Closed Form Solutions For Mixed Convection With Magnetohydrodynamic Effect in a Vertical Porous Annulus Surrounding an Electric Cable

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
A. Barletta ◽  
E. Magyari ◽  
S. Lazzari ◽  
I. Pop
Keyword(s):  
Mixed Convection ◽  
Closed Form ◽  
Heat Generation ◽  
Internal Heat ◽  
Boussinesq Approximation ◽  
Closed Form Solutions ◽  
Electric Cable ◽  
Special Cases ◽  
Magnetohydrodynamic Effect ◽  
The Magnetic Field

Mixed convection Darcy flow in a vertical porous annulus around a straight electric cable is investigated. It is assumed that the flow is fully developed and parallel. Moreover, the Boussinesq approximation is used. The magnetic field with a steady electric current in the cable is radially varying according to the Biot–Savart law. Two flow regimes are investigated. The first is mixed convection with negligible effects of internal heat generation due to Joule heating and viscous dissipation. The second is forced convection with important effects of heat generation. In these two special cases, closed form expressions of the velocity profile and of the temperature profile, as well as of the flow rate and the Nusselt number, are obtained. The main features of these solutions are discussed.

Mixed Convection From a Convectively Heated Vertical Plate to a Fluid With Internal Heat Generation

2011 ◽  
Vol 133 (12) ◽  
Author(s):  
O. D. Makinde ◽  
A. Aziz
Keyword(s):  
Mixed Convection ◽  
Heat Generation ◽  
Vertical Plate ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Local Similarity ◽  
Flowing Fluid ◽  
Cold Fluid ◽  
Special Cases ◽  
The Right

A numerical approach has been adopted to study steady mixed convection from the right face of a vertical plate of finite thickness. Cold fluid flowing over the right face of the plate contains a heat generation that decays exponentially with a dimensionless distance from the wall. The left face of the plate is in contact with a hot flowing fluid. The heating process on that side is characterized by a convective boundary condition that takes into account the conduction resistance of the plate as well as a possible contact resistance between the hot fluid and the left face of the plate. Using a pseudo similarity approach, the continuity, momentum, and energy equations for mixed convective flow over the right face of the plate are transformed into a set of coupled ordinary differential equations. It is found that for a true similarity solution, the convective heat transfer coefficient associated with the hot fluid must be proportional to x−1/2, and both the thermal expansion coefficient and the internal heat generation rate for the cold fluid must be proportional to x−1, where x is the upward distance along the plate. The equations give local similarity solutions. The effects of local Grashof number (defined to represent a mixed convection parameter), Prandtl number, Biot number, and the internal heat generation parameter on the velocity and temperature profiles are illustrated and interpreted in physical terms. The present results agree closely with the existing results for the special cases of the problem. This close agreement lends support to the validity of the present analysis and the accuracy of the numerical computations. The paper also contains a table in which the data for the local skin friction and local Nusselt number are provided for various combination values of the parameters that govern the momentum and energy transport in the mixed boundary layer.

scholarly journals Chaotic Convection in a Ferrofluid with Internal Heat Generation

CFD letters ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 62-74
Author(s):  
Nor Halawati Senin ◽  
Nor Fadzillah Mohd Mokhtar ◽  
Mohamad Hasan Abdul Sathar
Keyword(s):  
Heat Generation ◽  
Mathematical Formulation ◽  
Internal Heat ◽  
Boussinesq Approximation ◽  
Internal Heat Generation ◽  
Lorenz Attractor ◽  
Nonlinear Stability Analysis ◽  
Layer System ◽  
Chaotic Convection ◽  
The Impact

The nonlinear stability analysis of a ferrofluid layer system is formulated mathematically. This system considered the upper and lower free isothermal boundary with the system heated from below. A mathematical formulation is produced to study the behaviour of the chaotic convection in a ferrofluid layer system using Galerkin truncated expansion. The Boussinesq approximation is opted with the existence of internal heating and the magnetic number. It is found that the transition to chaos in this present study is identical to the Lorenz attractor and thus validate the method and analysis of this study. The impact of elevating the internal heat generation is found to hasten the instability of the system and as for the magnetic number, at M1 = 2.5 the homoclinic bifurcation occurs and thus accelerates the convection process.

Closed-form formula of magnetic gradient tensor for a homogeneous polyhedral magnetic target: A tetrahedral grid example

Geophysics ◽  
2017 ◽  
Vol 82 (6) ◽  
pp. WB21-WB28 ◽  
Author(s):  
Zhengyong Ren ◽  
Chaojian Chen ◽  
Jingtian Tang ◽  
Huang Chen ◽  
Shuanggui Hu ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Closed Form ◽  
Magnetization Vector ◽  
Gradient Tensor ◽  
Magnetic Gradient ◽  
Closed Form Solutions ◽  
Tetrahedral Grid ◽  
Pipeline Model ◽  
The Magnetic Field ◽  
Closed Form Formula

A closed-form formula is developed for the full magnetic gradient tensor of a polyhedral body with a homogeneous magnetization vector. It is based on the direct derivative technique on the closed form of the magnetic field. These analytical expressions are implemented into an easy-to-use C++ package which simultaneously calculates the magnetic potential, the magnetic field, and the full magnetic gradient tensor for magnetic targets. Modern unstructured tetrahedral grids are adopted to represent the polyhedral body so that our code can deal with arbitrarily complicated magnetic targets. A prismatic body is tested to verify the accuracies of our closed-form formula. Excellent agreements are obtained between our closed-form solutions and solutions of a prismatic magnetic body with differences up to machine precision. A pipeline model is used to demonstrate its capability to deal with complicated magnetic targets. This C++ code is freely available to the magnetic exploration community.

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