Axisymmetric Stagnation Point Flow Research Articles

The Falkner–Skan flow of a viscoelastic fluid

We study the Falkner–Skan flow of a viscoelastic fluid governed by the FENE-P model, as the Reynolds number Re → ∞ and the Weissenberg number We is such that WeRe 1 / 2 = O ( 1 ) . We obtain a differential–algebraic system for the stream function and the stress. We show that a similarity solution exists only for axisymmetric stagnation point flow which corresponds to an angle of attack m = 1 3 . The stream function is then governed by a generalized Falkner–Skan equation. The effect of extensibility on the skin friction and first normal stress coefficients are discussed.

Axisymmetric Stagnation—Point Flow and Heat Transfer of a Viscous Fluid on a Rotating Cylinder With Time-Dependent Angular Velocity and Uniform Transpiration

The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite rotating circular cylinder with transpiration U0 are investigated when the angular velocity and wall temperature or wall heat flux all vary arbitrarily with time. The free stream is steady and with a strain rate of Γ. An exact solution of the Navier-Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by the use of appropriate transformations for the most general case when the transpiration rate is also time-dependent but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the angular velocity of the cylinder and its wall temperature or its wall heat flux vary as specified time-dependent functions. In particular, the cylinder may rotate with constant speed, with exponentially increasing/decreasing angular velocity, with harmonically varying rotation speed, or with accelerating/decelerating oscillatory angular speed. For self-similar flow, the surface temperature or its surface heat flux must have the same types of behavior as the cylinder motion. For completeness, sample semi-similar solutions of the unsteady Navier-Stokes equations have been obtained numerically using a finite-difference scheme. Some of these solutions are presented for special cases when the time-dependent rotation velocity of the cylinder is, for example, a step-function. All the solutions above are presented for Reynolds numbers, Re=Γa2∕2υ, ranging from 0.1 to 1000 for different values of Prandtl number and for selected values of dimensionless transpiration rate, S=U0∕Γa, where a is cylinder radius and υ is kinematic viscosity of the fluid. Dimensionless shear stresses corresponding to all the cases increase with the increase of Reynolds number and suction rate. The maximum value of the shear stress increases with increasing oscillation frequency and amplitude. An interesting result is obtained in which a cylinder rotating with certain exponential angular velocity function and at particular value of Reynolds number is azimuthally stress-free. Heat transfer is independent of cylinder rotation and its coefficient increases with the increasing suction rate, Reynolds number, and Prandtl number. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration.

Diffusion of a line vortex in a second grade fluid embedded in a stagnation point flow

The diffusion of a line vortex embedded in a radially inward axisymmetric stagnation point flow for a second grade fluid is considered. This flow is the diffusion of a line vortex for a second grade fluid when a stagnation point flow is superimposed. The velocity field is in the form of υ r = − αr, υ θ = υ( r, t), υ z = 2 αz, where r, θ, z are cylindrical polar coordinates and α is a constant related to the inward flow. An exact solution of the governing equation is given. It is found that for small values of the time the solution approaches the diffusion of a line vortex in a second grade fluid and for large values of the time the solution tends to the flow of a line vortex in a second grade fluid embedded in a radially inward axisymmetric stagnation point flow.

A self-similar solution for forced convection boundary layer flow of a FENE-P fluid

Boundary layer flow of a viscoelastic fluid subjected to an external pressure gradient, surface mass flux and nonisothermal surface temperature is considered. We show that for a FENE-P fluid, the flow is governed by a generalized Falkner–Skan equation which admits a similarity solution only when the streamwise coordinate exponent of the inviscid velocity m = 1 / 3 (i.e. for axisymmetric stagnation point flow). We find that skin friction drag decreases while heat transfer at the plate increases as polymer extensibility decreases.

Axisymmetric Stagnation-Point Flow and Heat Transfer of a Viscous Fluid on a Moving Cylinder With Time-Dependent Axial Velocity and Uniform Transpiration

