Time-periodic Body Force Research Articles

Effect of Gravity Modulation on the Onset of Heat Transfer by Buoyancy Induced Convection in Boussinesq- Stokes Suspension with Temperature

The current paper considers the Boussinesq-Stokes suspensions and the temperature-dependent viscosity with the influence of gravity modulation to analyze a weak non-linear stability problem of Rayleigh Benard magnetoconvection. In the study of convective instability problems, the impact of time-periodic body force also known as gravity modulation or g-gitter is essential. In the problem of gravity modulation, the gravity field has two components: one is the constant part and another an externally imposed time periodic part, which can be produced by oscillating the fluid layer. The effect of varying frequency of gravitational oscillation on convection is examined. The truncated form of the Fourier series is used in the non-linear analysis. The effects of numerous factors on the onset of convection have been discussed in this paper. The thermal Nusselt number is computed and shown for slow time periods using non-linear theory. The impacts of gravity modulation frequency and amplitude have been investigated in order to study heat transport in the system, as well as other aspects that exist in the problem.

Thermally-driven flows and turbulence in vibrated liquids

Within the vast array of variants encompassed by thermal (buoyancy) convection, the particular interest of the present work is on the specific dynamics which are enabled when standard steady gravity is replaced by a time-periodic body force induced by vibrations. The study is designed as a set of separate problems, where each exemplar aims to unravel the implications of the fundamental properties of this type of flow. These include the symmetry of the emerging pattern as perceived by a real observer and as seen in a “time-averaged space”, the synchronous or non-synchronous response of the velocity field to the applied forcing, the magnitude of the so-called Thermofluid-dynamic (TFD) distortions and the peculiar route of evolution towards chaos. A kaleidoscope of previously unknown solutions is reported giving emphasis to some still poorly known aspects such as the complex nature of the textural transitions that take place in the flow as the Gershuni number is increased (from 3.30 × 102 to 5.00 × 107 for Pr = 15). It is shown that the low-frequency regime is relatively stable over this range. In addition to the standard quadrupolar pattern, in such a case peculiar convective structures emerge where the time-averaged rolls display a very regular columnar arrangement, which has been rarely observed in earlier studies. Chaotic states are enabled when larger frequencies of vibration are considered. While for intermediate frequencies concurrent aspects of the Feigenbaum and Manneville and Pomeau mechanisms can be recognized, the hallmark of the high frequency regime is its adherence to the standard Ruelle-Takens scenario.

Linear and nonlinear triple diffusive convection in the presence of sinusoidal/non-sinusoidal gravity modulation: A comparative study

The non-uniform vertical vibrations (gravity modulation) or g-jitter or time-periodic body force, can be realized by oscillating the system vertically. The time periodic variations are considered to vary sinusoidally (trigonometric cosine) or non-sinusoidally (square, sawthooth and triangular) to study three-component convection in a Newtonian fluid using linear and non-linear analyses. In the linear theory, the expressions for the Rayleigh number and the correction Rayleigh number are obtained by using a regular perturbation method. The eigen value is obtained by adopting the classical Venezian approach. The generalized Lorenz model is derived using a Fourier series with additional modes and under the assumption of the Boussinesq approximation and small-scale convection motion. The resulting non-autonomous Lorenz model is solved numerically to quantify the heat and mass transports. The results are considered against the background of the results of no modulation. Study reveals that all the four types of gravity modulation delay the onset of convection and thereby lead to the diminished heat transfer situation. It is also found that in the case of trigonometric type of gravity modulation heat and mass transports are between those of the cases of triangular and square types. The results in respect of saw-tooth and triangular wave modulations are identical.

Localized stress amplification in inertialess channel flows of viscoelastic fluids

Nonmodal analysis typically uses square-integrated quantities to characterize amplification of disturbances. However, such measures may be misleading in viscoelastic fluids, where polymer stresses can be strongly amplified over a small region. Here, we show that when using a localized measure of disturbance amplification, spanwise-constant polymer-stress fluctuations can be more amplified than streamwise-constant polymer-stress fluctuations, which is the opposite of what is observed when a square-integrated measure of disturbance amplification is used. To demonstrate this, we consider a model problem involving two-dimensional pressure-driven inertialess channel flow of an Oldroyd-B fluid subject to a localized time-periodic body force. Nonmodal analysis of the linearized governing equations is performed using recently developed well-conditioned spectral methods that are suitable for resolving sharp stress gradients. It is found that polymer-stress fluctuations can be amplified by an order of magnitude while there is only negligible amplification of velocity fluctuations. The large stress amplification arises from the continuous spectrum of the linearized problem, and may put the flow into a regime where nonlinear terms are no longer negligible, thereby triggering a transition to elastic turbulence. The results suggest an alternate mechanism that may be useful for understanding recent experimental observations of elastic turbulence in microchannel flows of viscoelastic fluids. • Nonmodal amplification of localized time-periodic body forces is examined. • Poiseuille flow of Oldroyd-B fluids is considered. • Stress fluctuations can be amplified much more than velocity fluctuations. • Stress amplification is highly localized in space. • Results suggest a mechanism for transition to elastic turbulence in microchannels.

