Non-zero Reynolds Numbers Research Articles

Odd viscous flow past a sphere at low but non-zero Reynolds numbers

The measurement of lift on symmetrically shaped obstacles immersed in low Reynolds number flow is the quintessential way to signal odd viscosity. For flow past cylinders, such a lift force does not arise if incompressibility and no-slip boundary conditions are fulfilled, whereas for spheres, a lift force has been found in Stokes flow, which is valid for cases where the Reynolds numbers are negligible and convection can be ignored. When considering the role of convection at low but non-zero Reynolds numbers, two hurdles arise, the Whitehead paradox and the breaking of axial symmetry, which are overcome by the method of matched asymptotic expansions and the Lorentz reciprocal theorem, respectively. We also consider the case where axial symmetry is preserved because the translation of the sphere is aligned with the axis of chirality of odd viscosity. We find that while lift vanishes, the interplay between odd viscosity and convection gives rise to a stream-induced torque.

Novel study of inertial forces on MHD peristaltically driven micropolar fluid through porous-saturated asymmetric channel: Finite Galerkin approach

This focused study investigates the peristaltic motion of a micropolar fluid within an uneven channel filled with a porous medium, incorporating an orthogonal magnetic field to the flow. This research diverges from the traditional assumptions of lubrication theory. The governing equations, encompassing the physical characteristics of micropolar fluid peristalsis, are transformed into nonlinear coupled partial differential equations. These equations are solved using the finite element method, considering inertial effects, such as non-zero wave and Reynolds numbers. This study delves into the influence of various crucial parameters on axial velocity, pressure gradient, microrotation, and stream function, presenting graphical representations. Notably, the incremental phase shift causes an intermingling of upper and lower streamlines within both halves of the channel. As the Reynolds number increases, there is an observed reduction in bolus size, particularly at maximum phase shifts, with a tendency to move toward the central region. An increase in Hartmann number leads the bolus formation to vanish in both channels, reduces microrotation, and leads to increased pressure. Vorticity lines intensify and incline toward the peristaltic walls. An increase in the permeability parameter amplifies velocity, microrotation, volume, and bolus formation regardless of phase differences while countering pressure elevation per wavelength. Reduced concavity is observed as vorticity lines disperse across the entire area.

Numerical simulation of thixotropic–viscoelastic models for structured fluids in hierarchical grids

In the present work, we implement thixotropic–viscoelastic models in the HiGFlow system, which is a recently developed Computational Fluid Dynamics software that is able to simulate Newtonian, Generalized-Newtonian and viscoelastic flows using finite differences in tree-based grids. The system uses a moving least squares meshless interpolation technique, allowing to use more complex mesh configurations, still keeping the overall order of accuracy. We provide here a brief introduction to each of the implemented models (the Bautista–Manero–Puig, the Modified-Bautista, the NM_τp and NM_T models), where we describe their parameters, their strengths and their limitations. Unlike previous numerical works focused on this kind of fluids, non-zero Reynolds number numerical simulations of these models are carried out here in HiGFlow using different geometries like 2D channels, planar contraction 4:1 and 4:1:4 and our results are compared with those predicted by RheoTool, an open-source toolbox based on OpenFOAM. Although the transient numerical solutions differ, we observe that there is an excellent agreement in the steady-state profiles obtained in both software. A discussion of our results and their physical interpretation, a comparison with previously published results in the literature and a deep analysis of the corner vortex behaviour in the contraction geometries are also presented. Moreover, we show that our methodology offers flexibility and stability in the numerical simulations of thixotropic–viscoelastic flows for structured fluids, which allows the users to accurately simulate rheological behaviour of interest.

