Steady Axisymmetric Flow Research Articles

Newton-Krylov continuation of amplitude-modulated rotating waves in sheared annular electroconvection.

We present an approach for studying the primary, secondary, and tertiary flow transitions in sheared annular electroconvection. In particular, we describe a Newton-Krylov method based on time integration for the computation of rotating waves and amplitude-modulated rotating waves, and for the continuation of these flows as a parameter of the system is varied. The method exploits the rotational nature of the flows and requires only a time-stepping code of the model differential equations, i.e., it does not require an explicit code for the discretization of the linearized equations. The linear stability of the solutions is computed to identify the parameter values at which the transitions occur. We apply the method to a model of electroconvection that simulates the flow of a liquid crystal film in the smectic A phase suspended between two annular electrodes and subjected to an electric potential difference and a radial shear. Due to the layered structure of the smectic A phase, the fluid can be treated as two-dimensional (2D) and is modeled using the 2D incompressible Navier-Stokes equationscoupled with an equationfor charge continuity. The system is a close analog to laboratory-scale geophysical fluid experiments and thus represents an ideal system in which to apply the method before its application to these other systems that exhibit similar flow transitions. In the model for electroconvection, we identify the parameter values at which the primary transition from steady axisymmetric flow to rotating waves occurs, as well as at which the secondary transition from the rotating waves to amplitude-modulated rotating waves occurs. In addition, we locate the tertiary transition, which corresponds to a transition from the amplitude-modulated waves to a three-frequency flow. Of particular interest is that the method also finds a period-doubling bifurcation from the amplitude-modulated rotating waves and a subsequent transition from the flow resulting from this bifurcation.

MHD Couple stress fluid between two concentric spheres with slip regime

The problem of the axisymmetric steady flow of an incompressible couple stress fluid within two concentric spheres is analyzed analytically in this study. In the interior of the solid sphere, translation takes place with a uniform velocity, and in the exterior, translation takes place with a fixed velocity. A study of slip conditions was carried out on the surfaces of the two spheres. In terms of stream functions, modified Bessel functions provide an analytical solution to the problem. As the outer sphere remains stationary, the inner sphere experiences the normalized drag force. A graphic representation of drag is provided based on the slip and spin slip parameters, the couple stress parameter, the separation distance, and the MHD parameter. In addition, the results are compared with earlier results for special cases of translation of one rigid sphere through an unbounded magnetic medium, viscous fluid flow, and Couple stress fluid without MHD between two concentric spheres.

Swirling electrolyte. Part 1. Hierarchy of catastrophic transitions in steady annular flows

This study extends our previous work (McCloughan & Suslov, J. Fluid Mech., vol. 887, 2020, A23), where the existence of a saddle-node bifurcation of steady axisymmetric electrolyte flows driven by the Lorentz force in a shallow annular domain was first reported. Here we perform further weakly nonlinear analysis over a wider range of the governing parameters to demonstrate that the previously reported saddle-node bifurcation is a local feature of a global fold catastrophe, which, in turn, is a part of cusp catastrophe occurring as the thickness of the fluid layer increases. The amplitude equation characterising multiple flow solutions in the finite vicinity of catastrophe points is derived. The sensitivity of its coefficients and solutions to the distance from the catastrophe points is assessed demonstrating the robustness of the used analytical procedure. The asymptotic flow solution past the catastrophe point is subsequently obtained and its topology is explored confirming the existence of the secondary circulation in the bulk of flow (two-tori background flow structure). The latter is argued to lead to the appearance of experimentally observable vortices on the fluid surface. The rigorous justification of this conjecture is to be given in Part 2 of the study.

