Equations For non-Newtonian Fluids Research Articles

Strong solutions for the steady incompressible MHD equations of non-Newtonian fluids

In this paper we deal with a system of partial differential equations describing a steady motion of an incompressible magnetohydrodynamic fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). By using a fixed point argument in an appropriate functional setting, we proved the existence and uniqueness of strong solutions for the problem in a smooth domain Ω ⊂ Rn (n = 2, 3) under the conditions that the external force is small in a suitable norm.

Soft lubrication of cornstarch-based shear-thickening fluids

The behavior of complex fluids in thin films under the isoviscous elastohydrodynamic lubrication regime is a topic of current interest. A vast majority of the materials used to lubricate compliant contacts are shear-thinning fluids and, in comparison, very little is known about the lubrication properties of shear-thickening (ST) fluids. In this work we carried out both rheological and tribological experiments on ST fluids based on cornstarch suspensions. We investigated the influence of the polarity and viscosity of the carrier fluid and the cornstarch particles’ concentration. From a rheological perspective, the ST response is diminished (as the shear-thinning region enlarges) when non-polar carriers are used (e.g. silicone oil) while a marked thickening is observed for polar carriers (e.g. glycerol-water mixtures). From a tribological perspective, the friction coefficient at slow sliding speeds is strongly affected by the nature of the carrier fluid, and in particular its wettability in contact with hydrophobic PDMS tribopairs. We demonstrate that experimental data, plotted in terms of Stribeck curves, provide useful information on the entrainment of particles/fluid through the contact. Finally, experimental results are also compared with numerical solutions of the Reynolds equation for inelastic non-Newtonian fluids in the isoviscous elastohydrodynamic regime. A Carreau–Yasuda constitutive equation for the viscosity-shear rate relationship is used.

Existence of self-similar solutions of the two-dimensional Navier–Stokes equation for non-Newtonian fluids

The reduced problem of the Navier–Stokes and the continuity equations, in two-dimensional Cartesian coordinates with Eulerian description, for incompressible non-Newtonian fluids, is considered. The Ladyzhenskaya model, with a non-linear velocity dependent stress tensor is adopted, and leads to the governing equation of interest. The reduction is based on a self-similar transformation as demonstrated in existing literature, for two spatial variables and one time variable, resulting in an ODE defined on a semi-infinite domain. In our search for classical solutions, existence and uniqueness will be determined depending on the signs of two parameters with physical interpretation in the equation. Illustrations are included to highlight some of the main results.

Stability of the solutions of a convection–diffusion equation

A class of non-Newtonian fluid equations with a convection term is considered. If the diffusion coefficient is degenerate on the boundary, the interaction between the convection and diffusion is investigated, and the stability of solutions is established without any boundary condition.

H\xf6lder inequality applied on a non-Newtonian fluid equation with a nonlinear convection term and a source term

Consider a non-Newtonian fluid equation with a nonlinear convection term and a source term. The existence of the weak solution is proved by Simon’s compactness theorem. By the Hölder inequality, if both the diffusion coefficient and the convection term are degenerate on the boundary, then the stability of the weak solutions may be proved without the boundary value condition. If the diffusion coefficient is only degenerate on a part of the boundary value, then a partial boundary value condition is required. Based on this partial boundary, the stability of the weak solutions is proved. Moreover, the uniqueness of the weak solution is proved based on the optimal boundary value condition.

On the Non-Newtonian Fluid Equation with a Source Term and a Damping Term

A kind of non-Newtonian fluid equation with a damping term and a source term is considered. After giving a result of the existence, if the diffusion coefficient is degenerate on the boundary, the local stability of the weak solutions is established without any boundary condition. If the diffusion coefficient is degenerate on a part of the boundary, by imposing the homogeneous value condition on the other part of the boundary, the local stability of the weak solutions is proved. Moreover, if the equation is with a damping term, other than the finite propagation property, the results of this paper reveal the essential differences between the non-Newtonian fluid equation and the heat conduction equation in a new way.

The energy equality for weak solutions to the equations of non-Newtonian fluids

In this short paper, we extend the result of Shinbrot (1974) to an incompressible fluid with shear dependent viscosity. It is shown that a weak solution to the equations of non-Newtonian fluids lying in a space Lq(0,T;Lp) satisfies an energy equality, where 2rr−1≤p≤2rr−2 and 1p+1q≤r−1r, if r>2; p≥2rr−1 and r−1p+1q=r−12, if 2(n+1)n+2<r≤2. In particular, our result implies that the weak solution must satisfy the energy equality when r≥3n+2n+2, which is consistent with the known fact.

