The K-BKZ constitutive model is now 50 years old. The paper reviews the connections of the model and its variants with continuum mechanics and experiment, presenting an up-to-date recap of research and major findings in the open literature. In the Introduction a historical perspective is given on developments in the last 50 years of the K-BKZ model. Then a section follows on mathematical modeling of polymer flows, including governing equations of flow, rheological constitutive equations (with emphasis on viscoelastic integral constitutive equations of the K-BKZ type), dimensionless numbers, and boundary conditions. The Method of Solution section reviews the major developments of techniques necessary for particle tracking and calculation of the integrals for the viscoelastic stresses in flow problems. Finally, selected examples are given of successful application of the K-BKZ model in polymer flows relevant to rheology. 1. Introduction 1.1. Rheology Rheology is defined as the “study of deformation and flow of matter” [1, 2]. The term has ancient Greek roots that refer back to the 6th BC, when the Greek philosopher Heraclitus realized the relative change of all elements in his well-known motto “ ” or “everything flows” [3]. In our days the term “rheology” was first used in 1920 by the American chemistry professor Eugene Bingham in Lafayette College, Indiana, USA. Bingham consulted with colleagues in the Department of Classical Studies in his effort to explain the peculiar behavior of various colloidal solutions [4]. The term “rheology” and its above definition were accepted by the (American) Society of Rheology (SOR), founded in 1929 with its first president being Prof. Bingham. Many other national rheological societies have since come to being, with the European Society of Rheology (ESR) established in 1996 and encompassing many individual European societies. The various rheology societies celebrate every four years the advancements in rheology at the International Congress on Rheology. The last one took place in Lisbon, Portugal, in August 2012 [5]. According to the Heraclitian definition, the term “rheology” could be used for all materials, including the classical limit cases of Newtonian fluids, such as water, and elastic Hookean solids, such as metals. However, these limiting cases are often considered outside the scope of rheology, which deals mainly with materials characterized by complex behavior. As an example, the 1st annual meeting of SOR in 1929 in the USA included presentations on asphalt, lubricants, paint, plastics, and rubber, which gives
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