Design of a real prototype of a hybrid fuel cell suv: Vehicle "Hércules"

An aeromechanical evaluation of large (over 1 Mw of nominal power) Horizontal Axis Wind Turbines (HAWT’s) is performed is this paper. The strategy is based on the combination of an aerodynamic module, which provides the three-dimensional pressure distribution on the HAWT’s blades, an a structural module which takes such pressure forces as input data in order to compute both, blade deformation and strain and stress distributions over the blade. The aerodynamic module combines the three-dimensional non-linear lifting surface theory approach, which provides the effective incident velocity and angle of attack at each blade section, and a two-dimensional panel method for steady axisymmetric flow in order to obtain the 3D pressure distribution on the blade. Such pressure distribution constitutes the input data for the structural module, which is a finite element package whose output is the blade deformation and strain and stress distribution along the blade, as well as material induced fatigue. This methodology is applied to study a 50 m long blade able to provide a nominal power of 3 Mw.

High-Prandtl-number thermocapillary liquid bridges with dynamically deformed interface: effect of an axial gas flow on the linear stability

The linear stability of the axisymmetric steady flow in a thermocapillary liquid bridge made from 2-cSt silicone oil $({Pr}=28)$ is investigated numerically. The liquid bridge is heated either from above or below and exposed to an axial air flow which is confined to a concentric tube surrounding the bridge. At the annular inlet, the air flow is fully developed and has the same temperature as the adjacent support rod. Using an extended Oberbeck–Boussinesq approximation in which the density of both fluids depends linearly on the temperature in all equations, critical thermocapillary Reynolds numbers are obtained depending on the strength of the imposed axial air flow. The critical conditions are sensitive with respect to the direction of a weak air flow, because the air flow changes the plateau value of the interfacial temperature midway between the hot and cold ends. For stronger air flow the critical thermocapillary Reynolds number almost saturates at moderate values. Throughout, the instability arises as a hydrothermal wave with the gas phase being passive. The dynamic interface deformations for axisymmetric flow caused by the thermocapillarity flow in the liquid and by the stresses from the air flow are considered separately. Apart from turning points of the critical curve, the impact of dynamic surface deformations on the critical thermocapillary Reynolds number is moderate.

Flow instability in high-Prandtl-number liquid bridges with fully temperature-dependent thermophysical properties

The axisymmetric steady two-phase flow of a differentially heated thermocapillary liquid bridge in air and its linear stability is investigated numerically, taking into account dynamic interfacial deformations in the basic flow. Since most experiments require a high temperature difference to drive the flow into the three-dimensional regime, the temperature dependence of the material properties must be taken into account. Three different models are investigated for a high-Prandtl-number thermocapillary liquid bridge with nominal Prandtl number ${\textit {Pr}}=28.8$ : the Oberbeck–Boussinesq (OB) approximation, a linear temperature dependence of all material properties and a full nonlinear temperature dependence of all material properties. For all models, critical Reynolds numbers are computed as functions of the volume of the liquid bridge, its aspect ratio, its dimensional size and as a function of the strength of a forced axial flow in the ambient air. Under most circumstances the OB approximation overpredicts and the linear model underpredicts the critical Reynolds number, compared with the model based on the full temperature dependence of the material properties. Among the main influence factors are the proper selection of the reference temperature and, at larger temperature differences, the temperature dependence of the viscosity of the liquid.

Effect of fluid elasticity and shear thinning on heat transfer from a heated sphere at moderate Reynolds numbers

