Incompressible non-Newtonian Flow Research Articles

Lattice Boltzmann method for studying dynamics of single rising bubble in shear-thickening power-law fluids

Bubble motion in non-Newtonian fluids is widely present in various industrial processes such as crude oil extraction, enhancement of boiling heat transfer, CO<sub>2</sub> sequestration and wastewater treatment. System containing non-Newtonian liquid, as opposed to Newtonian liquid, has shear-dependent viscosity, which can change the hydrodynamic characteristics of the bubbles, such as their size, deformation, instability, terminal velocity, and shear rate, and ultimately affect the bubble rising behaviors. In this work, the dynamic behavior of bubble rising in a shear-thickened fluid is studied by using an incompressible lattice Boltzmann non-Newtonian gas-liquid two-phase flow model. The effects of the rheological exponent <i>n</i>, the Eötvös number (<i>Eo</i>), and the Galilei number (<i>Ga</i>) on the bubble deformation, terminal velocity, and the shear rate are investigated. The numerical results show that the degree of bubble deformation increases as <i>Eo</i> grows, and the effect of <i>n</i> on bubble deformation degree relates to <i>Ga</i>. On the other hand, the terminal velocity of the bubbles increases monotonically and nonlinearly with <i>Ga</i> for given <i>Eo</i> and <i>n</i>, and the effect of <i>n</i> on the terminal velocity of the bubbles turns stronger as <i>Ga</i> increases. When <i>Ga</i> is fixed and small, the terminal velocity of the bubble increases and then decreases with the increase of <i>n</i> at small <i>Eo</i>, and increases with the increase of <i>n</i> when <i>Eo</i> is large; but when <i>Ga</i> is fixed and large, the terminal velocity of the bubbles increases with the increase of <i>n</i> in a more uniform manner. In addition, regions with high shear rates can be found near the left end and right end of the bubble. The size of these regions grows with <i>Eo</i> and <i>Ga</i>, exhibiting an initial increase followed by a decrease as <i>n</i> increases. Finally, the orthogonal experimental method is used to obtain the influences of the aforementioned three factors on the shear rate and terminal velocity. The order of influence on shear rate is <i>n</i>, <i>Ga</i> and <i>Eo</i> which are arranged in descending order. For the terminal velocity, <i>Ga</i> has the greatest influence, followed by <i>n</i>, and <i>Eo</i> has the least influence.

Analysis of a Stabilised Finite Element Method for Power-Law Fluids

A low-order finite element method is constructed and analysed for an incompressible non-Newtonian flow problem with power-law rheology. The method is based on a continuous piecewise linear approximation of the velocity field and piecewise constant approximation of the pressure. Stabilisation, in the form of pressure jumps, is added to the formulation to compensate for the failure of the inf-sup condition, and using an appropriate lifting of the pressure jumps a divergence-free approximation to the velocity field is built and included in the discretisation of the convection term. This construction allows us to prove the convergence of the resulting finite element method for the entire range r>frac{2 d}{d+2} of the power-law index r for which weak solutions to the model are known to exist in d space dimensions, d in {2,3}.

Macroscopic model for unsteady generalized Newtonian fluid flow in homogeneous porous media

A macroscopic model for unsteady incompressible isothermal non-Newtonian flow in homogeneous porous media, taking into account inertial and slip effects at solid–fluid interfaces, is derived in this work. The development is carried out considering a general Newton’s law of viscosity for the fluid phase. Using the classical volume averaging method, the seepage velocity is shown to be solenoidal. The macroscopic momentum equation is derived in the Laplace domain, employing a simplified version of the volume averaging method, which calls upon Green’s formulas and adjoint problems for Green’s function pairs for the velocity and pressure. In the Laplace domain, the macroscopic momentum equation takes the form of Darcy’s law corrected by a term that accounts for the initial flow condition. Once transformed back into the time domain, this equation provides the macroscopic velocity that depends on two terms. The first one is under the form of a time convolution between the macroscopic pressure gradient and the time derivative of an apparent permeability tensor. The second one is a memory term that accounts for the effect of the initial flow conditions. These two effective quantities are determined from the solution of a single closure problem that naturally results from the derivations. The model is consistent with the unsteady model in the Newtonian case and simplifies to the steady versions of some non-Newtonian macroscopic flow models. The macroscopic model is validated with pore-scale simulations performed in 2D model porous structures, considering a Carreau fluid. The impact of inertia and non-Newtonian effects on the dynamics of the macroscopic coefficients is highlighted.

