Rigid-free Boundary Research Articles

On the onset of stationary convection on Jeffrey nanofluid layer saturated with a porous medium: Brinkman model

In this paper, we investigate the onset of convection in a Jeffrey nanofluid layer saturated with the porous medium using Darcy-Brinkmann model. Normal mode analysis and Galerkin type weighted residual method (GWRM) are used to analyse conservation equations. Effects of Brownian motion and thermophoresis are taken into account in the Jeffrey nanofluid model. The Buongiorno model deployed for nanoparticles incorporates the influences of thermophoresis and Brownian motion. Three cases of free-free, rigid-rigid and rigid-free boundaries are considered. For stationary convection, the effects of Darcy number, Jeffrey parameter, Lewis number, nanoparticle Rayleigh number, porosity and modified diffusivity ratio for all the above mentioned boundary conditions are investigated analytically and graphically. The numerical computed values of stationary thermal Rayleigh number are presented graphically for three distinct combinations of boundary conditions. The study is of great significance in many different areas such as automotive, pharmaceutical, geophysics, soil sciences, food processing, oceanography, limnology, etc., and excellent coincidence is found regarding the present paper and earlier published work.

Effect of magnetic field on the onset of thermal convection in a Jeffery nanofluid layer saturated by a porous medium: free-free, rigid-rigid and rigid-free boundary conditions

Effect of magnetic field on the onset of thermal convection in a Jeffery nanofluid layer saturated by a porous medium: free-free, rigid-rigid and rigid-free boundary conditions

ON THERMAL CONVECTION IN ROTATING CASSON NANOFLUID PERMEATED WITH SUSPENDED PARTICLES IN A DARCY-BRINKMAN POROUS MEDIUM

The present paper investigates the effect of suspended particles on thermal convection in rotating Casson nanofluids saturating a Darcy-Brinkman porous medium which has various applications in different sectors, including those that process food, paint, water generators, electricity generators, hydrology, and geophysics, heavily rely on rotation in thermal convection. With the aid of the Galerkin 1st approximation technique, the numerical examination is carried out. The Darcy-Brinkman porous media and particles suspension are taken into consideration throughout the conduct of this study. The non-Newtonian Casson nanofluid, Darcy-Brinkman porous medium, particle suspension and rotation parameter, and their impact on thermal convection have been analyzed and presented graphically for free-free, rigid-rigid, and rigid-free boundaries. It is found that for all boundary conditions the Casson nanofluid and suspended particle parameters have destabilizing impact on the stationary convection, whereas the parameter which occurred due to presence of rotation, i.e., Taylor number and Brinkman porous medium parameters, both delayed the stationary convection. In addition, we have discovered that for realistic rigid-rigid boundary condition, the system is determined to be more stable than for the other two boundary conditions. Also, on the basis of several approximations on the Taylor number and other parameters, the critical wave number and value for stationary convection are determined.

THERMAL INSTABILITY OF HYDROMAGNETIC JEFFREY NANOFLUIDS IN POROUS MEDIA WITH VARIABLE GRAVITY FOR FREE-FREE, RIGID-RIGID, AND RIGID-FREE BOUNDARIES

This article presents a numerical and graphical examination of thermal instability of hydromagnetic Jeffrey nanofluids in porous media with variable gravity for free-free, rigid-rigid, and rigid-free boundaries by using Galerkin technique and normal mode analysis. The Darcy model is employed. Four different gravity variable parameters: h(z) = -(e<sup>z</sup> - 1); exponential, h(z) = -z<sup>2</sup>; parabolic, h(z) = -z; and linear, h(z) = z are taken, and their effects on the Jeffrey parameter, magnetic field, moderated diffusivity ratio, porosity of porous media, Lewis number, and nanoparticle Rayleigh number on stationary convection have been calculated numerically and graphically shown for all three boundary conditions, namely free-free, rigid-rigid, and rigid-free. The necessary conditions for frequencies of the oscillatory mode under all three boundaries have been calculated. According to our research, positive linear gravity parameters make the system unstable for all three boundary conditions, but exponential gravity parameters are superior at stabilizing stationary convection for all three boundary conditions.

