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BrinkmanBard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions
The BrinkmanBard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number, (Formula presented.), and the wave number, (Formula presented.), are obtained in closed form. An asymptotic analysis is performed, to obtain (Formula presented.) for the case of adiabatic boundaries, for which (Formula presented.). A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the HopfRayleigh number are, however, opposite to each other. In view of the known analogy between Bard convection and TaylorCouette flow in the linear regime, it is imperative that the results of the latter problem, viz., BrinkmanTaylorCouette flow, become as well known. 2023 by the authors. -
A Comparative Study of Thermoconvective Flows of a Newtonian Fluid Over Three Horizontal Undulated Surfaces in a Porous Medium
This paper presents a comparison between the results of three thermoconvective flows of a Newtonian fluid over uniformly heated, undulated horizontal surfaces in a porous medium against the background of the results of a flat plate. The undulations are assumed to have sinusoidal, sawtooth, and triangular waveforms. A system of nonlinear coupled partial differential equations arising in the study is solved using the KellerBox method. Streamlines and isotherms have been plotted and analyzed to examine the effect of parameters on the fluid dynamics and heat transfer. At large surface amplitudes, secondary flow is observed in the cases of sinusoidal and triangular waveforms, but not in the cases of a sawtooth surface and a flat plate. The magnitude of the slip velocity at the horizontal surface is greatest for the sine waveform, while it is least in the case of triangular. The flat plate does not support slip in the velocity to the extent seen in the case of undulated surfaces. The variation of the mean Nusselt number and mean skin friction with surface amplitude and the Rayleigh number indicate that heat transfer and viscous friction at the boundary increase with individual and collective increases in the values of the amplitude and the Rayleigh number. Further, the mean Nusselt number and mean skin friction are found to be maximum for the sinusoidal surface and minimum for the triangular one. The heat transfer and skin friction by the flat surface are much less than that of all three undulated surfaces. Copyright VC 2022 by ASME. -
Nonlinear Rayleigh-Bard magnetoconvection of a weakly electrically conducting Newtonian liquid in shallow cylindrical enclosures
A study of nonlinear axisymmetric Rayleigh-Bard magnetoconvection in a cylindrical enclosure filled with a dilute concentration of carbon-based nanotubes in a weakly electrically conducting Newtonian liquid heated from below for various aspect ratios is carried out. Cylindrical geometry is the prototype for heat storage devices and thermal coolant systems with a controlled environment. There is an analogy between porous media and magnetohydrodynamic problems and hence Rayleigh-Bard magnetoconvection problem is practically important. The solution of the velocity and the temperature is in terms of the Bessel functions of the first kind and hyperbolic functions that are further used to study the marginal stability curves, heat transport, and the dynamical system. Symmetric and asymmetric boundaries of the realistic-type are considered on the horizontal and vertical bounding surfaces. The results of these boundaries are compared with those of the idealistic-type which are symmetric. A unified analysis approach is adopted for all boundary combinations in deriving the Lorenz model and studying the nonlinear dynamics. The time-dependent Nusselt numbers incorporating the effect of the curvature of the cylinder accurately captures the enhanced heat transport situation in the regular convective regime. Further, the influence of various parameters on the indicators of chaos such as the rH-plots, Lorenz attractor, bifurcation diagram, and the time series plot is investigated. The rH-plots clearly point to the appearance of chaos and also assist in determining its intensity and periodicity. The trapping region of the solution of the Lorenz model having the shape like that of a rugby-ball is highlighted in the paper. The size of the ellipsoid shrinks with increase in the strength of the magnetic field and also depends on the boundary conditions. 2024 -
Effect of a non-uniform basic temperature gradient on Rayleigh-Benard convection in a micropolar fluid
