A new mathematical technique for analysis of internal viscoplastic flows through rectangular ducts

Abstract

The flow of Bingham materials inside rectangular channels is characterized by a plug region located in the central region of the conduit and four dead zones in the vicinity of the corners of the geometry (i.e., the unyielded regions) that grow as the Bingham number is increased. In this paper, a semianalytical technique is presented for solving the flow of Bingham yield-stress materials inside straight rectangular ducts. To capture the topological shapes of the yielded and unyielded regions, a “squircle” mapping approach is proposed and the interfaces between the yielded and unyielded regions are captured based on minimizing the variational formulation for the velocity principle. The nonlinear governing equations are consequently solved using the Ritz method. To find the optimized solution, a genetic algorithm method is implemented, applying bounded constraints and a sufficient number of populations. The presented results are benchmarked against numerical works available in literature, revealing that the presented solution can accurately capture the positions of both the plug and dead regions. The critical Bingham number at which the plug region meets the dead regions is reported for different aspect ratios. The velocity profiles and the friction factor of flow with different Bingham numbers and aspect ratios are investigated in detail. Following this, in the limiting case in which the Bingham number is sufficiently small, an alternative, simpler analytical approach using elliptical mapping is also proposed. The authors believe that the proposed method could be employed as a useful tool to obtain fast, accurate approximate analytical solutions for similar classes of problem.