Peristaltic transport of a power-law fluid induced by a single wave: A numerical analysis using the cumulant lattice Boltzmann method

Peristaltic pumping is the primary mechanism of food transport in the human intestine. Intestinal contents are often modeled as power-law fluids with low-behavior indices (n < 1). Peristaltic flows were studied for periodic contraction waves (L/λ=1) with infinitely long wavelengths (λ/D→∞) in the Stokes flow regime (Re→0). However, the peristaltic flow generated by an isolated contraction wave with a short wavelength at nonzero Reynolds numbers is more relevant to physiological conditions. In this study, we investigated the peristaltic transport of a power-law fluid with a low behavior index of n = 0.21 at nonzero Reynolds numbers up to Re = 10, generated by a single short contraction wave. First, we investigated the analytical solution for the peristaltic transport of the power-law fluid for λ/D→∞ and Re→0. The analytical solution shows that the discharge flow rate of a power-law fluid generated by a single contraction wave is much smaller than that of a Newtonian fluid (n = 1). Next, we investigated the peristaltic transport for Re≤ 10 using the cumulant lattice Boltzmann method. The numerical results demonstrate that the discharge flow rate for the power-law fluid sharply increased owing to the inertia effect. The power-law fluid induces an asymmetric flow field with respect to the contraction wave at smaller Reynolds numbers than Newtonian fluids. The inertia effect was increased by the sharpness of the contraction wave. These results suggest that intestinal contents can be transported more quickly by an isolated contraction wave with a shorter wavelength when the contents have low consistency indices or when the contraction wave has a large propagation velocity.

Metachronal propulsion of non-Newtonian viscoelastic mucus in an axisymmetric tube with ciliated walls

Cilia-induced flow of viscoelastic mucus through an idealized two-dimensional model of the human trachea is presented. The cilia motion is simulated by a metachronal wave pattern which enables the mobilization of highly viscous mucus even at nonzero Reynolds numbers. The viscoelastic mucus is analyzed with the upper convected Maxwell viscoelastic formulation which features a relaxation time and accurately captures normal stress generation in shear flows. The governing equations are transformed from fixed to wave (laboratory) frame with appropriate variables and resulting differential equations are perturbed about wave number. The trachea is treated as an axisymmetric ciliated tube. Radial and axial distributions in axial velocity are calculated via the regular perturbation method and pressure rise is computed with numerical integration using symbolic software MATHEMATICA‘TM’. The influence of selected parameters which is cilia length, and Maxwell viscoelastic material parameter i.e. relaxation time for prescribed values of wave number are visualized graphically. Pressure rise is observed to increase considerably with elevation in both cilia length and relaxation time whereas the axial velocity is markedly decelerated. The simulations provide some insight into viscous-dominated cilia propulsion of rheological mucus and also serve as a benchmark for more advanced modeling.

Analytic structure of solutions of the one-dimensional Burgers equation with modified dissipation

We use the one-dimensional Burgers equation to illustrate the effect of replacing the standard Laplacian dissipation term by a more general function of the Laplacian—of which hyperviscosity is the best known example—in equations of hydrodynamics. We analyze the asymptotic structure of solutions in the Fourier space at very high wave-numbers by introducing an approach applicable to a wide class of hydrodynamical equations whose solutions are calculated in the limit of vanishing Reynolds numbers from algebraic recursion relations involving iterated integrations. We give a detailed analysis of their analytic structure for two different types of dissipation: a hyperviscous and an exponentially growing dissipation term. Our results, obtained in the limit of vanishing Reynolds numbers, are validated by high-precision numerical simulations at non-zero Reynolds numbers. We then study the bottleneck problem, an intermediate asymptotics phenomenon, which in the case of the Burgers equation arises when ones uses dissipation terms (such as hyperviscosity) growing faster at high wave-numbers than the standard Laplacian dissipation term. A linearized solution of the well-known boundary layer limit of the Burgers equation involving two numerically determined parameters gives a good description of the bottleneck region.