The qualitative effect of a non-uniform basic temperature gradient on the linear stability analysis of the Rayleigh-Benard convection in an Eringen's micropolar fluid is studied numerically using a single-term Galerkin technique. The eigenvalue is obtained for free-free, rigid-free, and rigid-rigid velocity boundary combinations with isothermal and adiabatic temperature conditions on the spin-vanishing boundaries. The eigenvalues are also obtained for lower rigid isothermal and upper free adiabatic boundaries with vanishing spin. The influence of various micropolar fluid parameters on the onset of stationary convection has been analysed. Six different basic temperature profiles are considered and their comparative influence on onset is discussed. It is observed that the Rayleigh number obtained is lower than that of the corresponding Newtonian fluid problem. Some important mechanisms of advancing or delaying convection are discussed. 1998 Elsevier Science Ltd. All rights reserved. -
Magnetoconvection in fluids with suspended particles under 1g and ?g
The role of magnetic field in the inhibition of natural convection driven by combined buoyancy and surface tension forces in a horizontal layer of an electrically conducting Boussinesq fluid with suspended particles confined between an upper free/adiabatic and a lower rigid/isothermal boundary is considered in 1g and ?g situations. The inhibition of convection is caused by a stationary and uniform magnetic field parallel to the gravity field. The magnetically-inert suspended particles are not directly influenced by the magnetic field but are influenced indirectly by the magnetically responding carrier fluid in which they are suspended. A linear stability analysis of the system is performed. The Rayleigh-Ritz technique is used to obtain the eigenvalues. The influence of various parameters on the onset of convection has been analysed. Six different reference steady-state temperature profiles are considered and their comparative influence on onset is discussed. Treating Marangoni number as the critical parameter it is shown that any particular infinitesimal disturbance can be stabilized with a sufficiently strong magnetic field. It is observed that the electrically conducting fluid layer with suspended particles heated from below is more stable compared to the classical electrically conducting fluid layer without suspended particles. The critical wave number is found to be insensitive to the changes in the suspension parameters but sensitive to the changes in the Chandrasekhar number. The problem has possible space applications. 2002 itions scientifiques et micales Elsevier SAS. All rights reserved. -
Effects of non-uniform temperature gradient and magnetic field on the onset of convection in fluids with suspended particles under microgravity conditions
The effects of a non-uniform temperature gradient and magnetic field on the onset of convection driven by surface tension in a horizontal layer of Boussinesq fluid with suspended particles confined between an upper free / adiabatic boundary and a lower rigid / isothermal boundary have been considered. A linear stability analysis is performed. The microrotation is assumed to vanish at the boundaries. The Galerkin technique is used to obtain the eigenvalues. The influence of various parameters on the onset of convection has been analysed. Six different non-uniform temperature profiles are considered and their comparative influence on onset is discussed. It is observed that the electrically conducting fluid layer with suspended particles heated from below is more stable compared to the classical electrically conducting fluid without suspended particles. The critical wave number is found to be insensitive to the changes in the parameters but sensitive to the changes in the Chandrasekhar number. The problem has possible applications in microgravity space situations. -
An analytical study of linear and non-linear convection in Boussinesq-Stokes suspensions
The Rayleigh-Benard situation in Boussinesq-Stokes suspensions is investigated using both linear and non-linear stability analyses. The linear and non-linear analyses are based on a normal mode solution and minimal representation of double Fourier series, respectively. The effect of suspended particles on convection is delineated against the background of the results of the clean fluid. The realm of non-linear convection warrants the quantification of heat transfer and this has been achieved on the Rayleigh-Nusselt plane. Possibility of aperiodic convection is discussed. 2003 Elsevier Science Ltd. All rights reserved. -
Effect of temperature/gravity modulation on the onset of magneto-convection in electrically conducting fluids with internal angular momentum
The effect of time-periodic temperature/gravity modulation on the onset of magneto-convection in electrically conducting fluids with internal angular momentum is investigated by making a linear stability analysis. The results of the present study are presented against the background of the results of weak electrically conducting fluids. The qualitative findings of Siddheshwar and Pranesh are found to be true in the present case also except that the eigenvalue is found to be magnitudewise less than that obtained in the case of a weak electrically conducting fluid. -
