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A generalized rheological model of thixotropic materials
Authors:Z. Kembłowski  J. Petera
Affiliation:(1) Institute of Chemical Engineering, "Lstrok"ód"zacute" Technical University, ul. Wólcza"nacute"ska 175, PL-90-924 "Lstrok"ód"zacute", Poland
Abstract:Summary A generalization of the rheological model of thixotropic materials, presented previously, was carried out. In the generalized rheological equation of state the yield stress depending on the structural parameter was introduced. In the generalized rate equation the difference in the destruction and recovery rates of the material structure was taken into account. A procedure leading to the determination of nine rheological parameters of the generalized model was worked out. The model was checked experimentally for a thixotropic paint.
Zusammenfassung Eine früher dargestellte Theorie thixotroper Stoffe wird verallgemeinert, wobei eine von dem Strukturparameter abhängige Fließspannung eingeführt wird. Weiterhin wird der Unterschied zwischen der Zerstörungs-und der Wiederaufbaugeschwindigkeit der Stoffstruktur berücksichtigt. Eine Methode zur Bestimmung der neun benötigten Stoffparameter wird ausgearbeitet. Das Modell wird am Beispiel einer thixotropen Farbe experimentell geprüft.

Notation a rheological parameter in eq. [26], s–1 - A rheological parameter in eq. [16] - b rheological parameter in eq. [26] - c function in eq. [21] - 
$$bar c$$
averaged value of functionc in eq. [28] - cLambda function in the rate equation [23], defined by eq. [21] - G function [1] defining material of the rate type - h function [2] determining the state of thixotropic fluid - k rheological parameter in the Herschel-Bulkley equation [17] or, in special case, in eq. [8], Nsn/m2 - K function in eq. [18], Nsm/m2 - m rheological parameter in eq. [18] or, in special case, in eq. [10] - n rheological parameter in the Herschel-Bulkley model [17] or, in special case, in eq. [8] - s rheological parameter in eq. [16] - t time, s - x arbitrary real variable - agr rheological parameter in eq. [9], s - 
$$dot gamma $$
shear rate, s–1 - kappa structural parameter, defined by eq. [2] - 
$$dot k$$
substantial derivative of structural parameter, s–1 - kappae function [6] describing the equilibrium curve in the coordinate system (
$$left( {dot gamma , k} right)$$
) - kappa0 initial value of structural parameter (att = 0) - Lambda natural time function of the thixotropic material, defined by eq. [22] - tau shear stress, N/m2 - 
$$dot tau $$
substantial derivative of shear stress, N/m2 s - taue function describing equilibrium flow curve in the coordinate system (
$$left( {dot gamma , tau } right)$$
) - tau0 equilibrium yield stress, defined by eq. [12], N/m2 - tauy function of structural parameterkappa describing the yield stress - psgr function in eq. [11]Notation used in the algorithm:(Appendix) i,j,k integer - ke(i) ordinal number of the experimental point at which the line ofkappai = const intersects the equilibrium flow curve - li number of the experiments of the type ldquostepchange of the shear raterdquo - lj number of experimental points in one experiment of the type ldquostep-change of the shear raterdquo - ne number of experimental points on the equilibrium flow curve - nk number of experimental points on the line of constantkappa - ny number of lines of constantkappa - t(j) measured time interval (from the moment of the step-change of shear rate) - 
$$dot gamma left( {i, k} right)$$
abscissa of the experimental point of ordinal numberk on the line ofkappai = const, in the coordinate system (
$$left( {dot gamma , tau } right)$$
) - 
$$dot gamma _e (i)$$
abscissa of the experimental point of ordinal numberi on the equilibrium flow curve, in the coordinate system (
$$left( {dot gamma , tau } right)$$
) - 
$$dot gamma _s (i)$$
shear rate at which the experiment of the type ldquostep-change of shear raterdquo was carried out - taue(i) ordinate of the experimental point of ordinal numberi on the equilibrium flow curve, in the coordinate system (
$$left( {dot gamma , tau } right)$$
) - tauy(i) value of yield stress atkappa = kappai - taus(i,j) experimental value of shear stress at constant value of shear rate
$$bar rho $$
(2i) for time intervalt(j) - taukappa(i,k) ordinate of the experimental point of ordinal numberk on the line ofkappai = const, in the coordinate system (
$$left( {dot gamma , tau } right)$$
) - Deltatau0 the admissible value of the difference between the experimental and theoretical value of shear stressWith 4 figures and 1 table
Keywords:
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