State Of Rigid-body Rotation Research Articles

Assessment of Bödewadt flow over a stretchable porous surface with variable physical properties: A comparative study

Rotating flows occurring over stationary or revolving boundaries have widespread applications in engineering related processes including electrochemistry, turbomachinery (because it involves turbines), compressors and centrifugal pumps, rheometers etc. and in food and chemical processing industries. Bödewadt flow situation concerns a laminar three-dimensional motion taking place over a planar surface in a viscous fluid which is in a state of rigid body rotation at the far field. Present research applies two distinct viscosity-temperature relations to analyze the Bödewadt flow over a stretching porous surface. . A user friendly and robust computational tool bvp4c of MATLAB is utilized to inspect the behaviors of controlling parameters on the resulting boundary layer formations. Interestingly, results corresponding to the two variable viscosity models in case of water are close to each other. The results indicate that either sufficient wall permeability or sufficient surface stretch rate is necessary to have a meaningful solution for the temperature profile. In the case of variable properties, computational data for the drag exhibited at the disk substantially differs from that found for the case of constant fluid properties.

Series solution of the rotationally symmetric flow in the presence of an infinite rotating disk with uniform suction

Abstract Von Karman swirling viscous flow due to an infinite rotating disk with uniform suction is investigated analytically, when the fluid at infinity is in a state of rigid body rotation. Different from previous results by numerical techniques or an asymptotic expansion method which is dependent on small or big values of the suction parameter, the analytical solution is independent of the suction parameter and uniformly valid for various values of suction parameter a , which helps to enrich the solution family of von Karman swirling viscous flow.

The transient development of the flow in an impulsively rotated annular container

When a fluid-filled container is spun up from rest to a constant angular velocity the fluid responds in such a way that the fluid–container system is ultimately in a state of rigid-body rotation. The fluid can then be said to have traversed a trajectory in phase space from a simple stable equilibrium state of no motion to another stable equilibrium representing full rigid-body rotation. This simple statement belies the fact that during this process the fluid can undergo a series of transitions, from a laminar through a transient turbulent state, before attaining the stable motion that is rigid-body rotation. Using a combination of analytical and computational methods, we focus on the dynamics resulting from an impulsive change in the rotation rate of a fluid-filled annulus, specifically, the impulsive spin-up of a stationary annulus, or the impulsive spin-down of an annulus already in a state of rigid-body rotation. We explore the initial development of the impulsively generated axisymmetric boundary layer, its subsequent instability, and the larger-scale transient features within this class of flows, allowing us to look at the effect these features have on the time it takes for the system to spin up to a steady state, or spin down to rest.

A new analysis of the rotationally symmetric flow in the presence of an infinite rotating disk

Von Kármán swirling viscous flow produced by an infinite rotating disk is investigated, both analytically and numerically, when the fluid at infinity is in a state of rigid body rotation. The analytical solution branch of the problem considered are shown conveniently when the fluid at infinity is rotating with a different ratio, 0 ≤ s ≤ 1, of the angular velocity to that of the disk. As for − 0.1575 ≤ s < 0, the forward shooting technique by Runge–Kutta combined with Newton–Raphson method, different from previous conclusions, can still give effectively the first solution branch of dual solutions by Zandbergen and Dijkstra [11] who adopted a backward shooting technique. The solution branches of von Kármán swirling viscous flow are enriched. A possible explanation is made from both mathematical and physical perspectives.

Numerical study of thermal radiation and thermophoresis on peristalsis with rotational aspects

The present work concentrates for the impact of thermal radiation on peristaltic transport of viscous fluid in a rotating channel. Both fluid and channel are in a state of rigid body rotation. The influences of thermophoresis and chemical reaction are taken into account. Convective conditions for heat and mass transfer in the formulation are adopted. In addition, the non-uniform heat source/sink effect is included in heat transfer analysis. Exact solutions for stream function and temperature are obtained. Numerical solution for concentration of the developed mathematical model are obtained by considering low Reynolds number and long wavelength. The effects of emerging physical parameters are analyzed through graphical illustrations. It is found that the influence of thermophoretic and thermal radiation parameters on the temperature and concentration are quite opposite. Further heat transfer coefficient decays when rotation is increased.

