Slip flows of Newtonian and viscoelastic fluids in a 4:1 contraction

This work presents a numerical study of the 4:1 planar contraction flow of a viscoelastic fluid described by the simplified Phan-Thien–Tanner model under the influence of slip boundary conditions at the channel walls. The linear Navier slip law was considered with the dimensionless slip coefficient varying in the range ½0; 4500. The simulations were carried out for a small constant Reynolds number of 0.04 and Deborah numbers (De) varying between 0 and 5. Convergence could not be achieved for higher values of the Deborah number, especially for large values of the slip coefficient, due to the large stress gradients near the singularity of the reentrant corner. Increasing the slip coefficient leads to the formation of two vortices, a corner and a lip vortex. The lip vortex grows with increasing slip until it absorbs the corner vortex, creating a single large vortex that continues to increase in size and intensity. In the range De = 3–5 no lip vortex was formed. The flow is characterized in detail for De ¼ 1 as function of the slip coefficient, while for the remaining De only the main features are shown for specific values of the slip coefficient.

Modelling integral viscoelastic flows in OpenFOAM

[Excerpt] A large number of constitutive equations were developed for viscoelastic fluids, some empirical and other with strong physical foundations. The currently available macroscopic constitutive equations can be divided in two main types: differential and integral. Some of the constitutive equations, e.g. Maxwell are available both in differential and integral types. However, relevant in tegral models, like K - BKZ, just possesses the integral form. (...)

Finite integration methods for isospectral flows

In this paper we consider the approximate computation of isospectral flows based on finite integration methods( FIM) with radial basis functions( RBF) interpolation,a new algorithm is developed. Our method ensures the symmetry of the solutions. Numerical experiments demonstrate that the solutions have higher accuracy by our algorithm than by the second order Runge- Kutta( RK2) method.

Green's relations, regularity and abundancy for semigroups of quasi-onto transformations

Let V be an infinite-dimensional vector space and for every infinite cardinal n such that n≤dimV, let AE(V,n) denote the semigroup of all linear transformations of V whose defect is less than n. In 2009, Mendes-Gonçalves and Sullivan studied the ideal structure of AE(V,n). Here, we consider a similarly-defined semigroup AE(X,q) of transformations defined on an infinite set X. Quite surprisingly, the results obtained for sets differ substantially from the results obtained in the linear setting.