Equations For non-Newtonian Fluids Research Articles

Obtaining Constitutive Equations for Thickening and Filtration Non-Newtonian Fluids

The study of filtration and thickening of particulate systems are used in many industrial processes involving processes of solid-liquid separation, such as in sedimentation ponds, filters, the drilling of oil wells, among others. This paper aims to advance the empirical mechanisms involved in the processes of solid-liquid separation and obtain constitutive equations relating the pressure in solids and porous media permeability from non-Newtonian fluids. In the experiments used aqueous solutions of xanthan gum concentration of 0.1% in weight basis in order to ensure non-newtonian means. For the preparation of suspensions, was used calcium carbonate as particulate material in the separation process involved an initial concentration of 12% by volume. The concentrated sediment was maintained between 30 and 48% by volume. Settling tests were carried to term and sediments resulting from each test were evaluated by making use of the Gamma Rays Attenuation Technique (GRAT). The results show that GRAT is effective in determining sediment concentration distributions formed from non-Newtonian solutions, allowing the constitutive equations to obtain pressure and the solid porous medium permeability, very important for simulations of solid-liquid separation processes.

The Convergence of Coupled Boundary-Layer Equation in the Contact Interface

This paper introduces two equations of non-Newtonian boundary-layer fluid: Cauchy equation of flow field and P-T/T equation of stress field. Secondly, we analyze the convergence of this system of fluid-solid coupled equation with semi-discrete finite element method. We use Galerkin finite element method on the space and semi-implicit C-N difference scheme on the time. Thus, the convergent order of the coupled equations is O（h2+k2） .

The velocity field due to an oscillating plate in an Oldroyd-B fluid

Analytical solutions to the Navier–Stokes equations for non-Newtonian fluids are rare and tend to be mathematically complicated. We analytically solve Stokes’ second problem — the flow due to an infinite plate oscillating in-plane — for a viscoelastic fluid described by the Oldroyd-B constitutive relation, using mathematical techniques familiar to third- or fourth-year undergraduate students in physics, engineering, or mathematics. The solution is compared to the well-known solution of Stokes’ second problem for a Newtonian fluid to illustrate the effects of fluid elasticity on the flow, and we provide a straightforward interpretation of these effects in terms of the quality factor of the oscillations. This calculation provides a mathematically accessible introduction to non-Newtonian fluid flow that illustrates important physical effects while limiting the mathematical complications.

Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise

Motivated by applications to various semilinear and quasi-linear stochastic partial differential equations (SPDEs) appeared in real world models, we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Lévy processes. We illustrate the main results of our paper by showing how it can be applied to a large class of SPDEs such as stochastic reaction–diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature.

Acoustic detection of pressure-induced phase separation spinodals in polymer solutions

Acoustic detection of pressure-induced spinodal phase separation is predicted using an integrated compressible polymer solution thermodynamic-multi transport model and by calculating sound speed and attenuation under pressure conditions that gradually approach the pressure-induced (PIPS) phase separation conditions from the stable state. The integrated model consists of the thermodynamic pressure–concentration phase diagram computed using the Sanchez–Lacombe equation of state (polyethylene–hexane solution), momentum, mass and heat transport equations, and rheological constitutive equations for Newtonian and non-Newtonian fluids. The linearized model results in two coupled density-polymer concentration wave equations that describe the propagation of acoustic waves. Solving the set of equation leads to a six order polynomial that generalizes the Kirchhoff–Langevin biquadratic equation for an ideal single component gas, and that gives an expression for the attenuation and the phase velocity. For the Newtonian fluid, the behavior of the acoustic parameters, attenuation and phase velocity, is analyzed as a function of PIPS proximity and found to be sensitive at low frequency only. On the other hand for non-Newtonian fluids viscoelastic relaxation leads to pressure sensitivity of acoustic properties at all frequencies and hence providing a robust method to detect PIPS. The results provide new insights on the coupling between acoustics and phase separation thermodynamics as well as provide useful scaling pressure-attenuation-speed relations of utility to polymer processing.

An Explicit and Continuously Differentiable Flow Equation for Non-Newtonian Fluids in Pipes

Summary A simple model approximates Herschel-Bulkley (HB) non-Newtonian fluids in laminar flow in pipes. It is continuously differentiable and explicit in the frictional pressure gradient. Such properties are needed for control, real-time optimization, and prediction in drilling operations. The accuracy of the new model is evaluated by comparing it with a numerical implementation of the implicit HB model and some representative approximations used in the literature. Very little loss in accuracy is experienced compared with the implicit solution.

On the strong solutions of one-dimensional Navier-Stokes-Poisson equations for compressible non-Newtonian fluids

We study the strong solutions of 1D Navier-Stokes-Poisson equations for compressible non-Newtonian fluids in bounded intervals. The model is raised from the viscous isentropic gas flow under considering an external force and the non-Newtonian gravitational force term. By using the iterative method we prove the local existence and uniqueness of strong solutions based on some compatibility condition. The main condition is that the initial density vacuum is allowed.