The drag and heat transfer characteristics of an isothermal sphere in a FENE-P-type viscoelastic fluid have been studied in the steady axisymmetric flow regime. The governing equations, namely, mass, momentum, and energy equations, together with FENE-P type viscoelastic constitutive equations, have been solved using the finite volume method based opensource CFD code OpenFOAM over the following ranges of conditions: Reynolds number, 1 ≤ Re ≤ 100 ; Prandtl number, 1 ≤ Pr ≤ 50 ; Weissenberg number, 0 ​ ≤ Wi ≤ 10 and polymer extensibility parameter, 10 ≤ L 2 ≤ 500 for a fixed value of the polymer viscosity ratio of β = 0.5 . At low Reynolds numbers, fluid viscoelasticity causes an upward and downward shifting in the axial velocity profiles with reference to that in a Newtonian fluid along the upstream and downstream axis of the sphere, respectively. However, such a shift progressively diminishes with the increasing Reynolds number. Furthermore, a “velocity overshoot” and/or a “negative wake” is also observed in a viscoelastic polymer solution at low Reynolds numbers, and this is accentuated with the increasing Weissenberg number. The separation of boundary layers is seen to be suppressed with the increasing Weissenberg number. The drag coefficient decreases with the Reynolds number irrespective of the fluid type and the corresponding dependence on the Weissenberg number is found to be non-monotonic depending upon the values of Re and L 2. The average Nusselt number always increases with the increasing values of Re, Pr, and Wi and with the decreasing value of L 2. A limited number of simulations have also been carried out to show the effect of the temperature-dependent viscosity of the fluid on the flow dynamics and heat transfer characteristics. Finally, simple correlations for the drag coefficient and average Nusselt number are presented which not only capture the functional dependence but also can be used for the interpolation of the present results for the intermediate values of the governing parameters in a new application.

Linear stability analysis on the thermocapillary flow of molten silicon in a liquid bridge between unequal disks under a rotating magnetic field

This paper explores the stability of thermocapillary flow in a molten silicon liquid bridge situated between two unequal disks and subjected to a rotating magnetic field. Two different heating strategies are examined, and the static deformation of the free surface is taken into account by solving the Young–Laplace equation. Using linear stability analysis based on the Legendre spectral element method, the critical conditions are determined, and the underlying instability mechanism is uncovered through energy analysis. When heated from the bottom disk, three transitions between axisymmetric steady flow and three-dimensional oscillatory flow are observed within a narrow range of 7000 ≤ Ta ≤ 9000. In contrast, when heated from the top disk, the rotating magnetic field significantly enhances the flow stability. Furthermore, based on the energy analysis, we identify two types of instability caused by purely hydrodynamic mechanisms.

Linear global stability of a flow past a sphere under a streamwise magnetic field

The global linear stability analysis for the magnetohydrodynamic liquid metal flow past an insulated sphere subjected to a constant streamwise magnetic field is investigated in the range of the Reynolds number $Re\leq 400$ and the interaction number $N\leq 40$ coupled with direct numerical simulations, where $N$ stands for strength of the electromagnetic force. The stability of the steady axisymmetric base flow to independent time-azimuthal modes is discussed. Five critical curves associated with various wake transitions are obtained in the $\{Re, N\}$ phase diagram. These critical curves reveal the stabilising effect of a weak magnetic field, the destabilising effect of a strong magnetic field and re-stabilising effect of a much stronger magnetic field. To explore the impact of the magnetic field on flow instability, a sensitivity analysis utilizing an adjoint method is performed for the first regular bifurcation. Sensitivity functions of growth rate to base-flow modifications and Lorentz force are defined to identify the region that has the most significant influence on flow instability, such as the recirculation region responsible for the stabilising effect at a weak magnetic field and the shear layer region responsible for the destabilising effect at a strong magnetic field. Furthermore, a competition between the stabilising and shear destabilising effects of the magnetic field is discussed. This analysis provides valuable insights into the non-monotonic effect of the magnetic field on flow instability.

Numerical Analysis of Velocity and Thermal Wall Slip Effects on the Boundary Layer Flow Over an Exponentially Stretching Bullet-Shaped Object in Presence of Suction and Injection

A steady two-dimensional axisymmetric incompressible flow over an exponentially stretching bullet-shaped surface has been considered. The present work is mainly focused on fluid flow by the effect of multiple slips. The governing partial differential equations and auxiliary boundary conditions have been converted into higher-order equations by using assisting similarity transformations. These higher-order ODEs are then transformed into a 1st-order system of LDEs by the method of spectral quasi-linearization (SQLM). The validity, accuracy, and convergence of the solution have been performed by using SQLM. The fluid velocity, temperature, skin friction coefficient, and Nusselt number have been depicted graphically for the mentioned parameters as also the numerical values of velocity gradient, and Nusselt number in a table. The numerical investigation shows that the velocity gradient enhances due to the parameter of magnetic, thermal slip, and Prandtl number whereas the remaining parameters have a reverse effect on it. The heat transfer rate reduces for the parameters of magnetic, multiple slip, injection, and viscous dissipation but suction and heat generation have a reverse effect on it. The results of in this work have been justified due to the validity and accuracy of the present problem. Due to the endless application of Newtonian fluids in engineering and industry, no attempt has been taken to inspect the MHD flow with a dual slip effect along with exponential stretching bullet-shaped surface. Also, the current work is of immediate interest to those systems that are highly influenced by the heat transfer process and desired product quality.

