Non-Newtonian flow equations

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as... 

It is evident that application of Green s theorem cannot eliminate second-order derivatives of the shape functions in the set of working equations of the least-sc[uares scheme. Therefore, direct application of these equations should, in general, be in conjunction with C continuous Hermite elements (Petera and Nassehi, 1993 Petera and Pittman, 1994). However, various techniques are available that make the use of elements in these schemes possible. For example, Bell and Surana (1994) developed a method in which the flow model equations are cast into a set of auxiliary first-order differentia] equations. They used this approach to construct a least-sciuares scheme for non-Newtonian flow equations based on equal-order C° continuous, p-version hierarchical elements. 

The momentum balance equations can be written in a form that is valid for the Navier-Stokes equations as well as low Reynolds number non-Newtonian flow equations ... 

Cross, M.M. (1965). Rheology of non-Newtonian flow equation for pseudoplastic systems. Journal of Colloid Science, Vol.20, pp. 417-437 de Waele, A. (1923). Oil Color, Chemical Association Journal, Vol.6, pp. 23-88 Edwards, M.F., Jadallah, M.S.M. and Smith, R. (1985). Head losses in pipe fittings at low Reynolds numbers. Chemical Engineering Research and Design, Vol.63, (January 1985) pp. 43-50... 

---