Keller-Box Scheme to Mixed Convection Flow Over a Solid Sphere with the Effect of MHD

2021 ◽  
Vol 6 (1) ◽  
pp. 97
Author(s):  
Mohammad Ghani ◽  
Wayan Rumite
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Mixed Convection ◽  
Numerical Solutions ◽  
Solid Sphere ◽  
Convection Flow ◽  
Boundary Layer Equations ◽  
Buoyancy Forces ◽  
Dimensional Boundary ◽  
External Forces

Mixed convection is the combination of a free convection caused by the buoyancy forces due to the different density and a forced convection due to external forces that increase the heat exchange rate. This means that, in free convection, the effect of external forces is significant besides buoyancy forces. In this study the fluid type with viscoelastic effect is non-Newtonian. The viscoelastic fluids that pass over a surface of a sphere form a thin layer, which due to their dominant viscosity is called by the border layer. The obtained limiting layer is analyzed with the thickness of the boundary layer-  near the lower stagnating point, then obtained dimensional boundary layer equations, continuity, momentum, and energy equations. These dimensional boundary layer equations are then transformed into non-dimensional boundary layer equations by using non-dimensional variables. Further, the non-dimensional boundary layer equations are transformed into ordinary differential equations by using stream function, so that obtained the non-similar boundary layer equations. These non-similar boundary layer equations are solved numerically by using finite difference method of Keller-Box. The discretization results are non-linear and it should be linearized using newton linearization technique. The numerical solutions are analyzed the effect of Prandtl number, viscoelastic, mixed convection, and MHD parameters towards velocity profile, temperature profile, and wall temperature.

scholarly journals Pressure Work Effect on Natural Convection Flow from a Vertical Circular Cone with Suction and Non-Uniform Surface Temperature

1970 ◽  
Vol 36 ◽  
pp. 6-11 ◽  
Author(s):  
MA Alim ◽  
Md M Alam ◽  
Md MK Chowdhury
Keyword(s):  
Boundary Layer ◽  
Temperature Gradient ◽  
Free Convection ◽  
Surface Temperature ◽  
Numerical Solutions ◽  
Circular Cone ◽  
Convection Flow ◽  
Transfer Velocity ◽  
Pressure Work ◽  
Boundary Layer Equations

Laminar free convection from a vertical circular cone maintained at non-uniform surface temperature with effects of suction and pressure work are considered. Non-similarity solutions for boundary layer equations are found to exist when the surface temperature follows the power law variations with the distance measured from the leading edge. The numerical solutions of the transformed non-similar boundary layer equations are obtained by using a finite difference method with Keller Box scheme. Solutions obtained in terms of skin-friction, local rate of heat transfer, velocity and temperature profiles for the values of Prandtl number, pressure work parameter and temperature gradient are displayed in both graphical and tabular forms.Keywords: Free convection, Viscous dissipation, Boussinesq approximation, Temperature gradient, Pressure work.Journal of Mechanical Engineering Vol.36 Dec. 2006 pp.6-11DOI = 10.3329/jme.v36i0.805

A New Approach for the Derivation of the Similarity Equation of the 2-D Unsteady Boundary Layer Equations

2012 ◽  
Author(s):  
Md. Abdus Sattar
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Length Scale ◽  
Local Similarity ◽  
Unsteady Boundary Layer ◽  
Similarity Equation ◽  
Boundary Layer Equations ◽  
Dimensional Boundary ◽  
The One

A local similarity equation for the hydrodynamic 2-D unsteady boundary layer equations has been derived based on a time dependent length scale initially introduced by the author in solving several unsteady one-dimensional boundary layer problems. Similarity conditions for the potential flow velocity distribution are also derived. This derivation shows that local similarity solutions exist only when the potential velocity is inversely proportional to a power of the length scale mentioned above and is directly proportional to a power of the length measured along the boundary. For a particular case of a flat plate the derived similarity equation exactly corresponds to the one obtained by Ma and Hui[1]. Numerical solutions to the above similarity equation are also obtained and displayed graphically.

Periodic momentum and thermal boundary layer mixed convection flow around the surface of a sphere in the presence of viscous dissipation

2017 ◽  
Vol 95 (10) ◽  
pp. 976-986 ◽  
Author(s):  
Muhammad Ashraf ◽  
Almas Fatima ◽  
R.S.R. Gorla
Keyword(s):  
Boundary Layer ◽  
Mixed Convection ◽  
Viscous Dissipation ◽  
Thermal Boundary Layer ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Convection Flow ◽  
Mixed Convection Flow ◽  
Implicit Finite Difference ◽  
Convection Parameter

Numerical solutions for the periodic laminar boundary layer mixed convection flow around the surface of a heated sphere in the presence of viscous dissipation have been obtained by solving the governing equations using an implicit finite difference numerical technique. The fluid under consideration is assumed to be viscous and incompressible. Periodic momentum and thermal boundary layer profiles for different positions of x around the surface of the sphere are evaluated. The features of the obtained results for different values of mixed convection parameter λ, Prandtl number Pr, viscous dissipation parameter N, and frequency parameter ω are shown graphically. The obtained results confirm significant effect of all these mentioned parameters on periodic momentum and thermal boundary layer mixed convection flow around different positions of the sphere.

