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This note is concerned with efficient Newton-multigrid FEM solver for multifield nonlinear flow problems. In our approach, for efficient FEM solver, we advantageously use the delicate symbiosis aspects of the problem settings for FEM approximations, and the algorithmic tools to obtain the numerical solutions. We concretize our ideas on thixoviscoplastic flow problems. It is a two-field coupled nonlinear problem. And beside the integrated nonlinearity within momentum and microstructure equations, thixoviscoplastic problems induce a nonlinear two-way coupling. As far as FEM numerical solutions are concerned, we set the problem in a suitable variational form to use the corresponding wellposedness analysis to develop FEM techniques for the solver. Indeed, the wellposedness study is not an intellectual exercise, rather it is the foundation for the approximate thixoviscoplastic problem as well as for the development of an efficient solver. We base our investigations for the solver on our wellposedness and error analysis results of thixoviscoplastic flow problems published in Proc. Appl. Math. Mech. [1, 2]. We continue our series, and proceed to develop a monolithic Newton-multigrid thixoviscoplastic solver. The solver is based on Newton’s method and geometric multigrid techniques to treat the coupling of the problem. So, we use Local Pressure Schur Complement (LPSC) concept to solve the coupled problem on mesh’s elements, and proceed with outer blocks Gauss-Seidel iteration to update the global solutions. Furthermore, we handle the nonlinearity of the problem with the combined adaptive discrete Newton’s and multigrid methods. The adaptivity within discrete Newton’s method is based on the adaptive step-length control for the discrete differencing in the Jacobian calculations, while the convergence of linear multigrid solver is made to match the convergence requirement of nonlinear solver, accordingly. And the solver’s update parameters are solely dependent on the actual convergence rate of the nonlinear problem. We provide the numerical results of solver performance for thixoviscoplastic lid-driven cavity flow., Ergebnisberichte des Instituts für Angewandte Mathematik;665

We are concerned, in this work, with Finite Element Method (FEM) for modeling and simulation of thixotropy in viscoplastic materials. We use a quasi-Newtonian approach to integrate the constitutive equation, which results in a new thixo-viscoplastic (TYP) generalized Navier-Stokes (N-8) equations. To solve the corresponding flow fields at once, we developed a FEM TYP solver based on monolithic Newton-multigrid method. The phenomenologi­cal process of competition of breakdown and buildup characteristics of thixotropic material is replicated throughout, localization and shear banding for Couette flow on one hand, and induction of more shear rejuvenation layers nearby walls for contraction flow on the other hand., Ergebnisberichte des Instituts für Angewandte Mathematik;663

In this contribution, we shall be concerned with the question of wellposedness of thixoviscoplastic flow problems in context of FEM approximations.We restrict our analysis to a quasiNewtonian modeling approach with the aim to set foundations for an efficient monolithic Newtonmultigrid solver. We present the wellposedness of viscoplastic subproblems and structure subproblems in parallel/independent fashion showing the possibility for a combined treatment. Then, we use the fixed point theorem for the coupled problem. For the numerical solutions, we choose 4:1 contraction configuration and use monolithic Newton-multigrid solver. We analyse the effect of taking into consideration thixotropic phenomena in viscoplastic material and opening up for more different coupling by inclusions of shear thickening and shear thinning behaviors for plastic viscosity and/or elastic behavior below the critical yield stress limit in more a general thixotropic models.

This note is concerned with error analysis of FEM approximations for quasi-Newtonian modelling of thixo-viscoplastic, TVP, flow problems. The developed FEM settings for thixotropic generalized Navier-Stokes equations is based on a constrained monotonicity and continuity for the coupled system, which is a cornerstone for an efficient monolithic Newton-multigrid solver. The manifested coarseness in the energy inequality by means of proportional dependency of its constants on regularization parameter, nonoptimal estimate for microstructure, and extra regularization requirement for velocity, is due to weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with higher order stabilized FEM. Furthermore, we showed the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical simulations of TVP flow problems in a 4:1 contraction configuration., Ergebnisberichte des Instituts für Angewandte Mathematik;656

[EN] This note is concerned with the application of Finite Element Method (FEM) and NewtonMultigrid solver to simulate thixo-viscoplastic flows. The thixo-viscoplastic stress dependent on material microstructure is incorporated via viscosity approach into generalized Navier-Stokes equations. The full system of equations is solved in a monolithic framework based on Newton-Multigrid FEM Solver. The developed solver is used to analyze the thixo-viscoplastic flow problem in a Lid-driven cavity configuration., The authors acknowledge the funding provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 446888252. Additionally, the authors acknowledge the financial grant provided by the Bundesministerium fr Wirtschaft und Energie aufgrund eines Beschlusses des Deutschen Bundestages through AiF-Forschungsvereinigung: Forschungs- Gesellschaft Verfahrens Technik e. V. - GVT under the IGF project number 20871 N. We would also like to gratefully acknowledge the support by LSIII and LiDO3 team at ITMC, TU Dortmund University, Germany.