Comparison of the effects of three types of time-periodic body force on linear and non-linear stability of convection in nanoliquids

Abstract The effect of three different types of time-periodic vertical oscillations of the Rayleigh–Benard system involving Newtonian nanoliquids is studied in the paper. Twenty nanoliquids are considered for investigation and it is found that enhanced heat transport is seen in all of them compared to what is observed in the corresponding carrier liquids without the nanoparticles. The increase in the amplitude of the vertical vibration is shown to augment convection in the nanoliquids whereas increase in frequency is to inhibit convection. On comparing the extent of heat transport in the presence of modulation with that in its absence, it was found that heat transport diminished in the former case. It was also found that the trigonometric sine type of gravity modulation transports heat intermediate to that of triangular and square types. The triangular type inhibits heat transport while square type enhances heat transport leading one to conclude that heat transport in a Rayleigh–Benard system can be enhanced by using a combination of dilute concentration of nanoparticles and gravity modulation using a square wave.

Faraday instability on a sphere: numerical simulation

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate–prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry.

Faraday waves on a cylindrical fluid filament – generalised equation and simulations

We perform linear stability analysis of an interface separating two immiscible, inviscid, quiescent fluids subject to a time-periodic body force. In a generalised, orthogonal coordinate system, the time-dependent amplitude of interfacial perturbations, in the form of standing waves, is shown to be governed by a generalised Mathieu equation. For zero forcing, the Mathieu equation reduces to a (generalised) simple harmonic oscillator equation. The generalised Mathieu equation is shown to govern Faraday waves on four time-periodic base states. We use this equation to demonstrate that Faraday waves and instabilities can arise on an axially unbounded, cylindrical capillary fluid filament subject to radial, time-periodic body force. The stability chart for solutions to the Mathieu equation is obtained through numerical Floquet analysis. For small values of perturbation and forcing amplitude, results obtained from direct numerical simulations (DNS) of the incompressible Euler equation (with surface tension) show very good agreement with theoretical predictions. Linear theory predicts that unstable Rayleigh–Plateau modes can be stabilised through forcing. This prediction is borne out by DNS results at early times. Nonlinearity produces higher wavenumbers, some of which can be linearly unstable due to forcing and thus eventually destabilise the filament. We study axisymmetric as well as three-dimensional perturbations through DNS. For large forcing amplitude, localised sheet-like structures emanate from the filament, suffering subsequent fragmentation and breakup. Systematic parametric studies are conducted in a non-dimensional space of five parameters and comparison with linear theory is provided in each case. Our generalised analysis provides a framework for understanding free and (parametrically) forced capillary oscillations on quiescent base states of varying geometrical configurations.

Effects of Gravity Modulation and Internal Heat Generation on the onset of Rayleigh-Benard convection in a Micropolar Fluid

The effect of gravity modulation (time periodic body force or g-jitter) on the onset of Rayleigh-Bénard convection in a micropolar fluid with internal heat generation is investigated by making a linear stability analysis. The stability of a horizontal layer of fluid heated from below is examined by assuming time periodic body force in the presence of internal heat source. A regular perturbation method is used to arrive at an expression to compute the critical Rayleigh number for small amplitude of modulation and dimensionless internal heat source. The Venezian approach is adopted to obtain the eigen value of the problem. The results obtained during the analysis have been presented graphically.

Effect of Gravity Modulation on the Onset of Rayleigh-Bénard Convection in a Weak Electrically Conducting Couple Stress Fluid with Saturated Porous Layer

The effect of time-periodic body force (TBF, also called gravity modulation) of small amplitude in a weak electrically conducting couple stress fluid with saturated porous layer is investigated by using a linear stability analysis. A regular perturbation method is used to arrive at an expression for the correction Rayleigh number. The Venezian approach is adopted in arriving at the critical Rayleigh and wave number for small amplitudes of TBF. The effect of roles of Couple stress parameter, Hartmann number, Darcy number, Porous parameter and Prandtl number on the onset of convection is studied. It is found that TBF leads to delay in convection. Also the results suggest that instead of taking electrically non-conducting fluid it is better to consider electrically conducting fluid with weak electrical conductivity as this ensures a stable environment in the presence of a magnetic field. The system is most stable with respect to TBF.

