Subcritical Motion Research Articles

Effect of rotation and internal heat generation on Darcy–Brinkman convection in Oldroyd‐B liquid

AbstractConvection in an Oldroyd‐B liquid saturated highly permeable porous medium is studied via both linear and nonlinear theories. Estimating a convection threshold is the objective of linear‐stability analysis whereas convection amplitudes and heat transfer are elucidated by performing nonlinear‐stability analysis. The eigenvalue problem is solved by the Galerkin method of weighted residuals. The oscillatory mode becomes dominant over the stationary mode. This is because of the race among diffusivity, viscoelasticity, internal‐heat generation, and rotation. The increasing permeability, internal heat generation coefficient, and stress‐relaxation parameter are liable to subcritical motions while the rotation, viscosities ratio, heat capacities ratio, and strain retardation parameter are responsible for the system attaining a supercritical state. The Runge–Kutta–Gill method presents the mechanism to evaluate the amount of heat transfer. The increasing Rayleigh number, internal Rayleigh number, Darcy number, Deborah number, Prandtl number, and the heat capacities ratio enhance the heat transfer. This offers a convenient mechanism for regulating convection. The results obtained in the present paper are expected to play a decisive role in some of the real‐life applications such as oil‐reservoir modeling, crude oil extraction, crystal growth, medicine industries, geothermal‐energy utilization, and so on.

Effect of Time-Dependent Sinusoidal Boundary Temperatures on the Onset Of Ferroconvection in a Porous Medium

<p>The effect of temperature modulation on the onset of Darcy ferroconvection in a horizontal porous layer heated from below is investigated. The analysis is based on the assumption that the amplitude of the temperature modulation is small enough compared with the imposed steady temperature difference. The effect of the oscillating temperature field is treated by a perturbation expansion in powers of the amplitude of the applied field. The effect of magnetic parameters, Vadasz number and temperature modulation in the cases of symmetric, asymmetric and bottom wall modulation, were discussed. The study divulges that subcritical motion exists for symmetric temperature modulation for low frequency. In the case of asymmetric and bottom wall modulation only supercritical motion exists.</p>

Study of Rayleigh–Bénard convection in a chemically reactive fluid using a generalized Lorenz model and the cubic–quintic Ginzburg–Landau equation

Analytical study of linear and weakly nonlinear stability analyses of Rayleigh–Bénard convection in a chemically reactive two-component fluid system is reported by considering physically realistic and idealistic boundaries. Analytical expression for the eigenvalue as functions of the chemical potential and the relaxation parameter is reported in the cases of stationary, oscillatory, and subcritical instabilities. The preferred stationary mode at onset is attributed to the assumption of a chemical reaction. The possibility of subcritical motion in a specific range of values of the relaxation parameter is reported and the threshold value of this parameter at which the transition from the subcritical to the critical motion takes place is documented for different values of the chemical reaction rate. The higher-order generalized Lorenz model leads to the reduced-order cubic–quintic, Ginzburg–Landau equation (GLE), and using its solution, the heat transport is quantified in steady and unsteady convective regimes. The drawback of the cubic GLE and the need for the cubic–quintic GLE for studying the heat transfer in the case of subcritical regime is explained. The possibility of having pitchfork and inverted bifurcations at various values of the relaxation parameter is highlighted. The steady Nusselt number plots clearly show that in the case of a supercritical bifurcation, a smooth transition takes place from unity in the conduction state to a higher value as the Rayleigh number exceeds its critical value predicted by linear theory. In the case of subcritical bifurcation, however, a discontinuous transition is observed.

Weakly nonlinear stability analysis of salt-finger convection in a longitudinally infinite cavity

This paper is a two-dimensional linear and weakly nonlinear stability analyses of the three-dimensional problem of Chang et al. [“Three-dimensional stability analysis for a salt-finger convecting layer,” J. Fluid Mech. 841, 636–653 (2018)] concerning salt-finger convection, which is seen when there is sideways heating and salting along the vertical walls along with a linear variation of temperature and concentration on the horizontal walls. A two-dimensional linear stability analysis is first carried out in the problem with the knowledge that the result could be different from those of a three-dimensional study. A two-dimensional weakly nonlinear stability analysis, that is, then performed points to the possibility of the occurrence of sub-critical motions. Stability curves are drawn to depict various instability regions. With the help of a detailed stability analysis, the stationary mode is shown to be the preferred one compared to oscillatory. Local nonlinear stability analysis of the system is done in a neighborhood of the critical Rayleigh number to predict a sub-critical instability region. The existence of a stable solution at the onset of a weakly nonlinear convective regime is indicated, allowing one to perform a bifurcation study in the problem. Heat and mass transports are discussed by analyzing the Nusselt number, Nu, and Sherwood number, Sh, respectively. A simple relationship is obtained between the Nusselt number and the Sherwood number exclusively in terms of the Lewis number, Le.

