Effect of Velocity and Temperature Boundary Conditions on Convective Instability in a Ferrofluid Layer

2008 ◽  
Vol 130 (10) ◽  
Author(s):  
C. E. Nanjundappa ◽  
I. S. Shivakumara
Keyword(s):  
Heat Flux ◽  
Boundary Conditions ◽  
Biot Number ◽  
Perturbation Technique ◽  
Lower Boundary ◽  
Thermal Conditions ◽  
Regular Perturbation ◽  
Fixed Temperature ◽  
Temperature Boundary ◽  
Upper Boundary

A variety of velocity and temperature boundary conditions on the onset of ferroconvection in an initially quiescent ferrofluid layer in the presence of a uniform magnetic field is investigated. The lower boundary of the ferrofluid layer is assumed to be rigid-ferromagnetic, while the upper boundary is considered to be either rigid-ferromagnetic or stress-free. The thermal conditions include a fixed heat flux at the lower boundary and a general convective, radiative exchange at the upper boundary, which encompasses fixed temperature and fixed heat flux as particular cases. The resulting eigenvalue problem is solved using the Galerkin technique and also by the regular perturbation technique when both boundaries are insulated to temperature perturbations. It is observed that an increase in the magnetic number and the nonlinearity of fluid magnetization as well as a decrease in Biot number are to destabilize the system. Further, the nonlinearity of fluid magnetization is found to have no effect on the onset of ferroconvection in the absence of the Biot number.

Effect of Boundary Conditions on the Onset of Thermomagnetic Convection in a Ferrofluid Saturated Porous Medium

2009 ◽  
Vol 131 (10) ◽  
Author(s):  
I. S. Shivakumara ◽  
C. E. Nanjundappa ◽  
M. Ravisha
Keyword(s):  
Heat Flux ◽  
Porous Medium ◽  
Boundary Conditions ◽  
Porous Layer ◽  
Saturated Porous Medium ◽  
Perturbation Technique ◽  
Lower Boundary ◽  
Fluid Viscosity ◽  
Thermomagnetic Convection ◽  
Upper Boundary

The onset of thermomagnetic convection in a ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated for a variety of velocity and temperature boundary conditions. The Brinkman–Lapwood extended Darcy equation, with fluid viscosity different from effective viscosity, is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-ferromagnetic, while the upper boundary is considered to be either rigid-ferromagnetic or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective-radiative exchange at the upper boundary, which encompasses fixed temperature and heat flux as particular cases. The resulting eigenvalue problem is solved using the Galerkin technique and also by using regular perturbation technique when both boundaries are insulated to temperature perturbations. It is found that the increase in the Biot number and the viscosity ratio, and the decrease in the magnetic as well as in the Darcy number is to delay the onset of ferroconvection. Besides, the nonlinearity of fluid magnetization has no effect on the onset of convection in the case of fixed heat flux boundary conditions.

The effects of thermal conditions on the cell sizes of two-dimensional convection

1994 ◽  
Vol 281 ◽  
pp. 33-50 ◽  
Author(s):  
Masaki Ishiwatari ◽  
Shin-Ichi Takehiro ◽  
Yoshi-Yuki Hayashi
Keyword(s):  
Heat Flux ◽  
Lower Boundary ◽  
Thermal Conditions ◽  
Heat Fluxes ◽  
Internal Cooling ◽  
Two Dimensional ◽  
Numerical Integrations ◽  
The Difference ◽  
Convective Cells ◽  
Upper Boundary

The effects of thermal conditions on the patterns of two-dimensional Boussinesq convection are studied by numerical integration. The adopted thermal conditions are (i) the heat fluxes through both upper and lower boundaries are fixed, (ii) the same as (i) but with internal cooling, (iii) the temperature on the lower boundary and the heat flux through the upper boundary are fixed, (iv) the same as (iii) but with internal cooling, and (v) the temperatures on both upper and lower boundaries are fixed. The numerical integrations are performed with Ra = 104 and Pr = 1 over the region whose horizontal and vertical lengths are 8 and 1, respectively.The results confirm that convective cells with the larger horizontal sizes tend to form under the conditions where the temperature is not fixed on any boundaries. Regardless of the existence of internal cooling, one pair of cells spreading all over the region forms in the equilibrium states. On the other hand, three pairs of cells form and remain when the temperature on at least one boundary is fixed. The formation of single pairs of cells appearing under the fixed heat flux conditions shows different features with and without internal cooling. The difference emerges as the appearance of a phase change, whose existence can be suggested by the weak nonlinear equation derived by Chapman & Proctor (1980).

