A remark on similarity analysis of boundary layer equations of a class of non-Newtonian fluids

A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluid in which the stress, an arbitrary function of rates of strain, is studied. It is shown that under any group of transformation, for an arbitrary stress function, not all non-Newtonian fluids possess a similarity solution for the flow past a wedge inclined at arbitrary angle except Ostwald-de-Waele power-law fluid. Further it is observed, for non-Newtonian fluids of any model only $90°$ of wedge flow leads to similarity solutions. Our results contain a correction to some flaws in Pakdemirli׳s [14] (1994) paper on similarity analysis of boundary layer equations of a class of non-Newtonian fluids.     

On the spatial localization of the laminar boundary layer in a non-Newtonian dilatant fluid

In dilatant fluids the shear perturbation propagation rate is finite, in contrast to Newtonian and pseudoplastic fluids in which it is infinite [1]. Therefore, in certain dilatant fluid flows, frontal surfaces separating regions with zero and nonzero shear perturbations may be formed. Since, in a sense, the boundary layer is a “time scan” of the nonstationary shear perturbation propagation process, in dilatant fluids the boundary layer should definitely be spatially localized. This was first mentioned in [2] where, however, it was mistakenly asserted that boundary layer spatial localization does not take place in all dilatant fluids and is absent in so-called “hardening” dilatant fluids. In [3], the solutions of the laminar boundary-layer equations for speudoplastic and “hardening” dilatant fluids were investigated qualitatively. The formation of frontal surfaces in dilatant fluid flows is usually mathematically related with the existence of singular solutions of the corresponding differential equations [4]. However, since the analysis performed in [3] was inaccurate, in that study singular solutions were not found and it was incorrectly concluded that in “hardening” dilatant fluids there is no spatial boundary layer localization. The investigation performed in [5] showed that in fact in “hardening” dilatant fluids boundary layers are spatially localized, since there exist singular solutions of the corresponding differential equations. Subsequently, this result was reproduced in [6], where an attempt was also made to carry out a qualitative investigation of the solutions of the laminar boundary-layer equations for other types of dilatant fluids. The author did not find singular solutions in this case and mistakenly concluded that in these fluids there is no spatial boundary layer localization. This misunderstanding was due to the fact that in [6] it was not understood that in dilatant fluid flows the formation of frontal surfaces can be mathematically described not only in relation to the existence of singular solutions.     

Self-similar solutions in the problem of generalized vortex diffusion

The analysis of the group properties and the search for self-similar solutions in problems of mathematical physics and continuum mechanics have always been of interest, both theoretical and applied [1–3]. Self-similar solutions of parabolic problems that depend only on a variable of the type η = x/√t are classical fundamental solutions of the one-dimensional linear and nonlinear heat conduction equations describing numerous physical phenomena with initial discontinuities on the boundary [4]. In this study, the term “generalized vortex diffusion” is introduced in order to unify the different processes in mechanics modeled by these problems. Here, vortex layer diffusion and vortex filament diffusion in a Newtonian fluid [5] can serve as classical hydrodynamic examples. The cases of self-similarity with respect to the variable η are classified for fairly general kinematics of the processes, physical nonlinearities of the medium, and types of boundary conditions at the discontinuity points. The general initial and boundary value problem thus formulated is analyzed in detail for Newtonian and non-Newtonian power-law fluids and a medium similar in behavior to a rigid-ideally plastic body. New self-similar solutions for the shear stress are derived.     

MHD equations for paramagnetic media

The conservation laws are used to obtain phenomenologically the complete system of equations of motion of a conductive paramagnetic fluid in a magnetic field. In addition to the usual MHD equations (with additional terms accounting for the magnetization of the medium), this system includes the equation for the rate of change of the magnetic moment.The hydrodynamic equations for a fluid with internal rotation have been obtained in [1] and extended in [2] to the case of the paramagnetic properties resulting from this rotation: here the fluid was considered nonconducting. The analysis of [2] is extended to the case of a fluid with nonzero electrical conductivity. This will be the same extension of MHD as the theory of [1, 2] is for conventional hydrodynamics.     

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.     

