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Newtonian fluid

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ANewtonian fluid is afluid in which theviscous stresses arising from itsflow are at every point linearly correlated to the localstrain rate — therate of change of itsdeformation over time.^[1]^[2]^[3]^[4] Stresses are proportional to magnitude of the fluid'svelocity vector.

A fluid is Newtonian only if thetensors that describe the viscous stress and the strain rate are related by a constantviscosity tensor that does not depend on the stress state and velocity of the flow. If the fluid is alsoisotropic (i.e., its mechanical properties are the same along any direction), the viscosity tensor reduces to two real coefficients, describing the fluid's resistance to continuousshear deformation and continuouscompression or expansion, respectively.

Newtonian fluids are the easiestmathematical 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 includeoobleck (which becomes stiffer when vigorously sheared) and non-drippaint (which becomesthinner when sheared). Other examples include manypolymer solutions (which exhibit theWeissenberg effect), molten polymers, many solid suspensions, blood, and most highly viscous fluids.

Newtonian fluids are named afterIsaac Newton, who first used thedifferential equation to postulate the relation between the shear strain rate andshear stress for such fluids.

Definition

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An element of a flowing liquid or gas will endure forces from the surrounding fluid, includingviscous stress forces that cause it to gradually deform over time. These forces can be mathematicallyfirst order approximated by aviscous stress tensor, usually denoted by $\tau$ .

The deformation of a fluid element, relative to some previous state, can be first order approximated by astrain tensor that changes with time. The time derivative of that tensor is thestrain rate tensor, that expresses how the element's deformation is changing with time; and is also thegradient of the velocityvector field $v {\displaystyle v}$ at that point, often denoted $\nabla v$ .

The tensors $\tau$ and $\nabla v$ can be expressed by 3×3matrices, relative to any chosencoordinate system. The fluid is said to be Newtonian if these matrices are related by theequation ${\boldsymbol {\tau }}={\boldsymbol {\mu }}(\nabla v)$ where $\mu$ 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

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For anincompressible and isotropic Newtonian fluid inlaminar flow only in the direction x (i.e. where viscosity is isotropic in the fluid), the shear stress is related to the strain rate by the simpleconstitutive equation $\tau =\mu {\frac {du}{dy}}$ where

$\tau$ is theshear stress ("skin drag") in the fluid,
$\mu$ is a scalar constant of proportionality, thedynamic viscosity of the fluid
${\frac {du}{dy}}$ is thederivative in the direction y, normal to x, of theflow velocity component u that is oriented along the direction x.

In case of a general 2D incompressibile flow in the plane x, y, the Newton constitutive equation become:

$\tau _{xy}=\mu \left({\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}\right)$ where:

$\tau _{xy}$ is theshear stress ("skin drag") in the fluid,
${\frac {\partial u}{\partial y}}$ is thepartial derivative in the direction y of theflow velocity component u that is oriented along the direction x.
${\frac {\partial v}{\partial x}}$ is the partial derivative in the direction x of the flow velocity component v that is oriented along the direction y.

We can now generalize to the case of anincompressible flow with a general direction in the 3D space, the above constitutive equation becomes $\tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)$ where

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

or written in more compact tensor notation ${\boldsymbol {\tau }}=\mu \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{T}\right)$ where $\nabla \mathbf {u}$ is the flow velocity gradient.

An alternative way of stating this constitutive equation is:

Stokes' stressconstitutive equation(expression used for incompressible elastic solids)

{\boldsymbol {\tau }}=2\mu {\boldsymbol {\varepsilon }}

where ${\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\mathbf {\nabla u} +\mathbf {\nabla u} ^{\mathrm {T} }\right)$ is the rate-of-strain tensor. So this decomposition can be made explicit as:^[5]

Stokes's stress constitutive equation(expression used for incompressible viscous fluids)

{\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right]

This constitutive equation is also called theNewton law of viscosity.

The totalstress tensor ${\boldsymbol {\sigma }}$ can always be decomposed as the sum of theisotropic stress tensor and thedeviatoric stress tensor ( ${\boldsymbol {\sigma }}'$ ):

${\boldsymbol {\sigma }}={\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} +{\boldsymbol {\sigma }}'$

In the incompressible case, the isotropic stress is simply proportional to the thermodynamicpressure $p {\displaystyle p}$ : $p=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})=-{\frac {1}{3}}\sum _{k}\sigma _{kk}$

and the deviatoric stress is coincident with the shear stress tensor ${\boldsymbol {\tau }}$ : ${\boldsymbol {\sigma }}'={\boldsymbol {\tau }}=\mu \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{T}\right)$

The stressconstitutive equation then becomes $\sigma _{ij}=-p\delta _{ij}+\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)$ or written in more compact tensor notation ${\boldsymbol {\sigma }}=-p\mathbf {I} +\mu \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{T}\right)$ where $\mathbf {I}$ is the identity tensor.