Unsteady flow in a rotating torus after a sudden change in rotation rate

AbstractWe consider the temporal evolution of a viscous incompressible fluid in a torus of finite curvature; a problem first investigated by Madden &amp; Mullin (J. Fluid Mech., vol. 265, 1994, pp. 265–217). The system is initially in a state of rigid-body rotation (about the axis of rotational symmetry) and the container’s rotation rate is then changed impulsively. We describe the transient flow that is induced at small values of the Ekman number, over a time scale that is comparable to one complete rotation of the container. We show that (rotationally symmetric) eruptive singularities (of the boundary layer) occur at the inner or outer bend of the pipe for a decrease or an increase in rotation rate respectively. Moreover, on allowing for a change in direction of rotation, there is a (negative) ratio of initial-to-final rotation frequencies for which eruptive singularities can occur at both the inner and outer bend simultaneously. We also demonstrate that the flow is susceptible to a combination of axisymmetric centrifugal and non-axisymmetric inflectional instabilities. The inflectional instability arises as a consequence of the developing eruption and is shown to be in qualitative agreement with the experimental observations of Madden &amp; Mullin (1994). Throughout our work, detailed quantitative comparisons are made between asymptotic predictions and finite- (but small-) Ekman-number Navier–Stokes computations using a finite-element method. We find that the boundary-layer results correctly capture the (finite-Ekman-number) rotationally symmetric flow and its global stability to linearised perturbations.

Asymptotic and numerical solutions of the initial value problem in rotating planetary fluid cores

P>An initial state of fluid motion in planetary cores or atmospheres, excited, for example, by the giant impact of an asteroid or an earthquake and then damped by viscous dissipation, decays towards the state of rigid-body rotation. The process of how the initial state approaches the final state, the initial value problem, is investigated both analytically and numerically for rotating fluid spheres. We derive an explicit asymptotic expression for the time-dependent solution of the initial value problem valid for an asymptotically small Ekman number E. We also perform a fully numerical analysis to simulate time-dependent solutions of the initial value problem for a small value of E. Comparison between the asymptotic solution and the corresponding numerical simulation shows a satisfactory quantitative agreement. For the purpose of illustrating why spherical geometry represents an intricate and exceptional case, we also briefly discuss the initial value problem in a rotating fluid channel. Geophysical and planetary physical implications of the result are also discussed.

On viscous decay factors for spherical inertial modes in rotating planetary fluid cores: Comparison between asymptotic and numerical analysis

The initial value problem of how an initial state of fluid motion, excited by earthquake or tide and then damped by viscous dissipation, decays toward the state of rigid-body rotation is considered for rapidly rotating fluid spheres like planetary fluid cores. An essential element in an asymptotic time-dependent solution for the initial value problem is the viscous decay factors for spherical inertial modes. We derive an analytical expression for the viscous decay factors valid for a broad range of the inertial modes that are required for an asymptotic solution of the initial value problem at an arbitrarily small but fixed Ekman number. We also perform fully numerical analysis to compute the viscous decay factors for several selected inertial modes, showing a quantitative agreement between the asymptotic and numerical analysis. It is argued that the correct viscous decay factors cannot be derived using an asymptotic expansion based on the half powers of a small Ekman number.

A variational analysis of a non-Newtonian flow in a rotating system

This study deals with the flow of a third grade fluid past an infinite plate. Both fluid and plate are in a state of rigid body rotation. The analysis presented here deals with similarity solutions of the problem and is done in the following way. We consider a Lagrangian formulation which allows us to directly determine conserved quantities leading to a reduction of the resultant systems.

Nonlinear Spin-Up of a Rotating Stratified Fluid: Theory

We consider the boundary layer that forms on the wall of a rotating container of stratified fluid when altered from an initial state of rigid body rotation. The container is taken to have a simple axisymmetric form with sloping walls. The introduction of a non-normal component of buoyancy into the velocity boundary-layer is shown to have a considerable effect for certain geometries. We introduce a similarity-type solution and solve the resulting unsteady boundary-layer equations numerically for three distinct classes of container geometry. Computational and asymptotic results are presented for a number of parameter values. By mapping the parameter space we show that the system may evolve to either a steady state, a double-structured growing boundary-layer, or a finite-time breakdown depending on the container type, rotation change and stratification. In addition to extending the results of Duck et al. (1997) to a more general container shape, we present evidence of a new finite-time breakdown associated with higher Schmidt numbers.

