The viscous stress tensor is a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
used in
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
to model the part of the
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
at a point within some material that can be attributed to the
strain rate
In materials science, strain rate is the change in strain (deformation) of a material with respect to time.
The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change wi ...
, the
rate at which it is
deforming around that point.
The viscous stress tensor is formally similar to the
elastic stress tensor (Cauchy tensor) that describes internal forces in an
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
material due to its deformation. Both tensors map the
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
of a
surface element to the density and direction of the stress acting on that surface element. However, elastic stress is due to the ''amount'' of deformation (
strain
Strain may refer to:
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* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
), while viscous stress is due to the ''rate'' of change of deformation over time (strain rate). In
viscoelastic
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 wi ...
materials, whose behavior is intermediate between those of liquids and solids, the
total stress tensor
Total may refer to:
Mathematics
* Total, the summation of a set of numbers
* Total order, a partial order without incomparable pairs
* Total relation, which may also mean
** connected relation (a binary relation in which any two elements are compa ...
comprises both viscous and elastic ("static") components. For a completely fluid material, the elastic term reduces to the
hydrostatic pressure
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
.
In an arbitrary coordinate system, the viscous stress and the strain rate at a specific point and time can be represented by 3 × 3
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
of real numbers. In many situations there is an approximately linear relation between those matrices; that is, a fourth-order
viscosity tensor such that . The tensor has four indices and consists of 3 × 3 × 3 × 3 real numbers (of which only 21 are independent). In a
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. Stresses are proportional to the rate of chang ...
, by definition, the relation between and is perfectly linear, and the viscosity tensor is independent of the state of motion or stress in the fluid. If the fluid is isotropic as well as Newtonian, the viscosity tensor will have only three independent real parameters: a
bulk viscosity Volume viscosity (also called bulk viscosity, or dilatational viscosity) is a material property relevant for characterizing fluid flow. Common symbols are \zeta, \mu', \mu_\mathrm, \kappa or \xi. It has dimensions (mass / (length × time)), and the ...
coefficient, that defines the resistance of the medium to gradual uniform compression; a
dynamic viscosity
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.
Viscosity quantifies the inter ...
coefficient that expresses its resistance to gradual shearing, and a
rotational viscosity coefficient which results from a coupling between the fluid flow and the rotation of the individual particles.
In the absence of such a coupling, the viscous stress tensor will have only two independent parameters and will be symmetric. In
non-Newtonian fluid
A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for exa ...
s, on the other hand, the relation between and can be extremely non-linear, and may even depend on other features of the flow besides .
Definition
Viscous versus elastic stress
Internal
mechanical stresses in a
continuous medium are generally related to deformation of the material from some "relaxed" (unstressed) state. These stresses generally include an
elastic ("static") stress component, that is related to the current ''amount'' of deformation and acts to restore the material to its rest state; and a
viscous stress component, that depends on the ''rate'' at which the deformation is changing with time and opposes that change.
The viscous stress tensor
Like the total and elastic stresses, the viscous stress around a certain point in the material, at any time, can be modeled by a stress tensor, a
linear relationship
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...
between the
normal direction vector of an ideal plane through the point and the local
stress density
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
on that plane at that point.
In any chosen
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
with axes numbered 1, 2, 3, this viscous stress tensor can be represented as a 3 × 3
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
of real numbers:
:
Note that these numbers usually change with the point and time .
Consider an
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
flat
surface element centered on the point , represented by a vector whose length is the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
of the element and whose direction is perpendicular to it. Let be the infinitesimal force due to viscous stress that is applied across that surface element to the material on the side opposite to . The components of along each coordinate axis are then given by
:
In any material, the total stress tensor is the sum of this viscous stress tensor , the elastic stress tensor and the hydrostatic pressure . In a perfectly fluid material, that by definition cannot have static shear stress, the elastic stress tensor is zero:
:
where is the
unit tensor, such that is 1 if and 0 if .
While the viscous stresses are generated by physical phenomena that depend strongly on the nature of the medium, the viscous stress tensor is only a description the local momentary forces between adjacent parcels of the material, and not a property of the material.
