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 ...

, including

fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...

, an upper-convected time derivative or Oldroyd derivative, named after James G. Oldroyd, is the rate of change of some

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 tens ...

property of a small parcel of fluid that is written in the coordinate system rotating and stretching with the fluid. The operator is specified by the following formula: :

\stackrel = \frac \mathbf - (\nabla \mathbf)^T \cdot \mathbf - \mathbf \cdot (\nabla \mathbf)

where: *

is the upper-convected time derivative of a tensor

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

\mathbf

\frac

is the

substantive derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material de ...

\nabla \mathbf=\frac

is the tensor of

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

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. ...

s for the fluid. The formula can be rewritten as: :

_ = \frac   + v_k \frac   - \frac   A_ - \frac   A_

By definition, the upper-convected time derivative of the

Finger tensor In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...

is always zero. It can be shown that the upper-convected time derivative of a spacelike vector field is just its

Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vecto ...

by the velocity field of the continuum. The upper-convected derivative is widely used in

polymer A polymer (; Greek ''poly-'', "many" + '' -mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic and ...

rheology Rheology (; ) is the study of the flow of matter, primarily in a fluid (liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an appli ...

for the description of the behavior of a

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 linear ...

fluid under large deformations.

Examples for the
symmetric tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of ord ...
''A''

Simple shear

For the case of simple shear: :

\nabla \mathbf = \begin 0 & 0 & 0 \\  & 0 & 0 \\ 0 & 0 & 0 \end

Thus, :

\stackrel = \frac \mathbf-\dot \gamma \begin 2 A_ & A_ & A_ \\ A_ & 0 & 0 \\ A_ & 0 & 0 \end

Uniaxial extension of incompressible fluid

In this case a material is stretched in the direction X and compresses in the directions Y and Z, so to keep volume constant. The gradients of velocity are: :

\nabla \mathbf = \begin \dot \epsilon & 0 & 0 \\ 0 & -\frac   & 0 \\ 0 & 0 & -\frac 2 \end

Thus, :

\stackrel = \frac \mathbf-\frac  2 \begin 4A_ & A_ & A_ \\ A_ & -2A_ & -2A_ \\ A_ & -2A_ & -2A_ \end

References

* {{cite book , author=Macosko, Christopher, title=Rheology. Principles, Measurements and Applications , publisher=VCH Publisher , year=1993 , isbn=978-1-56081-579-2 ;Notes Multivariable calculus Fluid dynamics Non-Newtonian fluids

Examples for the symmetric tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of ord ... ''A''

Simple shear

Uniaxial extension of incompressible fluid

See also

References

Examples for the
symmetric tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of ord ...
''A''