Rheology

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Continuum mechanics
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Navier–Stokes equations
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Rheology is the study of the deformation and flow of matter under the influence of an applied stress, which might be, for example, a shear stress or extensional stress. The experimental characterisation of a material's rheological behaviour is known as rheometry, although the term rheology is frequently used synonymously with rheometry, particularly by experimentalists. Theoretical aspects of rheology are the relation of the flow/deformation behaviour of material and its internal structure (e.g. the orientation and elongation of polymer molecules), and the flow/deformation behaviour of materials that cannot be described by classical fluid mechanics or elasticity. This is also often called Non-Newtonian fluid mechanics in the case of fluids.

The term rheology was coined by Eugene Bingham, a professor at Lafayette College, in 1920, from a suggestion by a colleague, Markus Reiner. The term was inspired by the quotation mistakenly attributed to Heraclitus, (actually coming from the writings of Simplicius) panta rei, "everything flows".

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[edit] Scope

In practice, rheology is principally concerned with extending the "classical" disciplines of elasticity and (Newtonian) fluid mechanics to materials whose mechanical behavior cannot be described with the classical theories. It is also concerned with establishing predictions for mechanical behavior (on the continuum mechanical scale) based on the micro- or nanostructure of the material, e.g. the molecular size and architecture of polymers in solution or the particle size distribution in a solid suspension.

Continuum mechanics Solid mechanics or strength of materials Elasticity
Plasticity Rheology
Fluid mechanics Non-Newtonian fluids
Newtonian fluids

Rheology unites the seemingly unrelated fields of plasticity and non-Newtonian fluids by recognizing that both these types of materials are unable to support a shear stress in static equilibrium. In this sense, a plastic solid is a fluid. Granular rheology refers to the continuum mechanical description of granular materials.

One of the tasks of rheology is to empirically establish the relationships between deformations and stresses, respectively their derivatives by adequate measurements. These experimental techniques are known as rheometry and are concerned with the determination with well-defined rheological material functions. Such relationships are then amenable to mathematical treatment by the established methods of continuum mechanics.

The characterisation of flow or deformation originating from a simple shear stress field is called shear rheometry (or shear rheology). The study of extensional flows is called extensional rheology. Shear flows are much easier to study and thus much more experimental data are available for shear flows than for extensional flows.

[edit] Applications

Rheology has important applications in engineering, geophysics and physiology. In particular, hemorheology, the study of blood flow, has an enormous medical significance. In geology, solid Earth materials that exhibit viscous flow over long time scales are known as rheids. In engineering, rheology has had its predominant application in the development and use of polymeric materials (plasticity theory has been similarly important for the design of metal forming processes, but in the engineering community is often not considered a part of rheology). Rheology modifiers are also a key element in the development of paints in achieving paints that will level but not sag on vertical surfaces.

[edit] Elasticity, viscosity, solid- and liquid-like behavior, plasticity

One generally associates liquids with viscous behaviour (a thick oil is a viscous liquid) and solids with elastic behaviour (an elastic string is an elastic solid). A more general point of view is to consider the material behaviour at short times (relative to the duration of the experiment/application of interest) and at long times.

Liquid and solid character are relevant at long times

We consider the application of a constant stress (a so-called creep experiment):

  • if the material, after some deformation , eventually resists further deformation, it is considered a solid
  • if, by contrast, the material flows indefinitely, it is considered a liquid
By contrast, elastic and viscous (or intermediate, viscoelastic) behaviour is relevant at short times (transient behaviour)

We again consider the application of a constant stress:

  • if the material deformation follows the applied stress, then the material is purely elastic
  • if the deformation increases linearly at constant stress, then the material is viscous
  • if neither the deformation with time, nor its derivative (deformation rate) follows the stress, the material is viscoelastic
Plasticity is equivalent to the existence of a yield stress

A material that behaves as a solid under low applied stresses may start to flow above a certain level of stress, called the yield stress of the material. The term plastic solid is often used when this plasticity threshold is rather high, while yield stress fluid is used when the threshold stress is rather low. There is no fundamental difference, however, between both concepts.

[edit] Dimensionless numbers in rheology

Deborah number

When the rheological behavior of a material includes a transition from elastic to viscous as the time scale increase (or, more generally, a transition from a more resistant to a less resistant behavior), one may define the relevant time scale as a relaxation time of the material. Correspondingly, the ratio of the relaxation time of a material to the timescale of a deformation is called Deborah number. Small Deborah numbers correspond to situations where the material has time to relax (and behaves in a viscous manner), while high Deborah numbers correspond to situations where the material behaves rather elastically.

Note that the Deborah number is relevant for materials that flow on long time scales (like a Maxwell fluid) but not for the reverse kind of materials (like the Voigt or Kelvin model) that are viscous on short time scales but solid on the long term.

Reynolds number

In fluid mechanics, the Reynolds number is the ratio of inertial forces (vsρ) to viscous forces (μ/L) and consequently it quantifies the relative importance of these two types of forces for given flow conditions. Thus, it is used to identify different flow regimes, such as laminar or turbulent flow.

It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining dynamic similitude. When two geometrically similar flow patterns, in perhaps different fluids with possibly different flow rates, have the same values for the relevant dimensionless numbers, they are said to be dynamically similar.

Typically it is given as follows:

\mathit{Re} = {\rho v_{s}^2/L \over \mu v_{s}/L^2} = {\rho v_{s} L\over \mu} = {v_{s} L\over \nu} = \frac{\mbox{Inertial forces}}{\mbox{Viscous forces}}

where:

  • vs - mean fluid velocity, [m s-1]
  • L - characteristic length, [m]
  • μ - (absolute) dynamic fluid viscosity, [N s m-2] or [Pa s]
  • ν - kinematic fluid viscosity: ν = μ / ρ, [m² s-1]
  • ρ - fluid density, [kg m-3].

[edit] External links

Journals covering rheology
Organizations concerned with the study of rheology
Rheology Conferences, Seminars, and Workshops
Other Resources
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General subfields within physics