma }}\right\vert ^{n-1}{\frac {dv_{x}}{dy}}}$,

where

• ${\displaystyle \left\vert {\dot {\gamma }}\right\vert ^{n-1}}$is the absolute value of the strain rate to the (n-1) power;
• ${\textstyle {\frac {dv_{x}}{dy}}}$ is the velocity gradient;
• n is the power law index.

If

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

### Fluid model

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

${\displaystyle {\sqrt {\tau }}={\sqrt {\tau _{0}}}+S{\sqrt {dV \over dy}}}$

where τ0 is the yield stress and

${\displaystyle S={\sqrt {\frac {\mu }{(1-H)^{\alpha }}}}}$,

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

## Examples

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.

## Related Research Articles

In physics, the Navier–Stokes equations are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

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. It arises from the shear force, the component of force vector parallel to the material cross section. 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

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 materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or forces to strains or deformations.

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.

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).

Apparent viscosity is the shear stress applied to a fluid divided by the shear rate. For a Newtonian fluid, the apparent viscosity is constant, and equal to the Newtonian viscosity of the fluid, but for non-Newtonian fluids, the apparent viscosity depends on the shear rate. Apparent viscosity has the SI derived unit Pa·s (Pascal-second), but the centipoise is frequently used in practice:.

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 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.

Biofluid dynamics may be considered as the discipline of biological engineering or biomedical engineering in which the fundamental principles of fluid dynamics are used to explain the mechanisms of biological flows and their interrelationships with physiological processes, in health and in diseases/disorder. It can be considered as the conjuncture of mechanical engineering and biological engineering. It spans from cells to organs, covering diverse aspects of the functionality of systemic physiology, including cardiovascular, respiratory, reproductive, urinary, musculoskeletal and neurological systems etc. Biofluid dynamics and its simulations in computational fluid dynamics (CFD) apply to both internal as well as external flows. Internal flows such as cardiovascular blood flow and respiratory airflow, and external flows such as flying and aquatic locomotion. Biological fluid Dynamics involves the study of the motion of biological fluids. It can be either circulatory system or respiratory systems. Understanding the circulatory system is one of the major areas of research. The respiratory system is very closely linked to the circulatory system and is very complex to study and understand. The study of Biofluid Dynamics is also directed towards finding solutions to some of the human body related diseases and disorders. The usefulness of the subject can also be understood by seeing the use of Biofluid Dynamics in the areas of physiology in order to explain how living things work and about their motions, in developing an understanding of the origins and development of various diseases related to human body and diagnosing them, in finding the cure for the diseases related to cardiovascular and pulmonary systems.

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.

## References

1. Panton, Ronald L. (2013). Incompressible Flow (Fourth ed.). Hoboken: John Wiley & Sons. p. 114. ISBN   978-1-118-01343-4.
2. Batchelor, G. K. (2000) [1967]. An Introduction to Fluid Dynamics. Cambridge Mathematical Library series, Cambridge University Press. ISBN   978-0-521-66396-0.
3. Kundu, P.; Cohen, I. Fluid Mechanics. p. (page needed).
4. "Newtonian and Non-Newtonian Fluids: The Importance in Processing". Central States Industrial. Retrieved 2021-04-27.
5. Volobuev, A. N. (2012). Basis of Nonsymmetrical Hydromechanics. New York: Nova Science Publishers, Inc. ISBN   978-1-61942-696-2.