Experiments on the stability of an impulsively-initiated circular Couette flow

Experiments were performed to study the stability characteristics of an unsteady circular Couette flow generated by an impulsive stop of the outer cylinder; the initial condition was a state of rigid-body rotation. Instability of the unsteady basic state is manifested by Gortler vortices, which themselves become unstable to longer-wavelength disturbances, or Taylor vortices which persist indefinitely. The quantities of primary interest are the onset time of instability, the axial vortex wavelength at onset, and the time-evolution of this wavelength. A one-dimensional photodiode array is used to gather data from the flow, which is seeded with flow-visualization material. At sufficiently high values of the Reynolds number, the influence of the inner cylinder on the onset of instability is negligible, based on comparisons with previous experimental data.

Marginal stability of impulsively initiated Couette flow and spin-decay

A viscous, incompressible fluid is contained within an infinitely long circular cylinder or between a pair of infinitely long concentric cylinders. In both cases, the entire system is in a state of rigid-body rotation. At time t = 0 the outer flow boundary is impulsively brought to rest, giving rise to a potentially unstable, unsteady swirl flow. The stability of these flows is examined by employing energy theory with a marginal stability criterion to obtain lower bounds on the onset times for instability. In some cases, the asymptotic steady-state flow will be stable, with any instability merely a transient, while in other cases, stability of the asymptotic state is not guaranteed.

Flow of a Dusty Gas with Suspended Particles in a Rotating Frame of Reference

The solution is given for the problem on the motion of a dusty gas under a rotating system of co-ordinates. The gas containing a uniform distribution of dust particles, occupies the infinite space above a rigid plane boundary. The motion of the dusty gas is induced by the motion of the plate moving with a velocity which decreases exponentially with time in a rotating frame of reference such that the plate and the gas are in a state of rigid body rotation about an axis normal to the plate. The velocity fields for the dusty gas, and the dust particles along and normal to the direction of motion of the plate are obtained in closed forms. Finally, some velocity distributions are calculated with particular reference to the effect of rotation at different times, and it is found that with the increase of the value of the rotation parameter omega, the velocity of the dusty gas along the direction of motion of the plate gradually increases while the velocity of the dust particle along the same direction gradually decreases.

Source-sink flow in a gas centrifuge

We consider steady non-axisymmetric source-sink flow of a perfect gas in a rapidly rotating circular cylinder for the case in which the sources and the sinks are distributed on the top and bottom. We apply a linearized analysis to a small perturbation from the state of rigid-body rotation. We show that the radial pressure gradient plays an important role in the determination of the axisymmetric part of the flow field and that the effects of viscosity and thermal conductivity govern the overall configuration of the non-axisymmetric part. We also show that the transport of gas in the main inner flow is axial and that radial transport is confined to the horizontal boundary layers.

Spiral structure and hydrodynamic gravitational resonance

A persistent and large scale galactic spiral structure might be the result of resonant excitation of density waves, forced in the low viscosity interstellar gas, by the gravitational field of an asymmetric stellar structure (galactic nucleus, condensation, companion galaxy, etc.) rotating like a rigid body around the galactic centre. This paper deals with this problem in the simple case of a homogeneous plane viscous medium of infinite extent, in a state of rigid body rotation. Numerical estimates are given in the case of density waves excited by the gravitational field of a central dumb-bell. They show that the mechanism of hydrodynamic gravitational resonance is worthy of a further, more realistic, treatment.

Spin-up of a stratified fluid: theory and experiment

Stratified spin-up, the process of adjustment of a uniformly rotating stratified fluid to an abrupt change in the rotation of the container, is important in many geophysical contexts. An experimental study of this process is presented here for the case where a linearly stratified salt solution is enclosed in a cylindrical container whose rotation rate is changed by a small amount. Results are presented for a limited range of values of B, the internal Froude number, which measures the ratio of the frequencies due to buoyancy and rotation. The experimental study is augmented by a theoretical treatment of idealized models which clarify the more fundamental physical processes that occur. The response of a stratified fluid is faster than that of a homogeneous fluid but the adjustment is limited to layers near the bottom and top boundaries the thickness of which is determined by the value of B. A comparison of the experimental results with the theories of Holton, Walin and Sakurai is also made and it is shown that for the present physical arrangement (insulated side walls) the theories of the latter two authors agree much more closely with experiment than does the theory of Holton. However, all three theories tend to over-estimate the azimuthal displacement in the regions near the upper and lower boundaries where the spin-up is most rapid. The Sweet-Eddington circulation, which accompanies the ideal state of rigid-body rotation, can be significant under normal laboratory conditions and it was necessary to correct some of the spin-up results for this effect. The circulation in the vertical plane is described qualitatively.