Onset of Convection in a Horizontal Porous Layer Saturated by a Power-Law Fluid

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Z. Alloui ◽  
N. Ben Khelifa ◽  
H. Beji ◽  
P. Vasseur
Keyword(s):  
Rayleigh Number ◽  
Newtonian Fluid ◽  
Power Law ◽  
Numerical Study ◽  
Flow Characteristics ◽  
Finite Amplitude ◽  
Parallel Flow ◽  
Governing Equations ◽  
Onset Of Convection ◽  
Power Law Index

This paper investigates the onset of motion, and the subsequent finite-amplitude convection, in a shallow porous cavity filled with a non-Newtonian fluid. A power-law model is used to characterize the non-Newtonian fluid behavior of the saturating fluid. Constant fluxes of heat are imposed on the horizontal walls of the layer. The governing parameters of the problem under study are the Rayleigh number R, the power-law index n, and the aspect ratio of the cavity A. An analytical solution, valid for shallow enclosures (A ≫ 1), is derived on the basis of the parallel flow approximation. In the range of the governing parameters considered in this study, a good agreement is found between the analytical predictions and the numerical results obtained by solving the full governing equations. For dilatant fluids (n > 1), it is found that the onset of motion is linearly unstable, i.e., always occurs provided that the supercritical Rayleigh number RCsup≥0. For pseudoplastic fluids (n < 1), the supercritical Rayleigh number for the onset of motion is RCsup=∞. However, it is demonstrated, on the basis of the nonlinear parallel flow theory, that the onset of motion occurs above a subcritical Rayleigh number RCsub which depends upon the power-law index n. For finite-amplitude convection, the heat and flow characteristics predicted by the analytical model are found to agree well with a numerical study of the full governing equations.

A Numerical Study of the Onset of Cellular Benard Convection in Shear Rate Dependent Non Newtonian Fluids in Porous Media: Non-Darcian Effects

2007 ◽  
Author(s):  
V. Dakshina Murty ◽  
David M. Pratt ◽  
Larry W. Byrd
Keyword(s):  
Porous Media ◽  
Rayleigh Number ◽  
Power Law ◽  
Stokes Equations ◽  
Numerical Study ◽  
Critical Rayleigh Number ◽  
Newtonian Fluids ◽  
Onset Of Convection ◽  
Bénard Convection ◽  
Benard Convection

A numerical method based on the finite element method is applied to the study of onset of Benard convection in porous media. The flow is described using the so-called Darcy Brinkman model, which has close resemblance to the Navier-Stokes equations. Itis found that for Darcy numbers less than 0.0001 the results are indistinguishable from regular Darcy flows. The non-Newtonain nature of the fluid is described by the so-called power law model, of which Newtonian fluid is a special case. Numerical results are presented for n varying from 0.4 to 1.5. The critical value of Rayleigh number for onset of convection for Newtonian fluids is found to be 40 which is close to the theoretical value of 4π2; boundary conditions on the horizontal walls have little effect in the sense that whether it is a slip or free (no shear condition) the results appear to be the same for onset of cellular motion. It is also found that the value of critical Rayleigh number increases with power law index.

A NUMERICAL STUDY OF PULSATILE POWER LAW FLUID FLOW IN A POROUS CHANNEL

2009 ◽  
Vol 09 (03) ◽  
pp. 437-447
Author(s):  
B. V. RATHISH KUMAR ◽  
SHALINI ◽  
MOHIT NIGAM ◽  
VIVEK SANGWAN ◽  
S. V. S. S. N. V. G. KRISHNA MURTHY
Keyword(s):  
Power Law ◽  
Numerical Study ◽  
Flow Characteristics ◽  
Time Discretization ◽  
Darcy Number ◽  
Flow Properties ◽  
Power Law Fluid ◽  
Discretization Schemes ◽  
Homogeneous Porous Medium ◽  
Power Law Index

The pulsatile flow characteristics of a power law fluid in a channel filled with a homogeneous porous medium are investigated by employing the Darcy–Brinkman–Forchheimer model. Finite element method in conjunction with β-family of time discretization schemes for parabolic equation have been used to numerically solve the model for analyzing the flow. Influence of various parameters, such as power law index (n), Darcy number (Da*), Forchheimer coefficient (Γ), pulsatile amplitude parameter (A), and Womerseley parameter (α), on the flow properties have been analyzed. Increasing Γ or decreasing Da* leads to decrease in velocities and shear stress for all values of n.