Concordancing in ESP Class Rooms

The effect of time periodic body force (or g-jitter or gravity modulation) on the onset of Rayleigh-Bnard electro-convention in a micropolar fluid layer is investigated by making linear and non-linear stability analysis. The stability of the horizontal fluid layer heated from below is examined by assuming time periodic body acceleration. This normally occurs in satellites and in vehicles connected with micro gravity simulation studies. A linear and non-linear analysis is performed to show that gravity modulation can significantly affect the stability limits of the system. The linear theory is based on normal mode analysis and perturbation method. Small amplitude of modulation is used to compute the critical Rayleigh number and wave number. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation. The non-linear analysis is based on the truncated Fourier series representation. The resulting non-autonomous Lorenz model is solved numerically to quantify the heat transport. It is observed that the gravity modulation leads to delayed convection and reduced heat transport.

Linear and Weakly Non-Linear Analyses of Gravity Modulation and Electric Field on the onset of Rayleigh-Bénard Convection in a Micropolar Fluid

The effect of time periodic body force (or g-jitter or gravity modulation) on the onset of Rayleigh-Benard electro-convention in a micropolar fluid layer is investigated by making linear and non-linear stability analysis. The stability of the horizontal fluid layer heated from below is examined by assuming time periodic body acceleration. This normally occurs in satellites and in vehicles connected with micro gravity simulation studies. A linear and non-linear analysis is performed to show that gravity modulation can significantly affect the stability limits of the system. The linear theory is based on normal mode analysis and perturbation method. Small amplitude of modulation is used to compute the critical Rayleigh number and wave number. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation. The non-linear analysis is based on the truncated Fourier series representation. The resulting non-autonomous Lorenz model is solved numerically to quantify the heat transport. It is observed that the gravity modulation leads to delayed convection and reduced heat transport.

A Note on the Existence and Uniqueness of Time-Periodic Electro-rheological Flows

We prove the existence and uniqueness of time-periodic motions of an electro-rheological fluid, subject to a time-periodic body force. Further if the force vanishes in its periodicity interval and if a suitable relation between the size of the force and its period holds, then the solution periodically becomes extinct.

Existence of time-periodic weak solutions to the stochastic Navier-Stokes equations around a moving body

The existence of time-periodic stochastic motions of an incompressible fluid is obtained. Here the fluid is subject to a time-periodic body force and an additional time-periodic stochastic force that is produced by a rigid body moves periodically stochastically with the same period in the fluid.

Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane

We consider the two-dimensional motion of a Navier-Stokes liquid in the whole plane, under the action of a time-periodic body force $F$ of period $T$, and tending to a prescribed nonzero constant velocity at infinity. We show that if the magnitude of $F$, in suitable norm, is sufficiently small, there exists one and only one corresponding time-periodic flow of period $T$ in an appropriate function class.

Effect of Gravity Modulation on the Onset of Ferroconvection in a Densely Packed Porous Layer

The stability of a horizontal porous layer of a ferromagnetic fluid heated from below is studied when the fluid layer is subject to a time-periodic body force. Modified Darcy law is used to describe the fluid motion. The effect of gravity modulation is treated by a perturbation expansion in powers of the amplitude of modulation. The stability of the system, characterized by a correction Rayleigh number, is determined as a function of the frequency of modulation, magnetic parameters, and Vadasz number. It is found that subcritical instability is possible for low frequency g-jitter and that the magnetic and g-jitter mechanisms work against each other for small and moderate values of frequency of modulation. The effect of Vadasz number is shown to be reinforcing the influence of gravity modulation for small and moderate values of frequency. The magnetic, porous and modulation effects disappear altogether for sufficiently large values of the frequency of modulation. Keywords-Ferromagnetic fluid, Gravity modulation, Perturbation method, Porous layer, Stability. Gupta and Gupta (3) investigated thermal instability in a layer of ferromagnetic fluid subject to coriolis force and permeated by a vertical magnetic field. It is substantiated that overstability cannot occur if the Prandtl number is greater than unity. Gotoh and Yamada (4) investigated the linear convective instability of a ferromagnetic fluid layer heated from below and confined between two horizontal ferromagnetic boundaries. The Galerkin technique is used and the Legendre polynomials are taken as the trial functions. It is shown that the magnetization of the boundaries and the nonlinearity of fluid magnetization reduce the critical Rayleigh number and the effects of magnetization and buoyancy forces are shown to compensate each other. Blums (5) examined the possibility of having convection in ferromagnetic fluids as a result of magneto- diffusion of colloidal particles which give rise to non-uniform magnetization. Stiles and Kagan (6) examined the thermoconvective instability of a horizontal layer of ferrofluid in a strong vertical magnetic field. Their work also questioned the satisfactory agreement claimed to exist between the experiments and the theoretical model of Finlayson which has been generalized by them. Odenbach (7) investigated the convective flow generated by the interaction of a magnetic field gradient with a gradient in magnetization in a magnetic fluid. This gradient was caused by the diffusion of the magnetic particles in the field gradient. Aniss et al. (8) investigated the effect of a time-sinusoidal magnetic field on the onset of convection in a horizontal magnetic fluid layer heated from above. The Floquet theory is used to determine the convective threshold for free-free and rigid-rigid cases. The possibility to produce a competition between the harmonic and subharmonic modes at the onset of convection is discussed. Abraham (9) investigated the Rayleigh-Benard problem in a micropolar ferromagnetic fluid layer in the presence of a vertical uniform magnetic field analytically. It is shown that the micropolar ferromagnetic fluid layer heated from below is more stable as compared with the classical Newtonian ferromagnetic fluid. The effect of radiative heat transfer on ferroconvection has been studied by Maruthamanikandan (10) using the linear stability analysis. Consideration is given to two asymptotic cases, viz., transparent and opaque layers of fluid. The critical values marking the onset of convection are obtained using the Galerkin technique. Bajaj (11) considered thermosolutal convection in magnetic fluids in the presence of a vertical magnetic field and bifrequency vertical vibrations. The regions of parametric instability have been obtained using the Floquet theory. Ramanathan and Muchikel (12) investigated the effect of temperature-dependent viscosity on ferroconvective instability in a porous medium. It is found that the stationary mode of instability is

