Stabilized lowest equal-order mixed finite element method for the Oseen viscoelastic fluid flow

Hussain, Shahid; Mahbub, Md. Abdullah Al; Nasu, Nasrin Jahan; Zheng, Haibiao

doi:10.1186/s13662-018-1916-0

Cited by 7 publications

(7 citation statements)

References 51 publications

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“…where τ denotes the polymeric stress tensor, u the velocity vector and λ denotes the Weissenberg number, which can be defined as the product of the relaxation time and a characteristic strain rate. While 0 < α < 1 is represented as the fraction of viscoelastic viscosity [36]. The rate of the strain tensor can be written as D(u) = 1 2 (∇u + (∇u) T ), and the g a (τ, ∇u) is given by…”

Section: Model Equationsmentioning

confidence: 99%

“…The dimensionless steady-state model equations under the open, bounded and connected domain Ω are considered, with the homogenous Dirichlet boundary condition for the velocity u. Since, in this case there is no inflow boundary, and thus, no boundary condition is required for stress [36]. To this end, we can re-write the OVFF model as problem (O) Find (τ,u, p) such that:…”

Section: Model Equationsmentioning

confidence: 99%

“…In this example, the theoretical convergence rates are examined by applying fluid flow across a domain Ω = [0, 1] × [0, 1] with parameters a = 0, λ = 5.0 and α = 0.5, respectively. Different authors used this experimental pattern for Stokes and Navier-Stokes equations [23,36,40,41], where the function b(x) was chosen to be the exact solution of velocity u. In this context, a right hand-side function is added to the (8) and f in (9) is studied with the help of following true solution.…”

Section: Analytical Solution Testmentioning

confidence: 99%

“…We consider a third example to test the proposed scheme, which is a well-known problem for viscoelastic flow "4-to-1 contraction channel flow problem" which has a huge application in polymeric liquid industries and studied by many authors [20,43]. Moreover, 4-to-1 has been widely used in the literature to show the convergence, stability, and behavior of the streamlines of the contraction channel and the behavior of pressure [36,41]. The geometry of the 4-to-1 contraction commonly occurs in the forming of 'die' for the viscoelastic fluid.…”

Section: -To-1 Contraction Channel Flowmentioning

confidence: 99%

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SUPG Approximation for the Oseen Viscoelastic Fluid Flow with Stabilized Lowest-Equal Order Mixed Finite Element Method

et al. 2019

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In this article, a stabilized mixed finite element (FE) method for the Oseen viscoelastic fluid flow (OVFF) obeying an Oldroyd-B type constitutive law is proposed and investigated by using the Streamline Upwind Petrov-Galerkin (SUPG) method. To find the approximate solution of velocity, pressure and stress tensor, we choose lowest-equal order FE triples P1-P1-P1, respectively. However, it is well known that these elements do not fulfill the in f -sup condition. Due to the violation of the main stability condition for mixed FE method, the system becomes unstable. To overcome this difficulty, a standard stabilization term is added in finite element variational formulation. The technique is applied herein possesses attractive features, such as parameter-free, flexible in computation and does not require any higher-order derivatives. The stability analysis and optimal error estimates are obtained. Three benchmark numerical tests are carried out to assess the stability and accuracy of the stabilized lowest-equal order feature of the OVFF.case [8], and the time-dependent case of the same continuous interpolation was analysed by Baranger in [9].In Newtonian fluid flow, the Oseen equations are abridged to linearize the system. This is because the Oseen fluid flow model is the reduced linearized form of the Newtonian fluid which is described by the Navier-Stokes equation [10,11]. Moreover, the viscoelastic fluid flow model equations are non-linear equations in many terms. Hence, by taking the assumption of creeping fluid flow, the inertial part (u · ∇ u) of the momentum equation can be ignored. In this sense, the non-linearity arise only in the constitutive equation [12,13] which may be introduce in a linear form by fixing u(x) with a known velocity field b(x). The resulting system of equations can explicitly described the parameter space for α, λ and ∇b ∞ , which ensure the well-posedness of the continuous problem and its numerical approximation [14].In the FE framework, the main difficulty arises in the viscoelastic fluid flow due to its hyperbolic constitutive equation. It needs a stabilization (upwinding ) technique to approximate the FE solution. There are two main approaches that are mainly used to solve the Oldroyd-B model by using the OVFF technique; the discontinuous Galerkin (DG) approximation and the (SUPG) estimate to deal with the spurious oscillations of the hyperbolic constitutive equation; see [15][16][17][18][19][20]. Herein, we consider SUPG method to solve the OVFF as an upwinding technique [21]. To the best of our knowledge, V.J. Ervin et al. introduced the SUPG method in [22] to approximate the model equations with the standard FE or Hood-Taylor FE pair (P2 for velocity, P1 for pressure and P1 for continuous stress), where the existence and uniqueness of a solution to the problem was shown. Later, the same authors studied a defect correction method in [23], where they used the same standard FE (P2-P1) to approximate the velocity and pressure but for discontinuous stress. They have used the conforming ...