Effect of wall deformability on the stability of shear-imposed film flow past an inclined plane

The linear stability of shear-imposed liquid film flow past an inclined plane is examined when a soft, deformable solid layer is attached to the inclined plane. A liquid film flowing past a rigid inclined plane exhibits a long-wave gas–liquid (GL) interfacial instability which is modified by the presence of an imposed shear stress at gas–liquid interface. This GL interfacial mode does not become unstable in creeping flow limit whether a GL interfacial shear is present or not. We demonstrate that the GL interface becomes unstable even at zero Reynolds number when the shear-imposed liquid film flows past an inclined plane which is coated with soft solid layer. This GL mode instability exists only when both shear and a deformable liquid–solid (LS) interface are simultaneously present. For non-zero Reynolds number, we show that there exists multiple unstable modes originating because of the presence of deformable LS interface. These unstable LS modes become important and practically realizable only for shear-imposed liquid film flow and become irrelevant for film flows in absence of imposed shear. We also show that this LS interfacial instability dominate the stability behavior of the composite fluid film-solid system in low Reynolds number regime. Our results suggest that the shear-imposed film flow can be made unstable by using a deformable solid coating in parameter regime where the film flow otherwise remains stable in rigid wall limit. Finally, we show that a deformable solid layer can be used to obtain a stable film flow configuration for the parameter regime where the GL interfacial mode is unstable for shear imposed film flow over a rigid incline. Thus, we demonstrate the capability of a soft solid layer in manipulation and control of instabilities for shear-imposed film flows.

Peristaltic transport of thixotropic fluids: A numerical simulation

Peristaltic flow of a thixotropic fluid obeying the Moore model is numerically studied using the multiple-relaxation-time lattice Boltzmann method (MRT-LBM). Converged results could be obtained in a planar two-dimensional channel at large Reynolds numbers for arbitrary wavelengths and amplitude ratios for nonzero Reynolds numbers. It is shown that depending on the Reynolds number and the parameters of the propagating wave, the time constant introduced through a fluid’s thixotropy may increase the mean flow rate of peristaltic pumps. Our numerical results suggest that for thixotropic fluids there exists a threshold wav-enumber for the peristaltic wave above which thixotropy can boost fluid transport but below which it can have an opposite effect.

Numerical Analysis of Peristaltic Transport of Casson Fluid for non-zero Reynolds Number in Presence of the Magnetic Field

Abstract In this study, the peristaltic flow of a Casson fluid in a channel is considered in the presence of an applied magnetic field. Flow is considered in the moving frame of reference with constant velocity along the wave. The developed mathematical model is presented by a set of partial differential equations. A numerical algorithm based on finite element method is implemented to evaluate the numerical solution of the governing partial differential equations in the stream-vorticity formulation. The obtained results are independent of low Reynolds number and long wavelength assumptions, so the effects of non-zero moderate Reynolds number are presented. The expression for the pressure is also calculated implicitly and discussed through graphs. Computed solutions are presented in the form of contours of streamlines and vorticity. Velocity profile and pressure distribution for variation of different involved parameters are also presented through graphs. The investigation shows that the strength of circulation for stream function increases by increasing the Reynolds and Hartmann numbers. Enhancement in longitudinal velocity is noted by increasing the Reynolds number and Casson parameter while increasing Hartmann number reduces the longitudinal velocity. Comparison of the present results with the available results in literature is also included and found in good agreement.

Numerical Study of Mixed Convective Peristaltic Flow through Vertical Tube with Heat Generation for Moderate Reynolds and Wave Numbers

The current study focuses on the numerical investigation of the mixed convective peristaltic mechanism through a vertical tube for non-zero Reynolds and wave number. In the set of constitutional equations, energy equation contains the term representing heat generation parameter. The problem is formulated by dropping the assumption of lubrication theory that turns the model mathematically into a system of the nonlinear partial differential equations. The results of the long wavelength in a creeping flow are deduced from the present analysis. Thus, the current study explores the neglected features of peristaltic heat flow in the mixed convective model by considering moderate values of Reynolds and wave numbers. The finite element based on Galerkin’s weighted residual scheme is applied to solve the governing equations. The computed solution is presented in the form of contours of streamlines and isothermal lines, velocity and temperature profiles for variation of different involved parameters. The investigation shows that the strength of circulation for stream function increases by increasing the wave number and Reynolds number. Symmetric isotherms are reported for small values of time-mean flow. Linear behavior of pressure is noticed by vanishing inertial forces while the increase in pressure is observed by amplifying the Reynolds number.