Bénard-Taylor ferroconvection with time-dependent sinusoidal boundary temperatures

The combined effect of centrifugal acceleration and time-varying boundary temperatures on the onset of convective instability of a rotating magnetic fluid layer is investigated by means of the regular perturbation method. A perturbation expansion in terms of the amplitude of applied temperature field is implemented to effectively deal with the effects of temperature modulation. The criterion for the threshold is established based on the condition of stationary instability manifesting prior to oscillatory convection. The modulated critical Rayleigh number is computed in terms of Prandtl number, magnetic parameters, Taylor number and the frequency of thermal modulation. It is shown that subcritical motion exists only for symmetric excitation and the destabilizing effect of magnetic mechanism is perceived only for asymmetric and bottom wall excitations. It is also delineated that, for bottom wall modulation, rotation tends to stabilize the system at low frequencies and the opposite is true for moderate and large frequencies. Furthermore, it is established that, notwithstanding the type of thermal excitation, the modulation mechanism attenuates the influences of both magnetic stresses and rotation for moderate and large frequencies.

Convective heat and mass transports and chaos in two-component systems: comparison of results of physically realistic boundary conditions with those of artificial ones

Linear and weakly nonlinear stability analyses of double-diffusive convection in two-component liquids with either potassium chloride (KCl) or sodium chloride (NaCl) aqueous solution, and heat being present is investigated in the paper for free, and rigid, isothermal, iso-solutal boundaries. Using the thermophysical values of the aqueous solutions, we have shown that the stationary convection is the preferred mode at onset and that sub-critical motion is possible. We found that the critical thermal Rayleigh number for water + NaCl + heat is higher compared to that of water + KCl+ heat. The study shows that for water + KCl + heat, the transition from convective motion to chaotic motion occurs at $$r_{\mathrm{H}}=27.2$$ for free boundaries and at 48.5 for rigid boundaries. Here, $$r_{\mathrm{H}}$$ denotes the Hopf thermal Rayleigh number. Further, the existence of windows of mildly chaotic points and fully periodic intervals are reported using Lyapunov exponents and bifurcation diagrams. Chaotic motions in both the aqueous solutions are nearly identical. The percentage increase in heat transport in the double-diffusive system involving NaCl is nearly 1% more than that of KCl in the case of free boundaries, whereas in the case of realistic boundaries it is nearly 1.6%. The comparison of the Nusselt and the Sherwood numbers between water + KCl and water + NaCl leads us to the conclusion that the aqueous solution with lower Lewis number transports maximum heat in the case of free boundaries and opposite is seen in the case of rigid boundaries due to the boundary effect. The many qualitative similarities between the results of artificial and realistic boundaries are highlighted.

LINEAR AND NONLINEAR STUDY OF RAYLEIGH-BÉNARD CONVECTION IN OLDROYD-B FLUIDS UNDER ROTATIONAL MODULATION

The behavior of a fluid system undergoing Rayleigh-Bénard convection when subjected to a modulation of rotation is studied. The fluid considered is Oldroyd-B liquid. This paper consists of a linear as well as a nonlinear study. The thermal Rayleigh number is arrived at by employing a regular perturbation technique. The stress relaxation parameter is shown to destabilize the system, whereas the strain retardation parameter stabilizes it. The nonlinear analysis is carried out using a truncated Fourier series expansion, which results in a system of Lorenz equations. The Nusselt number quantifies the amount of heat transfer in the system. Rotation modulation gives rise to subcritical motion.

Finite-amplitude ferro-convection and electro-convection in a rotating fluid

The onset of Rayleigh–Bénard convection in a horizontal ferrofluid layer subjected to rotation with respect to the vertical axis is theoretically investigated in the paper. Using a truncated Fourier series representation, an analytically intractable fifth-order Lorenz model that has two quadratic nonlinearities is derived and then reduced to the analytically tractable Landau equation with cubic nonlinearity using the method of multi-scales. The critical Rayleigh number of the linear stability theory and that of the energy stability method is drawn from other works and compared with that obtained by the weakly nonlinear stability method reported in the paper. It is found that the critical Rayleigh number obtained by the two nonlinear theories predicts subcritical motions. Further, the results on electro-convection in a rotating fluid layer are extracted from the corresponding problem of ferro-convection by establishing an one-to-one correspondence between the governing equations of the two problems.

