Steady Axisymmetric Flow Research Articles

Effect of the Prandtl number on the instabilities of the thermocapillary flow in an annular pool

This paper is devoted to the instability mechanisms of the thermocapillary flow in an annular pool. The stability limit of the axisymmetric basic state was studied over a wide range of Prandtl numbers (0.001 ≤ Pr ≤ 6.7) using linear stability analysis based on the spectral element method. The results demonstrate five types of instabilities, and the corresponding instability mechanisms were revealed by disturbance energy analysis. In particular, in the narrow range 1.4 ≤ Pr ≤ 1.53, with the increase in the Marangoni number, three transitions between the axisymmetric steady flow and the three-dimensional oscillatory flow were found, owing to the coupling and interaction of the hydrodynamic and hydrothermal instability mechanisms.

Effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool

This paper presents a series of three-dimensional numerical simulations on the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in a shallow annular pool. The annular pool with a fixed aspect ratio of 0.05 and radius ratio of 0.5 is filled with 0.65 cSt silicone oil. Its Prandtl number is 6.7. Biot number of surface heat dissipation ranges from 0 to 50. The results show that when Marangoni number is small, thermocapillary convection is the axisymmetric steady flow. When Marangoni number exceeds the critical value, the basic flow will destabilize and transit to the hydrothermal waves that propagate along the clockwise or anticlockwise direction. The critical Marangoni number of the flow destabilization increases with the increase of Biot number. Furthermore, when Biot number increases, the radial temperature gradient on the free surface decreases near the inner cylindrical wall, but increases near the outer cylindrical wall, which results in the thermocapillary convective cell moving gradually from the inner to the outer cylindrical walls. After the flow destabilizes, the temperature fluctuation on the free surface mainly appears near the outer cylinder at a large Biot number.

Surface Heat Dissipation Dependence of Thermocapillary Convection of Moderate Prandtl Number Fluid in an Annular Pool

In order to understand surface heat dissipation dependence of thermocapillary convection for moderate Prandtl number fluid in a deep annular pool, a series of three-dimensional numerical simulations have been carried out by using the finite volume method. The radius ratio and the aspect ratio of an annular pool are fixed at 0.5 and 1.0, respectively. The working fluid is 0.65cSt silicone oil with Prandtl number of 6.7. Surface heat dissipation Biot (Bi) number is varied from 0 to 50. Results indicate that with the increase of Biot number, the radial temperature gradient near the inner cylindrical wall decreases, and near the outer cylindrical wall it increases, so the flow is enhanced. When 0 < Bi < 10, with the increase of Marangoni number, the axisymmetric steady flow first transits to the standing wave, and then to the azimuthal waves. The standing wave should be attributed to Marangoni-Benard instability. However, the azimuthal waves should be corresponded to hydraulic instability, which is mainly driven by the azimuthal motion of temperature fluctuation from the sudden change of flow direction near the bottom and the inner cylindrical wall. When Bi ≥ 10, when the flow destabilizes, the axisymmetric steady flow transits directly to the azimuthal waves. With the increase of Biot number, the critical Marangoni number of the flow destabilization increases. Furthermore, the fundamental frequency and the wave number of three-dimensional oscillatory flow increase gradually with the increase of Biot number.

Effect of the crucible/crystal rotation on thermocapillary instability in a shallow Czochralski configuration

To understand the effects of crucible/crystal rotation on the instability of thermocapillary flow in a shallow Czochralski configuration, a series of linear stability analysis were performed by using Legendre spectral element method, and the energy analysis was applied to explore the instability mechanism. The results indicate that there are two types of instabilities with the increase of dimensionless crucible rotation rate ωc. Thermocapillary force and crucible rotation are the main causes of flow instability for low and high crucible rotation rate, respectively. In the interval 1627<ωc < 1822, with the increase of Marangoni number, three turning points were observed, indicating three transitions between two-dimensional axisymmetric steady flow and three-dimensional oscillatory flow due to the interaction between thermocapillary force and crucible rotation. For crystal rotation, two oscillatory modes for flow instability were observed. When the dimensionless crystal rotation rate is small (ωs < 3600), the oscillatory flow rotates in the opposite direction of the crystal rotation, but in the same direction in the case of large crystal rotation rate (ωs > 3600), and the thermocapillary force is the primary inducement of flow instability.