The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite moving cylinder with time-dependent axial velocity and with uniform normal transpiration Uo are investigated. The impinging free stream is steady and with a constant strain rate k¯. An exact solution of the Navier–Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by use of appropriate transformations for the most general case when the transpiration rate is also time-dependent but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the axial velocity of the cylinder and its wall temperature or its wall heat flux vary as specified time-dependent functions. In particular, the cylinder may move with constant speed, with exponentially increasing–decreasing axial velocity, with harmonically varying axial speed, or with accelerating–decelerating oscillatory axial speed. For self-similar flow, the surface temperature or its surface heat flux must have the same types of behavior as the cylinder motion. For completeness, sample semisimilar solutions of the unsteady Navier–Stokes and energy equations have been obtained numerically using a finite-difference scheme. Some of these solutions are presented for special cases when the time-dependent axial velocity of the cylinder is a step-function, and a ramp function. All the solutions above are presented for Reynolds numbers, Re=ka¯2/2υ, ranging from 0.1 to 100 for different values of dimensionless transpiration rate, S=Uo/ka¯, where a is cylinder radius and υ is kinematic viscosity of the fluid. Absolute value of the shear-stresses corresponding to all the cases increase with the increase of Reynolds number and suction rate. The maximum value of the shear- stress increases with increasing oscillation frequency and amplitude. An interesting result is obtained in which a cylinder moving with certain exponential axial velocity function at any particular value of Reynolds number and suction rate is axially stress-free. The heat transfer coefficient increases with the increasing suction rate, Reynolds number, Prandtl number, oscillation frequency and amplitude. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration. It is shown that a cylinder with certain type of exponential wall temperature exposed to a temperature difference has no heat transfer.

Stream function, flow separation and force equation for stagnation flow passing a small solid sphere touching a rising gas bubble

This paper considers an axisymmetric stagnation flow past a small solid sphere touching an air bubble, which is significantly larger than the particle but smaller than the capillary length so that the deformation can be neglected. The disturbed flow due to the presence of the particle at the bubble surface was modelled by considering an axisymmetric stagnation point flow about a sphere and a plane. The stream function for the disturbed flow was derived based on the tangent-sphere coordinates. It is shown that the flow separation having an infinite set of nested ring vortices exists in a finite region enclosing the contact point. The presence of such vortices might have considerable influence on the bubble-particle attachment as well as on the transport of surfactants during the bubble-particle interaction. The force equation was obtained from the derived stream function and was equated to the modified Stokes drag force equation, and then an expression for the correction factor was obtained. The model for the correction factor includes the effect of particle size, and predicts finite force values at zero separation distance between the particle and bubble surfaces. The models presented in this paper should provide a better estimate for calculating the normal (stagnation flow) fluid force acting on particles in contact with bubbles in the multiphase flows found in mineral flotation and other metallurgical operations.

Stagnation‐point Flow towards a Stretching Surface

AbstractAn exact similarity solution of the Navier‐Stokes equations is obtained. The solution represents steady axisymmetric stagnation‐point flow towards a stretching surface. It is shown that the flow displays a boundary‐layer structure when the stretching velocity of the surface is less than the free stream velocity. On the other hand, an inverted boundary layer is formed when the surface stretching velocity exceeds the free stream velocity. Temperature distribution in the flow is found when the surface is held at a constant temperature. It turns out that when the surface temperature exceeds the ambient temperature, heat flows from the surface to the fluid near the stagnation point but further away from the stagnation point, heat flows from the fluid to the stretching surface.

Unsteady axisymmetric stagnation-point flow of a viscous fluid on a cylinder

The unsteady viscous flow in the vicinity of an axisymmetric stagnation point of an infinite circular cylinder is investigated when both the free stream velocity and the velocity of the cylinder vary arbitrarily with time. The cylinder moves either in the same direction as that of the free stream or in the opposite direction. The flow is initially ( t=0) steady and then at t>0 it becomes unsteady. The semi-similar solution of the unsteady Navier–Stokes equations has been obtained numerically using an implicit finite-difference scheme. Also the self-similar solution of the Navier–Stokes equations is obtained when the velocity of the cylinder and the free stream velocity vary inversely as a linear function of time. For small Reynolds number, a closed form solution is obtained. When the Reynolds number tends to infinity, the Navier–Stokes equations reduce to those of the two-dimensional stagnation-point flow. The shear stresses corresponding to stationary and the moving cylinder increase with the Reynolds number. The shear stresses increase with time for the accelerating flow but decrease with increasing time for the decelerating flow. For the decelerating case flow reversal occurs in the velocity profiles after a certain instant of time.

An analysis of flow, temperature, and chemical composition distortion in gas sampling through an orifice during chemical vapor deposition

Measurement of the chemical composition of gases sampled through a small hole in the substrate can be a useful diagnostic for investigations of the chemistry of chemical vapor deposition (CVD) processes. Ideally, one would measure the composition of the gas at the growth surface. However, the flow disturbance due to sampling causes the conditions at the mouth of the orifice to be different from those at the growth surface. Unless the orifice diameter is sufficiently small, relative to the thickness of chemical and thermal boundary layers above the growth surface, the sampled composition will differ from the composition at the growth surface. In this work, we present results of two-dimensional simulations of the flow, heat transfer, and chemical reactions in an axisymmetric stagnation point flow with gas sampling through a small orifice in the substrate on the symmetry axis of the flow field. Detailed results are given for atmospheric-pressure radio-frequency plasma CVD of diamond, corresponding to experiments performed in our laboratory. We also present more general results, approximate analytical representations of the flow field, and scaling rules for the size of the disturbance due to the sampling orifice.

