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A finite element algorithm is presented for the simulation of steady incompressible fluid flow with heat transfer using triangular meshes. The continuity equation is modified by employing the artificial compressibility concept to provide coupling between the pressure and velocity fields of the fluid. A standard Galerkin finite element method is used for spatial discretization and an explicit multistage Runge-Kutta scheme is used to march in the time domain. The resulting procedure is stabilized using an artificial dissipation technique. To demonstrate the performance of the proposed algorithm a wide range of test cases is solved including applications with and without heat transfer. Both natural and forced convection applications are studied.

PurposeThis paper sets out to present a fully explicit smoothed particle hydrodynamics (SPH) method to solve non‐Newtonian fluid flow problems.Design/methodology/approachThe governing equations are momentum equations along with the continuity equation which are described in a Lagrangian framework. A new treatment similar to that used in Eulerian formulations is applied to viscous terms, which facilitates the implementation of various inelastic non‐Newtonian models. This approach utilizes the exact forms of the shear strain rate tensor and its second principal invariant to calculate the shear stress tensor. Three constitutive laws including power‐law, Bingham‐plastic and Herschel‐Bulkley models are studied in this work. The imposition of the incompressibility is fulfilled using a penalty‐like formulation which creates a trade‐off between the pressure and density variations. Solid walls are simulated by the boundary particles whose positions are fixed but contribute to the field variables in the same way as the fluid particles in flow field.FindingsThe performance of the proposed algorithm is assessed by solving three test cases including a non‐Newtonian dam‐break problem, flow in an annular viscometer using the aforementioned models and a mud fluid flow on a sloping bed under an overlying water. The results obtained by the proposed SPH algorithm are in close agreement with the available experimental and/or numerical data.Research limitations/implicationsIn this work, only inelastic non‐Newtonian models are studied. This paper deals with 2D problems, although extension of the proposed scheme to 3D is straightforward.Practical implicationsThis study shows that various types of flow problems involving fluid‐solid and fluid‐fluid interfaces can be solved using the proposed SPH method.Originality/valueUsing the proposed numerical treatment of viscous terms, a unified and consistent approach was devised to study various non‐Newtonian flow models.

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SUMMARYA modified weakly compressible smoothed particle hydrodynamics (WCSPH) is presented, which utilizes consistent discretization schemes for spatial derivatives in the flow equations. Here, each SPH particle is considered as a computational point that represents a specific part of the fluid. To overcome non-physical oscillations that usually arise in standard WCSPH, we modified the mass conservation equation by using a numerical filter. This modification is based on the difference between two discretization schemes used for the term r rP Á. Furthermore, a new implementation of wall boundary condition in SPH is introduced. This condition is imposed on the pressure of wall boundary particles to ensure that the acceleration of each boundary particle in normal direction to the wall is zero. Thus, no penetration through walls will occur. To examine the performance of the modified method, we solved a series of two-dimensional incompressible internal flow benchmark problems. By comparing the result with analytical solutions and the results of the standard WCSPH, we show that the use of consistent schemes in conjunction with the proposed numerical filter improves both accuracy and speed of the numerical method.

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The flow of non-Newtonian fluids through two-dimensional porous media is analyzed at the pore scale using the smoothed particle hydrodynamics (SPH) method. A fully explicit projection method is used to simulate incompressible flow. This study focuses on a shear-thinning power-law model (n < 1), though the method is sufficiently general to include other stress-shear rate relationships. The capabilities of the proposed method are demonstrated by analyzing a Poiseuille problem at low Reynolds numbers. Two test cases are also solved to evaluate validity of Darcy's law for power-law fluids and to investigate the effect of anisotropy at the pore scale. Results show that the proposed algorithm can accurately simulate non-Newtonian fluid flows in porous media.