Newtonian fluid
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A Newtonian fluid is a
A fluid is Newtonian only if the
Newtonian fluids are the easiest
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 .
The deformation of a fluid element, relative to some previous state, can be first order approximated by a
The tensors and can be expressed by 3×3
Incompressible isotropic case
For an
- is the shear stress ("skin drag") in the fluid,
- is a scalar constant of proportionality, the dynamic viscosity of the fluid
- 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:
- is the shear stress ("skin drag") in the fluid,
- is the partial derivative in the direction y of the flow velocity component u that is oriented along the direction 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 an
- 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 . It is the ij-th component of the shear stress tensor
or written in more compact tensor notation
An alternative way of stating this constitutive equation is:
where
This constitutive equation is also called the Newtonian law of viscosity.
The total stress tensor can always be decomposed as the sum of the
In the incompressible case, the isotropic stress is simply proportional to the thermodynamic pressure :
and the deviatoric stress is coincident with the shear stress tensor :
The stress constitutive equation then becomes
General compressible case
The Newton's constitutive law for a compressible flow results from the following assumptions on the Cauchy stress tensor:[5]
- the 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 , or more simply the rate-of-strain tensor:
- the deviatoric stress is linear in this variable: , where is independent on the strain rate tensor, is the fourth-order tensor representing the constant of proportionality, called the viscosity or elasticity tensor, and : is the double-dot product.
- the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, bysecond viscosityand thedynamic viscosity, as it is usual in linear elasticity:Linear stress constitutive equation (expression similar to the one for elastic solid)
where is the identity tensor, and is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as:
Since the trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow:
Given this relation, and since the trace of the identity tensor in three dimensions is three:
the trace of the stress tensor in three dimensions becomes:
So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics:[6]
Introducing the bulk viscosity ,
we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics:[5]
which can also be arranged in the other usual form:[7]
Note that in the compressible case the pressure is no more proportional to the isotropic stress term, since there is the additional bulk viscosity term:
and the
Note that the incompressible case correspond to the assumption that the pressure constrains the flow so that the volume of
Both bulk viscosity and dynamic viscosity 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 these transport coefficient in the conservation variables is called an equation 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 the dispersion. In some cases, the second viscosity can be assumed to be constant in which case, the effect of the volume viscosity is that the mechanical pressure is not equivalent to the thermodynamic pressure:[10] as demonstrated below.
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
More generally, in a non-isotropic Newtonian fluid, the coefficient that relates internal friction stresses to the
There is general formula for friction force in a liquid: The vector
Newtonian law of viscosity
The following equation illustrates the relation between shear rate and shear stress for a fluid with laminar flow only in the direction x:
- 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)
- is the viscosity, and
- is the flow velocity gradient along the direction y, that is normal to the flow velocity .
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:
- 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 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:
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.
See also
- Fluid mechanics
- Non-Newtonian fluid
- Strain rate tensor
- Viscosity
- Viscous stress tensor
References
- ISBN 978-1-118-01343-4.
- ISBN 978-0-521-66396-0.
- ^ Kundu, P.; Cohen, I. Fluid Mechanics. p. (page needed).
- ISBN 978-0-521-11903-0– via kirbyresearch.com.
- ^ a b c Batchelor (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.
- ISBN 978-1-61942-696-2.