Study of Non-Newtonian biomagnetic blood flow in a stenosed bifurcated artery having elastic walls

Fluid structure interaction (FSI) gained attention of researchers and scientist due to its applications in science fields like biomedical engineering, mechanical engineering etc. One of the major application in FSI is to study elastic wall behavior of stenotic arteries. In this paper we discussed an incompressible Non-Newtonian blood flow analysis in an elastic bifurcated artery. A magnetic field is applied along x direction. For coupling of the problem an Arbitrary Lagrangian–Eulerian formulation is used by two-way fluid structure interaction. To discretize the problem, we employed P_{2} P_{1} finite element technique to approximate the velocity, displacement and pressure and then linearized system of equations is solved using Newton iteration method. Analysis is carried out for power law index, Reynolds number and Hartmann number. Hemodynamic effects on elastic walls, stenotic artery and bifurcated region are evaluated by using velocity profile, pressure and loads on the walls. Study shows there is significant increase in wall shear stresses with an increase in Power law index and Hartmann number. While as expected increase in Reynolds number decreases the wall shear stresses. Also load on the upper wall is calculated against Hartmann number for different values of power law index. Results show load increases as the Hartmann number and power law index increases. From hemodynamic point of view, the load on the walls is minimum for shear thinning case but when power law index increased i.e. for shear thickening case load on the walls increased.

Homogenization of an incompressible non-Newtonian flow through a thin porous medium

In this paper, we consider a non-Newtonian flow in a thin porous medium \(\Omega _{\varepsilon }\) of thickness \(\varepsilon \) which is perforated by periodically solid cylinders of size \(a_{\varepsilon }\). The flow is described by the 3D incompressible Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index \(1<p<+\infty \). We consider the limit when domain thickness tends to zero, and we obtain different models depending on the magnitude \(a_{\varepsilon }\) with respect to \(\varepsilon \).

A Unified Theory for Some Non-Newtonian Fluids Under Singular Forcing

We consider a model of steady, incompressible non-Newtonian flow with neglected convective term under external forcing. Our structural assumptions allow for certain nondegenerate power-law or Carreau-type fluids. Within our setting, we provide the full-range theory, namely, existence, optimal regularity, and uniqueness of solutions, not only with respect to forcing belonging to Lebesgue spaces, but also with respect to their refinements, namely, the weighted Lebesgue spaces, with weights in a respective Muckenhoupt class. The analytical highlight is derivation of existence and uniqueness theory for forcing with its regularity well below the natural duality exponent, via estimates in weighted spaces. It is a generalization of [M. Bulíček, L. Diening, and S. Schwarzacher, Anal. PDE, 9 (2016), pp. 1115--1151] to incompressible fluids. Moreover, two technical results, needed for our analysis, may be useful for further studies. They are the solenoidal, weighted, biting div-curl lemma and the solenoidal Lipschitz approximations on domains.

On Preconditioning of Incompressible Non-Newtonian Flow Problems

On Preconditioning of Incompressible Non-Newtonian Flow Problems

A Numerical Method for Simulation of Incompressible Three-Dimensional Newtonian and Non-Newtonian Flows

Three-dimensional Newtonian and non-Newtonian flows are found in numerous engineering applications. Simulation of this class of problems requires robust mathematical and computational modeling. The main goal of this article is to propose a unified numerical methodology for solving three-dimensional, laminar, incompressible Newtonian and non-Newtonian steady flows. A second-order, fully implicit finite-difference approximation is used to discretize the governing equations in a collocated mesh, in which Jacobian matrices are derived to account for distinct constitutive relations for Newtonian and non-Newtonian fluids. The strategy also uses a Euler implicit pseudo-transient time stepping aiming at steady-state solutions. Examples illustrating Newtonian and non-Newtonian (polymer melts) fluid flow in 3-D geometries are discussed.

Modeling of Generalized Newtonian lid driven cavity flow using SPH method

In this paper a smoothed particle hydrodynamics (SPH) method is introduced for simulating two-dimensional incompressible non-Newtonian fluid flows, and the non-Newtonian effects in the flow of a fluid which can be modelled by generalized Newtonian constitutive equations are investigated. Two viscoplastic models including Bingham-plastic and power-law models are considered along with the Newtonian model. The governing equations include the conservation of mass and momentum equations in a pseudo-compressible form. The spatial discretization of these equations is achieved by using the SPH method. The temporal discretization algorithm is a fully explicit two-step predictor–corrector scheme. In the prediction step, an intermediate velocity field is obtained using a forward scheme in time without enforcing incompressibility. The correction step consists of solving a pressure Poisson equation to satisfy incompressibility by providing a trade-off between the pressure and density variables. The performance of the proposed scheme is evaluated by studying a benchmark problem including flow of viscoplastic fluids in a lid-driven cavity. Both Newtonian and non-Newtonian cases are investigated and the results are compared with available numerical data. It was shown that in all cases the method is stable and the results are in very good agreement with available data.