Effect of viscoelasticity on the rotation of a sphere in shear flow

When particles are dispersed in viscoelastic rather than Newtonian media, the hydrodynamics will be changed entailing differences in suspension rheology. The disturbance velocity profiles and stress distributions around the particle will depend on the viscoelastic material functions. Even in inertialess flows, changes in particle rotation and migration will occur. The problem of the rotation of a single spherical particle in simple shear flow in viscoelastic fluids was recently studied to understand the effects of changes in the rheological properties with both numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346] and experiments [Snijkers et al., J. Rheol. 53 (2009) 459–480]. In the simulations, different constitutive models were used to demonstrate the effects of different rheological behavior. In the experiments, fluids with different constitutive properties were chosen. In both studies a slowing down of the rotation speed of the particles was found, when compared to the Newtonian case, as elasticity increases. Surprisingly, the extent of the slowing down of the rotation rate did not depend strongly on the details of the fluid rheology, but primarily on the Weissenberg number defined as the ratio between the first normal stress difference and the shear stress.In the present work, a quantitative comparison between the experimental measurements and novel simulation results is made by considering more realistic constitutive equations as compared to the model fluids used in previous numerical simulations [D’Avino et al., J. Rheol. 52 (2008) 1331–1346]. A multimode Giesekus model with Newtonian solvent as constitutive equation is fitted to the experimentally obtained linear and nonlinear fluid properties and used to simulate the rotation of a torque-free sphere in a range of Weissenberg numbers similar to those in the experiments. A good agreement between the experimental and numerical results is obtained. The local torque and pressure distributions on the particle surface calculated by simulations are shown.     

A power-form method for dynamic systems: investigating the steady-state response of strongly nonlinear oscillators by their equivalent Duffing-type equation

This paper aims to apply a transformation method that replaces the elastic forces of the original equation of motion with a power-form elastic term. The accuracy obtained from the derived equivalent equations of motion is evaluated by studying the finite-amplitude damped, forced vibration of a vertically suspended load body supported by incompressible, homogeneous, and isotropic viscohyperelastic elastomer materials. Numerical integrations of the original equations of two oscillators described by neo-Hookean and Mooney–Rivlin viscohyperelastic elastomer material models, and their equivalent equations of motion, are compared to the frequency–amplitude steady-state solutions obtained from the harmonic balance and the averaging methods. It is shown from numerical integrations and approximate steady-state solutions that the equivalent equations predict well the original system dynamic response despite having higher system nonlinearities.

     

Lump,soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics

This article investigates a nonlinear fifth-order partial differential equation (PDE) in two-mode waves. The equation generalizes two-mode Sawada-Kotera (tmSK), two-mode Lax (tmLax), and two-mode Caudrey–Dodd–Gibbon (tmCDG) equations. In 2017, Wazwaz [1] presented three two-mode fifth-order evolutions equations as tmSK, tmLax, and tmCDG equations for the integrable two-mode KdV equation and established solitons up to three-soliton solutions. In light of the research above, we examine a generalized two-mode evolution equation using a logarithmic transformation concerning the equation’s dispersion. It utilizes the simplified technique of the Hirota method to obtain the multiple solitons as a single soliton, two solitons, and three solitons with their interactions. Also, we construct one-lump solutions and their interaction with a soliton and depict the dynamical structures of the obtained solutions for solitons, lump, and their interactions. We show the 3D graphics with their contour plots for the obtained solutions by taking suitable values of the parameters presented in the solutions. These equations simultaneously study the propagation of two-mode waves in the identical direction with different phase velocities, dispersion parameters, and nonlinearity. These equations have applications in several real-life examples, such as gravity-affected waves or gravity-capillary waves, waves in shallow water, propagating waves in fast-mode and the slow-mode with their phase velocity in a strong and weak magnetic field, known as magneto-sound propagation in plasmas.

     

Existence of Weak Solutions for the Three-Dimensional Motion of an Elastic Structure in an Incompressible Fluid

We study here the three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid. The structure and the fluid are contained in a fixed bounded connected set Ω. We show the existence of a weak solution for regularized elastic deformations as long as elastic deformations are not too important (in order to avoid interpenetration and preserve orientation on the structure) and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural (and it corresponds to many physical applications) to consider that its rigid motion (translation and rotation) may be large. The existence result presented here has been announced in [4]. Some improvements have been provided on the model: the model considered in [4] is a simplified model where the structure motion is modelled by decoupled and linear equations for the translation, the rotation and the purely elastic displacement. In what follows, we consider on the structure a model which represents the motion of a structure with large rigid displacements and small elastic perturbations. This model, introduced by [15] for a structure alone, leads to coupled and nonlinear equations for the translation, the rotation and the elastic displacement.     

