On the Bingham Flow in a Thin Y-Like Shaped Structure

Abstract

We consider the steady Bingham flow in a two-dimensional thin Y-like shaped structure, with no-slip boundary conditions and under the action of given external forces. After passage to the limit with respect to a small parameter related to the thickness of the domain, we obtain three uncoupled problems. Each of these problems describes an anisotropic flow, corresponding to a lower-dimensional “Bingham-like” constitutive law. These results are in accordance with the engineering models.

Numerical Study of the Equation on the Graph for the Steady State Non-Newtonian Flow in Thin Tube Structure

“…For the reader's convenience we remind the definitions of the tube structure and its graph given in [13]. 3, and e 1 , e 2 , ..., e M be M closed segments each connecting two of these points (i.e., each e j = O ij O kj , where i j , k j ∈ {1, ..., N }, i j ̸ = k j ). All points O i are supposed to be the ends of some segments e j .…”

“…Nonlinear equations on the graph are generated by the problems in thin tube structures for viscous flows with non-Newtonian rheology. The case of the Stokes equations with the shear rate dependent viscosity was considered in the set of three papers [21,22,23], where the complete asymptotic expansion of the solution was constructed (also see [3,4] for the Bingham law rheology and [9] for the power law rheology). In particular, in [22] a nonlinear equation on the graph is studied.…”

Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

“…A review of works related to problems of diffusion-convection in thin tubular structures is given in [24]. Here we mention the works [5,7,21] that we discovered while working on this article. The steady Bingham flow was studied in a two-dimensional thin Y -like shaped structure and three uncoupled problems were obtained in the limit in [7]; in the other ones, the incompressible stationary fluids flowing through multiple pipe systems were studied via asymptotic analysis.…”

“…Here we mention the works [5,7,21] that we discovered while working on this article. The steady Bingham flow was studied in a two-dimensional thin Y -like shaped structure and three uncoupled problems were obtained in the limit in [7]; in the other ones, the incompressible stationary fluids flowing through multiple pipe systems were studied via asymptotic analysis. Another related scenario occurs for thin debris-filled wellbores or when modelling effective surface roughness in any kind of tubes and channels with varying aperture (see e.g.…”