Hydromagnetic free convection flow with induced magnetic field effects

An exact solution is presented for the hydromagnetic natural convection boundary layer flow past an infinite vertical flat plate under the influence of a transverse magnetic field with magnetic induction effects included. The transformed ordinary differential equations are solved exactly, under physically appropriate boundary conditions. Closed-form expressions are obtained for the non-dimensional velocity (u), non-dimensional induced magnetic field component (B _x ) and wall frictional shearing stress i.e. skin friction function (τ _x ) as functions of dimensionless transverse coordinate (η), Grashof free convection number (G _r ) and the Hartmann number (M). The bulk temperature in the boundary layer (Θ) is also evaluated and shown to be purely a function of M. The Rayleigh flow distribution (R) is derived and found to be a function of both Hartmann number (M) and the buoyant diffusivity parameter (ϑ ^*). The influence of Grashof number on velocity, induced magnetic field and wall shear stress profiles is computed. The response of Rayleigh flow distribution to Grashof numbers ranging from 2 to 200 is also discussed as is the influence of Hartmann number on the bulk temperature. Rayleigh flow is demonstrated to become stable with respect to the width of the boundary layer region and intensifies with greater magnetic field i.e. larger Hartman number M, for constant buoyant diffusivity parameter ϑ ^*. The induced magnetic field (B _x ), is elevated in the vicinity of the plate surface with a rise in free convection (buoyancy) parameter G _r , but is reduced over the central zone of the boundary layer regime. Applications of the study include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.     

Heat transfer in nucleate pool boiling of aqueous SDS and triton X-100 solutions

Variation in degree of surface wettability is presented through the application of Cooper’s correlative approach (h ∝ M ^−0.5 q _w ^″0.67) for computing enhancement (ϕ) in nucleate pool boiling of aqueous solutions of SDS and Triton X-100 and its presentation with Marangoni parameter (χ) that represents the dynamic convection effects due to surface tension gradients. Dynamic spreading coefficient defined as σ _dyn N _a , which relates spreading and wetting characteristics with the active nucleation site density on the heated surface and bubble evolution process, represents cavity filling and activation process and eliminates the concentration dependence of nucleate pool boiling heat transfer in boiling of aqueous surfactant solutions. Using the dynamic spreading coefficient (σ _dyn N _a  = 0.09q _w ^″0.71), correlation predictions within ±15% for both SDS and Triton X-100 solutions for low heat flux boiling condition (q _w^″ ≤ 100 kW/m²) characterised primarily by isolated bubble regime are presented.     

Radiation effect on mixed convection over an isothermal cone in porous media

The radiation effect on the mixed convection flow of an optically dense viscous fluid adjacent to an isothermal cone embedded in a saturated porous medium with Rosseland diffusion approximation is numerically investigated. The entire regime of the mixed convection is included, as the mixed convection parameter of χ varies from 0 (pure free convection) to 1 (pure forced convection). The transformed nonlinear system of equations is solved by using an implicit finite difference method. Numerical results are given for the dimensionless temperature profiles and the local Nusselt number for various values of the mixed convection parameter χ, the cone angle parameter m, the radiation-conduction parameter R _d and the surface temperature parameter H. The local Nusselt number decreases initially, reaches a minimum in the intermediate value of χ and then increases gradually. It is apparent that increasing the cone angle parameter m enhances the local Nusselt number. The local Nusselt number is significantly increased for the large values of the radiation-conduction parameter R _d and the surface temperature parameter H, i.e., radiation effect becomes pronounced. Received on 25 October 1999     

Mixed Convection Boundary Layer Flow from a Horizontal Circular Cylinder Embedded in a Porous Medium Filled with a Nanofluid

Steady mixed convection boundary layer flow from an isothermal horizontal circular cylinder embedded in a porous medium filled with a nanofluid has been studied for both cases of a heated and cooled cylinder. The resulting system of nonlinear partial differential equations is solved numerically using an implicit finite-difference scheme. The solutions for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction φ and the mixed convection parameter λ. Three different types of nanoparticles are considered, namely Cu, Al₂O₃ and TiO₂. It is found that for each particular nanoparticle, as the nanoparticle volume fraction φ increases, the magnitude of the skin friction coefficient decreases, and this leads to an increase in the value of the mixed convection parameter λ which first produces no separation. On the other hand, it is also found that of all the three types of nanoparticles considered, for any fixed values of φ and λ, the nanoparticle Cu gives the largest values of the skin friction coefficient followed by TiO₂ and Al₂O₃. Finally, it is worth mentioning that heating the cylinder (λ > 0) delays separation of the boundary layer and if the cylinder is hot enough (large values of λ > 0), then it is suppressed completely. On the other hand, cooling the cylinder (λ < 0) brings the boundary layer separation point nearer to the lower stagnation point and for a sufficiently cold cylinder (large values of λ < 0) there will not be a boundary layer on the cylinder.     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