Heat transfer and axisymmetric stagnation point flow due to a shrinking vertical plate in a nanofluid with slip effects

The problem on steady axisymmetric stagnation point flow with velocity slip due to a shrinking vertical plate in a nanofluid was studied. This problem was focussing on the first-order and second-order velocity slip effects on the governing parameters, such as mixed convection parameter σ and nanoparticle volume fraction φ. Three types of nanofluids were considered in this study which known as Copper (Cu), Alumina (Al2O3) and Titania (TiO2) with the Prandtl number, Pr=6.2. In order to solve the problem, solver bvp4c in Matlab has been applied to solve the numerical part. Before the numerical phase, the governing system of partial differential equations was transformed first into ordinary differential equations by similarity transformation. Then, the observation was done to study the effects of first and second order velocity slip parameter, Δ and Λ, mixed convection parameter σ and nanoparticle volume fraction φ on heat transfer and fluid flow. Dual solutions exist for a certain range of mixed convection parameter σ. It is observed that when the nanoparticle volume fraction increase, the shear stress on the shrinking sheet also increase, same goes for heat transfer rate regarding the first-order and the second-order velocity slip parameters, Δ and Λ. The rate of heat transfer can be raised when the magnitude of the first-order and the second-order velocity slip parameter, Δ and Λ, decrease.

Lensing properties of rotational gas flow

A negative lens comprising a gas in steady axisymmetric flow is demonstrated experimentally and analyzed. The lens has potential applications in high-intensity laser optics and presents the possibility of adjusting the focusing properties on a submillisecond time scale. It can be operated in environments where conventional optical elements are vulnerable.

Critical Reynolds numbers of shear-thinning fluids flow past unbounded spheres

Three-dimensional simulations on the flow and wake behaviour of shear-thinning non-Newtonian fluids past spherical particles are carried out using a computational fluid dynamics based solver with the aim to investigate the critical Reynolds numbers for the onset of steady axisymmetric flow separation point and steady non-axisymmetric flow separation point. The present solver is thoroughly benchmarked through the conventional steps of domain independence, grid independence, and validation of the existing literature results. Before obtaining the critical Reynolds numbers for the case of shear-thinning fluids, the same for the case of Newtonian fluids are obtained and found to be in excellent agreement with literature values. The major findings of this work indicate that for the case of shear-thinning fluids, the first critical Reynolds number for the onset of steady axisymmetric flow separation increases with the decreasing power-law index of the non-Newtonian fluids; whereas the reverse is true for the case of second critical Reynolds number at which steady non-axisymmetric flow separation occurs. Furthermore, the non-axisymmetry is observed in both xy and xz planes for power-law fluids when n = 0.61 and 0.54; whereas in the case of fluids with n = 0.87, 0.73 and 0.66, the non-axisymmetry is observed only along the xz plane. Finally, correlations are proposed for both the critical Reynolds numbers as function of the power-law behaviour index.

Viscous flow past a porous sphere within a nonconcentric fictitious spherical cell

The quasi steady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous sphere situated at an arbitrary position within a virtual spherical cell along the line connecting their centers is analyzed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman model governs the flow inside the porous particle and the flow in the fictitious envelope medium is governed by Stokes equations. A general solution is constructed from the superposition of the fundamental solutions in the two spherical coordinate systems based on both the porous particle and virtual spherical cell. Boundary conditions on the particle’s surface and the fictitious spherical envelope for four known cell models are satisfied by a collocation method for truncated series. Numerical solutions for the drag force exerted on the porous sphere in the presence of the cell are obtained for various cases of the effective distance between the center of the porous particle and the fictitious envelope, the volume ratio of the porous particle over the surrounding sphere, the viscosity ratio, the stress jump coefficient, and a coefficient that is proportional to the permeability. Streamlines in and around porous sphere are presented for the unit cell models at different values of relevant physical parameters. In the limits of the motions of the porous particle in the concentric position with the cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature.

