The problem of forced convection along an isothermal moving plate is a classical problem of fluid mechanics that has been solved for the first time in 1961 by Sakiadis (1961). It appears that the first work concerning mixed convection along a moving plate is that of Moutsoglou and Chen (1980). Thereafter, many solutions have been obtained for different aspects of this class of boundary layer problems. In the previous works the fluid properties have been assumed constant. Ali (2006) in a recent paper treated, for the first time, the mixed convection problem with variable viscosity. He used the local similarity method to solve this problem but there are doubts about the validity of his results. For that reason we resolved the above problem with the direct numerical solution of the boundary layer equations without any transformation.
In this paper we employ renormalized viscosity and thermal diffusivity to construct a subgrid-scale model for large eddy simulation (LES) of turbulent thermal convection. For LES, we add $ν_\mathrm{ren} \propto Π_u^{1/3} (π/Δ)^{-1/3}$ to the kinematic viscosity; here $Π_u$ is the turbulent kinetic energy flux, and $Δ$ is the grid spacing. In our model, the turbulent Prandtl number is unity. We performed LES of turbulent thermal convection on a $128^3$ grid and compare the results with direct numerical simulation (DNS) on a $512^3$ grid. There is a good agreement between the LES and DNS results…
We investigate the effect of a temperature dependent viscosity on the onset of thermal convection in a horizontal layer of a binary fluid mixture that is heated from below. For an exponential temperature dependence of the viscosity, we find in binary mixtures as a function of a positive separation ratio and beyond a certain viscosity contrast a discontinuous transition between two stationary convection modes having a different wavelength. In the range of negative values of the separation ratio, a (continuous or discontinuous) transition from an oscillatory to a stationary onset of convection o…
The onset of thermal convection in a horizontal layer of a colloidal suspension is investigated in terms of a continuum model for binary-fluid mixtures where the viscosity depends on the local concentration of colloidal particles. With an increasing difference between the viscosity at the warmer and the colder boundary the threshold of convection is reduced in the range of positive values of the separation ratio psi with the onset of stationary convection as well as in the range of negative values of psi with an oscillatory Hopf bifurcation. Additionally the convection rolls are shifted downwa…
Numerical experiments of convection with grain-damage are used to develop scaling laws for convective heat flow, mantle velocity, and plate velocity across the stagnant lid and plate-tectonic regimes. Three main cases are presented in order of increasing complexity: a simple case wherein viscosity is only dependent on grainsize, a case where viscosity depends on temperature and grainsize, and finally a case where viscosity is temperature and grainsize sensitive, and the grain-growth (or healing) is also temperature sensitive. In all cases, convection with grain-damage scales differently than N…
In the above paper [Chaos, Solitons & Fractals, 13 (2002) 1843-1850] the author treats the boundary layer flow along a vertical isothermal plate of an electrically conducting fluid. The plate lies in a vertical free stream of constant velocity $u_\infty$, constant temperature $T_\infty$ and constant mass concentration $C_\infty$. The plate moves vertically with a velocity which is a linear function of distance $x$ ($u_w=cx$). A transverse magnetic field with constant strength $B_0$ is imposed on the flow. The boundary layer equations are integrated numerically. However, there is a fundamental …
Tidal interactions are important in driving spin and orbital evolution in planetary and stellar binary systems, but the fluid dynamical mechanisms responsible remain incompletely understood. One key mechanism is the interaction between tidal flows and convection. Turbulent convection is thought to act as an effective viscosity in damping large-scale tidal flows, but there is a long-standing controversy over the efficiency of this mechanism when the tidal frequency exceeds the turnover frequency of the dominant convective eddies. This high frequency regime is relevant for many applications, suc…
We focus the study of a convection problem in a 2D setup in the presence of the O(2) symmetry. The viscosity in the fluid depends on the temperature as it changes its value abruptly in an interval around a temperature of transition. The influence of the viscosity law on the morphology of the plumes is examined for several parameter settings, and a variety of shapes ranging from spout to mushroom shaped is found. We explore the impact of the symmetry on the time evolution of this type of fluid, and find solutions which are greatly influenced by its presence: at a large aspect ratio and high Ray…
The scaling of plate-tectonic convection is investigated by simulating thermal convection with pseudoplastic rheology and strongly temperature-dependent viscosity. The effect of mantle melting is also explored with additional depth-dependent viscosity. Heat-flow scaling can be constructed with only two parameters, the internal Rayleigh number and the lithospheric viscosity contrast, the latter of which is determined entirely by rheological properties. The critical viscosity contrast for the transition between plate-tectonic and stagnant-lid convection is found to be proportional to the square …