Dynamics of capsules enclosing viscoelastic fluid in simple shear flow

Previous studies on capsule dynamics in shear flow have dealt with Newtonian fluids, while the effect of fluid viscoelasticity remains an unresolved fundamental question. In this paper, we report a numerical investigation of the dynamics of capsules enclosing a viscoelastic fluid and which are freely suspended in a Newtonian fluid under simple shear. Systematic simulations are performed at small but non-zero Reynolds numbers (i.e. $Re=0.1$) using a three-dimensional front-tracking finite-difference model, in which the fluid viscoelasticity is introduced via the Oldroyd-B constitutive equation. We demonstrate that the internal fluid viscoelasticity presents significant effects on the deformation behaviour of initially spherical capsules, including transient evolution and equilibrium values of their deformation and orientation. Particularly, the capsule deformation decreases slightly with the Deborah number De increasing from 0 to $O(1)$. In contrast, with De increasing within high levels, i.e. $O(1{-}100)$, the capsule deformation increases continuously and eventually approaches the Newtonian limit having a viscosity the same as the Newtonian part of the viscoelastic capsule. By analysing the viscous stress, pressure and viscoelastic stress acting on the capsule membrane, we reveal that the mechanism underlying the effects of the internal fluid viscoelasticity on the capsule deformation is the alterations in the distribution of the viscoelastic stress at low De and its magnitude at high De, respectively. Furthermore, we find some new features in the dynamics of initially non-spherical capsules induced by the internal fluid viscoelasticity. Particularly, the transition from tumbling to swinging of oblate capsules can be triggered at very high viscosity ratios by increasing De alone. Besides, the critical viscosity ratio for the tumbling-to-swinging transition is remarkably enlarged with De increasing at relatively high levels, i.e. $O(1{-}100)$, while it shows little change at low De, i.e. below $O(1)$.

Steady bilayer channel and free-surface isothermal film flow over topography

Two flow configurations, involving bilayers, are explored systematically: through an inclined channel comprised of two planar surfaces in parallel alignment and down an inclined plane. Both liquid layers are assumed to be perfectly immiscible and the lower rigid bounding surface contains locally defined steep-sided topographical features – either a step-up/-down or a trench/peak. A common governing equation-set based on the long-wave approximation and depth-averaging is derived, embodying the more general case of a system of n-layers, and solved numerically.Results for the particular case of flow in a vertically aligned channel are found to be indistinguishable from corresponding solutions in the literature; those for the case of a step-up and non-zero Reynolds number having not been reported hitherto. New to this, the case of flow in a inclined channel is investigated as is the situation when, in horizontal alignment, the channel’s upper planar bounding surface moves with a constant speed inducing a shear flow.Gravity-driven bilayer film flow reveals a number of interesting features dependent on the fluid properties, the Reynolds number and the ratio of the upper to lower liquid layer thickness, with parallels drawn to the practically relevant cascade/slide-coating mode of multi-layer thin-film deposition. In the limit when both layers have identical properties the corresponding equivalent single layer solution is recovered exactly.

Numerical study at moderate Reynolds number of peristaltic flow of micropolar fluid through a porous-saturated channel in magnetic field

This paper analyzes the MHD flow of micropolar fluid induced by peristaltic waves passing through the porous saturated channel at large Reynolds number. The flow model is formulated in the absence of assumptions of lubrication theory which yields the governing equations into a non-linear set of coupled partial differential equations which allows studying the peristaltic mechanism at non-zero Reynolds and wave numbers. The influence of other involved parameters on velocity, stream function and microrotation are discussed through graphs plotted by using Galerkin’s finite element method. Besides that, the phenomena of pumping and trapping are also analyzed in the later part of the paper. To ensure the accuracy of the developed code, obtained results are compared with the results available in the literature and found in excellent agreement. It is found that the peristalsis mixing can be enhanced by increasing Hartmann number while it reduces by increasing permeability of the porous medium.