Dynamics of non-autonomous equations of non-Newtonian fluid on 2D unbounded domains

Dynamics of non-autonomous equations of non-Newtonian fluid on 2D unbounded domains

On the Class of Three Dimensional Unsteady Incompressible Magneto-Hydrodynamic Boundary Layer Equations of Non-Newtonian Power Law Fluids

On the Class of Three Dimensional Unsteady Incompressible Magneto-Hydrodynamic Boundary Layer Equations of Non-Newtonian Power Law Fluids

On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids

We investigate the exponential long-time behaviour of the stochastic evolution equations describing the motion of a non-Newtonian fluids excited by multiplicative noise. Some results on the exponential convergence in mean square and with probability one of the weak probabilistic solution to the stationary solutions are given. We also prove an interesting result related to the stabilization of these stochastic evolution equations.

On the Navier-Stokes equations: Existence theorems and energy equalities

Currently available results on the solvability of the Navier-Stokes equations for incompressible non-Newtonian fluids are presented. The order of nonlinearity in the equations may be variable; the only requirement is that it must be a measurable function. Unsteady and steady equations are considered. A lot of attention is paid to the recovery of energy balance, whose violation is theoretically admissible, in particular, in the three-dimensional classical unsteady Navier-Stokes equation. When constructing a weak solution by a limit procedure, a measure arises as a limit of viscous energy densities. Generally speaking, the limit measure contains a nonnegative singular (with respect to the Lebesgue measure) component. It is this singular component that maintains energy balance. Sufficient conditions for the absence of a singular component are studied: in this case, the standard energy equality holds. In many respects, only the regular component of the limit measure is important: in the natural form it is equal to the product of the viscous stress tensor and the gradient of a solution; if this natural form is retained, then the problem is solvable. Conditions are found for the validity of the indicated fundamental representation of the absolutely continuous component of the limit measure.

Steady flow for shear thickening fluids with arbitrary fluxes

We solve the stationary Navier–Stokes equations for non-Newtonian incompressible fluids with shear dependent viscosity in domains with unbounded outlets, in the case of shear thickening viscosity, i.e. the viscosity is given by the shear rate raised to the power p − 2 where p > 2 . The flux assumes arbitrary given values and the Dirichlet integral of the velocity field grows at most linearly in the outlets of the domain. Under some smallness conditions on the “energy dispersion” we also show that the solution of this problem is unique. Our results are an extension of those obtained by O.A. Ladyzhenskaya and V.A. Solonnikov [O.A. Ladyzhenskaya, V.A. Solonnikov, Determination of the solutions of boundary value problems for steady-state Stokes and Navier–Stokes equations in domains having an unbounded Dirichlet integral, J. Soviet Math. 21 (1983) 728–761] for Newtonian fluids ( p = 2 ).

Similarity and Boubaker Polynomials Expansion SchemeBPEScomparative solutions to the heat transfer equation for incompressible non-Newtonian fluids: case of laminar boundary energy equation

In this paper, a new model is proposed for the heat transfer characteristics of power law non- Newtonian fluids. The effects of power law viscosity on temperature field were taken into account by assuming that the temperature field is similar to the velocity field with modified Fourier’s law of heat conduction for power law fluid media. The solutions obtained by using Boubaker Polynomials Expansion Scheme (BPES) technique are compared with those of the recent related similarity method in the literature with good agreement to verify the protocol exactness.

The Navier-Stokes equations: Existence theorems and energy equalities

Currently available results on the solvability of the Navier-Stokes equations for incompressible non-Newtonian fluids are presented. The order of nonlinearity in the equations may be variable; the only requirement is that it must be a measurable function. Unsteady and steady equations are considered. A lot of attention is paid to the recovery of energy balance, whose violation is theoretically admissible, in particular, in the three-dimensional classical unsteady Navier-Stokes equation. When constructing a weak solution by a limit procedure, a measure arises as a limit of viscous energy densities. Generally speaking, the limit measure contains a nonnegative singular (with respect to the Lebesgue measure) component. It is this singular component that maintains energy balance. Sufficient conditions for the absence of a singular component are studied: in this case, the standard energy equality holds. In many respects, only the regular component of the limit measure is important: in the natural form it is equal to the product of the viscous stress tensor and the gradient of a solution; if this natural form is retained, then the problem is solvable. Conditions are found for the validity of the indicated fundamental representation of the absolutely continuous component of the limit measure.

