Numerical Partial Differential Equations: Finite Difference Methods

Thomas, J. W.

doi:10.1007/978-1-4899-7278-1

Cited by 924 publications

(740 citation statements)

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“…Equations (12), (18) and (22), together with the boundary conditions (15)(16)(17), (19)(20), (23) and (24), are solved numerically using an implicit, iterative tri-diagonal finite difference method similar to that discussed in Refs. [47,48]. Although not shown here, the full solution of the momentum equations in three dimensions reveals that the component G is rather negligible (see Ref.…”

Section: Self-similar Solutionsmentioning

confidence: 81%

“…This was such that when the difference between the two consecutive iterations became less than 10 -7 , the solution was assumed to have converged and hence the iterative process was terminated. On the basis of the implemented numerical scheme, the numerical error is of O Dg ð Þ 2 [46,47]. The solutions developed in Sects.…”

Section: Grid Independency and Validationmentioning

confidence: 99%

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Mixed convection and thermodynamic irreversibilities in MHD nanofluid stagnation-point flows over a cylinder embedded in porous media

Alizadeh

Karimi

Arjmandzadeh

et al. 2018

J Therm Anal Calorim

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The impingement of CuO-water nanofluid flows upon a cylinder subject to a uniform magnetic field with constant surface temperature and embedded in porous media is investigated for the first time in literature. The surface of the cylinder can feature uniform or non-uniform mass transpiration and is hotter than the incoming nanofluid flow. The gravitational effects are taken into account and the three-dimensional governing equations of mixed convection in curved porous media, under magnetohydrodynamic effects, are reduced to those solvable by a finite difference scheme. Through varying a mixed convection parameter, the situations dominated by forced, mixed and free convection are examined systematically. The numerical solutions of these equations reveal the flow velocity and temperature fields as well as the Nusselt number and induced shear stress. These are then used to calculate the rate of entropy generation within the system by viscous and heat transfer irreversibilities. The results show that Nusselt number increases with increasing the concentration of nanoparticles, while it slightly deceases through intensifying the magnetic parameter. Non-uniform transpiration is shown to strongly affect the average rate of heat transfer. Importantly, it is demonstrated that the specific mode of heat convection can majorly influence the intensity of entropy generation and that the irreversibilities are much larger under natural convection compared to those in mixed and forced convection. Calculation of Bejan number shows that this is due to more pronounced relative contribution of viscous irreversibilities when free convection effects dominate the mixed convection process.

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Section: Self-similar Solutionsmentioning

confidence: 81%

Section: Grid Independency and Validationmentioning

confidence: 99%

Mixed convection and thermodynamic irreversibilities in MHD nanofluid stagnation-point flows over a cylinder embedded in porous media

Alizadeh

Karimi

Arjmandzadeh

et al. 2018

J Therm Anal Calorim

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“…The equation for can be derived in a similar manner. Using , a finite-difference Crank-Nicolson scheme (Thomas, 1995) for Eqn. (6) can be obtained as:…”

Section: Bpm Equationsmentioning

confidence: 99%

Effect of submicron deformations on the transmission of all-solid photonic bandgap fibre

et al. 2016

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A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk  Abstract Waveguide losses in all-solid photonic bandgap fibre are studied numerically using both vectorial mode solver and vectorial beam propagation methods. Confinement loss, including material losses, is comprehensively evaluated for defect modes of hexagonal lattice photonic bandgap fibre. The excitation of the index-guiding modes at the bandgap edges leads to a narrowing of the transmission bands. Submicron deformations of the transverse structure of the fibre lead to a significant reduction in the width of the transmission bands, yet the minimum value of the losses within these bandgaps remains practically unchanged. Longitudinal variation of the fibre profile increases the effective losses by up to tens of dB/m.

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“…The Neumann-type boundary condition is imposed on the boundary of the current block-centered grid system because the outmost nodes of the grid system are not located on the boundary of the considered domain but at the central points of the boundary blocks (Wang and Anderson, 1982;Thomas, 1995). The impervious boundary is simulated using the added imaginary blocks outside the considered domain.…”

Section: Boundary Conditionsmentioning

confidence: 99%

Application of block-centered finite difference formulation for non-linear finite strain consolidation

et al. 2014

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A one-dimensional block-centered finite-difference model has been developed to estimate the rate of non-linear finite strain consolidation. The governing equations including the hydrodynamic and constitutive equations are presented. The hypothesis of the uniqueness of the End-of-Primary (EOP) void ratio -effective stress relationship is adopted to calculate the primary consolidation settlement. The explicit block-centered finite difference formulations and boundary conditions are presented and discussed. The developed model was compared with a point-centered finite-difference program, ILLICON to show the efficiency of the blockcentered model. The block-center model provides an efficient tool to deal with interface boundaries and has advantageous ability to take into consideration the time-dependent loading, layered soil systems, and variable soil properties.

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Numerical Partial Differential Equations: Finite Difference Methods

Cited by 924 publications

References 0 publications

Mixed convection and thermodynamic irreversibilities in MHD nanofluid stagnation-point flows over a cylinder embedded in porous media

Mixed convection and thermodynamic irreversibilities in MHD nanofluid stagnation-point flows over a cylinder embedded in porous media

Effect of submicron deformations on the transmission of all-solid photonic bandgap fibre

Application of block-centered finite difference formulation for non-linear finite strain consolidation

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