scholarly journals On the Use of Asymptotic Solutions to Plane Ice—Water Problems

1969 ◽  
Vol 8 (53) ◽  
pp. 285-300
Author(s):  
G. S. H. Lock
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Solutions ◽  
Variable Properties ◽  
Governing Equations ◽  
Freezing And Thawing ◽  
One Dimensional ◽  
Wide Range ◽  
Order Of Magnitude ◽  
Water Problems ◽  
Ice Water

The paper considers one-dimensional freezing and thawing of ice–water systems for the conditions first examined by Stefan. An order-of-magnitude analysis applied to the governing equations and boundary conditions quantifies the error resulting from the neglect of various factors. Principal among these are density difference, initial superheat and variable properties.Asymptotic solutions for the temperature distribution and interface history are developed for a wide range of boundary conditions: prescribed temperature or heat flux, prescribed convection and prescribed radiation. Comparison with known results reveals the general adequacy of the asymptotic solutions and an estimate of the error incurred.

scholarly journals On the Use of Asymptotic Solutions to Plane Ice—Water Problems

1969 ◽  
Vol 8 (53) ◽  
pp. 285-300 ◽  
Author(s):  
G. S. H. Lock
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Solutions ◽  
Variable Properties ◽  
Governing Equations ◽  
Freezing And Thawing ◽  
One Dimensional ◽  
Wide Range ◽  
Order Of Magnitude ◽  
Water Problems ◽  
Ice Water

The paper considers one-dimensional freezing and thawing of ice–water systems for the conditions first examined by Stefan. An order-of-magnitude analysis applied to the governing equations and boundary conditions quantifies the error resulting from the neglect of various factors. Principal among these are density difference, initial superheat and variable properties.Asymptotic solutions for the temperature distribution and interface history are developed for a wide range of boundary conditions: prescribed temperature or heat flux, prescribed convection and prescribed radiation. Comparison with known results reveals the general adequacy of the asymptotic solutions and an estimate of the error incurred.

Lubrication Theory for Micropolar Fluids

1971 ◽  
Vol 38 (3) ◽  
pp. 646-650 ◽  
Author(s):  
S. J. Allen ◽  
K. A. Kline
Keyword(s):  
Flow Field ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Two Dimensional ◽  
Slider Bearing ◽  
Governing Equations ◽  
Micropolar Fluids ◽  
Linear Ordinary Differential Equations ◽  
Two Dimensional Flow ◽  
Order Of Magnitude

The equations governing the flow of a fluid with rigid, spherical substructure are summarized. A two-dimensional flow field is considered and applied to the geometry of a slider bearing. Order-of-magnitude arguments are used which reduce the governing equations to a system of coupled, linear, ordinary differential equations. The equations are solved subject to appropriate boundary conditions and the effects of substructure discussed with the help of a specific numerical example.

scholarly journals Effect of Variable Properties and Moving Heat Source on Magnetothermoelastic Problem under Fractional Order Thermoelasticity

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Chunbao Xiong ◽  
Ying Guo
Keyword(s):  
Heat Source ◽  
Fractional Order ◽  
Axial Direction ◽  
Moving Heat Source ◽  
Variable Properties ◽  
Governing Equations ◽  
Temperature Dependent ◽  
One Dimensional ◽  
Temperature Dependent Properties ◽  
Nondimensional Temperature

A one-dimensional generalized magnetothermoelastic problem of a thermoelastic rod with finite length is investigated in the context of the fractional order thermoelasticity. The rod with variable properties, which are temperature-dependent, is fixed at both ends and placed in an initial magnetic field, and the rod is subjected to a moving heat source along the axial direction. The governing equations of the problem in the fractional order thermoelasticity are formulated and solved by means of Laplace transform in tandem with its numerical inversion. The distributions of the nondimensional temperature, displacement, and stress in the rod are obtained and illustrated graphically. The effects of the temperature-dependent properties, the velocity of the moving heat source, the fractional order parameter, and so forth on the considered variables are concerned and discussed in detail, and the results show that they significantly influence the variations of the considered variables.