Similarity analysis of a forced uniform flow impinging on a rotating disk in a vapor deposition process

A theoretical study for a forced uniform flow impinging on a rotating disk, typically involved in Chemical Vapor Deposition(CVD) and Vapor-phase Axial Deposition(VAD) processes, has been carried out. A set of exact solutions for flow and temperature fields are developed by employing a similarity variable obtained from force balance on a control volume near the disk. The solutions depend on the rotating speed of the disk, .omega., and the forced flow speed toward the disk, a. For constant forced flow speed, the overall boundary layer thickness decreases when the rotating speed increases. Approximately 5%, 15%, and 30% decreases of the thickness are obtained for .omega./a = 2, 5, and 10, respectively, compared to the case of .omega./a = 0 (axisymmetric stagnation point flow). For constant rotating disk speed the boundary layer thickness immediately decreases as the forced flow speed increases, compared to the case of .omega./a .rarw. .inf. (induced flow near a rotating disk). Effects of .omega. and a on heat transfer coefficient are studied and explained with the boundary layer characteristics.

Axisymmetric stagnation-point flow impinging on a transversely oscillating plate with suction

The viscous fluid motion generated by axisymmetric stagnation-point flow of strain rate a impinging on a flat plate oscillating in its own plane with velocity amplitude U0 and frequency ω, including uniform suction of strength W0 is considered. A coordinate decomposition transforms the full Navier-Stokes equations into a primary equation describing the steady flow and a secondary equation describing the unsteady motion coupled to the primary solution. The solution to the boundary-value problem is governed by two dimensionless groups: the suction parameter S = W0 √aν and the frequency parameter Ω = ω/a, where ν is the kinematic viscosity. Numerical integrations performed with a Runge-Kutta routine provide an exact solution to the Navier-Stokes equations. Values of the steady shear stress are found to agree with asymptotic results for large values of |S|, with S>0 representing suction and S<0 representing blowing. The magnitude and phase of the unsteady shear stress are given over a range of frequencies sufficient to recover analytical asymptotic results at large values of Ω. The unsteady shear stress lags the wall motion by π radians for Ω → 0 and by 5π/4 radians when Ω → ∞. Velocity profiles at selected parameter values during a period of plate oscillation are presented and discussed.

Chebyshev solution of axisymmetric stagnation flow on a cylinder

A numerical procedure, based on Chebyshev polynomials, has been developed for a class of problems in the boundary layer theory of a micropolar fluid. Here, the results are presented for the steady micropolar fluid in the vicinity of an axisymmetric stagnation point flow on a cylinder. This method is accomplished by starting with a Chebyshev approximation for the highest order derivatives and generating approximations to the lower order derivatives. The numerical solutions are presented for a range of values of material parameters and Reynolds number. Furthermore, the results have been compared with the results of the corresponding flow of a Newtonian fluid. This type of stagnation flow has applications in certain cooling processes.

On a class of non-axisymmetric flux rope solutions to the electromagnetic induction equation

Abstract We display non-axisymmetric generalisations of a known exact axisymmetric flux rope solution to the induction equation. The assumed velocity field is a simple axisymmetric stagnation point flow. Solutions for magnetic fields with non-zero azimuthal wavenumber are given. These provide local models for fluxrope features seen in numerical computations of probable fast dynamos. Remarkably, they can have positive growth rates for any magnetic Reynolds number, and have finite energy (per unit length along the axis) provided only that the growth rate is less than one, in velocity-based units. Thus although not chaotic, this flow functions as a fast dynamo, albeit one with the unphysical feature that its velocity tends to infinity far from the origin. It is therefore desirable to exclude such a feature when choosing velocity fields as candidates for a realistic fast dynamo, to avoid finding one for a spurious reason.