Free Convection in Viscoplastic Fluid due to Partial Bi-Heating From Bottom

2016 ◽  
Author(s):  
Naushad Hasin Khan ◽  
M. A. Hassan
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Rayleigh Number ◽  
Free Convection ◽  
Yield Stress ◽  
Power Law ◽  
Viscoplastic Fluid ◽  
Onset Of Convection ◽  
Square Enclosure ◽  
Power Law Index

The numerical investigation of laminar natural convection of viscoplastic fluid in a two dimensional square enclosure has been reported in this work. The enclosed fluid is subjected to partial bi-heating from the bottom wall and symmetrical cooling from the sides under steady condition. Yield stress fluid has been heated through two heaters symmetrically placed on the either side of the centre of the bottom wall of the square enclosure. The viscoplastic fluid is the one which requires a minimum critical stress called yield stress to flow otherwise behave as a solid, have been modeled with Herschel–Bulkley model. Such fluids have significant technological relevance due to its wide application ranging from cosmetics products, food processing industries, pharmaceuticals to natural occurring like flow of debris and lava. The solution of governing partial differential equations has been approached using finite volume based formulation. Non uniform set of grid has been used. The effects of yield stress, heat flux, and power law index on the flow and thermal characteristics of the free convection of Herschel-Bulkley fluids have been studied for a particular value of Prandtl number. The flow and thermal fields have been investigated for the following ranges of conditions: Rayleigh number varies between 103 and 106 whereas power law index ranges from 0 to 1. The heat transfer characteristic has been depicted with the help of isotherms and the flow field has been illustrated by streamlines. The onset of convection is substantially delayed due to presence of yield stress of the fluid. This results in enhanced critical Rayleigh number for onset of convection. With increase in the Yield number in turn yield stress, results in the weakening of heat transfer through convection and at a particular relatively higher value of Yield number the heat transfer is solely taken place by conduction mode. Due to the symmetry in both heating and boundary conditions, the obtained isotherms and streamlines of the right half are symmetrical to the left half of the square enclosure. The conductive mode of heat transfer becomes dominated by increasing yield stress and reducing Ra and vice versa. The simultaneous presence of yielded and unyielded region presents an interesting pattern in the convection zone. Furthermore, it can be seen that rise in heat flux, in turn Ra, promotes the buoyancy driven circulation of viscoplastic fluid i.e. enhances natural convective heat transfer. In addition, the effect of power law index has been investigated. Power law index has little effect on thermal distribution and flow field.

Numerical simulation of two-dimensional compressible convection

1975 ◽  
Vol 70 (4) ◽  
pp. 689-703 ◽  
Author(s):  
Eric Graham
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Fluid Velocity ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Onset Of Convection ◽  
Local Sound ◽  
Constant Viscosity ◽  
Convection Problem

A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.

Effect of a stabilizing gradient of solute on thermal convection

1968 ◽  
Vol 34 (2) ◽  
pp. 315-336 ◽  
Author(s):  
George Veronis
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Prandtl Number ◽  
Critical Rayleigh Number ◽  
Effective Diffusion ◽  
Stratified Fluid ◽  
Finite Amplitude ◽  
Oscillatory Motion ◽  
Linear Stability Theory ◽  
Onset Of Convection

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

A Numerical Study of Dean Instability in Non-Newtonian Fluids

2005 ◽  
Vol 128 (1) ◽  
pp. 34-41 ◽  
Author(s):  
H. Fellouah ◽  
C. Castelain ◽  
A. Ould El Moctar ◽  
H. Peerhossaini
Keyword(s):  
Aspect Ratio ◽  
Cross Section ◽  
Power Law ◽  
Numerical Study ◽  
Rectangular Cross Section ◽  
Newtonian Fluids ◽  
Dean Number ◽  
Bingham Number ◽  
Curved Channels ◽  
Power Law Index

We present a numerical study of Dean instability for non-Newtonian fluids in a laminar 180deg curved-channel flow of rectangular cross section. A methodology based on the Papanastasiou model (Papanastasiou, T. C., 1987, J. Rheol., 31(5), pp. 385–404) was developed to take into account the Bingham-type rheological behavior. After validation of the numerical methodology, simulations were carried out (using FLUENT CFD code) for Newtonian and non-Newtonian fluids in curved channels of square or rectangular cross section and for a large aspect and curvature ratios. A criterion based on the axial velocity gradient was defined to detect the instability threshold. This criterion was used to optimize the grid geometry. The effects of curvature and aspect ratio on the Dean instability are studied for all fluids, Newtonian and non-Newtonian. In particular, we show that the critical value of the Dean number decreases with increasing curvature ratio. The variation of the critical Dean number with aspect ratio is less regular. The results are compared to those for Newtonian fluids to emphasize the effect of the power-law index and the Bingham number. The onset of Dean instability is delayed with increasing power-law index. The same delay is observed in Bingham fluids when the Bingham number is increased.

scholarly journals Free Convection in a Triangular Cavity Filled with Hybrid-Nanofluid along with Sinusoidal Heat