Electroconvection in Rotating Jeffrey Nanofluid Saturating Porous Medium: Free–Free, Rigid-Free, Rigid–Rigid Boundaries

The impact of rotation and the boundaries on the initiation of convective instability in a rheological nanofluid layer heated beneath saturated by a porous media with the inclusion of an AC electric field (vertical) is studied employing linear stability analysis. The stationary convective stability of rheological nanofluid is customarily established utilizing Buongiorno model for nanoparticles and Jeffrey model for rheological behavior of regular fluid. The Buongiorno model deployed for nanofluids incorporates the influence of thermophoresis and Brownian motion. Using the normal mode technique, the set of coupled differential equations is solved analytically for both stress-free boudaries and numerically by using the Galerkin-type Weighted Residual Method (GWRM) for top-free, bottom-rigid and rigid–rigid bounding surfaces. The numerical computed values of stationary thermal Rayleigh number are presented graphically for three distinct combinations of boundary conditions. The Taylor number accounting for rotation parameter, Jeffrey parameter, and nanofluid Lewis number delay the start of stationary convection, whereas electric field and concentration Rayleigh number destabilize a system for three groups of boundaries. The bottom-/top-heavy nanofluids are found to be more/less stable. Rigid–rigid boundaries augment the stability in a more pronounced manner than that of the stress-free and rigid-free boundaries. The conditions for non-occurrence of over stability are also derived. This study is of great significance in many metallurgical processes including megma flow, deep convective chimneys, polymer solutions, microfluidic devices and blood flow in micro circulatory systems. An excellent coincidence is found admist present paper and the earlier published work.

Gravity-modulated Rayleigh–Bénard convection in a Newtonian liquid bounded by rigid–free boundaries: a comparative study with other boundary conditions

Gravity-modulated Rayleigh–Bénard convection in a Newtonian liquid bounded by rigid–free boundaries: a comparative study with other boundary conditions

ONSET OF DARCY-BRINKMAN CONVECTION IN A ROTATING BIDISPERSIVE POROUS MEDIUM

The onset of Darcy-Brinkman convection in a rotating bidispersive porous layer is studied for different boundary configurations. Rotating bidispersive porous layer manifests in many applications such as food and chemical processes, rotating machinery, the petroleum industry, solidification and centrifugal casting of metals, biomechanics, and geophysical problems. Three types of boundary configurations are considered, namely rigid-rigid, rigid-free, and free-free boundaries. The fluid motion is characterized using the Brinkman-Darcy equation with a single temperature in macro and micro phases. The linear stability theory is used, and the obtained eigenvalue problem is addressed analytically for the free-free boundary conditions. The eigenvalue problems for the rigid-rigid and rigid-free boundaries are solved numerically. The effects of the Taylor number, momentum transfer coefficient, permeability ratio, Darcy number, and porosity ratio on the system's stability are graphically explored. The Darcy number, permeability ratio, and Taylor number are found to have a stabilizing influence on the system. Further, it is found that the momentum transfer coefficient shows destabilizing effect. The porosity ratio is found to have a stabilizing impact on the system.

Nonlinear stability analysis of penetrative convection in ferrofluids via internal heating

The phenomenon of natural convection driven by internal heating (also known as penetrative convection) has widespread applications in several branches of engineering. The present article investigates the onset of penetrative convection in a horizontal layer of ferrofluids. We assume four different supply functions generate uniform/non-uniform internal heating in the layer. Two sets of realistic boundary conditions (rigid–rigid and rigid-free) are chosen for analysis instead of commonly used artificial boundary conditions of free–free boundaries. The Rayleigh number serves as the eigenvalue of the system of governing equations and is obtained using linear and energy stability analyses. The Chebyshev Tau method is employed to numerically solve the eigenvalue problems obtained through both analyses. It was observed that the energy bounds are lower than the linear ones, indicating the existence of subcritical instabilities for the present problem. The quantitative results obtained for both linear and nonlinear analyses reveal that the Rayleigh number for rigid-free boundaries is always less than that for rigid–rigid boundaries. Moreover, it was also observed that the magnetic parameters advance the onset of penetrative convection.