Numerical Analysis of Mixed Convective Peristaltic Flow in a Vertical Channel in Presence of Heat Generation without using Lubrication Theory

In this paper, heat transfer analysis of peristaltic mixed convection flow through a vertical channel is presented in addition, effects of heat generation are also investigated. The mathematical model is represented by the system of non-linear partial differential equations. The analysis is made in the presence of non-zero wave and Reynolds numbers. The results of the long wavelength assumption in a creeping flow can be deduced. These results thus predict new features in the peristaltic transport in the absence of the approximation of long wave length and low Reynolds number. The moderate finite elements based technique has been used to compute the highly accurate solution of the governing problem. To ensure the accuracy of the computed solution, the results obtained are validated against the available results in the literature and found good agreement. The obtained result are presented through graphs and the influence of involved pertinent parameters is analyzed.

Global m-Equivariant Solutions of Nematic Liquid Crystal Flows in Dimension Two

In this article we construct a global solution of the simplified Ericksen-Leslie system. We show that the velocity of the solution can be decomposed into the sum of three parts. The main flow is governed by the Oseen vortex with the same circulation Reynolds number as the initial fluid. The secondary flow has finite kinetic energy and decay in the speed (1 + t)−2 as $${t \rightarrow \infty}$$ . The third part is a minor flow whose kinetic energy decays faster than the secondary flow. As for the orientation variable, our solution has a phase function which diverges logarithmically to $${\infty}$$ as $${t \rightarrow \infty}$$ . This indicates that the orientation variable will keep rotating around the z-axis while $${t \rightarrow \infty}$$ . This phenomenon results from a non-trivial coupling between the orientation variable and a fluid with a non-zero circulation Reynolds number.

Peristaltic transport of solid particles suspended in a viscoplastic fluid: A numerical study

Peristaltic transport of circular rigid particle(s) suspended in a viscoplastic fluid is numerically investigated in a planar two-dimensional channel using the lattice Boltzmann method (LBM) coupled with the smoothed profile method (SPM). Numerical results could be obtained at non-zero Reynolds numbers for a wide range of the propagating wave's parameters. Our numerical results reveal that a fluid's yield stress has a dramatic effect on the peristaltic transport of solid particles. For a single particle suspended in a viscoplastic fluid, which obeys the bi-viscosity (Bingham) model as its constitutive equation, it is found that there exists a threshold wave number below which the yield stress slows down the particle's transport but above which it has an accelerating effect. A fluid's yield stress is also predicted to lower the average velocity of an initially-centered single particle at any given confinement ratio. For a given Bingham number, particle's velocity is predicted to increase by an increase in the confinement ratio provided that it is larger than a critical value. An increase in the Reynolds number may increase or decrease the particle's average velocity depending on the Bingham number. For the two-particle and four-particle scenarios, our numerical results show that the initial positioning of the particles in the channel has a decisive role in their eventual fate.

Extension of a viscous thread with temperature-dependent viscosity and surface tension

We consider the evolution of a long and thin vertically aligned axisymmetric viscous thread that is composed of an incompressible fluid. The thread is attached to a solid wall at its upper end, experiences gravity and is pulled at its lower end by a fixed force. As the thread evolves, it experiences either heating or cooling by its environment. The heating affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We develop a framework that can deal with threads that have arbitrary initial shape, are non-uniformly preheated and experience spatially non-uniform heating or cooling from the environment during the pulling process. When inertia is completely neglected and the temperature of the environment is spatially uniform, we obtain analytic solutions for an arbitrary initial shape and temperature profile. In addition, we determine the criteria for whether the cross-section of a given fluid element will ever become zero and hence determine the minimum stretching force that is required for pinching. We further show that the dynamics can be quite subtle and leads to surprising behaviour, such as non-monotonic behaviour in time and space. We also consider the effects of non-zero Reynolds number. If the temperature of the environment is spatially uniform, we show that the dynamics is subtly influenced by inertia and that the location at which the thread will pinch is selected by a competition between three distinct mechanisms. In particular, for a thread with initially uniform radius and a spatially uniform environment but with a non-uniform initial temperature profile, pinching can occur either at the hottest point, at the points near large thermal gradients or at the pulled end, depending on the Reynolds number. Finally, we show that similar results can be obtained for a thread with initially uniform radius and uniform temperature profile but exposed to a spatially non-uniform environment.