Magnetoconvection in a micropolar fluid
The problem of Rayleigh-Bard convection in an electrically conducting micropolar fluid layer permeated by a uniform, vertical magnetic field is investigated with free-free, isothermal, spin-vanishing boundaries. The influence of the various micropolar fluid parameters and magnetic field on the onset of stationary convection has been analysed. It is observed that the electrically conducting micropolar fluid layer heated from below is more stable as compared with the classical electrically conducting Newtonian fluid. The critical wave number is found to be insensitive to the changes in the micropolar fluid parameters, but sensitive to the Chandrasekhar number. 1998 Elsevier Science Ltd. All rights reserved. -
Suction-injection effects on the onset of Rayleigh-Bard-Marangoni convection in a fluid with suspended particles
The effect of Suction-Injection-Combination (SIC) on the linear stability of Rayleigh-Bard Marangoni convection in a horizontal layer of an Boussinesq fluid with suspended particles confined between an upper free adiabatic boundary and a lower rigid isothermal/adiabatic boundary is considered. The Rayleigh-Ritz technique is used to obtain the eigenvalues. The influence of various parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pre-sensitivity is found in respect of the critical wave number. It is observed that the fluid layer with suspended particles heated from below is more stable compared to the classical fluid layer without suspended particles. The problem has possible applications in microgravity situations. -
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. 1999 Elsevier Science B.V. -
Effect of gravity modulation on linear, weakly-nonlinear and local-nonlinear stability analyses of stationary double-diffusive convection in a dielectric liquid
The paper deals with the study of effect of gravity modulation on double-diffusive convection in a dielectric liquid for the cases of rigid-rigid and free-free boundaries. Using a modified Venezian approach, expressions for the Rayleigh number and its correction are determined. FourierGalerkin expansion is employed for a weakly nonlinear stability analysis and this results in a fifth-order Lorenz system that retains the structure of the classical one in the limiting case. A local nonlinear stability analysis using the method of multiscales leads to the time-periodic GinzburgLandau equation from the time-periodic generalized Lorenz system and the numerical solution of this simpler equation helps in quantifying unsteady heat and mass transports. Influence of various non-dimensional parameters (Lewis number, solutal Rayleigh number, electrical Rayleigh number and Prandtl number), amplitude and frequency of gravity modulation on onset of convection and heat and mass transports is discussed. The study reveals that the influence of gravity modulation is to stabilize the system and enhance heat and mass transports. The results from free-free boundaries are qualitatively similar to that of rigid-rigid boundaries. Further, it is shown that in the case of free-free boundaries the heat and mass transports are less compared to those of rigid-rigid boundaries. 2020, Springer Nature B.V. -
Effect of rotation on Brinkman-Bard convection of a Newtonian nanoliquid using local thermal non-equilibrium model
Rayleigh-Bard-Taylor convection in a Newtonian, nanoliquid-saturated high porous medium using the local thermal non-equilibrium model (LTNE) is studied analytically using the single term Galerkin technique. The Bousinessq approximation is considered to be valid and the exerted centrifugal force due to rotation is taken. A high porosity porous material glass reinforced fiber with porosity 0.88% is considered and hence the Brinkman model is adopted. The rate of rotation is quantified by the Taylor number and the stability of the system is controlled by thermal Rayleigh number. The expression for the critical eigenvalue (Rayleigh number) is obtained for both idealistic and realistic boundary conditions, that is, stress-free, isothermal and rigid-rigid, isothermal boundary conditions. The presumption of LTNE advances the inception of convection and increases the transport of heat in comparison with that of the local thermal equilibrium (LTE) assumption whereas the opposite phenomenon is seen with the effect of rotation. The effect of various non-dimensional parameters on the convection onset and on transport of heat is also investigated. The results of Rayleigh-Bard-Taylor convection using the LTE assumption are obtained as limiting cases of the present study for infinite values of the ratio of thermal conductivities and the interphase heat transfer coefficient. 2021 Elsevier Ltd -
Unsteady natural convection in a liquid-saturated porous enclosure with local thermal non-equilibrium effect