Effects of viscous dissipation on forced convective heat transfer in a channel embedded in a power-law fluid saturated porous medium

The convective heat transfer analysis in a channel embedded in a power-law fluid saturated porous medium subject to uniform heat flux is presented and compared with a Newtonian fluid concerning the effects of viscous dissipation. Governing momentum and energy equations for non-Newtonian fluids which accounts for the viscous dissipation effects are solved numerically. The temperature profiles of the non-Newtonian fluids are found to relate closely to the velocity profiles. When viscous dissipation is taken account of, Nusselt numbers for non-Newtonian fluid are found to deviate more from Newtonian fluid with increasing Brinkman number for a certain range of the Darcy number.

Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space

We consider a generalization of the compressible barotropic Navier–Stokes equations to the case of non-Newtonian fluid in the whole space. The viscosity tensor is assumed to be coercive with an exponent q > 1 . We prove that if the total mass and momentum of the system are conserved, then one can find a constant q γ > 1 depending on the dimension of space n and the heat ratio γ such that for q ∈ [ q γ , n ) there exists no global in time smooth solution to the Cauchy problem. We prove also an analogous result for solutions to equations of magnetohydrodynamic non-Newtonian fluid in 3D space.

Shallow Water equations for Non-Newtonian fluids

The purpose of this paper is to provide a consistent thin layer theory for some Non-Newtonian fluids that are incompressible and flowing down an inclined plane under the effect of gravity. We shall provide a better understanding of the derivation of Shallow Water models in the case of power-law fluids and Bingham fluids. The method is based on asymptotic expansions of solutions of the Cauchy Momentum equations in the Shallow Water scaling and in the neighbourhood of steady solutions so that we can close the average equations on the fluid height h and the total discharge rate q. Such a method has been first introduced in the case of Newtonian fluids where the computations are proved to be rigorous (Vila, in preparation [20]; Bresch and Noble, 2007 [9]) whereas the more complex case of arbitrary topography has been treated formally (Boutounet et al., 2008 [5]). The well posedness of the free surface Cauchy Momentum equations for these Non-Newtonian fluids is still an open problem: the computations carried out here are only formal.

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

This work presents a three-dimensional finite-element algorithm, based on the phase-field model, for computing interfacial flows of Newtonian and complex fluids. A 3D adaptive meshing scheme produces fine grid covering the interface and coarse mesh in the bulk. It is key to accurate resolution of the interface at manageable computational costs. The coupled Navier–Stokes and Cahn–Hilliard equations, plus the constitutive equation for non-Newtonian fluids, are solved using second-order implicit time stepping. Within each time step, Newton iteration is used to handle the nonlinearity, and the linear algebraic system is solved by preconditioned Krylov methods. The phase-field model, with a physically diffuse interface, affords the method several advantages in computing interfacial dynamics. One is the ease in simulating topological changes such as interfacial rupture and coalescence. Another is the capability of computing contact line motion without invoking ad hoc slip conditions. As validation of the 3D numerical scheme, we have computed drop deformation in an elongational flow, relaxation of a deformed drop to the spherical shape, and drop spreading on a partially wetting substrate. The results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate. Excellent agreement is achieved provided that the 3D interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid. Since our model involves several coupled partial differential equations and we use a fully implicit scheme, the matrix inversion requires a large memory. This puts a limit on the scale of problems that can be simulated in 3D, especially for viscoelastic fluids.

Dynamic response of wall shear stress on the stenosed artery

The present study deals with an appropriate mathematical model of an artery in the presence of constriction in which the generated wall shear stress due to blood flow is analysed. The geometry of the stenosed arterial segment in the diseased state, causing malfunction of the cardiovascular system, is formed mathematically. The flowing blood contained in the stenosed artery is treated as non-Newtonian and the flow is considered to be two-dimensional. The motion of the arterial wall and its effect on local fluid mechanics is not ruled out from the present pursuit. The flow analysis applies the time-dependent, two-dimensional incompressible nonlinear Navier–Stokes equations for non-Newtonian fluids. The flow-field can be obtained primarily following the radial coordinate transformation, using the appropriate boundary conditions and finally adopting a suitable finite difference scheme numerically. The influences of flow unsteadiness, the arterial wall distensibility and the presence of stenosis on the flow-field and the wall shear stresses are quantified in order to indicate the susceptibility to atherosclerotic lesions and thereby to validate the applicability of the present theoretical model.

An Iterative Method for the Stokes-Type Problem with Variable Viscosity

This paper concerns an iterative technique for solving discretized Stokes-type equations with varying viscosity coefficient. We build a special block preconditioner for the discrete system of equations and perform an analysis revealing its properties; the theoretical analysis is based on the weighted Necas inequality. The subject of this paper is motivated by numerical solution of incompressible non-Newtonian fluid equations. In particular, the general analysis is applied to the linearized equations of the regularized Bingham model of viscoplastic fluid. Numerical experiments show that the suggested preconditioner leads to an iterative method insensitive to the variation of mesh size and the regularization parameter of the fluid model.