MODELLING OF GENERALIZED NEWTONIAN LID-DRIVEN CAVITY FLOW USING AN SPH METHOD

In this paper a smoothed particle hydrodynamics (SPH) method is introduced for simulating two-dimensional incompressible non-Newtonian fluid flows, and the non-Newtonian effects in the flow of a fluid which can be modelled by generalized Newtonian constitutive equations are investigated. Two viscoplastic models including Bingham-plastic and power-law models are considered along with the Newtonian model. The governing equations include the conservation of mass and momentum equations in a pseudo-compressible form. The spatial discretization of these equations is achieved by using the SPH method. The temporal discretization algorithm is a fully explicit two-step predictor–corrector scheme. In the prediction step, an intermediate velocity field is obtained using a forward scheme in time without enforcing incompressibility. The correction step consists of solving a pressure Poisson equation to satisfy incompressibility by providing a trade-off between the pressure and density variables. The performance of the proposed scheme is evaluated by studying a benchmark problem including flow of viscoplastic fluids in a lid-driven cavity. Both Newtonian and non-Newtonian cases are investigated and the results are compared with available numerical data. It was shown that in all cases the method is stable and the results are in very good agreement with available data.

Theorems about the attractor for incompressible non-Newtonian flow driven by external forces that are rapidly oscillating in time but have a smooth average

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces g ɛ ( x , t ) = g ( x , t , t / ɛ ) possessing the average g 0 ( x , t ) as ɛ → 0 + , where 0 < ɛ ⩽ ɛ 0 < 1 . Firstly, with assumptions ( A 1 ) – ( A 5 ) on the functions g ( x , t , ξ ) and g 0 ( x , t ) , we prove that the Hausdorff distance between the uniform attractors A ɛ and A 0 in space H, corresponding to the oscillating equations and the averaged equation, respectively, is less than O ( ɛ ) as ɛ → 0 + . Then we establish that the Hausdorff distance between the uniform attractors A ɛ V and A 0 V in space V is also less than O ( ɛ ) as ɛ → 0 + . Finally, we show A ɛ ⊆ A ɛ V for each ɛ ∈ [ 0 , ɛ 0 ] .

A numerical method for incompressible non-Newtonian fluid flows based on the lattice Boltzmann method

A new numerical method for incompressible non-Newtonian fluid flows based on the lattice Boltzmann method (LBM) is proposed. The essence of the present method lies in the determination of shear-dependent viscosity of the fluid by using a variable parameter related to the local shear rate. Also, the relaxation time in the BGK collision term is kept at unity taking account of numerical stability. The method is applied to two representative test case problems, power-law fluid flows in a reentrant corner geometry and non-Newtonian fluid flows in a three-dimensional porous structure. These simulations indicate that the method can be useful for practical non-Newtonian fluid flows, such as shear-thickening (dilatant) and shear-thinning (pseudoplastic) fluid flows.

Numerical Simulation of Semi-Solid Magnesium Alloy in Continuous Roll-Casting Process

It is significative to link the semi-solid process and continuous roll-casting process (CRP), through which we can get high quality magnesium alloy strips with non-dendritic structure and improve the overall properties of the products. Furthermore, because CRP of semi-solid metal combines casting and forming, it shortens the period of technological process. In general, several technical parameters such as the inlet temperature of semi-solid slurry, the heat transfer between roll and molten alloy, the roll-casting speed, the position of solidification final point influence on the strip quality in continuous roll-casting process. Therefore, the effects of the parameter to continuous roll-casting process should be investigated. In this paper, the finite volume method is used in the analysis of the CRP, a two-dimensional incompressible non-Newtonian flow with heat transfer is considered. Based on the simulated results, the optimizational technical parameters are obtained and the experiment of CRP was carried out. An ideal AZ91D strip with non-dendritic structure has been obtained and also has been tested by cold rolling and hot rolling. The results show that microstructure and mechanical properties of the strip are fine.

Simulation of Incompressible Non-Newtonian Flows Through Channels with Sudden Expansion Using the Power-Law Model

The goal of this work is to analyze incompressible Newtonian and non-Newtonian flows through channels with sudden expansion. The governing equations are solved using the finite-differences explicit Runge-Kutta time-stepping scheme in nondimensionalized form in which continuity and momentum are solved simultaneously along the grid points. The power-law model is applied to predict pseudoplastic (shear-thinning) and dilatant (shear-thickening) behavior in such expansions. The critical Reynolds number, in which the solution becomes asymmetric, is analyzed. Numerical results for a 3 : 1 expansion show good agreement with other numerical tests found in the literature for Reynolds numbers ranging from 40 to 140 for Newtonian flow. For the non-Newtonian case, a comparison with an analytical solution is presented.