Mathematical model for axisymmetric steady flow within layered structures with finite or infinitely high bulk viscosity

A mathematical model was developed for viscous incompressible medium; this model comprises a small parameter for regularization of ill-posed problem. Starting from the analogy with strain problems of axisymmetric laminar structures which include the elastic orthotropic and elastic volume-incompressible layers with a connection to viscous laminar fluid flow, we have developed a method providing a single algorithm for computation of velocity field, stresses, and other parameters of laminar composite material. The example of calculation for the binding agent flow during pultrusion formation is presented.

Axisymmetric rotational stagnation-point flow impinging on a permeable stretching/shrinking rotating disk

This paper extends the recent work by Weidman (2015) on the steady axisymmetric rotational stagnation-point flow impinging on a rotating disk to the case of a permeable stretching/shrinking rotating disk. Similarity variables are used to convert the Navier–Stokes equations to a pair of coupled ordinary differential equations governed by a dimensionless rotation rate α, the velocity of the wall relative to the outer flow λ and the wall transfer rate S. These equations are then solved numerically using the bvp4c function from MATLAB software. The skin friction coefficients or shear stress in both the radial and azimuthal directions are determined. It is found that, for given parameter values, up to three solutions can exist. Asymptotic expressions are derived for large stretching rates λ and for strong fluid withdrawal and injection as well as for rapid rotation rates α, these being compared with the numerically determined values.

Flow pattern transition and destabilization mechanism of thermocapillary convection for low Prandtl number fluid in a deep annular pool with surface heat dissipation

In order to understand clearly the flow pattern transition and the destabilization mechanism of thermocapillary convection for low Prandtl number fluids in a deep annular pool with surface heat dissipation, we carried out a series of three-dimensional numerical simulations by using the finite volume method. The radius ratio and the aspect ratio of an annular pool are respectively fixed at 0.5 and 1.0. Prandtl number of the working fluid is 0.011. Because the total heat dissipation coefficient on the free surface for low Prandtl fluids is small, Biot number is varied from 0 to 1.0. Results indicate that thermocapillary convection experiences the transitions from axisymmetric steady state flow into three-dimensional steady flow, and then into three-dimensional oscillation flow with the increase of Marangoni number. The critical Marangoni number of flow pattern transition decreases slightly with the increase of Biot number, and the maximum temperature and velocity fluctuations appear near the lower part of outer wall. The azimuthal temperature fluctuation on the free surface gradually shrinks to the inner wall, and the temperature fluctuation region decreases. However, it remains almost unchanged near the bottom of the annular pool, but the fluctuation amplitude increases.

Aerodynamic Shape Influence and Optimum Thickness Distribution Analysis of Perceptive Wind Turbine Blade

The aerodynamic module combines the three-dimensional nonlinear lifting surface theory approach, which provides the effective propagated incident velocity and angle of attack at the blade section separately, and a two-dimensional panel method for steady axisymmetric and non-symmetric flow has to be involved to obtain the 3D pressure and velocity distribution on the wind mill model blade. Wind mill and turbines have become an economically competitive form of efficiency and renewable work generation. In the abroad analytical studies, the wind turbine blades to be the target of technological improvements by the use of highly possible systematic , aerodynamic and design, material analysis, fabrication and testing. Wind energy is a peculiar form of reduced form of density source of power. To make wind power feasible, it is important to optimize the efficiency of converting wind energy into productivity source. Among the different aspects involved, rotor aerodynamics is a key determinant for achieving this goal. There is a tradeoff between thin airfoil and structural efficiency. Both of which have a strong impact on the cost of work generated. Hence the design and analysis process for optimum design requires determining the load factor, pressure and velocity impact and optimum thickness distribution by finding the effect of blade shape by varying thickness on the basis of both the aerodynamic output and the structural weight.

