Well-posedness of the free surface problem on a Newtonian fluid between cylinders rotating at different speeds

2021 ◽  
pp. 1-26
Author(s):  
Jiaqi Yang
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Newtonian Fluid ◽  
Mathematical Analysis ◽  
Perturbation Series ◽  
Well Posedness ◽  
Centrifugal Forces ◽  
Analysis Theory ◽  
Concentric Cylinders ◽  
Rigorous Mathematical Analysis

When a liquid fills the semi-infinite space between two concentric cylinders which rotate at different steady speeds, how about the shape of the free surface on top of the fluid? The different fluids will lead to a different shape. For the Newtonian fluid, the meniscus descends due to the centrifugal forces. However, for the certain non-Newtonian fluid, the meniscus climbs the internal cylinder. We want to explain the above phenomenon by a rigorous mathematical analysis theory. In the present paper, as the first step, we focus on the Newtonian fluid. This is a steady free boundary problem. We aim to establish the well-posedness of this problem. Furthermore, we prove the convergence of the formal perturbation series obtained by Joseph and Fosdick in Arch. Ration. Mech. Anal. 49 (1973), 321–380.

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

An implicit scheme for simulation of free surface non-Newtonian fluid flows on dynamically adapted grids

2021 ◽  
Vol 36 (3) ◽  
pp. 165-176
Author(s):  
Kirill Nikitin ◽  
Yuri Vassilevski ◽  
Ruslan Yanbarisov
Keyword(s):  
Free Surface ◽  
Numerical Model ◽  
Newtonian Fluid ◽  
Viscoelastic Fluid ◽  
Fluid Flows ◽  
Numerical Results ◽  
Implicit Scheme ◽  
New Approach

Abstract This work presents a new approach to modelling of free surface non-Newtonian (viscoplastic or viscoelastic) fluid flows on dynamically adapted octree grids. The numerical model is based on the implicit formulation and the staggered location of governing variables. We verify our model by comparing simulations with experimental and numerical results known from the literature.

Study on a Method by Using VB for Failure Diagnosis of Mechatronical Equipments

2012 ◽  
Vol 546-547 ◽  
pp. 1532-1537
Author(s):  
Jiu Jun Zhen ◽  
Hong Ke Yang
Keyword(s):  
Mathematical Logic ◽  
Fault Diagnosis ◽  
Mathematical Analysis ◽  
Rapid Development ◽  
Science And Technology ◽  
Failure Diagnosis ◽  
Analysis Theory ◽  
Maintenance Work

In general, the failure diagnosis and maintenance work of mechatronical equipments have greater difficulty, especially for less experienced technician. However, with the rapid development of science and technology, the mathematical analysis and computer can help us. So, According to the principle, structures and fault characteristics of mechatronical equipments, a method based on combination of the mathematical logic analysis theory and VB software is proposed. The principle of fault diagnosis methods, procedures and examples are included in this article. The results show that the method can effectively improve the efficiency and reduce errors caused by the additional diagnosis of faults.

scholarly journals The Finite Element Numerical Investigation of Free Surface Newtonian and Non-Newtonian Fluid Flows in the Rectangular Tanks

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Puyang Gao
Keyword(s):  
Finite Element ◽  
Free Surface ◽  
Level Set ◽  
Newtonian Fluid ◽  
Mass Conservation ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Two Phase ◽  
Navier Stokes Equation ◽  
Correction Technique

In this paper, we develop a new computational framework to investigate the sloshing free surface flow of Newtonian and non-Newtonian fluids in the rectangular tanks. We simulate the flow via a two-phase model and employ the fixed unstructured mesh in the computation to avoid the mesh distortion and reconstruction. As for the solution of Navier–Stokes equation, we utilize the SUPG finite element method based on the splitting scheme. The same order interpolation functions are then used for velocity and pressure. Moreover, the moving interface is captured via the concise level set method. We take advantage of the implicit discontinuous Galerkin method to handle the solution of level set and its reinitialization equations. A mass correction technique is also added to ensure the mass conservation property. The dam break-free surface flow is simulated firstly to demonstrate the validity of our mathematical model. In addition, the sloshing Newtonian fluid in the tank with flat and rough bottoms is considered to illustrate the feasibility and robustness of our computational scheme. Finally, the development of free surface for non-Newtonian fluid is also studied in the two tanks, and the influence of power-law index on the sloshing fluid flow is analyzed.

scholarly journals Peristaltic Motion of Non-Newtonian Fluid with Heat and Mass Transfer through a Porous Medium in Channel under Uniform Magnetic Field

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Nabil T. M. Eldabe ◽  
Bothaina M. Agoor ◽  
Heba Alame
Keyword(s):  
Magnetic Field ◽  
Mass Transfer ◽  
Porous Medium ◽  
Heat And Mass Transfer ◽  
Newtonian Fluid ◽  
Pressure Rise ◽  
Perturbation Series ◽  
Peristaltic Motion ◽  
Method Of Solution ◽  
Permeability Parameter

This paper is devoted to the study of the peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in the channel under the effect of magnetic field. A modified Casson non-Newtonian constitutive model is employed for the transport fluid. A perturbation series’ method of solution of the stream function is discussed. The effects of various parameters of interest such as the magnetic parameter, Casson parameter, and permeability parameter on the velocity, pressure rise, temperature, and concentration are discussed and illustrated graphically through a set of figures.

Exact solutions for a class of nonlinear free surface flows

1991 ◽  
Vol 434 (1892) ◽  
pp. 677-682 ◽  
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Infinite System ◽  
Free Stream ◽  
Constant Pressure ◽  
Velocity Ratio ◽  
Mapping Function ◽  
Free Surface Flows ◽  
Surface Flows ◽  
Nonlinear Free Surface

We consider a class of inviscid free surface flows where the free surface is of finite length and in which the pressure on the free boundary p b is different from the free stream pressure p ∞ . The aim of the paper is to determine the shape of the free surface as a function of the velocity ratio parameter λ . The free boundary problem is tackled by seeking a mapping z ═ f (ζ) such that the flow past a circle in the ζ-plane maps to a flow with constant pressure p b on the free surface in the z -plane. The formulation leads to an infinite system of coupled nonlinear equations for the coefficients in the mapping function. Remarkably, the system can be solved exactly to yield two families of free surface flows of the form z ═ ζ + λ 2 /ζ + a ( λ ) ln (ζ + b ( λ )/ζ ─ b ( λ )). The nature of the solutions, their limitations and possible extensions to them are discussed.

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