Symmetry
Ignoring the torque on an element due to the flow ("extrinsic" torque), the viscous "intrinsic" torque per unit volume on a fluid element is written (as an antisymmetric tensor) as
:
and represents the rate of change of intrinsic angular momentum density with time. If the particles have rotational degrees of freedom, this will imply an intrinsic angular momentum and if this angular momentum can be changed by collisions, it is possible that this intrinsic angular momentum can change in time, resulting in an intrinsic torque that is not zero, which will imply that the viscous stress tensor will have an antisymmetric component with a corresponding
rotational viscosity coefficient.
If the fluid particles have negligible angular momentum or if their angular momentum is not appreciably coupled to the external angular momentum, or if the equilibration time between the external and internal degrees of freedom is practically zero, the torque will be zero and the viscous stress tensor will be symmetric. External forces can result in an asymmetric component to the stress tensor (e.g.
ferromagnetic fluids which can suffer torque by external
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s).
Physical causes of viscous stress
In a solid material, the elastic component of the stress can be ascribed to the deformation of the
bonds between the
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
s and
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s of the material, and may include
shear stress
Shear stress, often denoted by (Greek: tau), 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 ot ...
es. In a fluid, elastic stress can be attributed to the increase or decrease in the mean spacing of the particles, that affects their collision or interaction rate and hence the transfer of
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
across the fluid; it is therefore related to the microscopic
thermal
A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
random component of the particles' motion, and manifests itself as an isotropic
hydrostatic pressure
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
stress.
The viscous component of the stress, on the other hand, arises from the macroscopic ''mean'' velocity of the particles. It can be attributed to
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
or particle
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
between adjacent parcels of the medium that have different mean velocities.
The viscosity equation
The strain rate tensor
In a smooth flow, the rate at which the local deformation of the medium is changing over time (the strain rate) can be approximated by a
strain rate tensor
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 defi ...
, which is usually a function of the point and time . With respect to any coordinate system, it can be expressed by a 3 × 3 matrix.
The strain rate tensor can be defined as the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
strain tensor
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
with respect to time, or, equivalently, as the symmetric part of the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
(derivative with respect to space) of the flow velocity vector :
:
where denotes the velocity gradient. In Cartesian coordinates, is the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
,
:
and therefore
:
Either way, the strain rate tensor expresses the rate at which the mean velocity changes in the medium as one moves away from the point – except for the changes due to
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
of the medium about as a rigid body, which do not change the relative distances of the particles and only contribute to the rotational part of the viscous stress via the rotation of the individual particles themselves. (These changes comprise the
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wit ...
of the flow, which is the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
(rotational) of the velocity; which is also the antisymmetric part of the velocity gradient .)
General flows
The viscous stress tensor is only a linear approximation of the stresses around a point , and does not account for higher-order terms of its
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
. However in almost all practical situations these terms can be ignored, since they become negligible at the size scales where the viscous stress is generated and affects the motion of the medium. The same can be said of the strain rate tensor as a representation of the velocity pattern around .
Thus, the linear models represented by the tensors and are almost always sufficient to describe the viscous stress and the strain rate around a point, for the purpose of modelling its
dynamics. In particular, the local strain rate is the only property of the velocity flow that directly affects the viscous stress at a given point.
On the other hand, the relation between and can be quite complicated, and depends strongly on the composition, physical state, and microscopic structure of the material. It is also often highly non-linear, and may depend on the strains and stresses previously experienced by the material that is now around the point in question.
General Newtonian media
A medium is said to be
Newtonian if the viscous stress is a linear function of the strain rate , and this function does not otherwise depend on the stresses and motion of fluid around . No real fluid is perfectly Newtonian, but
many important fluids, including gases and water, can be assumed to be, as long as the flow stresses and strain rates are not too high.
In general, a linear relationship between two second-order tensors is a fourth-order tensor. In a Newtonian medium, specifically, the viscous stress and the strain rate are related by the
viscosity tensor :
:
The viscosity coefficient is a property of a Newtonian material that, by definition, does not depend otherwise on or .
