Closed-form solutions for unsaturated flow under variable flux boundary conditions

1996 ◽  
Vol 19 (4) ◽  
pp. 207-213 ◽  
Author(s):  
P. Broadbridge ◽  
M.P. Edwards ◽  
J.E. Kearton
Keyword(s):  
Boundary Conditions ◽  
Closed Form ◽  
Unsaturated Flow ◽  
Closed Form Solutions ◽  
Flux Boundary Conditions ◽  
Variable Flux

scholarly journals Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 226
Author(s):  
Efthimios Providas ◽  
Stefanos Zaoutsos ◽  
Ioannis Faraslis
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Closed Form ◽  
Linear Operators ◽  
General Boundary ◽  
Order Differential Operator ◽  
General Boundary Conditions ◽  
Symbolic Form ◽  
Closed Form Solutions

This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces.

Effects of thermal radiation and magnetohydrodynamics on Ree-Eyring fluid flows through porous medium with slip boundary conditions

2019 ◽  
Vol 15 (2) ◽  
pp. 492-507 ◽  
Author(s):  
K. Ramesh ◽  
Sartaj Ahmad Eytoo
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Porous Medium ◽  
Boundary Conditions ◽  
Closed Form ◽  
Parallel Plates ◽  
Slip Boundary Conditions ◽  
Content Type ◽  
Closed Form Solutions ◽  
Slip Boundary

Purpose The purpose of this paper is to investigate the three fundamental flows (namely, both the plates moving in opposite directions, the lower plate is moving and other is at rest, and both the plates moving in the direction of flow) of the Ree-Eyring fluid between infinitely parallel plates with the effects of magnetic field, porous medium, heat transfer, radiation and slip boundary conditions. Moreover, the intention of the study is to examine the effect of different physical parameters on the fluid flow. Design/methodology/approach The mathematical modeling is performed on the basis of law of conservation of mass, momentum and energy equation. The modeling of the present problem is considered in Cartesian coordinate system. The governing equations are non-dimensionalized using appropriate dimensionless quantities in all the mentioned cases. The closed-form solutions are presented for the velocity and temperature profiles. Findings The graphical results are presented for the velocity and temperature distributions with the pertinent parameters of interest. It is observed from the present results that the velocity is a decreasing function of Hartmann number. Temperature increases with the increase of Ree-Eyring fluid parameter, radiation parameter and temperature slip parameter. Originality/value First time in the literature, the authors obtained closed-form solutions for the fundamental flows of Ree-Erying fluid between infinitely parallel plates with the effects of magnetic field, porous medium, heat transfer, radiation and slip boundary conditions. Moreover, the results of this paper are new and original.

scholarly journals Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of Compressible Flow Similarity Solutions

2015 ◽  
Vol 1 (2) ◽  
Author(s):  
Scott D. Ramsey ◽  
Philip R. Ivancic ◽  
Jennifer F. Lilieholm
Keyword(s):  
Shock Wave ◽  
Wave Propagation ◽  
Boundary Conditions ◽  
Closed Form ◽  
Compressible Flow ◽  
Similarity Solutions ◽  
Test Problems ◽  
Closed Form Solutions ◽  
Self Similar ◽  
Temporal Extent

This work is concerned with the use of similarity solutions of the compressible flow equations as benchmarks or verification test problems for finite-volume compressible flow simulation software. In practice, this effort can be complicated by the infinite spatial/temporal extent of many candidate solutions or “test problems.” Methods can be devised with the intention of ameliorating this inconsistency with the finite nature of computational simulation; the exact strategy will depend on the code and problem archetypes under investigation. For example, self-similar shock wave propagation can be represented in Lagrangian compressible flow simulations as rigid boundary-driven flow, even if no such “piston” is present in the counterpart mathematical similarity solution. The purpose of this work is to investigate in detail the methodology of representing self-similar shock wave propagation as a piston-driven flow in the context of various test problems featuring simple closed-form solutions of infinite spatial/temporal extent. The closed-form solutions allow for the derivation of similarly closed-form piston boundary conditions (BCs) for use in Lagrangian compressible flow solvers. The consequences of utilizing these BCs (as opposed to directly initializing the self-similar solution in a computational spatial grid) are investigated in terms of common code verification analysis metrics (e.g., shock strength/position errors and global convergence rates).

An Extended Separation-of-Variable Method for Free Vibration of Rectangular Mindlin Plates

2021 ◽  
pp. 2150154
Author(s):  
Gen Li ◽  
Yufeng Xing ◽  
Zekun Wang
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Closed Form ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Arbitrary Boundary ◽  
Unknown Variable ◽  
Closed Form Solutions ◽  
Solution Methods ◽  
Separation Of Variable

For rectangular thick plates with non-Levy boundary conditions, it is important to explore analytical free vibration solutions because the classical inverse and semi-inverse exact solution methods are not applicable to this category of problems. This work is to develop an extended separation-of-variable (SOV) method to find closed-form analytical solutions for the free vibration of rectangular Mindlin plates with arbitrary homogeneous boundary conditions. In the extended SOV method, characteristic differential equations and boundary conditions in two directions are obtained by employing the Rayleigh principle and the assumption that the mode functions are in the SOV form, and two transcendental eigenvalue equations are achieved through boundary conditions. But these two eigenvalue equations cannot be solved simultaneously since there are two equations and only the natural frequency is the unknown variable. Considering this, the second assumption in this method is that the natural frequencies corresponding to two-direction mode functions are independent of each other in the mathematical sense, thus there are two unknowns in two transcendental eigenvalue equations, and the closed-form solutions for plates with arbitrary boundary conditions can be obtained non-iteratively. From the physical sense, the natural frequencies pertaining to different direction mode functions should be the same, and this conclusion is validated analytically and numerically. The present natural frequencies and mode shapes agree well with those obtained by other analytical and numerical methods. Especially, for the plates with at least two opposite sides simply supported, the present solutions are exact.

The Singularity in Multi-Material Wedges under Thermal Loading

2004 ◽  
Vol 261-263 ◽  
pp. 345-350
Author(s):  
Chyan Bin Hwu ◽  
Won Jun Lee
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Closed Form ◽  
Elastic Constants ◽  
Thermal Loading ◽  
Stroh Formalism ◽  
Different Boundary Conditions ◽  
Closed Form Solutions

By employing the Stroh formalism for plane anisotropic thermoelasticity, closed-form solutions for the orders of stress and heat flux singularities of multi-material wedges have been obtained. Several different boundary conditions are considered in this paper such as insulated or isothermal as well as free-free or fixed-fixed or free-fixed or fixed-free wedge boundaries. The solutions show that the singular orders are influenced by the wedge configurations (n wedge angles), boundary conditions, elastic constants and heat conduction coefficients, but are independent of the thermal moduli.

Sign in / Sign up

Export Citation Format

Share Document