PENALTY APPROXIMATIONS TO THE STATIONARY POWER-LAW NAVIER–STOKES PROBLEM

In this work, penalty approximations to the steady state Navier-Stokes problem governed by the the power-law model for viscous incompressible non-Newtonian flows in bounded convex domains in Rd (2 ≤ d) are studied. Existence and uniqueness of solutions to the penalty approximations are proved, convergence is shown, and rates of convergence are derived.

Exact solutions of non-Newtonian fluid flows with prescribed vorticity

The equations of motion of a non-Newtonian second-grade fluid flow are highly nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions. Due to the complexity of the equations, inverse methods described by Nemenyi [1] have become attractive in the study of non-Newtonian fluids. In these methods, certain physical or geometrical properties of the flow field are assumed a priori. Lin and Tobak [2] studied steady plane viscous incompressible flows for a chosen vorticity function by decomposing the nonlinear fourth-order partial differential equation in the streamfunction. This excellent approach yielded two second-order linear partial differential equations in the streamfunction. Hui [3] used this approach to study unsteady plane viscous incompressible flows. During the past decade, there has been substantial interest in flows of viscoelastic liquids due to the occurrence of these flows in industrial processes. In this paper, we study the steady and unsteady incompressible viscous non-Newtonian second-grade fluid flows in which the vorticity is proportional to the streamfunction perturbed by a uniform stream. The solutions obtained are exact solutions and represent various non-parallel flows of second-grade fluids. The plan of this paper is as follows: In Sect. 2, the equations of motion of an unsteady plane incompressible second-grade fluid are given, and the vorticity function is assumed to be∇ 2 ψ=A(ψ−Ux−BUy 2). In Sect. 3, solutions for the steady flow are obtained. In Sect. 4, solutions for unsteady flow are obtained.

Least-squares p-r finite element methods for incompressible non-Newtonian flows

A least-squares mixed p-type finite element method is developed for numerical solution of incompressible non-Newtonian flows. Hierarchical piecewise polynomials are introduced as element basis functions while singularities and boundary layers are treated by a combination of mesh redistribution and polynomial refinement. Scaling of the original differential equations is found to be important for the least-squares minimization process. We discuss both nonlinear algebraic and differential constitutive models and present numerical examples to illustrate benefits and shortcomings of the present approach.

Computation of the filtration laws through porous media for a non-Newtonian fluid obeying the power law

We consider a stationary flow of an incompressible non-Newtonian flow through a porous medium, induced by an injection velocity when inertial effects are negligible. At the pore scale, the governing equations are based on a nonlinear relation between the stress and the rate of deformation. In such a situation, the limit problem obtained when the pore size tends to zero, is called the homogenized problem that leads to the filtration law. This filtration law is given by a non-linear system coupling a local problem on a typical cell of the porous medium to a global problem at the scale of the whole porous medium. We propose, in this work, a numerical method to solve this homogenized problem and apply this method when the velocity dependent viscosity is given by the power law. Finally, we propose some numerical experiments to illustrate our approach.

Three-Dimensional Non-Newtonian Fluid Flow Analysis by Finite-Element Method Using Simultaneous Relaxatiuon Method of Velocity and Pressure. Flow of Power-Law Fluids through an Axisymmetric Abrupt Contraction.

A simple and low-storage finite-element method for three-dimensional incompressible non-Newtonian flow is presented. The present method employs the rational Runge-Kutta scheme for time integration and simultaneous relaxation method of velocity and pressure for the incompressible condition. In order to reduce computational storage, coefficient matrices are calculated by simple integration formulas which are applicable to large systems. Flow of power-law fluids through an axisymmetric abrupt contruction is calculated to confirm the present method. The present method is stable in numerical simulation of power-law fluids, and the numerical results agree with other numerical results.

The Treatment of Spurious Pressure Modes in Spectral Incompressible Flow Calculations

Algorithms for the transient and steady state simulation of incompressible Newtonian and non-Newtonian flows are described for the primitive variable formulation of the governing equations. Spectral approximations are used for the spatial discretization. Attention is given to the satisfaction of the incompressibility constraint and the determination of the pressure. Spurious pressure modes are removed by means of a singular value decomposition. The corresponding velocity field is divergence free at all the collocation points. Numerical results are presented for Newtonian flow in a regularized driven square cavity and for non-Newtonian flow in a planar channel and in a journal bearing for a realistic range of material parameters.