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Section: Model Equationsmentioning

confidence: 99%

Section: Model Equationsmentioning

confidence: 99%

Section: Analytical Solution Testmentioning

confidence: 99%

Section: -To-1 Contraction Channel Flowmentioning

confidence: 99%

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SUPG Approximation for the Oseen Viscoelastic Fluid Flow with Stabilized Lowest-Equal Order Mixed Finite Element Method

et al. 2019

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“…In order to solve the above-described problems, this work aims to develop a stabilization mechanism. We refer to [11,12] for the related stabilization technique, which has been implemented in the literature for different model issues.…”

Section: Introductionmentioning

confidence: 99%

A Linear Stabilized Incompressible Magnetohydrodynamic Problem with Magnetic Pressure

Hussain,

Bakhet,

AlNemer

et al. 2024

Mathematics

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The objective of this article is to examine, stabilize, and linearize the incompressible magnetohydrodynamic model equations. The approximate solutions are carried out through the lowest equal order mixed finite element (FE) approach, involving variables such as fluid velocity, hydro pressure, magnetic field, and magnetic pressure. The formulation of the variational form for the approximate solution necessitates the use of a pair of approximating spaces. However, these spaces cannot be arbitrarily chosen; they must adhere to strict stability conditions, notably the Ladyzhenskaya–Babuska–Brezzi (LBB) or inf-sup condition. This study addresses the absence of stabilization and linearization techniques in the incompressible magnetohydrodynamic model equations using the lowest equal order mixed finite element approach. The article introduces a stabilization technique to meet two stability conditions, proving its existence and uniqueness. This novel approach was not previously explored in the literature. The proposed stabilized technique does not necessitate parameters or computing higher-order derivatives, making it computationally efficient. The study offers numerical tests demonstrating optimal convergence and effectiveness of the revised approach in two-dimensional settings.

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Two-Grid Stabilized Lowest Equal-Order Finite Element Method for the Dual-Permeability-Stokes Fluid Flow Model

Haque,

Nasu,

Al Mahbub

et al. 2024

J Sci Comput

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This paper proposes and investigates the two-grid stabilized lowest equal-order finite element method for the time-independent dual-permeability-Stokes model with the Beavers-Joseph-Saffman-Jones interface conditions. This method is mainly based on the idea of combining the two-grid and the two local Gauss integrals for the dual-permeability-Stokes system. In this technique, we use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the dual-permeability-Stokes model using the lowest equal-order finite element quadruples. In the two-grid scheme, the global problem is solved using the standard $$ P_1-P_1-P_1-P_1 $$ P 1 - P 1 - P 1 - P 1 finite element approximations only on a coarse grid with grid size H. Then, a coarse grid solution is applied on a fine grid of size h to decouple the interface terms and the mass exchange terms for solving the three independent subproblems such as the Stokes equations, microfracture equations, and the matrix equations on the fine grid. On the other hand, microfracture and matrix equations are decoupled through the mass exchange terms. The weak formulation is reported, and the optimal error estimate is derived for the two-grid schemes. Furthermore, the numerical results validate that the two-grid stabilized lowest equal-order finite element method is effective and has the same accuracy as the coupling scheme when we choose $$ h=H^2 $$ h = H 2 .

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Stabilized lowest equal-order mixed finite element method for the Oseen viscoelastic fluid flow

Cited by 7 publications

References 51 publications

SUPG Approximation for the Oseen Viscoelastic Fluid Flow with Stabilized Lowest-Equal Order Mixed Finite Element Method

SUPG Approximation for the Oseen Viscoelastic Fluid Flow with Stabilized Lowest-Equal Order Mixed Finite Element Method

A Linear Stabilized Incompressible Magnetohydrodynamic Problem with Magnetic Pressure

Two-Grid Stabilized Lowest Equal-Order Finite Element Method for the Dual-Permeability-Stokes Fluid Flow Model

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