The strain rate tensor is symmetric by definition, so it has only six linearly independent elements. Therefore, the viscosity tensor has only 6 × 9 = 54 degrees of freedom rather than 81. In most fluids the viscous stress tensor too is symmetric, which further reduces the number of viscosity parameters to 6 × 6 = 36.
Shear and bulk viscous stress
Absent of rotational effects, the viscous stress tensor will be symmetric. As with any symmetric tensor, the viscous stress tensor can be
expressed as the sum of a
traceless
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix ().
It can be proved that the trace o ...
symmetric tensor , and a scalar multiple of the identity tensor. In coordinate form,
:
This decomposition is independent of the coordinate system and is therefore physically significant. The constant part of the viscous stress tensor manifests itself as a kind of pressure, or bulk stress, that acts equally and perpendicularly on any surface independent of its orientation. Unlike the ordinary hydrostatic pressure, it may appear only while the strain is changing, acting to oppose the change; and it can be negative.
The isotropic Newtonian case
In a Newtonian medium that is isotropic (i.e. whose properties are the same in all directions), each part of the stress tensor is related to a corresponding part of the strain rate tensor.
:
where and are the scalar isotropic and the zero-trace parts of the strain rate tensor , and and are two real numbers.
Thus, in this case the viscosity tensor has only two independent parameters.
The zero-trace part of is a symmetric 3 × 3 tensor that describes the rate at which the medium is being deformed by shearing, ignoring any changes in its volume. Thus the zero-trace part of is the familiar viscous shear stress that is associated to progressive
shearing
Sheep shearing is the process by which the woollen fleece of a sheep is cut off. The person who removes the sheep's wool is called a '' shearer''. Typically each adult sheep is shorn once each year (a sheep may be said to have been "shorn" or ...
deformation. It is the viscous stress that occurs in fluid moving through a tube with uniform
cross-section (a
Poiseuille flow
The poiseuille (symbol Pl) has been proposed as a derived SI unit of dynamic viscosity, named after the French physicist Jean Léonard Marie Poiseuille (1797–1869).
In practice the unit has never been widely accepted and most international s ...
) or between two
parallel
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moving plates (a
Couette flow
In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces flow ...
), and resists those motions.
The part of acts as a scalar multiplier (like ), the average ''expansion rate'' of the medium around the point in question. (It is represented in any coordinate system by a 3 × 3 diagonal matrix with equal values along the diagonal.) It is numerically equal to of the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the velocity
:
which in turn is the relative rate of change of
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of the fluid due to the flow.
Therefore, the scalar part of is a stress that may be observed when the material is being compressed or expanded at the same rate in all directions. It is manifested as an extra
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
that appears only while the material is being compressed, but (unlike the true hydrostatic pressure) is proportional to the rate of change of compression rather the amount of compression, and vanishes as soon as the volume stops changing.
This part of the viscous stress, usually called bulk viscosity or volume viscosity, is often important in
viscoelastic
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 wi ...
materials, and is responsible for the
attenuation
In physics, attenuation (in some contexts, extinction) is the gradual loss of flux intensity through a medium. For instance, dark glasses attenuate sunlight, lead attenuates X-rays, and water and air attenuate both light and sound at variable att ...
of
pressure waves in the medium. Bulk viscosity can be neglected when the material can be regarded as incompressible (for example, when modeling the flow of water in a channel).
The coefficient , often denoted by , is called the coefficient of
bulk viscosity Volume viscosity (also called bulk viscosity, or dilatational viscosity) is a material property relevant for characterizing fluid flow. Common symbols are \zeta, \mu', \mu_\mathrm, \kappa or \xi. It has dimensions (mass / (length × time)), and the ...
(or "second viscosity"); while is the coefficient of common (shear) viscosity.
See also
*
Vorticity equation
*
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are 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 Geo ...
References
{{reflist
Tensor physical quantities
Viscosity