The Onset of Darcy–Brinkman Electroconvection in a Dielectric Fluid Saturated Porous Layer

The combined effect of a vertical AC electric field and the boundaries on the onset of Darcy–Brinkman convection in a dielectric fluid saturated porous layer heated either from below or above is investigated using linear stability theory. The isothermal bounding surfaces of the porous layer are considered to be either rigid or free. It is established that the principle of exchange of stability is valid irrespective of the nature of velocity boundary conditions. The eigenvalue problem is solved exactly for free–free (F/F) boundaries and numerically using the Galerkin technique for rigid–rigid (R/R) and lower-rigid and upper-free (F/R) boundaries. It is observed that all the boundaries exhibit qualitatively similar results. The presence of electric field is emphasized on the stability of the system and it is shown that increasing the AC electric Rayleigh number R _ea is to facilitate the transfer of heat more effectively and to hasten the onset of Darcy–Brinkman convection. Whereas, increase in the ratio of viscosities Λ and the inverse Darcy number Da ⁻¹ is to delay the onset of Darcy–Brinkman electroconvection. Besides, increasing R _ea and Da ⁻¹ as well as decreasing Λ are to reduce the size of convection cells.     

Stability of a fluid-saturated porous medium contained in a vertical cylinder heated from below by forced convection

A linear stability analysis determining the critical Rayleigh number R _c for onset of convection in a bounded vertical cylinder containing a fluid-saturated porous medium is performed for insulated sidewalls, isothermal top surface, and bottom surface heated by forced convection. This Newtonian heating of the bottom surface involves a Biot number Bi that allows consideration of the continuum of boundary conditions ranging from constant heat flux, with global minimum R _min=27.096 found as Bi→0, to isothermal, with global minimum R _min=4π² found as Bi→ ∞. In both cases and for most cylinder aspect ratios, incipient convection sets in as an asymmetric mode, though islands of aspect ratio exist where the onset mode is symmetric. Sample three-dimensional renderings of disturbance temperature distributions showing preferred modes at onset of convection for fixed Bi are provided and an analytical fit to R _min as a function of Bi is given.     

Ferromagnetic Convection in a Rotating Ferrofluid Saturated Porous Layer

The effect of Coriolis force on the onset of ferromagnetic convection in a rotating horizontal ferrofluid saturated porous layer in the presence of a uniform vertical magnetic field is studied. The boundaries are considered to be either stress free or rigid. The modified Brinkman–Forchheimer-extended Darcy equation with fluid viscosity different from effective viscosity is used to characterize the fluid motion. The condition for the occurrence of direct and Hopf bifurcations is obtained analytically in the case of free boundaries, while for rigid boundaries the eigenvalue problem has been solved numerically using the Galerkin method. Contrary to their stabilizing effect in the absence of rotation, increasing the ratio of viscosities, Λ, and decreasing the Darcy number Da show a partial destabilizing effect on the onset of stationary ferromagnetic convection 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 and rigid–rigid boundaries in the presence of buoyancy and magnetic forces together or in isolation are emphasized in triggering the onset of ferromagnetic convection in a rotating ferrofluid saturated porous layer. For smaller Taylor number domain, the stress-free boundaries are found to be always more unstable than in the case of rigid 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 rigid case.     

Analytical and numerical study of buoyancy-driven convection in a vertical enclosure filled with nanofluids

This paper presents an analytical and numerical study of natural convection of nanofluids contained in a rectangular enclosure subject to uniform heat flux along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number Ra, the Prandtl number Pr, the aspect ratio of the cavity A and the solid volume fraction of nanoparticles, Φ. Three types of nanoparticles are taken into consideration: Cu, Al₂O₃ and TiO₂. Various models are used for calculating the effective viscosity and thermal conductivity of nanofluids. In the first part of the analytical study, a scale analysis is made for the boundary layer regime situation. In the second part, an analytical solution based on the parallel flow approximation is reported for tall enclosures (A ≫ 1). In the boundary layer regime a good agreement is obtained between the predictions of the scale analysis and those of the analytical solution. Solutions for the flow fields, temperature distributions and Nusselt numbers are obtained explicitly in terms of the governing parameters of the problem. A numerical study of the same phenomenon, obtained by solving the complete system of the governing equations, is also conducted. A good agreement is found between the analytical predictions and the numerical simulations.     