Biological reactor retrofitting using CFD-ASM modelling

In recent years, the interest in modelling activated sludge (AS) systems by means of Computational Fluid Dynamics (CFD) techniques has significantly increased. This work shows a successful case study combining CFD hydrodynamics and biokinetic modelling. The hydrodynamics is analysed by using the Reynolds-averaged Navier-Stokes equation for incompressible non-Newtonian fluids and SST turbulence model. Biokinetics has been included in the CFD as transport equations with source and sink terms defined by the Activated Sludge Model n°1 (ASM1). Furthermore, a strategy for reducing the computational cost while maintaining accuracy of the results of these calculations has been proposed. This strategy is based on a two-step solver configuration and the definition of a variable timestep scheme. The resulting CFD-ASM approach permits a proper evaluation of denitrification in the anoxic tanks as well as the reproduction of nitrate and readily biodegradable substrate distributions. To demonstrate the strength of the proposed CFD-ASM, it has been used to evaluate the operation of a full-scale AS system and optimize its performance through changes in the biological reactor anoxic zone. The original configuration has been retrofitted and modified after detecting intrinsic defects in the fluid behaviour within the tank. This study has been assessed by analysing hydrodynamics in detail and validating the simulation results with tracer tests and flow velocity measurements. Substantial variations on the Residence Time Distribution have been confirmed when modifying the internal elements of the tank configuration: the wall-bushing and the stirrer positioning. As a result of this work, an influential short circuiting was corrected improving hydrodynamics and increasing mean residence time, all favouring denitrification efficiency. Outcomes of this study show the benefit of CFD when applied to AS tanks.

Degenerate non-Newtonian fluid equation on the half space

Degenerate non-Newtonian fluid equation on the half space

Control for well-posedness about a class of non-Newtonian incompressible porous medium fluid equations

Considering the non-Newtonian fluid equation of incompressible porous media, using the properties of operator semigroup and measure space and the principle of squeezed image, Fourier analysis and a priori estimate in the measurement space are used to discuss the non-compressible porous media, the properness of the solution of the equation, its gradual behavior and its topological properties. Through the diffusion regularization method and the compressed limit compact method, we study the overall decay rate of the solution of the equation in a certain space when the initial value is sufficient. The decay estimation of the solution of the incompressible seepage equation is obtained, and the asymptotic behavior of the solution is obtained by using the double regularization model and the Duhamel principle.

Transformations, properties, and exact solutions of unsteady axisymmetric boundary layer equations for non-Newtonian fluids

Unsteady axisymmetric boundary layer equations for power-law non-Newtonian fluids are analyzed. A number of new exact solutions containing arbitrary functions and free parameters are constructed using generalized or functional separation of variables. The solutions are obtained using a Crocco-type transformation reducing the order of the equations examined and simpler point transformations. Along with the exact solutions to axisymmetric boundary layer equations, some new exact solutions to planar boundary layer equations for non-Newtonian fluids are constructed. Several properties have been discovered that allow the exact solutions of the unsteady axisymmetric boundary layer equations to be generalized by including additional arbitrary functions therein. All results refer to an arbitrarily shaped streamlined solid of revolution.

A New Kind of Weak Solution of Non-Newtonian Fluid Equation

If the non-Newtonian fluid equation with a diffusion coefficient is degenerate on the boundary, the weak solution lacks the regularity to define the trace on the boundary. By introducing a new kind of weak solutions, the stability of the solutions is established without any boundary condition.

Physiological Flow of Jeffrey Six Constant Fluid Model due to Ciliary Motion

The main purpose of this article is to present a mathematical model of ciliary motion in an annulus. In this analysis, the peristaltic motion of non-Newtonian Jeffrey six constant fluid is observed in an annulus with ciliated tips in the presence of heat and mass transfer. The effects of viscous dissipation are also considered. The flow equations of non-Newtonian fluid for the two-dimensional tube in cylindrical coordinates are simplified using the low Reynolds number and long wave-length approximations. The main equations for Jeffrey six constant fluid are considered in cylindrical coordinates system. The resulting nonlinear problem is solved using the regular perturbation technique in terms of a variant of small dimensionless parameter α. The results of the solutions for velocity, temperature and concentration field are presented graphically. Bk is Brinkman number, ST is soret number, and SH is the Schmidth number. Outcome for the longitudinal velocity, pressure rise, pressure gradient and stream lines are represented through graphs. in the history, the viscous-dissipation effect is usually represented by the Brinkman number.

Steady flow for shear thickening fluids in domains with unbounded sections

We solve the stationary Stokes and Navier–Stokes equations for non-Newtonian incompressible fluids with shear dependent viscosity in domains with outlets containing unbounded cross sections, in the case of shear thickening viscosity. The flux assumes arbitrary given values and the growth of the cross sections are analyzed under different convergence hypotheses, inclusive the growth of Dirichlet's integral of the velocity field is deeply related the convergence hypotheses of such sections. We extend the results of the section 4 of [12, Ladyzhenskaya and Solonnikov] (for Newtonian fluids) to non-Newtonian fluids using the techniques found in [3, Dias and Santos].