General compressible case

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The Newton's constitutive law for a compressible flow results from the following assumptions on the Cauchy stress tensor:^[5]

the stress isGalilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient ${\textstyle \nabla \mathbf {u} }$ , or more simply the rate-of-strain tensor: ${\textstyle {\boldsymbol {\varepsilon }}\left(\nabla \mathbf {u} \right)\equiv {\frac {1}{2}}\nabla \mathbf {u} +{\frac {1}{2}}\left(\nabla \mathbf {u} \right)^{T}}$
the deviatoric stress islinear in this variable: ${\textstyle {\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})=-p\mathbf {I} +\mathbf {C} :{\boldsymbol {\varepsilon }}}$ , where ${\textstyle p}$ is independent on the strain rate tensor, ${\textstyle \mathbf {C} }$ is the fourth-order tensor representing the constant of proportionality, called the viscosity orelasticity tensor, and : is thedouble-dot product.
the fluid is assumed to beisotropic, as with gases and simple liquids, and consequently ${\textstyle \mathbf {C} }$ is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, byHelmholtz decomposition it can be expressed in terms of two scalarLamé parameters, thesecond viscosity ${\textstyle \lambda }$ and thedynamic viscosity ${\textstyle \mu }$ , as it is usual inlinear elasticity:
Linear stressconstitutive equation(expression similar to the one for elastic solid)
${\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})=-p\mathbf {I} +\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}$
where ${\textstyle \mathbf {I} }$ is theidentity tensor, and ${\textstyle \operatorname {tr} ({\boldsymbol {\varepsilon }})}$ is thetrace of the rate-of-strain tensor. So this decomposition can be explicitly defined as: ${\boldsymbol {\sigma }}=-p\mathbf {I} +\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right).$

Since thetrace of the rate-of-strain tensor in three dimensions is thedivergence (i.e. rate of expansion) of the flow: $\operatorname {tr} ({\boldsymbol {\varepsilon }})=\nabla \cdot \mathbf {u} .$

Given this relation, and since the trace of the identity tensor in three dimensions is three: $\operatorname {tr} ({\boldsymbol {I}})=3.$

the trace of the stress tensor in three dimensions becomes: $\operatorname {tr} ({\boldsymbol {\sigma }})=-3p+(3\lambda +2\mu )\nabla \cdot \mathbf {u} .$

So by alternatively decomposing the stress tensor intoisotropic anddeviatoric parts, as usual in fluid dynamics:^[6] ${\boldsymbol {\sigma }}=-\left[p-\left(\lambda +{\tfrac {2}{3}}\mu \right)\left(\nabla \cdot \mathbf {u} \right)\right]\mathbf {I} +\mu \left(\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{\mathrm {T} }-{\tfrac {2}{3}}\left(\nabla \cdot \mathbf {u} \right)\mathbf {I} \right)$

Introducing thebulk viscosity ${\textstyle \zeta }$ , $\zeta \equiv \lambda +{\tfrac {2}{3}}\mu ,$

we arrive to the linearconstitutive equation in the form usually employed inthermal hydraulics:^[5]

Linear stress constitutive equation(expression used for fluids)

${\boldsymbol {\sigma }}=-[p-\zeta (\nabla \cdot \mathbf {u} )]\mathbf {I} +\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]$

which can also be arranged in the other usual form:^[7] ${\boldsymbol {\sigma }}=-p\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right)+\left(\zeta -{\frac {2}{3}}\mu \right)(\nabla \cdot \mathbf {u} )\mathbf {I} .$

Note that in the compressible case the pressure is no more proportional to theisotropic stress term, since there is the additional bulk viscosity term: $p=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})+\zeta (\nabla \cdot \mathbf {u} )$

and thedeviatoric stress tensor ${\boldsymbol {\sigma }}'$ is still coincident with the shear stress tensor ${\boldsymbol {\tau }}$ (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which is proportional to the shear viscosity:

${\boldsymbol {\sigma }}'={\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]$

Note that the incompressible case correspond to the assumption that the pressure constrains the flow so that the volume offluid elements is constant:isochoric flow resulting in asolenoidal velocity field with ${\textstyle \nabla \cdot \mathbf {u} =0}$ .^[8]So one returns to the expressions for pressure and deviatoric stress seen in the preceding paragraph.