An Analytical Study of Thermal Instability in a Nanofluid Layer with a Vertical Magnetic Field

The paper re-examines analytically the work of Yadav et al. ([43], [44]) (referred to as Y hereafter) wherein they have examined the combined effect of vertical magnetic field and the free-free, rigid-rigid and rigid-free boundaries on the onset of convection in an electrically conducting nanofluid layer heated from below. The key feature of the present paper is that it explains that the case M=0 ([43], [44]) is not justified either on physical or on mathematical grounds. Moreover, a number of sufficient conditions regarding the non-existence of oscillatory convection have been found.

Rayleigh-Benard Convection in Nanofluids Layer saturated in a Rotating Anisotropic Porous Medium with Feedback Control and Internal Heat Source

Control strategy on Rayleigh-Benard convection in rotating nanofluids saturated in anisotropic porous layer heated from below is studied in the presence of uniformly internal heat source for rigid-rigid, free-free, and lower-rigid and upper-free boundaries. Feedback control strategy with an array of sensors situated at the top plate and actuators located at the bottom plate of the nanofluids layer are considered in this study. Linear stability analysis based on normal mode technique has been performed, the eigenvalue problem is obtained numerically by implementing the Galerkin method and computed by using Maple software. Model employed for the nanofluids includes the mechanisms of Brownian motion and thermophoresis. The problem of the onset of convective rolls instabilities in a horizontal porous layer with isothermal boundaries at unequal temperatures known as Horton-Roger-Lapwood model based on the Darcy model for the fluids flow is used. The influences of internal heat source’s strength, modified diffusivity ratio, nanoparticles concentration Darcy-Rayleigh number and nanofluids Lewis number are found to advance the onset of convection, meanwhile the mechanical anisotropy parameter, thermal anisotropy parameter, porosity, rotation, and controller effects are to slow down the process of convective instability. No visible observation on the modified particle density increment and rigid-rigid boundaries are the most stable system compared to free-free and rigid-free boundaries.

On the Linear Stability of Thermal Convection with Three Different Imposed Shear Flows

The problem of convection in a fluid with temperature dependent viscosity and imposed shear flow, driven by pressure gradients and by a top moving wall, is studied for the case of poorly thermal conducting horizontal walls. Analytical expressions accounting for temperature dependent viscosity effects were obtained for the critical Rayleigh number and frequency of oscillation under a shallow water approximation for Poiseuille, Couette and returning primary flows. The results of this investi- gation contirbute and extend previous findings showing that the onset of convection can be achieved at smaller critical Rayleigh and wavenumbers. The results include approximations of weak and strong shear flows along with conditions for rigid-rigid and rigid-free boundaries. It was found that the imposed shear flow does not influence the critical wavenumber but it does increases the critical Rayleigh number. In this case convection sets in as oscillatory.

Thermal Convection in a Nanofluid Layer with Thermorheological Effect

Thermorheological effect on the onset of convection in a horizontal nanofluid layer heated from below is studied for rigid-rigid, free-free and rigid-free boundaries by incorporating the effects of Brownian motion and thermophoresis. The viscosity of nanofluid is allowed to vary exponentially with temperature. The boundaries are considered to be isothermal and the normal flux of nanoparticle volume fraction is taken to be zero on the boundaries. The resulting eigenvalue problem is solved numerically by the Galerkin method and observed that the instability sets in only as stationary convection irrespective of the velocity boundary conditions. The stability characteristics of the system are strongly dependent on the viscosity parameter. At low values of the viscosity parameter, the variation in the critical Rayleigh number is found to be nominal but increases with increasing viscosity parameter, reaches a maximum and then decreases. Results also indicate that increase in the modified diffusivity ratio, concentration Rayleigh number, and Lewis number is to hasten the convection while the modified particle density increment has no observable effect on the onset of convection.