Free-surface film flow over topography: Full three-dimensional finite element solutions

An efficient Bubnov–Galerkin finite element formulation is employed to solve the Navier–Stokes and continuity equations in three-dimensions for the case of surface-tension dominated film flow over substrate topography, with the free-surface location obtained using the method of spines. The computational challenges encountered are overcome by employing a direct parallel multi-frontal method in conjunction with memory-efficient out-of-core storage of matrix co-factors. Comparison is drawn with complementary computational and experimental results for low Reynolds number flow where they exist, and a range of new benchmark solutions provided. These, in turn, are compared with corresponding solutions, for non-zero Reynolds number, from a simplified model based on the long-wave approximation; the latter is shown to produce comparatively acceptable results for the free-surface disturbance experienced, when the underpinning formal restrictions on geometry and capillary number are not exceeded.

On the stability of internally heated natural convection due to the absorption of radiation in a laterally confined fluid layer with a horizontal throughflow

This study focuses on the marginal stability of the natural convection flow induced by the absorption of radiation. Non-uniform internal heating associated with the absorption of radiation following Beer’s law and the absorption and re-emission of residual radiation by the bottom boundary wall lead to a nonlinear base temperature profile that evolves with time. A horizontal throughflow and lateral confinement are incorporated in the analysis, and two forms of convective instability, i.e. longitudinal and transverse rolls, are investigated. It is shown that the formation of longitudinal rolls, with their axes parallel to the throughflow direction, is not affected by the presence of the throughflow. For transverse rolls, with their axes perpendicular to the throughflow direction, however, the throughflow is shown to have a stabilising effect, with the critical Rayleigh number increasing with the increasing throughflow Reynolds number. It follows that longitudinal rolls are the preferred form of instability in laterally unconfined domains for any nonzero Reynolds number and in confined domains if the Reynolds number is greater than a critical value. The value of the critical Reynolds number, which separates the preferred mode of instability, is found to rapidly reduce with increasing lateral extent and approaches zero for a normalised lateral extent greater than 2.0. The lateral confinement is also shown to have a stabilising effect for both longitudinal and transverse rolls, with the critical Rayleigh number increasing with the reducing lateral extent. It is shown that the lateral confinement only substantially affects the wave number of the rolls with their axes parallel to the confining walls but not the rolls with their axes perpendicular to the confining walls.

Numerical simulations of the competition between the effects of inertia and viscoelasticity on particle migration in Poiseuille flow

In this work, we present 2D numerical simulations on the migration of a particle suspended in a viscoelastic fluid under Poiseuille flow at finite Reynolds numbers, in order to clarify the simultaneous effects of viscoelasticity and inertia on the lateral particle motion.The governing equations are solved through the finite element method by adopting an Arbitrary Lagrangian–Eulerian (ALE) formulation to handle the particle motion. The high accuracy provided by such a method even for very small particle–wall distances, combined with proper stabilization techniques for viscoelastic fluids, allows obtaining convergent solutions at relatively large flow rates, as compared to previous works. As a result, the detailed non-linear dynamics of the migration phenomenon in a significant range of Reynolds and Deborah numbers is presented.The simulations show that, in agreement with the previous literature, a mastercurve relating the migration velocity of the particle to its ‘vertical’ position completely describes the phenomenon. Remarkably, we found that, for comparable values of the Deborah and Reynolds numbers, inertial effects are negligible: migration is in practice driven by fluid viscoelasticity only. At moderate Reynolds numbers (20<Re<200) and by lowering De, the transition from viscoelasticity-driven to inertia-driven regimes occurs through two intermediate regimes characterized by multiple stable solutions, i.e. attractors of particle trajectories at different vertical positions across the gap. At low but non-zero Reynolds numbers, only two stable solutions are found for any non-zero Deborah number in the investigated range. In particular, the wall is always an attractor for the migrating particle.