Approximate analytical solutions for particle deposition in viscous stagnation-point flow in the inertial-diffusion regime with external forces

In a previous paper (1) the effects of particle inertia on the diffusional deposition of Brownian particles onto a surface in viscous, axisymmetric stagnation-point flow were investigated using the method of Brownian dynamics. Following Fernandez de la Mora and Rosner (2), under the conditions of large Schmidt numbers and small Stokes numbers (less than the critical Stokes number), an approximate analytical solution is obtained here that also includes an arbitrary external force acting on the particles. This solution is shown to be in excellent agreement with the results from Brownian dynamics simulations. An important characteristic of these solutions is inertial particle concentration “enrichment” outside the diffusion boundary layer near the adsorbing surface at subcritical Stokes numbers. In general, it is shown that attractive external particle forces only slightly reduce the amount of this enrichment. On the other hand, in the absence of inertia, attractive external particle forces increase the surface flux of particles beyond that for solely diffusional deposition. The combination of inertial particle concentration enrichment and an attractive external particle force leads to a synergistic-type increase in the surface flux of particles beyond that for solely diffusional deposition. Finally, Robinson's theorem (6) is extended to formally prove these results for inviscid-type flows.

Oblique vortices on a solid wall and on an interface between two immiscible viscous fluids

A local three-dimensional flow analysis has been made about a singular point at the interface between two immiscible viscous fluids. A generalization has been found of a ‘‘refraction law’’ for dividing streamlines that was derived in a previous study [Phys. Fluids 30, 3647 (1987)] for two-dimensional flows. It is shown here that the inclination angle of a vortex axis with an interface lies in a plane perpendicular to the shear flow that causes the inclination. An axisymmetric stagnation-point flow is tilted in the direction of the shear flow. Superposition of both vortex and stagnation-point flow yields an angle of inclination somewhere between the planes perpendicular and parallel to the shear flow. These results are valid for slip and nonslip surface conditions.

The effects of electrostatic and inertial forces on the diffusive deposition of small particles onto large disks: Viscous axisymmetric stagnation point flow approximations

Fine particles can deposit on microelectronic parts during their manufacture and cause damage. Prediction of the rates of deposition is important in planning to reduce such losses. Analytic expressions were developed and Monte Carlo simulations (Brownian dynamics) were carried out to predict deposition rates of sub-micron particles in viscous axisymmetric stagnation point flow. Diffusive, inertial and electrostatic deposition mechanisms were incorporated in the analysis. The electrostatic forces considered included the Coulombic force on a charged particle in an electric field and the image force on a charged particle near a conducting plane. Analytical solutions to the convective-diffusion equation, in the absence of inertial effects, showed that for a Boltzmann charge equilibrium, the electrostatic image force did not appreciably enhance deposition for a free-stream velocity of 50 cm s −1. However, the Coulomb electrostatic attraction of a charged particle towards a charged surface greatly enhanced deposition, assuming a field from the disk approximately equivalent to having a disk 20 cm in diameter (80 pF capacitance) at 2000 V. Some areas for future study are suggested by these results. Brownian dynamics simulations carried out for particles of the order of 1 μm in diameter at a much higher velocity (30 m s −1) demonstrated an interaction between inertial and diffusive deposition that produced inertially enhanced deposition at Stokes values below the critical value, St < 0.15. Where inertia and electrostatic effects were negligible, the simulations matched the known solution for stagnation flow deposition due to diffusion. In the Appendix is presented a derivation of the Langevin equation governing single-particle motion in a fluid that is not uniform in temperature as well as in velocity.

Flow induced morphological instabilities: Stagnation-point flows

We consider a planar stagnation-point flow on a directionally-solidifying interface and determine its effects on its morphological instability. We find that when Schmidt numbers are large, two-dimensional long-wave disturbances are unstable for any degree of constitutional undercooling. These modes propagate against the basic-state flow and are destabilized by the component of this flow normal to the interface. Analogous results are given for the axi-symmetric stagnation-point flow subject to axi-symmetric disturbances. Comparisons are made of the two cases.

A complete analytical solution to laminar heat transfer in axisymmetric stagnation point flow

Application du principe variationnel de Gyarmati qui permet le traitement de la thermodynamique des processus irreversibles, pour obtenir une solution analytique rapide aux problemes d'ecoulement laminaire et de transfert de chaleur dans les ecoulements de couche limite axisymetrique a point d'arret

Quasilinearized solution of axisymmetric hydromagnetic stagnation-point flow and heat transfer

Les equations de quantite de mouvement et de Maxwell sont resolues a l'aide d'une quasi-linearisation combinee avec un schema de Runge-Kutta du quatrieme ordre

Axisymmetric stagnation point flow and mass transfer for power-law fluids II. Moderate schmidt number correction

The theory for axisymmetric stagnation point flow of power-law fluids has been extended to include the correction terms for convective diffusion at moderate Schmidt numbers. The dimensionless mass transfer rate is expressed as an asymptotic series that is valid for Re (1 − n)/3(1 + n) Sc − 1 3 < 1. The result can be used to predict accurate diffusion coefficients for dilute species in fluids with specified power-law characteristics.