Stress-Dependent Porosity and Permeability of Porous Rocks Represented by a Mechanistic Elastic Cylindrical Pore-Shell Model

The stress dependency of the porosity and permeability of porous rocks is described theoretically by representing the preferential flow paths in heterogeneous porous rocks by a bundle of tortuous cylindrical elastic tubes. A Lamé-type equation is applied to relate the radial displacement of the internal wall of the cylindrical elastic tubes and the porosity to the variation of the pore fluid pressure. The variation of the permeability of porous rocks by effective stress is determined by incorporating the radial displacement of the internal wall of the cylindrical elastic tubes into the Kozeny–Carman relationship. The fully analytical solutions of the mechanistic elastic pore-shell model developed by combining the Lamé and Kozeny–Carman equations are shown to lead to very accurate correlations of the stress dependency of both the porosity and the permeability of porous rocks.

     

Thermomechanical modeling of nonlinear internal hysteresis due to incomplete phase transformation in pseudoelastic shape memory alloys

This paper presents a thermomechanical model for pseudoelastic shape memory alloys (SMAs) accounting for internal hysteresis effect due to incomplete phase transformation. The model is developed within the finite-strain framework, wherein the deformation gradient is multiplicatively decomposed into thermal dilation, rigid body rotation, elastic and transformation parts. Helmholtz free energy density comprises three components: the reversible thermodynamic process , the irreversible thermodynamic process and the physical constraints of both. In order to capture the multiple internal hysteresis loops in SMA, two internal variables representing the transition points of the forward and reverse phase transformation, \(\phi _s^f\) and \(\phi _s^r\), are introduced to describe the incomplete phase transformation process. Evolution equations of the internal variables are derived and linked to the phase transformation. Numerical implementation of the model features an Euler discretization and a cutting-plane algorithm. After validation of the model against the experimental data, numerical examples are presented, involving a SMA-based vibration system and a crack SMA specimen subjected to partial loading–unloading case. Simulation results well demonstrate the internal hysteresis and free vibration behavior of SMA.

     

A note on longitudinal oscillations of a generalized Burgers fluid in cylindrical domains

The starting solutions for the oscillating motion of a generalized Burgers fluid due to longitudinal oscillations of an infinite circular cylinder, as well as those corresponding to an oscillating pressure gradient, are established as Fourier–Bessel series in terms of some suitable eigenfunctions. These solutions, presented as sum of steady-state and transient solutions, describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the motion of the fluid is described by the steady-state solutions which are periodic in time and independent of the initial conditions. These solutions are also presented in simpler but equivalent forms in terms of modified Bessel functions of first and second kind. In both forms, the steady-state solutions can be specialized to give the similar solutions for Burgers, Oldroyd-B, Maxwell, second grade and Newtonian fluids performing the same motions. Finally, the required time to reach the steady-state for cosine and sine oscillations of the boundary is obtained by graphical illustrations.     

Internal waves of finite amplitude

Much research has been devoted to unsteady fluid flow with a free boundary. For example, Ovsyannikov [1] and Nalimov [2] have proven theorems on the existence and uniqueness of a solution, and a number of papers have proposed algorithms for numerical solution, based on various chain methods [3–6] or potential-theory methods [7–9]. In the present article we consider two-dimensional potential waves of finite amplitude on the interface between two heavy fluids of different densities. The initial problem is reduced to the Cauchy problem for a system of two integrodifferential equations. An algorithm for the numerical solution of this system is constructed, and the results of calculations are presented.     

An algorithm for rotating axisymmetric flows: model,validation and industrial applications

The study of axisymmetric flows is of interest not only from an academic point of view, due to the existence of exact solutions of Navier–Stokes equations, but also from an industrial point of view, since these kind of flows are frequently found in several applications. In the present work the development and implementation of a finite element algorithm to solve Navier–Stokes equations with axisymmetric geometry and boundary conditions is presented. Such algorithm allows the simulation of flows with tangential velocity, including free surface flows, for both laminar and turbulent conditions. Pseudo‐concentration technique is used to model the free surface (or the interface between two fluids) and the k–ε model is employed to take into account turbulent effects. The finite element model is validated by comparisons with analytical solutions of Navier–Stokes equations and experimental measurements. Two different industrial applications are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Determination of the magnitude of the separation criterion for a three-dimensional laminar boundary layer

