Newtonian fluid

A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time.^[1]^[2]^[3]^[4] Stresses are proportional to the rate of change of the fluid's velocity vector.

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 (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 easiest 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) and 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.

Definition

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

The deformation of a fluid element, relative to some previous state, can be first order approximated 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 $v$ at that point, often denoted $\nabla v$ .

The tensors $\tau$ and $\nabla v$ 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 ${\boldsymbol {\tau }}={\boldsymbol {\mu }}(\nabla v)$ 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

For an incompressible and isotropic Newtonian fluid in laminar flow only in the direction x (i.e. where viscosity is isotropic in the fluid), and with a flow velocity constant along the flow direction x, the viscous stress is related to the strain rate by the simple constitutive equation $\tau =-\mu {\frac {du}{dy}}$ where

$\tau$ is the shear stress ("skin drag") in the fluid,
$\mu$ is a scalar constant of proportionality, the dynamic viscosity of the fluid
${\frac {du}{dy}}$ is the derivative in the direction y, normal to x, of the flow 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 the shear stress ("skin drag") in the fluid,
${\frac {\partial u}{\partial y}}$ is the partial derivative in the direction y of the flow 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.

However, there are in this case two more components of the shear stress, that are described by Newton's constitutive equation: $\tau _{xx}=-2\mu {\frac {\partial u}{\partial x}}$ and: $\tau _{yy}=-2\mu {\frac {\partial v}{\partial y}}$ Note that in the previous case these components were identically 0: $\tau _{xx}=0$ since the x component of the velocity was constant along the direction x, and $\tau _{yy}=0$ since the y component of the velocity was zero in every point of the flow (the flow was only in the x direction).

We can now generalize to the case of an incompressible 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 $j$ th spatial coordinate
$v_{i}$ is the fluid's velocity in the direction of axis $i$
$\tau _{ij}$ is the $j$ -th component of the stress acting on the faces of the fluid element perpendicular to axis $i$ . 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.

One also defines a total stress tensor ${\boldsymbol {\sigma }}$ , that combines the shear stress tensor with conventional (thermodynamic) pressure $p$ . The stress constitutive 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.

The incompressible momentum Navier–Stokes equation results from the following assumptions on the deviatoric component of the Cauchy stress tensor:^[5]

the deviatoric stress is Galilean 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 ${\textstyle \nabla \mathbf {u} }$ .
the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently ${\textstyle {\boldsymbol {\tau }}}$ is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity ${\textstyle \mu }$ :
Stokes' stress constitutive 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 is constitutive equation is also called the Newtonian law of viscosity. Dynamic viscosity $μ$ need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state.^[6]

The divergence of the deviatoric stress in case of uniform viscosity is given by: $\nabla \cdot {\boldsymbol {\tau }}=2\mu \nabla \cdot {\boldsymbol {\varepsilon }}=\mu \nabla \cdot \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{\mathrm {T} }\right)=\mu \,\nabla ^{2}\mathbf {u}$ because ${\textstyle \nabla \cdot \mathbf {u} =0}$ for an incompressible fluid.

Incompressibility rules out density and pressure waves like sound or shock waves, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well with all fluids at low Mach numbers (say up to about Mach 0.3), such as for modelling air winds at normal temperatures.^[7]

Compressible isotropic case

For a generic compressible and isotropic Newtonian fluid, one additional term is involved in describing the shear stress constitutive equation. This term depends on the flow compressibility, i.e. the flow velocity divergence^[8]:

$\Delta \tau _{ij}=\left({\frac {2}{3}}\mu -\kappa \right)\delta _{ij}\sum _{a}{\frac {\partial v_{a}}{\partial x_{a}}}$

where

$x_{j}$ is the $j$ th spatial coordinate
$v_{i}$ is the fluid's velocity in the direction of axis $i$
$\Delta \tau _{ij}$ is the $j$ -th component of the stress acting on the faces of the fluid element perpendicular to axis $i$ due to the flow compressibility. It is the ij-th component of the shear stress tensor part due to the flow compressibility
$\kappa$ is the bulk viscosity of the flow, that is zero for example for monoatomic gases at low density
$\delta _{ij}$ is the Kronecker delta of i and j

Moreover, note that the sum:

$\sum _{a}{\frac {\partial v_{a}}{\partial x_{a}}}$

is the divergence of the flow velocity.