Using a Magnetic Field to Reduce Thermocapillary Convection in Thin Annular Pools

This paper presents the investigation of thermocapillary convection in three-dimensional (3-D) thin pools with three cases of annular gaps containing silicon melt under a vertical magnetic field. The model was composed of two vertical walls: the inner is cold, and the outside is hot. Radiation is emitted upward from the free top surface, and the bottom is heated vertically. Both cases are considered in this study, electrically insulating all walls; and only the bottom wall is electrically conducting for three annular gaps. The governing equations were solved numerically through the finite volume method. The effects of different parameters such as the Hartmann number, annular gaps on the temperature distribution, the hydrothermal wave number, and azimuthal patterns, as well as the transition from 3-D steady to axisymmetric flows, were investigated. The results showed three hydrothermal waves are observed in a 3-D steady flow. It was also found that with the increasing Hartmann number, the azimuthal velocity, the temperature fluctuation, and the electric potential decreased. The results also revealed that a stronger magnetic field was needed for the transition from unsteady flow to a nonaxisymmetric steady flow and at the end to steady axisymmetric flow. The findings revealed that electromagnetic damping is more prominent when the bottom wall is electrically conducting than when all walls are insulating.

Effect of rotating magnetic field on the stability of thermocapillary flow in a gallium arsenide liquid bridge between unequal ends

In this study, we investigated the impact of a rotating magnetic field on the stability of a thermocapillary flow in a gallium arsenide liquid bridge (Prandtl number Pr = 0.068) situated between two unequal disks, considering two different scenarios with radius ratios of Γr = 0.98 and Γr = 0.60 for the upper heated disk. By utilizing linear stability analysis based on the Legendre spectral element method, we first identified the critical parameters of the onset of flow instability, including critical Marangoni number (Mac), dimensionless oscillation frequency (fc), and azimuthal wavenumber (m). Then, we employed kinetic energy budget analysis to uncover the underlying instability mechanism. For radius ratio Γr = 0.98, three transitions between axisymmetric steady flow and three-dimensional oscillatory flow in the narrow range of Taylor number Ta (8700≤Ta ≤ 9500) are observed; these transitions arise due to the interplay between the flow induced by rotating magnetic field and thermocapillary flow. For the Γr = 0.60 scenario, the rotating magnetic field is observed to significantly enhance the flow stability. Additionally, our analysis identifies four instability types dominated by the hydrodynamic mechanism. In the meantime, the thermocapillary mechanism also contributes to flow instability in the specific region of Taylor number Ta (1250≤Ta ≤ 8000) for radius ratio Γr = 0.98.

Cattaneo-Christov heat flux model in radiative flow of (Fe3O4– TiO2/Transformer oil) and (Cu– TiO2/Transformer oil) magnetized hybrid nanofluids past through double rotating disks