2019 ◽  
Vol 4 (12) ◽  
pp. 48-52
Author(s):  
Md.Rakibul Hasan ◽  
Md. Borhan Uddin ◽  
Ahmed M. U.
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Hartmann Number ◽  
Volume Fraction ◽  
Numerical Study ◽  
Hybrid Nanofluid ◽  
Cool Temperature ◽  
Governing Equations ◽  
Uniform Cool ◽  
Generation Parameter

A numerical study on convective heat transfer of hybrid nanofluid packed in a right angled triangular cavity heated by a sinusoidal temperature maintained from lower side and subjected to a constant magnetic field have been studied in this work. The hypotenuse side of the triangular cavity has been kept in uniform cool temperature while the remaining side is insulated. The governing equations of the problem have been discretized numerically with help of finite element method. A fixed Prandtl number Pr=6.2 has been used for the numerical solution. Several values of Rayleigh number Ra=102-106 , and Hartmann number Ha=0-100 which are the non-dimensional governing parameters have been examined. The volume fraction  =0.01, 0.05, 0.1 and the heat generation parameter Q = 1 have been taken for this work. Calculate and the graph of Nusselt number corresponding to different parameters have been presented. The results show that Nusselt number has been decreasing function of nanoparticles Rayleigh number and also it is a decreasing function of Hartmann number. Obtained results has been compared with previously obtained data by other authors.

scholarly journals Finite Volume Simulation of Natural Convection for Power-Law Fluids with Temperature-Dependent Viscosity in a Square Cavity with a Localized Heat Source

2021 ◽  
Vol 39 (5) ◽  
pp. 1405-1416
Author(s):  
Hamza Daghab ◽  
Mourad Kaddiri ◽  
Said Raghay ◽  
Ismail Arroub ◽  
Mohamed Lamsaadi ◽  
...  
Keyword(s):  
Natural Convection ◽  
Heat Source ◽  
Finite Volume ◽  
Power Law ◽  
Numerical Study ◽  
Staggered Grid ◽  
Cooling Performance ◽  
Coupled Equations ◽  
Power Law Fluids ◽  
Power Law Index

In this paper, numerical study on natural convection heat transfer for confined thermo-dependent power-law fluids is conducted. The geometry of interest is a fluid-filled square enclosure where a uniform flux heating element embedded on its lower wall is cooled from the vertical walls while the remaining parts of the cavity are insulated, without slipping conditions at all the solid boundaries. The governing partial differential equations written in terms of non-dimensional velocities, pressure and temperature formulation with the corresponding boundary conditions are discretized using a finite volume method in a staggered grid system. Coupled equations of conservation are solved through iterative Semi Implicit Method for Pressure Linked Equation (SIMPLE) algorithm. The effects of pertinent parameters, which are Rayleigh number (103 ≤ Ra ≤ 106), power-law index (0.6 ≤ n ≤ 1.4), Pearson number (0 ≤ m ≤ 20) and length of the heat source (0.2 ≤ W ≤ 0.8) on the cooling performance are investigated. The results indicate that the cooling performance of the enclosure is improved with increasing Pearson and Rayleigh numbers as well as with decreasing power-law index and heat source length.

scholarly journals Numerical Solution of Biomagnetic Power-Law Fluid Flow and Heat Transfer in a Channel

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1959
Author(s):  
Adrian S. Halifi ◽  
Sharidan Shafie ◽  
Norsarahaida S. Amin
Keyword(s):  
Heat Transfer ◽  
Shear Stress ◽  
Power Law ◽  
Vortex Formation ◽  
Shear Thickening ◽  
Power Law Model ◽  
Governing Equations ◽  
Power Law Fluid ◽  
Transfer Parameters ◽  
Power Law Index

The effect of non-Newtonian biomagnetic power-law fluid in a channel undergoing external localised magnetic fields is investigated. The governing equations are derived by considering both effects of Ferrohydrodynamics (FHD) and Magnetohydrodynamics (MHD). These governing equations are difficult to solve due to the inclusion of source term from magnetic equation and the nonlinearity of the power-law model. Numerical scheme of Constrained Interpolation Profile (CIP) is developed to solve the governing equations numerically. Extensive results carried out show that this method is efficient on studying the biomagnetic and non-Newtonian power-law flow. New results show that the inclusion of power-law model affects the vortex formation, skin friction and heat transfer parameter significantly. Regardless of the power-law index, the vortex formation length increases when Magnetic number increases. The effect of this vortex however decreases with the inclusion of power-law where in the shear thinning case, the arising vortex is more pronounced than in the shear thickening case. Furthermore, increasing of power-law index from shear thinning to shear thickening, decreases the wall shear stress and heat transfer parameters. However for high Magnetic number, the wall shear stress and heat transfer parameters increase especially near the location of the magnetic source. The results can be used as a guide on assessing the potential effects of radiofrequency fields (RF) from electromagnetic fields (EMF) exposure on blood vessel.

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