We explore the instabilities developed in a fluid in which viscosity depends on temperature. In particular, we consider a dependency that models a very viscous (and thus rather rigid) lithosphere over a convecting mantle. To this end, we study a 2D convection problem in which viscosity depends on temperature by abruptly changing its value by a factor of 400 within a narrow temperature gap. We conduct a study which combines bifurcation analysis and time-dependent simulations. Solutions such as limit cycles are found that are fundamentally related to the presence of symmetry. Spontaneous plate-l…
When two fluids of different electrical conductivities are transported side by side in a microfluidic device under the influence of an electric field, an electrokinetic instability (EKI) is often generated after some critical values of the applied electric field strength and conductivity ratio. Many prior experimental and numerical studies show that this phenomenon results in a chaotic flow field inside a microdevice, thereby facilitating the mixing of two fluids if they are Newtonian in behaviour. However, the present numerical study shows that this chaotic convection arising due to the elect…
We have conducted laboratory experiments to model important aspects of plumes in mantle convection. We focus on the role of the viscosity ratio U (between the ambient fluid and the plume fluid) in determining the plume structure and dynamics. In our experiments, we are able to capture geophysical convection regimes relevant to mantle convection both for hot spots (when U > 1) and plate-subduction (when U < 1) regimes. The planar laser induced fluorescence (PLIF) technique is used for flow visualization and characterizing the plume structures. The convection is driven by compositional buoyancy …
Turbulent convection is thought to act as an effective viscosity ($ν_E$) in damping tidal flows in stars and giant planets. However, the efficiency of this mechanism has long been debated, particularly in the regime of fast tides, when the tidal frequency ($ω$) exceeds the turnover frequency of the dominant convective eddies ($ω_c$). We present the results of hydrodynamical simulations to study the interaction between tidal flows and convection in a small patch of a convection zone. These simulations build upon our prior work by simulating more turbulent convection in larger horizontal boxes, …
Numerical simulation of rotating convection in plane layers with free slip boundaries show that the convective flows can be classified according to a quantity constructed from the Reynolds, Prandtl and Ekman numbers. Three different flow regimes appear: Laminar flow close to the onset of convection, turbulent flow in which the heat flow approaches the heat flow of non-rotating convection, and an intermediate regime in which the heat flow scales according to a power law independent of thermal diffusivity and kinematic viscosity.
A low-dimensional model (LDM) for turbulent Rayleigh-Benard convection in a Cartesian cell with square domain, based on the Galerkin projection of the Boussinesq equations onto a finite set of empirical eigenfunctions, is presented. The empirical eigenfunctions are obtained from a joint Proper Orthogonal Decomposition (POD) of the velocity and temperature fields using the Snapshot Method on the basis of a direct numerical simulation (DNS). The resulting LDM is a quadratic inhomogeneous system of coupled ordinary differential equations which we use to describe the long-time temporal evolution o…
Suspended fibres significantly alter fluid rheology, as exhibited in for example solutions of DNA, RNA and synthetic biological nanofibres. It is of interest to determine how this altered rheology affects flow stability. Motivated by the fact thermal gradients may occur in biomolecular analytic devices, and recent stability results, we examine the problem of Rayleigh-Bénard convection of the transversely-isotropic fluid of Ericksen. A transversely-isotropic fluid treats these suspensions as a continuum with an evolving preferred direction, through a modified stress tensor incorporating four vi…
The vertical structure of stationary thin accretion discs is calculated from the energy balance equation with heat generation due to microscopic ion viscosity η and electron heat conductivity κ, both depending on temperature. In the optically thin discs it is found that for the heat conductivity increasing with temperature, the vertical temperature gradient exceeds the adiabatic value at some height, suggesting convective instability in the upper disc layer. There is a critical Prandtl number, Pr = 4/9, above which a Keplerian disc become fully convective. The vertical density distribution of …
The original Oberbeck-Boussinesq (OB) equations which are the coupled two dimensional Navier-Stokes(NS) and heat conduction equations have been investigated by E.N. Lorenz half a century ago with Fourier series and opened the way to the paradigm of chaos. In our former study-Chaos, Solitons and Fractals78, 249 (2015)-we presented fully analytic solutions for the velocity, pressure and temperature fields with the aim of the self-similar Ansatz and gave a possible explanation of the Rayleigh--Bènard convection cells. Now we generalize the Oberbeck-Boussinesq hydrodynamical system, going beyond t…