The onset of electrothermoconvection in a rotating Brinkman porous layer

The criterion for the onset of electrothermoconvection in a rotating horizontal layer of Brinkman porous medium is investigated for different types of velocity boundary conditions namely, (i) both boundaries stress-free, (ii) both boundaries rigid and (iii) lower rigid and upper free boundaries. Results indicate that the nature of boundaries and speed of rotation significantly influence the stability characteristics of the system. In the case of stress-free condition, it is shown that the necessary conditions for the occurrence of oscillatory onset are independent of an external electric field. Contrary to their stabilizing effect in the absence of rotation, increasing the ratio of viscosities Λ and decreasing the Darcy number Da show some destabilizing effect on the onset of stationary electrothermoconvection in the presence of rotation and some important observations are made on the stability characteristics of the system. Moreover, the similarities and differences between free–free, rigid–rigid and rigid–free boundaries are emphasized in triggering the onset of electrothermoconvection in a rotating porous layer. For small Taylor number domain, the stress-free boundaries is found to be always unstable than that of rigid–rigid and rigid–free boundaries. However, this trend is reversed at higher Taylor number domain because the stability of the stress-free case is increased more quickly than the other boundaries.

The onset of convection in a horizontal nanofluid layer of finite depth

This paper presents a linear stability analysis for the onset of natural convection in a horizontal nanofluid layer. The employed model incorporates the effects of Brownian motion and thermophoresis. Both monotonic and oscillatory convection for free–free, rigid–rigid, and rigid–free boundaries are investigated. The oscillatory instability is possible when nanoparticles concentrate near the bottom of the layer, so that the density gradient caused by such a bottom-heavy nanoparticle distribution competes with the density variation caused by heating from the bottom. It is established that the instability is almost purely a phenomenon due to buoyancy coupled with the conservation of nanoparticles. It is independent of the contributions of Brownian motion and thermophoresis to the thermal energy equation. Rather, the Brownian motion and thermophoresis enter to produce their effects directly into the equation expressing the conservation of nanoparticles so that the temperature and the particle density are coupled in a particular way, and that results in the thermal and concentration buoyancy effects being coupled in the same way.

Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.

Stability analysis of thermosolutal second-order fluid in porous Bénard layer

Linear and nonlinear stability of the motionless state of thermosolutal second-order fluid in porous Benard layer is investigated via Lyapunov direct method on the basis of Brinkman’s modification of the Darcy’s model. Critical Rayleigh numbers for linear and nonlinear stability are obtained for free-free, rigid-rigid and rigid-free boundaries. The stabilizing effect of solute concentration and the destabilizing effect of medium permeability and porosity on the basic motion are proved. In particular, for certain range of system parameters the sufficient and necessary condition for stability coincide. Keywords: Second-order fluid, Nonlinear stability, Porous medium, Solute concentration Mathematics Subject Classification (2000): 35Q35, 46N20, 76E06