Recent progresses in nanotechnologies and nanoscience have led to the creation of hybrid nanofluids, which are a complicated category of fluids with superior thermal features to regular nanofluids. The current framework demonstrates the importance of a two-dimensional steady incompressible axisymmetric flow of Maxwell hybrid nanofluid over double disks with thermal radiation. This investigation analyzes a novel idea regarding the execution of the Cattaneo-Christov heat theory and melting phenomenon by considering transformer oil as a base fluid. Two dissimilar classes of hybrid nanofluid, Iron–Titanium oxide/Transformer oil (Fe3O4–TiO2/TO) and Copper–Titanium oxide/Transformer oil (Cu–TiO2/TO) have been taken into our research work. The main equations (PDEs)are translated into a present set of ODEs using the necessary similarity variables. In MATLAB, the shooting scheme is utilized to evaluate the numerical and graphical outcomes of physical flow parameters. The radial velocity rose as the volume fraction of nanoparticles increased. The radial velocity field is increased as the porosity parameter is enhanced. The temperature profile is decreased with increasing the values of the thermal redaction parameter. Furthermore, because of the higher compactness of the copper nanoparticles, the addition of the volume fraction of nanoparticles slows the flow profile, and because copper is an excellent conductor of heat, it raises the fluid temperature throughout the domain. The mathematical fallouts also tackle the idea of employing magnetized spinning discs in space engines and nuclear propulsion, and such a model carries useful applications in heat transfer enhancement in a wide range of industrial thermal management devices and renewable energy generation systems.

Changing interface conditions in a two-fluid rotating flow

This experimental study reveals a striking nonlinear-physics phenomenon of fundamental and practical interest—changing conditions at the interface of two swirling immiscible fluids filling a vertical cylindrical container. To this end, we use a new measurement technique significantly advanced compared with prior studies. The rotating bottom disk drives a steady axisymmetric flow of both fluids. The lower fluid makes the centrifugal circulation (CC): It spirals on toroid surfaces going to the periphery near the bottom and going back to the axis near the interface. At a slow rotation (Re = 100), the upper fluid makes the anti-centrifugal circulation. As the rotation intensifies (Re = 175), the upper-fluid flow reverses into CC near the interface-axis intersection. For strong swirl (Re = 500), the CC occurs at the entire interface. In prior studies, the spatial resolution (1 mm) was insufficient to resolve the near-interface velocity field. Here, we use the advanced (light field) measurement technique, which has significantly better resolution (0.14 mm) and clearly shows that the radial velocity at the interface is negative for small Re and becomes zero for large Re. During these metamorphoses, the topology of the lower-fluid flow remains invariant, the interface has no visible deformation, and the flow is steady and axisymmetric.

Parallel and perpendicular flows of a couple stress fluid past a solid cylinder in cell model: Slip condition

The axisymmetric steady flow of a couple stress fluid between two concentric cylinders with a slip effect is investigated with the help of the cell model technique. Here, the inner cylinder is rigid, and the outer cylinder is fictitious. The tangential slip, vanishing of normal velocity, and zero couple stress conditions are applied on the inner cylindrical surface. In addition, zero shear stress (Happel's model), continuity of normal velocity component, and zero couple stress conditions are used on the outer cylindrical surface. We consider two flow problems: the first is the parallel flow, and the second is the perpendicular flow to the cylinder in the cell model. Also, we have discussed the random case. For all the cases, the Kozeny constant is calculated. We described some special cases and compared them with well-known results. The effects of slip and couple stress parameters on the Kozeny constant with fixed value of couple stress viscosity parameter are presented graphically. The influence of the couple stress viscosity parameter on the Kozeny constant with fixed values of couple stress, and slip parameters for parallel flow are expressed graphically. The numerical values for the Kozeny constant for different values of fractional void volume are tabulated. We also obtained the results of the consistent couple stress theory as a special case.

The Free Boundary of Steady Axisymmetric Inviscid Flow with Vorticity I: Near the Degenerate Point

In this paper, we investigate the singularity near the degenerate points of the steady axisymmetric flow with general vorticity of an inviscid incompressible fluid acted on by gravity and with a free surface. We called the points on the free boundary at which the gradient of the stream function vanishes as the degenerate points. The main results in this paper give the different classifications of the singularity near the degenerate points on the free surface. More precisely, we obtained that at the stagnation points, the possible profiles must be a Stokes corner, or a horizontal cusp, or a horizontal flatness. At the degenerate points on the symmetric axis except the origin, the wave profile must be a cusp. At the origin, the possible wave profiles must be a Garabedian pointed bubble, or a horizontal cusp, or a horizontal flatness.