Laminar natural convection from two horizontally aligned isothermal cylinders in unconfined Power-law fluids has been investigated numerically. The effect of horizontal spacing (0<=(S/D)<=10) on both momentum and heat transfer characteristics has been delineated under the following pertinent parameters: Grashof number (10<=Gr<=1e3), Prandtl number (0.71<=n<=100), and Power-law index (0.4<=n<=1.6). The heat transfer characteristics are elucidated in terms of isotherms, local Nusselt number (Nu) distributions and average Nusselt number values, whereas the flow characteristics are interpreted in …
In this study, we performed highly resolved large-eddy simulations (LES) to investigate the influence of variable properties on the forced turbulent convection in a channel. The constant heat flux boundary condition permits wall temperature fluctuations and thus induces variations of fluid properties. Since the effect of viscosity on the flow exhibits $Re_τ^{-1}$ scaling, we only considered $Re_τ= 180$ in the present study. Compared to the flow with constant properties, results indicate that the variable properties have trivial effects on the mean velocity and temperature profiles, Reynolds sh…
This paper presents a lattice Boltzmann model to simulate the aqueous humor (AH) dynamics in the human eyes by involving incompressible Navier-Stokes flow, heat convection and diffusion, and Darcy seepage flow. Verifying simulations indicate that the model is stable, convergent and robust. Further investigations were carried out, including the effects of heat convection and buoyancy, AH production rate, permeability of trabecular meshwork, viscosity of AH and anterior chamber angle on intraocular pressure (IOP). The heat convection and diffusion can significantly affect the flow patterns in th…
The problem of forced convection along an isothermal, constantly moving plate is a classical problem of fluid mechanics that has been solved for the first time in 1961 by Sakiadis (1961). Thereafter, many solutions have been obtained for different aspects of this class of boundary layer problems. Solutions have been appeared including mass transfer, varying plate velocity, varying plate temperature, fluid injection and fluid suction at the plate. The work by Hassanien (1997) belongs to the above class of problems, including a linearly varying velocity and the variation of fluid viscosity with …
The study investigates the problem of magnetohydrodynamic (MHD) free convection over a non-isothermal axisymmetric body under the action of transverse magnetic field. The body is embedded in a porous medium saturated with electrically conducting non-Newtonian power law fluid. In order to obtain similarity solution, it is assumed that the viscosity of the fluid decays exponentially with temperature. The qualitative results are illustrated for a vertical flat plate, horizontal cylinder and sphere.
Simple scalings suggest that super-Earths are more likely than an equivalent Earth-sized planet to be undergoing plate tectonics. Generally, viscosity and thermal conductivity increase with pressure while thermal expansivity decreases, resulting in lower convective vigor in the deep mantle. According to conventional thinking, this might result in no convection in a super-Earth's deep mantle. Here we evaluate this. First, we here extend the density functional theory (DFT) calculations of post-perovskite activation enthalpy of to a pressure of 1 TPa. The activation volume for diffusion creep bec…
Convection is thought to act as a turbulent viscosity in damping tidal flows and in driving spin and orbital evolution in close convective binary systems. This turbulent viscosity should be reduced, compared to mixing-length predictions, when the forcing (tidal) frequency $|ω_t|$ exceeds the turnover frequency $ω_{cv}$ of the dominant convective eddies. However, two contradictory scaling laws have been proposed and this issue remains highly disputed. To revisit this controversy, we conduct the first direct numerical simulations (DNS) of convection interacting with the equilibrium tidal flow in…
Laminar natural convection heat transfer from two horizontally aligned square cylinders has been investigated numerically using a finite-volume method (FVM) approach. Computations were performed to delineate the momentum and heat transfer characteristics under the following ranges of parameters: horizontal spacing between the cylinders (0 <= S/W <= 10), tilt angle of the square cylinder (0^0 <= δ <= 60^0), and Grashof number (10 <= Gr <= 10^5) for some specific Newtonian fluids having Prandtl number (0.71 <= Pr <= 7). The comprehensive results are represented in terms of temperature contours a…
Interfacial tension gradients drive flow along liquid-liquid interfaces in a process known as the Marangoni effect. Such gradients can be caused by surfactants, which has been extensively studied in the literature. Less is known of its nanoscale properties, where molecular interfaces display properties separate from bulk liquid matter such as interfacial viscosity. In this report we study the molecular solutal Marangoni effect using molecular dynamics simulations. We show that molecular interfacial effects are important and should be accounted for in nanofluidic regimes. Hydrodynamic models ca…