Axisymmetric Poiseuille flow of a Bingham plastic with rheological parameters varying linearly with pressure

We consider the steady axisymmetric Poiseuille flow of a Bingham plastic under the assumption that both the plastic viscosity and the yield stress vary linearly with pressure. An analytical solution is derived for the case where the growth coefficients of both rheological parameters are equal, which is indeed a reasonable assumption for certain oil-drilling fluids allowing the existence of a separable solution with a cylindrical unyielded core. The conditions for the occurrence of flow and the effects of the growth coefficient on the radius of the unyielded plug, the velocity profiles and the pressure distributions are discussed.

Emergence of Fluid Rotation in the Marangoni Boundary Layer in the Region of Local Cooling of the Free Boundary

Rotation bifurcation in a steady axisymmetric thermocapillary flow of an incompressible fluid filling a semi-infinite space bounded by the free surface with a nonuniform distribution of temperature is studied. The fluid flow is calculated on the basis of Navier–Stokes equations under the assumption of small diffusion coefficients. It is shown that the bifurcation triggers rotational motion in a thin Marangoni boundary layer in the case of local cooling of the free boundary near the axis of symmetry and in the presence of an external flow; there is no rotation outside this layer. In the case of local heating of the free boundary, rotation is not observed.

Study of polymeric liquid between stretching disks with chemical reaction

An analysis has been carried out to investigate the heat and mass transfer effects in steady axi-symmetric flow of a polymeric liquid (Maxwell fluid) between two infinite stretching disks in the presence of chemical reaction. Particular attention is to compute similarity solution of the governing nonlinear partial differential equations. The series solution of transformed nonlinear ordinary differential equations are computed via homotopy analysis method. The convergence of the developed series solution is also discussed. The quantities of interest like fluid velocity, fluid pressure, temperature, concentration, skin friction coefficient, local Nusselt and local Sherwood numbers are analyzed for the influence of embedded parameters. It is observed that the skin friction coefficient is an increasing function of Deborah number and Reynolds number. It is also observed that effects of stretching parameter on skin friction coefficient at upper and lower disks are opposite. Moreover, the heat transfer rate at lower disk is increases by increasing Deborah number. It is also observed that the surface mass transfer at both disks decreases by increasing Schmidt number.

Thermocapillary Convection in a Binary Mixture with Moderate Prandtl Number in a Shallow Annular Pool

This paper presented a series of numerical simulations on thermocapillary convection for mixed toluene/hexane solution with mass fraction of $c_{0}= 26.27$ % in a shallow annular pool. The Prandtl number of the binary solution is 5.54. Results indicate that when the annular pool subjects to a radial temperature gradient, the solute shifts toward the inner wall under the Soret effect, which leads to a concentration gradient with the opposite direction to the temperature gradient. With the increase of surface heat dissipation, thermocapillary convection is enhanced and the flow is more prone to destabilization. Therefore, the critical thermocapillary Reynolds number of the flow destabilization and the corresponding critical oscillation frequency all decrease, but the critical wave number increases. After flow destabilization, the concentration fluctuation similar to the temperature fluctuation on the free surface appears. No matter the free surface is adiabatic or not, the flow always undergoes a transition from two-dimensional steady axisymmetric flow to the hydrothermal waves, and then to chaos with the increase of thermocapillary Reynolds number.