Existence and regularity of solutions of a supersonic-sonic patch arising in axisymmetric relativistic transonic flow with general equation of state

In this article, we prove the existence and regularity of a smooth solution for a supersonic-sonic patch arising in a modified Frankl problem in the study of three-dimensional axisymmetric steady isentropic relativistic transonic flows over a symmetric airfoil. We consider a general convex equation of state which makes this problem complicated as well as interesting in the context of the general theory for transonic flows. Such type of patches appear in many transonic flows over an airfoil and flow near the nozzle throat. Here the main difficulty is the coupling of nonhomogeneous terms due to axisymmetry and the sonic degeneracy for the relativistic flow. However, using the well-received characteristic decompositions of angle variables and a partial hodograph transformation we prove the existence and regularity of solution in the partial hodograph plane first. Further, by using an inverse transformation we construct a smooth solution in the physical plane and discuss the uniform regularity of solution up to the associated sonic curve. Finally, we also discuss the uniform regularity of the sonic curve.

The MHD graphene nanofluid flow between two stretching discs

Consider a two-dimensional steady axisymmetric flow of an incompressible fluid with graphene nanoparticles between two infinite stretching discs with accelerated velocity and a magnetic field of strength B 0 is applied to the flow at the angle τ . The similarity transformation technique is adopted to find the analytical solution for the considered Navier–Stokes equation presented in a cylindrical coordinate form. The exact solution is analysed for the effect of stretching the Reynolds number and stretching velocity ratio. Pressure is high at the upper disc than at the lower disc when the stretching velocity ratio is less than unity. Then the effect of inclined MHD on the 2-D flow between stretching sheet discs yielding the solution in terms of an analytical method. The potential applications of these problems are used in the field of engineering and medical field, such as cancer tumour treatment and magnetic devices for cell separation.

Instability mechanisms of thermocapillary liquid bridges between disks of unequal radii

In this paper, we explore thermocapillary liquid bridges between two disks of unequal radii with Prandtl numbers Pr of 0.0258 (mercury) and 0.068 (gallium arsenide) to gain insight into the underlying instability mechanism. In the context of Legendre's spectral element method, we determine critical conditions via linear stability analysis and then identify the instability mechanism through energy analysis. For the mercury bridge (Pr = 0.0258), our analysis suggests that the flow instability undergoes an oscillatory bifurcation for radius ratios in the range of 0.5 ≤ Γr ≤ 0.66. In particular, we found three transitions between two-dimensional steady axisymmetric flow and three-dimensional stationary flow by further increasing the radius ratio to 0.73 ≤ Γr ≤ 0.76. For the gallium arsenide liquid bridge (Pr = 0.068), the instability is always an oscillatory bifurcation in the whole computational interval. Furthermore, our observations identify six instability modes with different mechanisms. All instability modes in the mercury bridge (Pr = 0.0258) are purely hydrodynamic, but the thermocapillary mechanism cannot be ignored in the gallium arsenide liquid bridge (Pr = 0.068) because of the enhanced Pr effect.

Stability of thermocapillary flow in liquid bridges fully coupled to the gas phase

The linear stability of the axisymmetric steady thermocapillary flow in a liquid bridge made from 2 cSt silicone oil (Prandtl number 28) is investigated numerically in the framework of the Boussinesq approximation. The flow and temperature fields in the surrounding gas phase (air) are taken into account for a generic cylindrical container hosting the liquid bridge. The flows in the liquid and in the gas are fully coupled across the hydrostatically deformed liquid–gas interface, neglecting dynamic interface deformations. Originating from a common reference case, the linear stability boundary is computed varying the length of the liquid bridge (aspect ratio), its volume and the gravity level, providing accurate critical data. The qualitative dependence of the critical threshold on these parameters is explained in terms of the characteristics of the critical mode. The heat exchange between the ambient gas and the liquid bridge that is fully resolved has an important influence on the critical conditions.