Steady flow for incompressible fluids in domains with unbounded curved channels

We give an overview on the solution of the stationary Navier-Stokes equations for non newtonian incompressible fluids established by G. Dias and M.M. Santos (Steady flow for shear thickening fluids with arbitrary fluxes, J. Differential Equations 252 (2012), no. 6, 3873-3898), propose a definition for domains with unbounded curved channels which encompasses domains with an unbounded boundary, domains with nozzles, and domains with a boundary being a punctured surface, and argue on the existence of steady flowfor incompressible fluids with arbitrary fluxes in such domains.

One-dimensional reductions and functional separable solutions to unsteady plane and axisymmetric boundary-layer equations for non-Newtonian fluids

The paper deals with equations describing the unsteady axisymmetric boundary layer of power-law non-Newtonian fluids on a body of revolution. The axisymmetric boundary-layer equation for the stream function is shown to reduce to a single third-order PDE of the formwtz+wzwxz−wxwzz=κrn+1(x)wzzn−1wzzz+F(t,x),where n is a rheological parameter of the fluid; the function r(x), determining the shape of the body, is assumed arbitrary. For this non-linear PDE, we describe one-dimensional reductions as well as a number of new generalized and functional separable solutions, which depend on two to five arbitrary functions. The solutions are obtained with the direct method of functional separation of variables by using particular solutions to an auxiliary ODE and systems of first-order PDEs. Many of the solutions are expressed in terms of elementary functions, which, together with significant arbitrariness, makes them especially useful for solving certain model problems and testing numerical and approximate analytical methods in fluid dynamics. Apart from power-law fluids, the paper looks at three-parameter polynomial and generalized Sisko models of non-Newtonian fluids. The unsteady plane boundary-layer equations for the general non-Newtonian fluid model are shown to admit a reduction to an ODE. The paper presents new exact solutions to the plane boundary-layer equations for power-law fluids as well as some other rheologically complex fluids.

On Objectivity, Irreversibility and Non-Newtonian Fluids

Early progress in non-Newtonian fluid mechanics was facilitated by the emergence of two fundamental and complementary principles: objective constitutive characterizations and unambiguous identification of irreversible processes. Motivated by practical and economic concerns in recent years, this line of fluid research has expanded to include debris flows, slurries, biofluids and fluid-solid mixtures; i.e., complex nonlinear fluids with disparate flow properties. Phenomenological descriptions of these fluids now necessarily include strong nonlinear coupling between the fluxes of mass, energy and momentum. Here, I review these principles, illustrate how they constrain the constitutive equations for non-Newtonian fluids and demonstrate how they have impacted other areas of fluid research.

Strong Solutions of Navier-Stokes-Poisson Equations for Compressible Non-Newtonian Fluids

This paper study the Navier-Stoke-Poisson equations for compressible non-Newtonian fluids in one dimensional bounded intervals. The motion of the fluid is driven by the compressible viscous isentropic flow under the self-gravitational and an external force. The local existence and uniqueness of strong solutions was proved based on some compatibility condition. The main condition is that the initial density vacuum is allowed.

Unsteady Squeezing Flow of Casson Fluid with Magnetohydrodynamic Effect and Passing through Porous Medium

An unsteady squeezing flow of Casson fluid having magnetohydrodynamic (MHD) effect and passing through porous medium channel is modeled and investigated. Similarity transformations are used to convert the partial differential equations (PDEs) of non-Newtonian fluid to a highly nonlinear fourth-order ordinary differential equation (ODE). The obtained boundary value problem is solved analytically by Homotopy Perturbation Method (HPM) and numerically by explicit Runge-Kutta method of order 4. For validity purpose, we compare the analytical and numerical results which show excellent agreement. Furthermore, comprehensive graphical analysis has been made to investigate the effects of various fluid parameters on the velocity profile. Analysis shows that positive and negative squeeze numberSqhave opposite effect on the velocity profile. It is also observed that Casson parameterβshows opposite effect on the velocity profile in case of positive and negative squeeze numberSq. MHD parameterMgand permeability constantMphave similar effects on the velocity profile in case of positive and negative squeeze numbers. It is also seen that, in case of positive squeeze number, similar velocity profiles have been obtained forβ,Mg, andMp. Besides this, analysis of skin friction coefficient has also been presented. It is observed that squeeze number, MHD parameter, and permeability parameter have direct relationship while Casson parameter has inverse relationship with skin friction coefficient.

A novel Reynolds equation of non-Newtonian fluid for lubrication simulation

The non-Newtonian phenomenon is significant in hydrodynamic lubrication of some special lubricants, or elastohydrodynamic lubrication (EHL). However, the conventional methods to solve the non-Newtonian lubrication problem are either too complicated or inaccurate. This paper puts forward a Reynolds equation for general lubrication problem of the non-Newtonian fluid by treating the lubricant flow as the superposition of the Poiseuille flow and Couette flow. Then, as examples, a set of simulations for EHL in the line contact are presented to investigate the feasibility of the method. Finally, comparisons with the conventional methods establish the validity and simplicity of the method.