Control of Thermo- and Solutocapillary Flows in FZ Crystal Growth by High-Frequency Vibrations

The paper deals with the numerical investigation of the possibilities to control convective flows in the liquid bridge in zero gravity conditions applying axial vibrations. The surface tension is assumed to be dependent both on the temperature and on the solute concentration. The free surface deformations and the curvature of the phase change surfaces are neglected but pulsational deformations of the free surface are accounted for. The first part of the paper concerns axisymmetric steady flows. The calculations show that the evolution of convective flow with the variation of thermal Marangoni number at a fixed value of the solutal Marangoni number is accompanied by the hysteresis phenomenon, which is related to the existence of two stable steady regimes in a certain parameter range. One of these regimes is thermocapillary dominated, it corresponds to the two-vortex flow, and the other is solutocapillary dominated, it corresponds to the single-vortex flow. Under vibrations, the range of the Marangoni numbers where hysteresis is observed becomes narrower and is shifted to the area of larger values. The second part of the paper concerns the stability of axisymmetric thermo-and solutocapillary flows and the transition to three-dimensional regimes. Significant mutual influence of flows generated by each process on the stability of the other is discovered. Stability maps in the parametric plane for the thermal Marangoni number, the solutal Marangoni number, are obtained for different values of vibration parameters. It is shown, that vibrations exert a stabilizing effect, increasing critical Marangoni numbers for all modes of instability. However, this effect is different for different modes and at high vibration intensity destabilization is possible. Consequently, vibrations can modify the scenario of the transition to the three-dimensional mode.

Flow bifurcation routes to chaos of thermocapillary convection for low Prandtl number fluid in shallow annular pool with surface heat dissipation

In order to understand clearly the transition characteristics of thermocapillary convection of low Prandtl number fluid in shallow annular pool with surface heat dissipation, some bifurcation routes to chaos of thermocapillary convection have been numerically investigated by using the finite volume method in this paper. The annular pool was filled with the low Prandtl (Pr) number fluid of Pr = 0.011. The range of Biot number is from 0 to 1. Results indicate that the flow bifurcation route of thermocapillary convection depends intensively on surface heat dissipation. When surface heat dissipation is smaller, with the increase of Marangoni number the flow bifurcation sequence of thermocapillary convection is two-dimensional axisymmetric steady flow → three-dimensional steady flow → coexisting hydrothermal waves and radial moving waves → radial moving waves → chaos. When surface heat dissipation is larger, the bifurcation sequence becomes simple as two-dimensional axisymmetric steady flow → hydrothermal waves → chaos. Furthermore, every bifurcation of thermocapillary convection is always accompanied by the variations of the wave number and the oscillatory frequency. Furthermore, the temperature fluctuation amplitude on the free surface increases gradually with the increase of Marangoni number.

A numerical study of the axisymmetric rotational stagnation point flow impinging radially a permeable stretching/shrinking surface in a nanofluid

PurposeThe purpose of this study is to analyze numerically the steady axisymmetric rotational stagnation point flow impinging on a radially permeable stretching/shrinking sheet in a nanofluid.Design/methodology/approachSimilarity transformation is used to convert the system of partial differential equations into a system of ordinary (similarity) differential equations. This system is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in MATLAB software.FindingsDual solutions exist when the surface is stretched, as well as when the surface is shrunk. For these solutions, a stability analysis is carried out revealing that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable and therefore not physically realizable.Originality/valueThe present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface, as they successfully extend the problem considered by Weidman (2016) to the case of nanofluids.

Entropy generation analysis and effects of slip conditions on micropolar fluid flow due to a rotating disk

AbstractThis article deals with the investigation of three-dimensional axisymmetric steady flow of micropolar fluid over a rotating disk in a slip-flow regime. Further, the generation of entropy due to heat transfer and fluid friction is identified. It is noticed that the entropy generation can be decreased and controlled in the presence of slip. The anisotropic slip has vital characteristics and it has a great influence on the flow field and heat transfer. The von Kármán similarity transformation is used to establish the equations governing the flow and heat transfer characteristics of the fluid. The impact of some important parameters on velocity profiles, angular velocity (microrotation) and energy distribution is discussed and illustrated through graphs and tables. The effects of physical parameters on the entropy generation and Bejan numbers are also presented graphically. In addition, the most favorable agreement is observed among the results of the present study and those of the earlier studies.