Hopf bifurcations of viscoelastic fluids heated from below

Nonlinear hydrodynamic stability analysis has been performed for viscoelastic fluids heated from below with rigid-rigid or rigid-free boundary conditions, which are compatible with experimental situations, for the range of viscoelastic parameters where Hopf bifurcations occur. Employing a general constitutive relation which encompasses the Maxwell model, the Jeffreys model, the Oldroyd-B model and the Phan-Thien-Tanner model, the effects of various viscoelastic parameters on the hydrodynamic stability are investigated. The results of nonlinear stability analysis using the power-series method reveal that values of viscoelastic parameters have significant effects on hydrodynamic stability and suggest that the system of Rayleigh-Bénard (R-B) convection may be used, at least in part, as a useful rheometric tool to assess the suitability of constitutive equations or to estimate the values of constitutive parameters of viscoelastic fluids. The results of the present work show that the variation of Nusselt number with respect to the Rayleigh number becomes rapid as either the Deborah number λ increases or the dimensionless retardation time ϵ decreases, and the oscillation frequency of the convection cell ω decreases as λ increases. Also investigated is the effect of λ and ϵ on the time constant of the Landau equation which represents the growth rate of convection cells at the onset of instability. The frontier curves separating the supercritical bifurcation and the subcritical bifurcation in the case of overstability are also presented in the ϵ-λ parameter plane. Though the Hopf bifurcation has not yet been found experimentally in the Rayleigh-Bénard convection of viscoelastic fluids since it requires a very large value of the Deborah number, the analysis of the present study is important in being one of the very few rigorous mathematical analyses of Hopf bifurcation based on the exact Navier-Stokes equation. Moreover, the results of the present paper may also be adopted in the analyses of various phenomena when the Hopf bifurcation is actually found experimentally.

A Numerical Investigation of Thermal Convection in a Heat-Generating Fluid Layer

Finite difference solutions of the equations governing thermal convection driven by uniform volumetric energy sources are presented for two-dimensional flows in a rectangular domain. The boundary conditions are a rigid, (i.e., zero slip), zero heat-flux lower surface, rigid adiabatic sides, and either a rigid or free (i.e., zero shear) isothermal upper surface. Computations are carried out for Prandtl numbers from 0.05 to 20 and Rayleigh numbers from 5 × 104 to 5 × 108. Nusselt numbers and average temperature profiles within the layer are in good agreement with experimental data for rigid-rigid boundaries. For rigid-free boundaries, Nusselt numbers are larger than in the former case. The structure of the flow and temperature fields in both cases is dominated by rolls, except at larger Rayleigh numbers where large-scale eddy transport occurs. Generally, low velocity upflows over broad regions of the layer are balanced by higher velocity downflows when the flow exhibits a cellular structure. The hydrodynamic constraint at the upper surface and the Prandtl number are found to influence only the detailed nature of flow and temperature fields. No truly steady velocity and temperature fields are found despite the fact that average Nusselt numbers reach steady values.

Hydrodynamic instability in fluid layers with uniform volumetric energy sources

Linear and energy theory stability criteria are presented for fluid layers of infinite horizontal extent heated internally by a uniform volumetric energy source. The thermal coupling between the layer and its environment is modeled by a general mixed boundary condition in both the conduction state and the disturbance temperature. Rigid-rigid, free-free, free-rigid, and rigid-free boundaries are considered in the computations. For a fixed ratio of upper surface Biot number to that at the lower surface, decreasing the Biot number is strictly destabilizing for both linear and energy theory criteria. A region of possible subcritical instability is found; its size is strongly dependent on Biot number and becomes small for small values of lower surface Biot number and large Biot number ratio. For two rigid surfaces and an upper and lower surface Biot number of 47.5, mean energy transport measurementswithin the convecting layer indicate a critical Reyleigh number close to that predicted by linear theory. Subcritical instability is not observed when finite amplitude disturbances are introduced at a Rayleigh number between the critical values predicted by the linear theory and the energy theory.

On the occurrence of cellular motion in Bénard convection

The interval of Rayleigh numbers in Bénard convection corresponding to cellular motion is determined in the case of free-free boundaries, rigid-free boundaries and rigid-rigid boundaries, taking into account the variation of the kinematic viscosity with temperature. Neglecting the effect of surface tension, it is shown that this interval is largest for the rigid-rigid case. The most important feature from the obtained formula (6.1) is, however, that the interval is extremely dependent on the depth of the fluid layer. To obtain a cellular pattern it is therefore necessary to have very small fluid depths. For example, with Silicone oil and a fluid depth of 7 mm, cellular motion will, according to the theory, be observed for Rayleigh numbers larger than the critical value and less than 1·07 times the critical value. For a fluid depth of 5 mm, however, the formula (6.1) gives that cellular motion will be observed for Rayleigh numbers up to 1·54 times the critical value.