Effects of thermal boundary conditions on rotating Rayleigh-Bénard convection with implications on geophysical and astrophysical systems

2021 ◽  
Author(s):  
Janet Peifer ◽  
Onno Bokhove ◽  
Steve Tobias
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Boundary Conditions ◽  
Biot Number ◽  
Thermal Flux ◽  
Thermal Boundary Condition ◽  
Actual Condition ◽  
Fixed Temperature ◽  
Thermal Boundary Conditions ◽  
Core Dynamics

<p>Rayleigh-Bénard convection (RBC) is a fluid phenomenon that has been studied for over a century because of its utility in simplifying very complex physical systems. Many geophysical and astrophysical systems, including planetary core dynamics and components of weather prediction, are modeled by including rotational forcing in classic RBC. Our understanding of these systems is confined by experimental and numerical limits, as well as theoretical assumptions. </p><p>The role of thermal boundary condition choice on experimental studies of geophysical and astrophysical systems has been often been overlooked, which could account for some lack of agreement between experimental and numerical models as well as the actual flows. The typical thermal boundary conditions prescribed at the top and the bottom of a convection system are fixed temperature conditions, despite few real geophysical systems being bounded with a fixed temperature. A constant heat flux is generally more applicable for real large-scale geophysical systems. However, when this condition is applied in numerical systems, the lack of fixed temperature can cause a temperature drift. In this study, we seek to minimize temperature drifting by applying a fixed temperature condition on one boundary and a fixed thermal flux on the other.</p><p>Experimental boundary conditions are also often assumed to be a fixed temperature. However, the actual condition is determined by the ratio of the height and thermal conductivity of the boundary material to that of the contained fluid, known as the Biot number. The relationship between the Biot number and thermal boundary condition behavior is defined by the Robin, or 'thin-lid', boundary condition such that low Biot number boundaries are essentially fixed thermal flux and high Biot number boundaries are essentially fixed temperature. </p><p>This study seeks to strengthen the link between numerical and experimental models and geophysical flows by investigating the effects of thermal boundary conditions and their relationship to real-world processes. Both fixed temperature and fixed flux boundary conditions are considered. In addition, the Robin boundary condition is studied at a range of Biot numbers spanning from fixed temperature to fixed flux, allowing intermediate conditions to be investigated. Each system is studied at increasingly rapid rotation rates, corresponding to decreasing Ekman numbers as low as Ek=10<sup>-5</sup> Heat transport is analyzed using the Nusselt number, Nu, and the form of the solution is described by the number of convection rolls and time-dependency. Further investigations will analyze Nu and fluid movement within a system with heterogeneous heat flux condition on the  sidewall boundary conditions, which is useful in the study of planetary core dynamics. The results of this study have implications for improvements in modeling geophysical systems both experimentally and numerically. </p>

scholarly journals Analytical Solution to the MHD Flow of Micropolar Fluid Over a Linear Stretching Sheet

2015 ◽  
Vol 20 (2) ◽  
pp. 397-406 ◽  
Author(s):  
P.G. Siddheshwar ◽  
U.S. Mahabaleshwar
Keyword(s):  
Numerical Methods ◽  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Micropolar Fluid ◽  
Stretching Sheet ◽  
Perturbation Technique ◽  
Mhd Flow ◽  
Regular Perturbation ◽  
Linear Differential

Abstract The flow due to a linear stretching sheet in a fluid with suspended particles, modeled as a micropolar fluid, is considered. All reported works on the problem use numerical methods of solution or a regular perturbation technique. An analytical solution is presented in the paper for the coupled non-linear differential equations with inhomogeneous boundary conditions.

Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile

1964 ◽  
Vol 18 (4) ◽  
pp. 513-528 ◽  
Author(s):  
E. M. Sparrow ◽  
R. J. Goldstein ◽  
V. K. Jonsson
Keyword(s):  
Heat Flux ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Temperature Profile ◽  
Fluid Layer ◽  
Bounding Surface ◽  
Fixed Temperature ◽  
Linear Temperature ◽  
Lower Bounding ◽  
Horizontal Fluid Layer

An investigation is carried out to determine the conditions marking the onset of convective motion in a horizontal fluid layer in which a negative temperature gradient occurs somewhere within the layer. In such cases, fluid of greater density is situated above fluid of lesser density. Consideration is given to a variety of thermal and hydrodynamic boundary conditions at the surfaces which bound the fluid layer. The thermal conditions include fixed temperature and fixed heat flux at the lower bounding surface, and a general convective-radiative exchange at the upper surface which includes fixed temperature and fixed heat flux as special cases. The hydrodynamic boundary conditions include both rigid and free upper surfaces with a rigid lower bounding surface. It is found that the Rayleigh number marking the onset of motion is greatest for the boundary condition of fixed temperature and decreases monotonically as the condition of fixed heat flux is approached. Non-linear temperature distributions in the fluid layer may result from internal heat generation. With increasing departures from the linear temperature profile, it is found that the fluid layer becomes more prone to instability, that is, the critical Rayleigh number decreases.

scholarly journals The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection

2015 ◽  
Vol 784 ◽  
Author(s):  
Michael A. Calkins ◽  
Kevin Hale ◽  
Keith Julien ◽  
David Nieves ◽  
Derek Driggs ◽  
...  
Keyword(s):  
Heat Flux ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Layer Structure ◽  
Plane Layer ◽  
Order System ◽  
Leading Order ◽  
Fixed Temperature ◽  
Rotating Convection ◽  
Thermal Boundary Conditions

The influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is investigated for the case of stress-free mechanical boundary conditions. It is shown that whereas the leading-order system satisfies fixed temperature boundary conditions implicitly, a double boundary layer structure is necessary to satisfy the fixed heat flux thermal boundary conditions. The boundary layers consist of a classical Ekman layer adjacent to the solid boundaries that adjust viscous stresses to zero, and a layer in thermal wind balance just outside the Ekman layers that adjusts the normal derivative of the temperature fluctuation to zero. The influence of these boundary layers on the interior geostrophically balanced convection is shown to be asymptotically weak, however. Upon defining a simple rescaling of the thermal variables, the leading-order reduced system of governing equations is therefore equivalent for both boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading-order influence on the interior convection, thus providing insight into geophysical and astrophysical flows where stress-free mechanical boundary conditions are often assumed.

Alternative Solutions for Longitudinal Fins of Rectangular, Trapezoidal, and Concave Parabolic Profiles

2006 ◽  
Author(s):  
A. Aziz
Keyword(s):  
Thermal Analysis ◽  
Heat Flux ◽  
Boundary Conditions ◽  
Fixed Temperature ◽  
Thermal Boundary Conditions ◽  
Convective Fin ◽  
The Relationship ◽  
Alternative Solutions ◽  
Dimensionless Heat Flux ◽  
Dimensionless Heat

The traditional thermal analysis of fins is based on the assumption of specified thermal boundary conditions at the base and tip of the fin. For situations when the fin base is in contact with a fluid experiencing condensation and the fin is required to remove the energy released by the fluid, the base is subjected to two boundary conditions: a fixed temperature and a fixed heat flux. This paper develops solutions for the temperature distribution in the fins under these conditions. Solutions are provided for rectangular, trapezoidal, and concave parabolic (finite tip thickness). Results illustrating the relationship between the dimensionless heat flux, the fin parameter, and dimensionless tip temperature are provided for all three geometries. The case of convective fin tip is also considered and lead to a relationship between the dimensionless heat flux, the fin parameter, and the Biot number at the tip. The results presented here provide tools that not only complement the traditional analyses but are believed to have more direct relevance for fin designers.

scholarly journals Fluid Flow Model with Mean Microtubule Pressure through Porous Media

2020 ◽  
pp. 38-44
Author(s):  
A. T. Ngiangia ◽  
P. O. Nwabuzor
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Flow Model ◽  
Perturbation Technique ◽  
Regular Perturbation ◽  
Fractional Model

We discussed in this paper a fractional model arising in flow of three different incompatible fluids through a porous medium with mean microtubule pressure. The method adopted for obtaining the solution is the regular perturbation technique for the analytical solution and for the transformation of the boundary conditions. The results are in decent agreement with the findings of researched work reviewed in this paper.

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