MHD mixed convective heat transfer flow about an inclined plate

Mixed convection heat transfer about a semi-infinite inclined plate in the presence of magneto and thermal radiation effects is studied. The fluid is assumed to be incompressible and dense. The nonlinear coupled parabolic partial differential equations governing the flow are transformed into the non-similar boundary layer equations, which are then solved numerically using the Keller box method. The effects of the mixed convection parameter R _i, the angle of inclination α, the magnetic parameter M and the radiation–conduction parameter R _d on the velocity and temperature profiles as well as on the local skin friction and local heat transfer parameters. For some specific values of the governing parameters, the results are compared with those available in the literature and a fairly good agreement is obtained.     

A proof of the Benjamin-Feir instability

The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on ℝ⁴. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used to prove the existence of a dispersion relation Λ(λ,σ, I ₁, I ₂)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I ₁ and mass flux I ₂ and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F₀) where F ₀ ≈ 0.8 and F is the Froude number.     

Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water at 4°C (maximum density) when the surface heat flux varies as x ^m and the velocity outside the boundary layer varies as x ^(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions when the mixed convection parameter λ is greater than some critical value λ _c (dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ _c is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and the limits as m → ∞ and as m → −1.     

Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet

An analysis is made for the steady two-dimensional magneto-hydrodynamic flow of an incompressible viscous and electrically conducting fluid over a stretching vertical sheet in its own plane. The stretching velocity, the surface temperature and the transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The transformed boundary layer equations are solved numerically for some values of the involved parameters, namely the magnetic parameter M, the velocity exponent parameter m, the temperature exponent parameter n and the buoyancy parameter λ, while the Prandtl number Pr is fixed, namely Pr = 1, using a finite difference scheme known as the Keller-box method. Similarity solutions are obtained in the presence of the buoyancy force if n = 2m−1. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. It is found that both the skin friction coefficient and the local Nusselt number decrease as the magnetic parameter M increases for fixed λ and m. For m = 0.2 (i.e. n = −0.6), although the sheet and the fluid are at different temperatures, there is no local heat transfer at the surface of the sheet except at the singular point of the origin (fixed point).     

Prediction of natural convection flow using network model and numerical simulations inside enclosure with distributed solid blocks

Steady state natural convection of a fluid with Pr ≈ 1 within a square enclosure containing uniformly distributed, conducting square solid blocks is investigated. The side walls are subjected to differential heating, while the top and bottom ones are kept adiabatic. The natural convection flow is predicted employing the nondimensional volumetric flow rate (Q_max^* Q_{\max }^{*} ) by using a network model and also using numerical simulations. For identical solid and fluid thermal conductivities (i.e. k _s  = k _f ), a parametric study of the effect of number of blocks (N ²), gap size (δ) and enclosure Rayleigh number (Ra) on Q_max^* Q_{\max }^{*} is performed using the two approaches. Network model predictions are observed to agree well with that from the simulations until Raδ³ ~ 12. Considering the enclosure with blocks as a porous medium, for a fixed enclosure Ra number, increasing the number of blocks for a fixed volumetric porosity leads to a decrease in enclosure permeability, which in turn reduces the flow rate. When the number of blocks is fixed, and for a given Ra number, the flow rate increases as the porosity increases by widening the gap between the blocks.     