Both bulk viscosity ${\textstyle \zeta }$ and dynamic viscosity ${\textstyle \mu }$ need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of thesetransport coefficient in theconservation variables is called anequation of state.^[9]

Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called thedispersion. In some cases, thesecond viscosity ${\textstyle \zeta }$ can be assumed to be constant in which case, the effect of the volume viscosity ${\textstyle \zeta }$ is that the mechanical pressure is not equivalent to the thermodynamicpressure:^[10] as demonstrated below. $\nabla \cdot (\nabla \cdot \mathbf {u} )\mathbf {I} =\nabla (\nabla \cdot \mathbf {u} ),$ ${\bar {p}}\equiv p-\zeta \,\nabla \cdot \mathbf {u} ,$ However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves,^[11] where second viscosity coefficient becomes important) by explicitly assuming ${\textstyle \zeta =0}$ . The assumption of setting ${\textstyle \zeta =0}$ is called as theStokes hypothesis.^[12] The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory;^[13] for other gases and liquids, Stokes hypothesis is generally incorrect.

Finally, note that Stokes hypothesis is less restrictive that the one of incompressible flow. In fact, in the incompressible flow both the bulk viscosity term, and the shear viscosity term in the divergence of the flow velocity term disappears, while in the Stokes hypothesis the first term also disappears but the second one still remains.

For anisotropic fluids

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

There is general formula for friction force in a liquid: The vectordifferential of friction force is equal the viscosity tensor increased onvector product differential of the area vector of adjoining a liquid layers androtor of velocity: $d\mathbf {F} =\mu _{ij}\,d\mathbf {S} \times \nabla \times \,\mathbf {u}$ where $\mu _{ij}$ is the viscositytensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components –turbulence eddy viscosity.^[14]

Newton's law of viscosity

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The following equation illustrates the relation between shear rate and shear stressfor a fluid with laminar flow only in the direction x: $\tau _{xy}=\mu {\frac {\mathrm {d} v_{x}}{\mathrm {d} y}},$ where:

$\tau _{xy}$ is the shear stress in the components x and y, i.e. the force component on the direction x per unit surface that is normal to the direction y (so it is parallel to the direction x)
$\mu$ is the dynamic viscosity, and
${\textstyle {\frac {\mathrm {d} v_{x}}{\mathrm {d} y}}}$ is the flow velocity gradient along the direction y, that is normal to the flow velocity $v_{x}$ .

If viscosity $\mu$ does not vary with rate of deformation the fluid is Newtonian.

Power law model

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In blue a Newtonian fluid compared to the dilatant and the pseudoplastic, angle depends on the viscosity.

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: $\tau _{xy}=-m\left|{\dot {\gamma }}\right|^{n-1}{\frac {dv_{x}}{dy}},$ where

$\left|{\dot {\gamma }}\right|^{n-1}$ 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.

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

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The relationship between the shear stress and shear rate in a casson fluid model is defined as follows: ${\sqrt {\tau }}={\sqrt {\tau _{0}}}+S{\sqrt {dV \over dy}}$ whereτ₀ is the yield stress and $S={\sqrt {\frac {\mu }{(1-H)^{\alpha }}}},$ whereα depends on protein composition andH is theHematocrit number.

Examples

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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.

References

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^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 – via kirbyresearch.com.
^^a ^b ^cBatchelor (1967) pp. 137 & 142.
^Chorin, Alexandre E.; Marsden, Jerrold E. (1993).A Mathematical Introduction to Fluid Mechanics. p. 33.
^Bird, Stewart, Lightfoot, Transport Phenomena, 1st ed., 1960, eq. (3.2-11a)
^Batchelor (1967) p. 75.
^Batchelor (1967) p. 165.
^Landau & Lifshitz (1987) pp. 44–45, 196
^White (2006) p. 67.
^Stokes, G. G. (2007). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids.
^Vincenti, W. G., Kruger Jr., C. H. (1975). Introduction to physical gas dynamic. Introduction to physical gas dynamics/Huntington.
^Volobuev, A. N. (2012).Basis of Nonsymmetrical Hydromechanics. New York:Nova Science Publishers, Inc.ISBN 978-1-61942-696-2.

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