A separation criterion, i.e., a definite relationship between the external flow and the boundary layer parameters [1], can be used to estimate the possibility of the origination of separation of a two-dimensional boundary layer. A functional form of the separation criterion has also been obtained for a three-dimensional boundary layer [2] on the basis of dimensional analysis. As in the case of the two-dimensional boundary layer, locally self-similar solutions can be used to determine the specific magnitude of the separation criterion as a function of the values of the governing parameters. Locally self-similar solutions of the two-dimensional laminar boundary-layer equations have been found at the separation point for a perfect gas with a linear dependence of the coefficient of viscosity on the temperature (Ω=1) and Prandtl number P=1 [3, 4]. The influence of blowing and suction has been studied for this case [5]. Self-similar solutions have been obtained for Ω=1, P=0.723 for the limit case of hypersonic perfect gas flow [6]. Locally self-similar solutions of the three-dimensional laminar boundary-layer equations at the separation point are presented in [7] for a perfect gas with Ω=1, P=1. There are no such computations for Ω≠1, P≠1; however, the results of computing several examples for a two-dimensional flow [8] show that the influence of the real properties of a gas can be significant and should be taken into account. Self-similar solutions of the two- and three-dimensional boundary-layer equations at the separation point are found in this paper for a perfect gas with a power-law dependence of the viscosity coefficient on the enthalpy (Ω=0.5, 0.75, 1.0) for different values of the Prandtl number (P=0.5, 0.7, 1.0) in a broad range of variation of the external stream velocity (v ₁ ² /2h₁* = 0–0.99) and the temperature of the streamlined surface. Magnitudes of the separation criterion for a laminar boundary layer have been obtained on the basis of these data.     

On stagnation point flow of Sisko fluid over a stretching sheet

The steady two-dimensional stagnation-point flow, represented by Sisko fluid constitutive model, over a stretching sheet is investigated theoretically. Using suitable similarity transformations, the governing boundary-layer equations are transformed into the self-similar non-linear ordinary differential equation. The transformed equation is then solved using a very efficient analytic technique namely the homotopy analysis method (HAM) and the HAM solutions are validated by the exact analytic solutions obtain in certain special cases. The influence of the power-law index (n), the material parameter (A) and the velocity ratio parameter (d/c) on the flow characteristics is studied and presented through several graphs. In addition, the local skin friction coefficient for several values of these parameters is tabulated and examined. The similarity solutions for both the Newtonian and the power-law fluids are presented as special cases of the analysis. The results obtained reveal that, in comparison with the Newtonian and the power-law fluids, the velocity profiles of the Sisko fluid are much faster (slower), for d/c<1 (d/c>1), respectively.     

Steady states of a continuous-flow adiabatic chemical reactor

Flow reactors are widely used in the chemical industry for purposes of catalytic reactions [1,2]. Calculation of reactors of this type, even in one-dimemional approximation, is complicated and possible only with the use of numerical methods [1, 3]. Such calculations make it possible to find the steady-state distribution of temperature and concentration in the chemical reactor if one exists; in general, however, there may be other steady-state regimes which may be preferable from the standpoint of obtaining a different degree of conversion of the starting product, operating stability, etc.In this connection special interest attaches to the question of the existence and number of steady-state solutions of the system of equations describing the reactor process.This problem was previously considered in [4–7]. Thus, in [4, 5] it was pointed out that in certain special cases more than one steady-state regime may exist. In [6, 7] the question of sufficient conditions of uniqueness was investigated. In [7] it was shown that the steady-state regime is unique in the ease of short reactors or a dilute mixture of reactants. In [8] the problem of the existence and uniqueness of the steady-state regime was examined for a chain reaction model with direct application of the general theorems of functional analysis.The present paper includes an analysis of a very simple mathematical model of an adiabatic chemical reactor in which an exothermic or endothermie reaction takes place. It is established that in the case of an endothermic process a unique steady-state regime always exists. In the exothermic case the problem of the steady-state regime also always has a solution which, however, may be nonunique; the possibility of the existence of several steady-state regimes, associated with the form of the temperature dependence of the heat release rate, is substantiated.The authors thank G. I. Barenblatt, A. I. Leonov, L. M. Pis'men, and Yu. I. Kharkats for discussing and commenting on the work.