Steady Finite-Amplitude Rayleigh–Bénard Convection in Nanoliquids Using a Two-Phase Model: Theoretical Answer to the Phenomenon of Enhanced Heat Transfer

Rayleigh–Bénard convection in liquids with nanoparticles is studied in the paper considering a two-phase model for nanoliquids with thermophysical properties determined from phenomenological laws and mixture theory. In the absence of nanoparticle-modified thermophysical properties as used in the paper, the problem is essentially binary liquid convection with Soret effect. The base liquids chosen for investigation are water, ethylene glycol, engine oil, and glycerine, and the nanoparticles chosen are copper, copper oxide, silver, alumina, and titania. Using data on these 20 nanoliquids, our theoretical model clearly explains advanced onset of convection in nanoliquids in comparison with that in the base liquid without nanoparticles. The paper sets to rest the tentativeness regarding the boundary condition to be chosen in the study of Rayleigh–Bénard convection in nanoliquids. The effect of thermophoresis is to destabilize the system and so is the effect of other parameters arising due to nanoparticles. However, Brownian motion effect does not have a say on onset of convection. In the case of nonlinear theory, the five-mode Lorenz model is derived under the assumptions of Boussinesq approximation and small-scale convective motions, and using it enhancement of heat transport due to the presence of nanoparticles is clearly explained for steady-state motions. Subcritical motion is shown to be possible in all 20 nanoliquids.

Effect of imposed time-periodic boundary temperature on on the onset of rayleigh-benard convection in a dielectric couple stress fluid

The effect of imposed time-periodic boundary temperature of small amplitude on electroconvection under AC electric field in dielectric couple stress liquids 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 sub-critical motions. The Venezian approach is adopted for obtaining eigen value of the problem. Three cases of oscillating temperature field are examined: (a) symmetric, so that the wall temperatures are modulated in phase, (b) asymmetric, corresponding to out-of phase modulation and (c) only the lower wall is modulated. It is shown that the system is most stable when the boundary temperatures are modulated out-of-phase.

Experimental investigation of bubbly flow and turbulence in hydraulic jumps

Many environmental problems are linked to multiphase flows encompassing ecological issues, chemical processes and mixing or diffusion, with applications in different engineering fields. The transition from a supercritical flow to a subcritical motion constitutes a hydraulic jump. This flow regime is characterised by strong interactions between turbulence, free surface and air–water mixing. Although a hydraulic jump contributes to some dissipation of the flow kinetic energy, it is also associated with increases of turbulent shear stresses and the development of turbulent eddies with implications in terms of scour, erosion and sediment transport. Despite a number of experimental, theoretical and numerical studies, there is a lack of knowledge concerning the physical mechanisms involved in the diffusion and air–water mixing processes within hydraulic jumps, as well as on the interaction between the free-surface and turbulence. New experimental investigations were undertaken in hydraulic jumps with Froude numbers up to Fr = 8.3. Two-phase flow measurements were performed with phase-detection conductivity probes. Basic results related to the distributions of void fraction, bubble frequency and mean bubble chord length are presented. New developments are discussed for the interfacial bubble velocities and their fluctuations, characterizing the turbulence level and integral time scales of turbulence representing a “lifetime” of the longitudinal bubbly flow structures. The analyses show good agreement with previous studies in terms of the vertical profiles of void fraction, bubble frequency and mean bubble chord length. The dimensionless distributions of interfacial velocities compared favourably with wall-jet equations. Measurements showed high turbulence levels. Turbulence time scales were found to be dependent on the distance downstream of the toe as well as on the distance to the bottom showing the importance of the lower (channel bed) and upper (free surface) boundary conditions on the turbulence structure.

External regulation of convection in a weak electrically conducting non-Newtonian liquid with g-jitter

The effect of time-periodic temperature/gravity modulation at the onset of magneto-convection in weak electrically conducting second order liquids is investigated by making a linear stability analysis. The Venezian [J. Fluid Mech. 35 (1969) 243] approach is adopted in arriving at the critical Rayleigh and wave numbers for small amplitude temperature/gravity modulation. The temperature modulation is shown to give rise to subcritical motion and gravity modulation leads to delayed convection. Subcriticality or super-criticality depends on the sign of a correction Rayleigh number being negative or positive. An asymptotic analysis is also presented for small/large frequencies. A similar role is shown to be played by the Prandtl number. Comparison is made between the effects of temperature, gravity and combined (temperature+gravity) modulations.

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.