Rayleigh‐Benard convection in a dielectric liquid: time‐periodic body force

AbstractWe discuss the thermal instability in a layer of dielectric fluid when the boundaries of the layer are subjected to small amplitude Time‐periodic body force (TBF). Perturbation solution in powers of the amplitude of the applied temperature field is obtained using the Venezian method (1969) [1]. A linear stability analysis is performed. The results of the study, indicate that TBF or g‐jitter leads to delayed convection Also for large frequencies the effect of TBF disappears.The problem has relevance in many dielectric fluid applications wherein regulation of thermal convection is called for. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Combined effect of temperature and gravity modulations on the onset of magneto-convection in weak electrically conducting micropolar liquids

The combined effect of time-periodic boundary temperature (TBT, also called temperature modulation) and time-periodic body force (TBF, also called gravity modulation) of small amplitude on magneto-convection in a micropolar liquid is investigated by making a linear stability analysis. A regular perturbation method is used to arrive at an expression for the correction Rayleigh number that throws light on the possibility of subcritical motions. The individual results of TBT and TBF are obtained as limiting cases of the present study. The Venezian [1] approach is adopted in arriving at the critical Rayleigh and wave numbers for small amplitudes of TBT and TBF. In the case of synchronous TBT and TBF for moderate frequency values, the role of Prandtl, Chandrasekhar, Rayleigh numbers, micropolar heat conduction parameters and ratio of amplitudes in inducing subcritical instability is delineated. The TBF and synchronous TBT are shown to give rise to subcritical instability. The system is shown to be most stable with respect to asynchronous TBT and TBF. Comparison is made between the effects of temperature, gravity and combined (temperature + gravity) modulations.

Existence of time-periodic solutions to the Navier–Stokes equations around a moving body

We demonstrate the existence of time-periodic motions of an incompressible Navier-Stokes fluid subject to a time-periodic body force, occupying the region exterior to a body that performs a periodic rigid motion of same period.

Effect of time-periodic boundary temperatures/body force on Rayleigh–Benard convection in a ferromagnetic fluid

We discuss the thermal instability in a layer of a ferromagnetic fluid when the boundaries of the layer are subjected to synchronous/asynchronous imposed time-periodic boundary temperatures (ITBT)/time–periodic body force (TBF). Only infinitesimal disturbances are considered. The Venezian approach is adopted in arriving at the critical Rayleigh and wave numbers for small amplitudes of ITBT. A perturbation solution in powers of the amplitude of the applied temperature field is obtained. When the ITBT at the two walls are synchronized then, for moderate frequency values, the role of magnetization in inducing sub-critical instabilities is delineated. A similar role is shown to be played by the Prandtl number. The magnetization parameters and Prandtl number have the opposite effect at large frequencies. The system is most stable when the ITBT is asynchronous. The effect of TBF on the onset of convection is found to be qualitatively similar to the effect of an asynchronous ITBT. Low Prandtl number fluids are shown to be more easily vulnerable to destabilization by TBF compared to very large Prandtl number fluids. The problem has relevance in many ferromagnetic fluid applications wherein regulation of thermal convection is called for.

Second Order Hyperbolic Equations with Small Nonlinearities

A second order partial differential equation which describes the propagation of one-dimensional nonlinear waves in a bounded, inhomogeneous, dissipative medium is analyzed using the method of multiple scales. The conditions under which the oppositely traveling components of the nonlinear motion uncouple to first order are given. The nonlinear interaction occurs at second order. Two examples are given: the first is a nonlinear free vibration problem; the second is a problem of resonant forced oscillations induced by a time-periodic body force. The nonlinear resonant motion is always bounded, but may contain shocks.