scholarly journals The Effect of Plate Oscillations on Horizontal Free Convection Flow

1977 ◽  
Vol 30 (3) ◽  
pp. 335 ◽  
Author(s):  
RL Verma ◽  
Punyatma Singh
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Free Convection ◽  
Skin Friction ◽  
Low Frequency ◽  
Convection Flow ◽  
Phase Lag ◽  
Mean Flow ◽  
Free Convection Flow ◽  
Boundary Layer Equations

The free convection flow along a semi-infinite horizontal plate oscillating in its own plane is analysed The basic flow is purely buoyancy induced, while the oscillations in the plate cause a time-dependent boundary layer flow and heat transfer. The boundary layer equations are linearized and the first two approximations are considered. Two separate solutions valid for high and low frequency ranges are obtained by a series expansion in terms of frequency parameters. The skin friction and the rate of heat transfer are studied for both frequency ranges. For very high frequencies, the oscillatory flow pattern is of a 'shear-wave' type, unaffected by the mean flow. It is found that the phase of the skin friction at the plate lags that of the plate oscillations by in and the rate of heat transfer has a phase lag of 1/2n.

Developing Laminar Free Convection in an Open Ended Vertical Annulus with a Rotating Inner Cylinder

1981 ◽  
Vol 103 (3) ◽  
pp. 552-558 ◽  
Author(s):  
M. A. I. El-Shaarawi ◽  
A. Sarhan
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Free Convection ◽  
Outer Cylinder ◽  
Convection Flow ◽  
Free Convection Flow ◽  
Boundary Layer Equations ◽  
Thermal Boundary Conditions ◽  
Displacement Thickness ◽  
Vertical Annulus

A finite-difference scheme is developed for solving the boundary layer equations governing the laminar free convection flow in open ended vertical concentric annuli with rotating inner walls. Numerical results are presented for a fluid of Pr = 0.7 in an annulus of radius ratio 0.5 under the thermal boundary conditions of one wall being isothermal and the opposite wall adiabatic. Comparing the present results for the development of δθ (tangential boundary layer displacement thickness) with those corresponding results of forced flows shows that heating the inner cylinder has always stabilizing effects while heating the outer cylinder has either destabilizing or stabilizing effects.

Laminar Boundary∼Layer Free Convection along an Inclined, Isothermal Surface

1972 ◽  
Vol 1 (4) ◽  
pp. 189-196 ◽  
Author(s):  
J.B. Lee ◽  
G.S.H. Lock
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Free Convection ◽  
Laminar Boundary Layer ◽  
Theoretical Consideration ◽  
Plane Surface ◽  
Convection Flow ◽  
Inclined Plane ◽  
Free Convection Flow ◽  
Boundary Layer Equations

This paper gives theoretical consideration to the problem of laminar, boundary-layer, free convection flow along a long, inclined, plane surface heated isothermally. Development of the appropriate boundary-layer equations is followed by their numerical solution for air. The effects of inclination and position on heat transfer and the temperature, pressure and velocity profiles are presented graphically for RaL ≤ 106.

Three-Dimensional Boundary Layers on Cones at Small Angles of Attack

1968 ◽  
Vol 35 (4) ◽  
pp. 634-640 ◽  
Author(s):  
O. Pinkus ◽  
S. B. Cousin
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Equivalent Radius ◽  
Wind Tunnel Model ◽  
Boundary Layer Equations ◽  
Tunnel Model ◽  
Dimensional Boundary ◽  
Heat Transfer Effects ◽  
Incipient Separation

Based on an expression given by Cooke, an equation is derived for a three-dimensional “equivalent radius” for cones at a small angle of attack. This function when used in any of the available axisymmetric boundary-layer equations yields corresponding solutions for yawed cones. Expressions for the streamlines along which the above equations are to be integrated are also derived. The method yields a line of possible incipient separation or wake formation in addition to the boundary-layer properties in both the longitudinal and circumferential direction. Numerical solutions including heat transfer effects are presented for a wind tunnel model and compared with experimental results.

scholarly journals Separation time analysis of transient magnetohydrodynamic mixed convection flow of nanofluid at lower stagnation point past a sphere

2017 ◽  
Vol 13 (3) ◽  
Author(s):  
Mohamad Alif Ismail ◽  
Nurul Farahain Mohammad ◽  
Sharidan Shafie
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Mixed Convection ◽  
Stagnation Point ◽  
Magnetic Particles ◽  
Volume Fraction ◽  
Convection Flow ◽  
Mixed Convection Flow ◽  
Boundary Layer Equations ◽  
Keller Box Method

In this paper, the unsteady magnetohydrodynamics (MHD) mixed convection flow of nanofluid at lower stagnation point past a sphere is studied. Nanoparticles Cu and TiO2 with water as a base fluid are considered. The separation times of the flow as the boundary layer start to separate at the surface of the sphere are given attention. The governing boundary layer equations in the form of partial differential equations are transformed into nonlinear coupled ordinary differential equations and solved numerically using an implicit finite-difference scheme known as Keller-box method. Results of the separation times of boundary layer flow for viscous and nanofluid influenced by magnetic parameter and volume fraction are shown in tabular form and analysed. This study concluded that the separation times can be delayed by added more magnetic particles and small amount the volume fraction.

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