Stability analysis of free convection in a liquid-saturated sparsely-packed porous medium with local-thermal-non-equilibrium (LTNE) effect is presented. For the vertical boundaries freefree, adiabatic and rigidrigid, adiabatic are considered while for horizontal boundaries it is the stress-free, isothermal and rigidrigid, isothermal boundary combinations we consider. From the linear theory, it is apparent that there is advanced onset of convection in a shallow enclosure followed by that in square and tall enclosures. Asymptotic analysis of the thermal Rayleigh number for small and large values of the inter-phase heat transfer coefficient is reported. Results of DarcyBard convection (DBC) and RayleighBard convection can be obtained as limiting cases of the study. LTNE effect is prominent in the case of BrinkmanBard convection compared to that in DBC. Using a multi-scale method and by performing a non-linear stability analysis the GinzburgLandau equation is derived from the five-mode Lorenz modal. Heat transport is estimated at the lower plate of the channel. The effect of the Brinkman number, the porous parameter and the inter-phase heat transfer coefficient is to favour delayed onset of convection and thereby enhanced heat transport while the porosity-modified ratio of thermal conductivities shows the opposite effect. 2020, Springer Nature B.V. -
Study of chaos in the Darcy-Bard convection problem with Robin boundary condition on the upper surface
Possibility of chaos is studied in Darcy-Bard convection using the Dirichlet and the Robin boundary condition at the lower and upper boundaries, respectively. Comparison is made with the results of Dirichlet (classical-Darcy-Bard convection, CDBC) and Neumann boundary condition (Barletta-Darcy-Bard convection, BDBC). It is found that the cell size at onset is bigger in the case of BDBC compared to the generalized-Darcy-Bard convection (GDBC) and much bigger compared to CDBC. The critical-Darcy-Rayleigh number of BDBC is found to be the least and that of CDBC is the largest. Nonlinear-stability-analysis is performed leading to the scaled-generalized-Vadasz-Lorenz model (SGVLM). In deriving this model, help is sought from a local-nonlinear-stability-analysis that yields the form of the convective-mode. The SGVLM is shown to be dissipative and conservative, with its bounded solution trapped within an ellipsoid. Onset of chaos and its characteristics are studied using the Hopf-Rayleigh-number, the Lorenz-butterfly-diagram, and the plot of the amplitude of the convective-mode vs the control-parameter, R, which is the eigenvalue. Chaos sets in earlier in CDBC and much later in BDBC when compared to that in GDBC. Beyond the onset of chaos is seen a sequence of chaotic and periodic motions, with the latter sometimes being present for an extended period. 2024 Author(s). -
Rheostatic effect of a magnetic field on the onset of chaotic and periodic motions in a five-dimensional magnetoconvective Lorenz system
This paper deals with a weakly nonlinear study of two-dimensional RayleighBard magnetoconvection using a simplified five-dimensional Lorenz model. The governing equations of the system are nondimensionalized and formulated in terms of the stream function and the scalar magnetic potential. A five-modal Fourier truncation scheme is employed and the resulting equations are scaled to obtain a five-dimensional autonomous dynamical system. The Hopf-Rayleigh number, signifying Hopf bifurcation, is numerically evaluated from the analysis of weakly nonlinear stability. Chaotic and periodic motions are depicted by plotting bifurcation diagrams, largest Lyapunov exponent (LLE) diagrams and three-dimensional projections of the phase-space. For a fixed set of parameter values, increasing the strength of the applied magnetic field is found to increase the Hopf-Rayleigh number, thereby delaying the destabilization of the system's equilibrium points. It is shown that while low magnetic field strengths favor the onset of chaotic motion directly from the steady state, stronger magnetic field strengths favor the onset of periodic convection from the steady state prior to the appearance of chaotic motion. We observe here that the applied magnetic field regulates the onset of chaotic and periodic motions in the system and therefore, has a rheostatic control over chaotic and periodic behaviors. 2025 Elsevier Ltd -
Reduction of a Tri-Modal Lorenz Model of Ferrofluid Convection to a CubicQuintic GinzburgLandau Equation Using the Center Manifold Theorem