As shown in earlier work (Ahlers et al., J. Fluid Mech. 569, p.409 (2006)), non-Oberbeck Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Benard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric the…
Turbulent convection is thought to act as an effective viscosity in damping equilibrium tidal flows, driving spin and orbital evolution in close convective binary systems. Compared to mixing-length predictions, this viscosity ought to be reduced when the tidal frequency $|ω_t|$ exceeds the turnover frequency $ω_{cν}$ of the dominant convective eddies, but the efficiency of this reduction has been disputed. We reexamine this long-standing controversy using direct numerical simulations of an idealized global model. We simulate thermal convection in a full sphere, and externally forced by the equ…
We perform a numerical study of the heat transfer and flow structure of Rayleigh-Bénard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity $ϕ$ in the flow properties during the transition process from the traditional RB convection with $ϕ=1$ (so no obstacles included) to Darcy-type porous-media convection with $ϕ$ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for the Rayleigh number $Ra$ from $10^5$ to $10^{10}$ and varying porosities…
Turbulent mixed convection in channel flows with heterogeneous surfaces is studied using direct numerical simulations. The relative importance between buoyancy and shear effects, characterized by the bulk Richardson number $Ri_b$, is varied in order to cover the flow regimes of forced, mixed and natural convection, which are associated with different large-scale flow organization. The heterogeneous surface consists of streamwise-aligned ridges, which are known to induce secondary motion in case of forced convection. The large-scale streamwise rolls emerging under smooth-wall mixed convection c…
We propose an analytical model to estimate the interface temperature $Θ_Γ$ and the Nusselt number $Nu$ for an evaporating two-layer Rayleigh-Bénard configuration in statistically stationary conditions. The model is based on three assumptions: (i) the Oberbeck-Boussinesq approximation can be applied to the liquid phase, while the gas thermophysical properties are generic functions of thermodynamic pressure, local temperature, and vapour composition, (ii) the Grossmann-Lohse theory for thermal convection can be applied to the liquid and gas layers separately, (iii) the vapour content in the gas …
We present mesoscale numerical simulations of Rayleigh-Bénard convection in a two-dimensional concentrated emulsion, confined between two parallel walls, heated from below and cooled from above, under the effect of buoyancy forces. The systems under study comprise finite-size droplets, whose concentration $Φ_0$ is varied, ranging from the dilute limit up to the point where the emulsion starts to be packed and exhibits non-Newtonian rheology. We focus on the characterisation of the convective heat transfer properties close to the transition from conductive to convective states. The convective f…
The effects of surfactant coating on a deformable viscous drop under the combined action of a shear flow and a uniform electric field, are investigated by solving the coupled equations of electrostatics, fluid flow and surfactant transport. Employing a comprehensive three-dimensional solution technique, the non-Newtonian shearing response of the bulk emulsion is analyzed in the dilute suspension regime. The present results reveal that the surfactant non-uniformity creates significant alterations in the flow disturbance around the drop, thereby influencing the viscous dissipation from the flowi…
We present mesoscale numerical simulations of Rayleigh-Bénard (RB) convection in a two-dimensional model emulsion. The systems under study are constituted of finite-size droplets, whose concentration Phi_0 is systematically varied from small (Newtonian emulsions) to large values (non-Newtonian emulsions). We focus on the characterisation of the heat transfer properties close to the transition from conductive to convective states, where it is known that a homogeneous Newtonian system exhibits a steady flow and a time-independent heat flux. In marked contrast, emulsions exhibit a non-steady dyna…
Convection is the predominant mechanism by which energy and angular momentum are transported in the outer portion of the Sun. The resulting overturning motions are also the primary energy source for the solar magnetic field. An accurate solar dynamo model therefore requires a complete description of the convective motions, but these motions remain poorly understood. Studying stellar convection numerically remains challenging; it occurs within a parameter regime that is extreme by computational standards. The fluid properties of the convection zone are characterized in part by the Prandtl numbe…
Numerical simulations of nonhydrostatic atmospheric flow, based on linearly decoupled semi-implicit or fully-implicit techniques, usually solve linear systems by a pre-conditioned Krylov method without preserving the skew-symmetry of convective operators. We propose to perform atmospheric simulations in such a fully-implicit manner that the difference operators preserve both the skew-symmetry and the tightly nonlinear coupling of the differential operators. We demonstrate that a symmetry-preserving Jacobian-free Newton-Krylov~(JFNK) method mimics a balance between convective transport and turb…