The Subcritical Motion of a Semisubmerged Body: Solvability of the Free Boundary Problem

We discuss existence and regularity of the solutions of the wave-resistance problem for a thin semisubmerged body moving at uniform subcritical velocity in a heavy fluid (e.g., water) of constant depth. The main assumption (on the geometry of the body) is that the flow is two-dimensional; i.e., it can be completely described in the vertical plane containing the direction of the motion. Then the problem can be formulated in terms of a boundary value problem for a holomorphic function (the complex velocity field) satisfying a nonlinear condition (the Bernoulli condition) on a free boundary (the free surface of the fluid). By a hodograph transformation and choosing an appropriate functional setting, we first reduce the problem to the resolution of a nonlinear functional equation depending on two unknown parameters, which are related to the positions in the hodograph plane of the points of contact between the free surface and the body. The main result of this paper is the proof of the existence, under mild as...

The subcritical motion of a surface-piercing cylinder: existence and regularity of waveless solutions of the linearized problem

We consider a boundary value problem in a strip, which is obtained by linearization of the two-dimensional problem of a stream about a slender cylinder, semisubmerged in a heavy fluid of finite depth; it is assumed that the cylinder has uniform, subcritical speed in the direction orthogonal to its generators. We discuss in particular the waveless statement of the problem, characterized by the asymptotic condition of vanishing flow oscillations at infinity. By suitable variational formulations of the problem, we find two classes of solutions, differing by their regularity properties. The most regular solutions exist for particular shapes of the cylinder's section and provide a velocity field which is everywhere continuous and bounded in the strip. The solutions of the other class exist for every (reasonably smooth) symmetric cylinder's profile and have finite energy, but are singular at two points on the strip boundary, representing the points where the free surface meets the cylinder's hull. The relevance of these results for the solvability of the nonlinear problem is discussed.

Subcritical Motions of a Binary Mixture with Anomalous Thermodiffusion in a Vertical Layer

The finite difference method is used to analyze the stability of a binary mixture with anomalous thermodiffusion relative to perturbations of finite amplitude. In the region of monotonic instability, subcritical motions are detected. The amplitude curves, the limits of main-flow stability relative to the most dangerous perturbations, and the stream function, temperature and concentration isolines are obtained.

Effect of temperature/gravity modulation on the onset of magneto-convection in weak electrically conducting fluids with internal angular momentum

The effect of time-periodic temperature/gravity modulation at the onset of magneto-convection in weak electrically conducting fluids with internal angular momentum is investigated by making a linear stability analysis. The Venezian approach is adopted in arriving at the critical Rayleigh and wave numbers for small amplitude temperature/gravity modulation. The temperature modulation is shown to give rise to sub-critical motion and gravity modulation leads to delayed convection. An asymptotic analysis is also presented for small and large frequencies.

Nonlinear analysis on the natural convection between vertical plates in the presence of a horizontal magnetic field

Finite-amplitude secondary motions of an electrically conducting fluid between two vertical parallel plates heated differntially in the presence of a horizontal magnetic field are obtained numerically in the limit of a small Prandtl number and a small magnetic Prandtl number. We find that, as in the purely hydrodynamic case with the Hartmann number H = 0 examined by Nagata and Busse (1983), the bifurcation of the secondary flows is supercritical at the critical Grashof number G = G c when H is small. Subcritical bifurcations occur at higher wavenumbers. As H is increased, the occurrence of the subcritical bifurcations moves gradually towards a smaller wavenumber region along the neutral curve. For H = 10 subcritical motions dominate across the whole range of wavenumbers. The stability analysis shows that the stability of the supercritical motions for small H is bounded by an onset of monotonically growing instability at a higher Grashof number G = G 2. c . The stability region ( G c < G < G 2. c ) becomes smaller with G 2. c approaching G c from above as H is increased. The region vanishes completely at H ⋍ 7.

Finite-amplitude convection in mixtures with heat sources proportional to the concentration of one of the components

A numerical investigation has been made into the equilibrium stability with respect to finite perturbations of a mixture with heat sources proportional to the concentration of an active component. The convective motions that develop after the loss of stability were also studied. The equations of thermoconcentration convection were solved by the grid method for a planar region of rectangular shape simulating a convective cell in the horizontal layer. Neutral curves for finite-amplitude perturbations are constructed, the regions of existence of subcritical motions are found, and a comparison with the results of linear theory is made.

Branching of solutions in the problem of the wavy flow of a viscous liquid with a free boundary

The linear problem of the stability of plane-parallel flow with a free boundary was studied on the basis of the Navier--Stokes equations in [5-9]. The nonlinear problem was investigated in [10-14] within the framework of the equations of P. L. Kapitsa; the Korteweg--de Vries equation and equations similar to it were used in [15, 16] to describe the flow in a liquid film. Nonlinear flow conditions in a liquid film and nonlinear stability were studied in [17-20] on the basis of the Navier--Stokes equations in a long-wave approximation. In the two latter works the Landau constant was calculated using a modified Reynolds--Potter method [21] and a conclusion was drawn with respect to the absence of long-wave subcritical motions.