The differential geometric method of the center manifold theorem is applied to the magnetic-Lorenz model of ferrofluid convection in an electrically non-conducting ferrofluid. The analytically intractable tri-modal nonlinear autonomous system (magnetic-Lorenz model) is reduced to an analytically tractable uni-modal cubicquintic GinzburgLandau equation. The inadequacy of the cubic GinzburgLandau equation and the need for the cubicquintic one is shown in the paper. The heat transport is quantified using the solution of the cubicquintic equation and the effect of ferrofluid parameters on it is demonstrated. The stable and unstable regions in the conductive regime and the conductive-convective regime is depicted using a bifurcation diagram. The noticeable discrepancy between the results of the two models is highlighted and the quintic non-linearity effects are delineated. 2021, Foundation for Scientific Research and Technological Innovation. -
Study of Natural Convection with Local Thermal Non Equilibrium Effects in Nanoliquid-Saturated Low Porosity Enclosures
Natural convection of nanoliquid in densely packed vertical porous enclosure is studied by subjecting the vertical walls to constant heat flux under local thermal non-equilibrium (LTNE) assumptions. Water, copper nanoparticles and porous material made of aluminum foam, glass balls and sand are considered for the study. The governing equations are modelled using single-phase model. Thermophysical properties of nanoliquid and nanoliquid-saturated porous medium are calculated using phenomenological laws and mixture theory. An analytical expression for velocity and temperature profiles of nanoliquid (base liquid+nanoparticles) and solid (porous medium) phases has been obtained. Weighted average Nusselt number is expressed as a function of aspect ratio, volume fraction, and properties concerning LTNE effects. LTNE effect is shown to be a heat transfer enhancing mechanism. The presence of nanoparticles is to enhance the heat transfer in water. Local thermal equilibrium results are obtained as a limiting case of the present study and so obtained results are compared with previously published paper in the literature. 2022, The Author(s), under exclusive licence to Springer Nature India Private Limited. -
Effect of Non-inertial Acceleration on BrinkmanBard Convection in Water-Copper Nanoliquid-Saturated Porous Enclosures
In the present paper we have considered rotating porous tall, square and shallow enclosures heated from below. Linear and non-linear analyses are made using a minimal representation by Fourier trigonometric series. The study is done for realistic boundary condition. Thermophysical properties of water-copper nanoliquid as a function of properties of water as base liquid, copper as nanoparticle and 30% glass fiber reinforced polycarbonate as porous medium are obtained from either phenomenological laws or mixture theory. Non-existence of oscillatory convection is discussed. The range for the existence of unicellular convection is mentioned. The effects of Brinkman number (?), porous parameter (?2), aspect ratio (A) and volume fraction (?) in the presence of rotation on the onset of convection and heat transfer are studied and illustrated graphically. The analytically intractable Lorenz model is derived and transformed into the tractable GinzburgLandau equation using the multiscales method. The definition of Ozoe heat transfer parameter is introduced to discuss the rate of heat transfer enhancement or reduction. It is observed that Ta, ? and ?2 have stabilizing effect on the system and thereby leading to diminished heat transfer whereas A and ? have destabilizing effect on the system and thereby leading to increased heat transfer. Among the three enclosures considered in the study enhanced heat transfer takes place in tall enclosure followed by square and shallow enclosures respectively. It is further observed that presence of nanoparticles advances the onset of convection and enhances the heat transfer. The results of the paper are compared with previous existing results in the absence of rotation and the good agreement is found between them. 2021, The Author(s), under exclusive licence to Springer Nature India Private Limited. -
Study of BrinkmanBard nanofluid convection with idealistic and realistic boundary conditions and by considering the effects of shape of nanoparticles
This study deals with linear and weakly non-linear stability analyses of BrinkmanBard convection in nanoliquid-saturated porous enclosures. Water with a dilute concentration of molybdenum disulfide nanoparticles with 0.06 volume fraction and 30% glass fiber-reinforced polycarbonate as a porous medium with porosity 0.88 are considered to be a working medium. The analytical solution is obtained in the present study for idealistic and realistic boundary conditions, and their results are compared. An analytically intractable Lorenz model with quadratic nonlinearities is reduced to a tractable GinzburgLandau amplitude equation with cubic nonlinearity using the multiscale method. Nanoparticles with different shapesare considered in the study, and their effects on the onset and heat transfer are discussed in great detail graphically in the presence of other parameters arising in the problem. 2021 Wiley Periodicals LLC