This contribution is concerned with the application of Finite Element Method (FEM) and Newton-Multigrid solvers to simulate thixotropic flows. The thixotropic stress dependent on material microstructure is incorporated via viscosity approach into generalized Navier-Stokes equations. The full system of equations is solved in a monolithic framework based on Newton-Multigrid FEM Solver. The developed solver is used to analyse the thixotropic viscoplastic flow problem in 4:1 contraction configuration., Proceedings in applied mathematics and mechanics;21(1)

The purpose of present study is to numerically investigate the natural convection flow of Ostwalde-de Waele type power law non-Newtonian fluid along the surface of rotating axi-symmetric round-nosed body. For computational purpose rotating hemisphere is used as a case study in order to examine the heat transfer mechanism near such transverse curvature geometries. The numerical scheme is applied after converting the dimensionless system of equations into primitive variable formulations. Implicit finite difference method is used to integrate the equations numerically. Its worth mentioning that all the numerical simulations performed here are valid particularly for the class of shear thickening fluid with wide range of Prandtl number, i.e. (10:0 ≤ Pr ≤ 1500:0). A detailed discussion is done to understand the effects of buoyant forces and power-law exponents on the rate of heat transfer and skin friction coefficient at the surface of the hemisphere. Comparison of present numerical results for different values of buoyancy ratio parameter λ with other published data has been shown in graphical form. For the first time the velocity profiles are plotted at the point of separation, which occurs when the portion of the boundary layer closest to the wall or leading edge reverses in flow direction. It is recorded that an increase in the power-law index n and Prandtl number Pr leads to an increase in the friction factor as well as in the rate of heat transfer., Ergebnisberichte des Instituts für Angewandte Mathematik;582

The intent of this paper is to establish the detailed parametric study for laminar natural convection ow along a vertical wavy plate. Typical sinusoidal surface is used to elu- cidate the heat transport phenomena for the gas having variable thermophysical properties. From the present analysis, we will interrogate whether the presence of roughness element disturbs the gas ow and alter the physical characteristics associates with the wavy surface or not? The numerical solutions are obtained after converting the governing equations into a suitable coordinate system. The results are interpreted for the parameters which emerge from the temperature dependent physical properties of the gases and transverse curvature of the surface. In order to ensure the accuracy, the present numerical results are also compared with some special cases and are found in good agreement. The key observation from the present analysis is that the amplitude of wavy surface parameter, a contributes in reduction of heat transfer rate., Ergebnisberichte des Instituts für Angewandte Mathematik;560

The paper considers the inuence of thermal Maragoni convection on boundary layer flow of two-phase dusty fluid along a vertical wavy surface. The dimensionless boundary layer equations for two-phase problem are reduced to a convenient form by primitive variable transformation (PVF) and then integrated numerically by employing the implicit finite difference method along with the Thomas Algorithm. The effect of thermal Maragoni convection, dusty water and sinusoidal waveform are discussed in detail in terms of local heat transfer rate, skin friction coefficient, velocity and temperature distributions. This investigation reveals the fact that the water-particle mixture reduces the rate of heat transfer, significantly., Ergebnisberichte des Instituts für Angewandte Mathematik;561

The purpose of the present study is to establish the detailed parametric solutions for laminar natural convection flow of two-phase dusty fluid moving along a vertical wavy plate. Typical sinusoidal surface is used to elucidate the heat transport phenomena for the carrier gas having variable thermophysical properties. The governing equations are cast into a system of parabolic partial differential equations by using set of continuous transformations and then the resulting system is integrated through implicit finite difference method. In order to ensure the accuracy, the present numerical results are also compared with the available published results and good compatibility is found between the present and previous results. It is showed that mass concentration parameter, D_p, and the variable thermophysical properties extensively promotes the rate of heat transfer near the uneven surface. It is also established that amplitude of the wavy surface enhances very drastically, which is also due to the variable properties of the fluid., Ergebnisberichte des Instituts für Angewandte Mathematik;559

This article is concerned with the class of solutions of gas boundary layer containing uniform, spherical solid particles over the surface of rotating axi-symmetric roundnosed body. By using the method of transformed coordinates, the boundary-layer equations for two-phase flow are mapped into a regular and stationary computational domain and then solved numerically by using implicit finite difference method. In this study, a rotating hemisphere is used as a particular example to elucidate the heat transfer mechanism near the surfaces of round nosed bodies. We will also investigate whether the presence of dust particles in carrier fluid disturb the flow characteristics associated with rotating hemisphere or not. A comprehensive parametric analysis is presented to show the influence of the particle loading, the buoyancy ratio parameter and the surface of rotating hemisphere on the numerical findings. In the absence of dust particles, the results are graphically compared with existing data in the open literature and an excellent agreement has been found. It is noted that, the concentration of dust particles parameter, D_p, strongly influence the heat transport rate near the leading edge., Ergebnisberichte des Instituts für Angewandte Mathematik;562

An analysis is performed to study two-phase natural convection flow with heat transfer of nanofluid along a vertical way surface. The model includes equations expressing conservation of total mass, momentum and thermal energy for two-phase nano fluid. Primitive variable formulations (PVF) are used to transform the dimensionless boundary layer equations into convenient coordinate system and resulting equations are integrated numerically via implicit finite difference iterative scheme. The effects of the controlling parameters on the dimensionless quantities such as skin friction coeffcient, rate of heat transfer and rate of mass transfer are explored. It is concluded from present analysis, that the diffusivity ratio parameter, NA and particle-density increment number, NB has pronounced effect in reduction of heat transfer rate., Ergebnisberichte des Instituts für Angewandte Mathematik;557

The prime purpose of this analysis is to investigate the effect of variable thermophysical properties of nano fluid on bioconvection boundary layer flow past a uniformly heated vertical cone. The governing equations with associated boundary conditions for this phenomenon are cast into a non-dimensional form via continuous transformation, which are then solved by using the implicit finite difference method. How does the phenomena of heat transfer is effected due to the temperature-dependent thermophysical properties of the fluid is the question investigated in this manuscript. Comprehensive numerical computations are carried out for wide range of the parameters that are describing the flow characteristics. The influence of the governing physical parameters on the distribution of velocity and temperature profiles, wall skin friction and local rate of heat transfer is explored. Tabular comparison with some special cases of published data is done and good agreement is found. It is recorded that, the variable thermophysical properties of the fluid are more likely to promote the rate of heat transfer, Q_w as compared to fluid having constant properties., Ergebnisberichte des Instituts für Angewandte Mathematik;552