This additional compressibility term of the shear stress, written in more compact tensor notation is: $\Delta {\boldsymbol {\tau }}=\left({\frac {2}{3}}\mu -\kappa \right)(\nabla \cdot \mathbf {v} )\mathbf {I}$ where:

$\nabla \mathbf {v}$ is the flow velocity gradient
$\nabla \cdot \mathbf {v}$ is the flow velocity divergence
$\mathbf {I}$ is the identity matrix

Then the Newton's constitutive equation for the shear stress becomes:

$\tau _{ij}=-\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)+\left({\frac {2}{3}}\mu -\kappa \right)\delta _{ij}\sum _{a}{\frac {\partial v_{a}}{\partial x_{a}}}$

Or, written in more compact tensor notation, it is: ${\boldsymbol {\tau }}=-\mu \left(\nabla \mathbf {v} +\nabla \mathbf {v} ^{T}\right)+\left({\frac {2}{3}}\mu -\kappa \right)(\nabla \cdot \mathbf {v} )\mathbf {I}$

The general Newton's constitutive equation for the total stress is correspondingly:

$\sigma _{ij}=\left(p+\left({\frac {2}{3}}\mu -\kappa \right)\sum _{a}{\frac {\partial v_{a}}{\partial x_{a}}}\right)\delta _{ij}-\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)$ or written in more compact tensor notation: ${\boldsymbol {\sigma }}=\left(p+\left({\frac {2}{3}}\mu -\kappa \right)\nabla \cdot \mathbf {v} \right)\mathbf {I} -\mu \left(\nabla \mathbf {v} +\nabla \mathbf {v} ^{T}\right)$ .

Note that in the incompressible case, the continuity equation for mass simply states that the flow velocity divergence is zero. Then one returns to the equations seen above for the incompressible case.

For anisotropic fluids

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

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: $d\mathbf {F} =\mu _{ij}\,d\mathbf {S} \times \mathrm {rot} \,\mathbf {u}$ where $\mu _{ij}$ is the viscosity tensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components – turbulence eddy viscosity.^[9]

Newtonian law of viscosity

The following equation illustrates the relation between shear rate and shear stress for an incompressible 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 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 is constant, the fluid is Newtonian.

Power law model

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.

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: ${\sqrt {\tau }}={\sqrt {\tau _{0}}}+S{\sqrt {dV \over dy}}$ where τ₀ is the yield stress and $S={\sqrt {\frac {\mu }{(1-H)^{\alpha }}}},$ 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.

References

^ 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 Batchelor (1967) pp. 142–148.
^ Batchelor (1967) p. 165.
^ See Acheson (1990).
^ Bird, Stewart, Lightfoot, Transport Phenomena, equation 3.2-11a
^ Volobuev, A. N. (2012). Basis of Nonsymmetrical Hydromechanics. New York: Nova Science Publishers, Inc. ISBN 978-1-61942-696-2.

[1] Panton, Ronald L. (2013). Incompressible Flow (Fourth ed.). Hoboken: John Wiley & Sons. p. 114. ISBN 978-1-118-01343-4.

[Batchelor-2] Batchelor, G. K. (2000) [1967]. An Introduction to Fluid Dynamics. Cambridge Mathematical Library series, Cambridge University Press. ISBN 978-0-521-66396-0.

[Kundu-3] Kundu, P.; Cohen, I. Fluid Mechanics. p. (page needed).

[Kirby-4] 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.

[Batchelor_142_148-5] Batchelor (1967) pp. 142–148.

[Batchelor_1967_p._165-6] Batchelor (1967) p. 165.

[Ach-7] See Acheson (1990).

[8] Bird, Stewart, Lightfoot, Transport Phenomena, equation 3.2-11a

[Volobuev-9] Volobuev, A. N. (2012). Basis of Nonsymmetrical Hydromechanics. New York: Nova Science Publishers, Inc. ISBN 978-1-61942-696-2.

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