Viscoelastic properties of ultra-high viscosity alginates

Ultra-high viscosity alginates were extracted from the brown seaweeds Lessonia nigrescens (UHV_N, containing 61% mannuronate (M) and 2% guluronate (G)) and Lessonia trabeculata (UHV_T, containing 22% M and 78% G). The viscoelastic behavior of the aqueous solutions of these alginates was determined in shear flow in terms of the shear stress σ ₂₁, the first normal stress difference N ₁, and the shear viscosity η in isotonic NaCl solutions (0.154 mol/L) at T = 298 K in dependence of the shear rate [(g)\dot]\dot{\gamma} for solutions of varying concentrations and molar masses (3–10 × 10⁵ g/mol, homologous series was prepared by ultrasonic degradation). Data obtained in small-amplitude oscillatory shear (SAOS) experiments obey the Cox–Merz rule. For comparison, a commercial alginate with intermediate chemical composition was additionally characterized. Particulate substances which are omnipresent in most alginates influenced the determination of the material functions at low shear rates. We have calculated structure–property relationships for the prediction of the viscosity yield, e.g., η–M _w–c–[(g)\dot]\dot{\gamma} for the Newtonian and non-Newtonian region. For the highest molar masses and concentrations, the elasticity yield in terms of N ₁ could be determined. In addition, the extensional flow behavior of the alginates was measured using capillary breakup extensional rheometry. The results demonstrate that even samples with the same average molar mass but different molar mass distributions can be differentiated in contrast to shear flow or SAOS experiments.     

Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer

Linear and nonlinear stability analyses were performed on a fluid layer with a concentration-based internal heat source. Clear bimodal behaviour in the neutral curve (with stationary and oscillatory modes) is observed in the region of the onset of oscillatory convection, which is a previously unobserved phenomenon in radiation-induced convection. The numerical results for the linear instability analysis suggest a critical value γ _c of γ, a measure for the strength of the internal heat source, for which oscillatory convection is inhibited when γ > γ _c . Linear instability analyses on the effect of varying the ratio of the salt concentrations at the upper and lower boundaries conclude that the ratio has a significant effect on the stability boundary. A nonlinear analysis using an energy approach confirms that the linear theory describes the stability boundary most accurately when γ is such that the linear theory predicts the onset of mostly stationary convection. Nevertheless, the agreement between the linear and nonlinear stability thresholds deteriorates for larger values of the solute Rayleigh number for any value of γ.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

Explicit equations for transient heat conduction in finite solids for <Emphasis Type="Italic">Bi</Emphasis> > 2

A set of correlations is developed for transient heat conduction in finite solids (plates, cylinders and spheres) exposed to high Biot number convection boundary conditions. For Bi ≥ 3, the error of the approximate solutions is well below 1% of the initial temperature difference driving the transient.     

Two-Phase Flow in Porous Media: Predicting Its Dependence on Capillary Number and Viscosity Ratio

Motivated by the need to determine the dependencies of two-phase flow in a wide range of applications from carbon dioxide sequestration to enhanced oil recovery, we have developed a standard two-dimensional, pore-level model of immiscible drainage, incorporating viscous and capillary effects. This model has been validated through comparison with several experiments. For a range of stable viscosity ratios (M = μ _injected,nwf/μ _{defending, wf} ≥ 1), we had increased the capillary number, N _c and studied the way in which the flows deviate from fractal capillary fingering at a characteristic time and become compact for realistic capillary numbers. This crossover has enabled predictions for the dependence of the flow behavior upon capillary number and viscosity ratio. Our results for the crossover agreed with earlier theoretical predictions, including the universality of the leading power-law indicating its independence of details of the porous medium structure. In this article, we have observed a similar crossover from initial fractal viscous fingering (FVF) to compact flow, for large capillary numbers and unstable viscosity ratios M < 1. In this case, we increased the viscosity ratio from infinitesimal values, and studied the way in which the flows deviate from FVF at a characteristic time and become compact for non-zero viscosity ratios. This crossover has been studied using both our pore-level model and micro-fluidic flow-cell experiments. The same characteristic time, τ = 1/M ^0.7, satisfactorily describes both the pore-level results for a range of large capillary numbers and the micro-fluidic flow cell results. This crossover should lead to predictions similar to those mentioned above.     

Further Comments on “Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work”

This note is concerned with the assertion of Barletta and Nield (2009a) that “a fluid with a thermal expansion coefficient greater than that of a perfect gas (β > β _perfect gas) is of marginal or no interest in the framework of convection in porous media”, and that for a remark of Magyari (Transp. Porous Media, 2009) about the forced convection eigenflow solutions, the circumstance β > β _perfect gas does not represent “a sound physical basis”. Here, it is shown, however, that these assertions are in contradiction with the experimentally measured values of β for important technical fluids as e.g., air, nitrogen, carbon dioxide, and ammonia where, in the temperature range between −20 and +100°C, just the inequality β > β _perfect gas holds.