Upper Convected Maxwell Article Index for
Upper 		Website Links For
Upper 		 

Information About

Upper Convected Maxwell
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



The model can be written as:
: \mathbf{T} + \lambda \mathbf{T}^{
abla} = 2\eta_0 \mathbf {D}
where:

: \mathbf{T}^{
abla} = rac{\partial}{\partial t} \mathbf{T} + \mathbf{v} \cdot
abla \mathbf{T} - (
abla \mathbf{v})^T \cdot \mathbf{T} - \mathbf{T} \cdot (
abla \mathbf{v})
  • \mathbf{v} is the fluid velocity

  • \eta_0 is material Viscosity at steady Simple Shear ;

  • \mathbf {D} is the tensor of the deformation rate.



CASE OF THE STEADY SHEAR

For this case only two components of the shear stress became non-zero:
:T_{12}=\eta_0 \dot \gamma
and
:T_{11}=2 \eta_0 \lambda {\dot \gamma}^2
where \dot \gamma is the shear rate.

Thus, the Upper Convected Maxwell model predicts for the simple shear that Shear Stress to be proportional to the shear rate and the First Difference Of Normal Stresses (T_{11}-T_{22}) is proportional to the square of the shear rate, the Second Difference Of Normal Stresses (T_{22}-T_{33}) is always zero. In other words, UCM predicts appearance of the first difference of normal stresses but does not predict Non-Newtonian Behavior of the shear viscosity nor the second difference of the normal stresses.

Usually quadratic behavior of the first difference of normal stresses and no second difference of the normal stresses is a realistic behavior of polymer melts at moderated shear rates, but constant viscosity is unrealistic and limits usability of the model.


CASE OF START-UP OF STEADY SHEAR

For this case only two components of the shear stress became non-zero:
:T_{12}=\eta_0 \dot \gamma (1-\exp(- rac t \lambda))
and
:T_{11}=2 \eta_0 \lambda {\dot \gamma}^2 (1 -\exp(- rac t \lambda)(1+ rac t \lambda ))

The equations above describe stresses gradually risen from zero the steady-state values.


CASE OF THE STEADY STATE UNIAXIAL EXTENSION OR UNIAXIAL COMPRESSION

For this case UCM predicts the normal stresses \sigma=T_{11}-T_{22}=T_{11}-T_{33} calculated by the following equation:
: \sigma= rac {2 \eta_0} {1-2\lambda \dot \epsilon} + rac {\eta_0} {1+ \lambda \dot \epsilon}
where \dot \epsilon is the elongation rate.

The equation predicts the elongation viscosity approaching 3 \eta_0 (the same as for the Newtonian Fluid s) for the case of low elongation rate ( \dot \epsilon \ll rac 1 \lambda) with fast deformation thickening with the steady state viscosity approaching infinity at some elongational rate (\dot \epsilon_\infty = rac 1 {2 \lambda}) and at some compression rate (\dot \epsilon_{-\infty} = - rac 1 {\lambda}). This behavior seems to be realistic.


CASE OF SMALL DEFORMATION

For the case of small deformation the nonlinearities introduced by the Upper Convected Derivative disappear and the model became an ordinary model of Maxwell Material .


REFERENCES