The effects of shear deformation on the free vibration of elastic beams with general boundary conditions

2009 ◽  
Vol 224 (1) ◽  
pp. 71-84 ◽  
Author(s):  
J Li ◽  
H Hua
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Dynamic Stiffness ◽  
Beam Theory ◽  
Mode Shapes ◽  
Governing Equations ◽  
One Dimensional ◽  
Elastic Beams ◽  
Shear Deformable

Free vibration characteristics of shear deformable elastic beams subjected to different sets of boundary conditions are investigated. The analysis is based on a unified one-dimensional shear deformation beam theory. The governing equations of the elastic beams are obtained by means of Hamilton's principle. Four different boundary conditions are considered. The natural frequencies and mode shapes are obtained by applying the dynamic stiffness method, where the elements of the exact dynamic stiffness matrix are derived by using the analytical solutions of the governing equations of the beam in free vibration. The numerical results for the particular beams with different slenderness ratios are presented and compared with those available in the literature.

Freezing Flow in a Subcooled Permeable Medium

1992 ◽  
Vol 114 (4) ◽  
pp. 1036-1041 ◽  
Author(s):  
S. K. Griffiths ◽  
R. H. Nilson
Keyword(s):  
Freezing Point ◽  
Freezing Temperature ◽  
Similarity Solutions ◽  
Inlet Temperature ◽  
Governing Equations ◽  
Phase Zone ◽  
Two Phase ◽  
Liquid Zone ◽  
One Dimensional ◽  
Order Of Magnitude

Analytical similarity solutions are derived for the problem of transient one-dimensional flow and freezing of a liquid in an initially dry permeable half-space. The structure of the flow consists of three regions: a liquid zone in which the temperature decreases to the freezing temperature, a central two-phase zone where the temperature is at the freezing point, and a leading gas-filled region in which the temperature is nearly undisturbed. The propagation velocity of this intrusion is determined as a function of the subcooling, latent heat, and other process parameters. As the inlet temperature approaches the freezing temperature, the governing equations admit a pair of solutions having propagation velocities that sometimes differ by more than an order of magnitude.

Length scale-dependent natural frequencies of piezoelectric microplates

2017 ◽  
Vol 24 (13) ◽  
pp. 2749-2759 ◽  
Author(s):  
M Jafari ◽  
E Jomehzadeh ◽  
M Rezaeizadeh
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Couple Stress Theory ◽  
Free Vibration Analysis ◽  
Length Scale ◽  
Governing Equations ◽  
Classical Plate Theory ◽  
Wide Range ◽  
Piezoelectric Layers

The length-scale free vibration analysis of a rectangular microplate coupled with piezoelectric layers is presented. The modified couple stress theory is used to describe the size effect of the system. The governing equations of motion are obtained using Hamilton’s principle based on the classical plate theory. The transverse part of the electric potential for the piezoelectric layers is considered to satisfy the Maxwell’s equation and the electrical boundary conditions. A new procedure is introduced to decouple the governing equations and then an analytical Levy-type solution is obtained. The exact natural frequencies are established for a wide range of length scales, various plate dimensions, several piezoelectric layer thicknesses, and different boundary conditions. The results show that the effect of length scale parameter is decreased by the piezoelectric electrical field.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

scholarly journals Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffnesses

2020 ◽  
Vol 11 (1) ◽  
pp. 127
Author(s):  
Fuchun Yang ◽  
Dianrui Wang
Keyword(s):  
High Speed ◽  
Differential Operators ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Distribution Area ◽  
Vibration Modes ◽  
Governing Equations ◽  
Vibration Properties ◽  
Wide Range ◽  
Forced Responses

Vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were studied. The governing equations were obtained by Hamilton’s principle based on a rotating frame on the ring. The governing equations were cast in matrix differential operators and discretized, using Galerkin’s method. The eigenvalue problem was dealt with state space matrix, and the natural frequencies and vibration modes were computed in a wide range of rotation speed. The properties of natural frequencies and vibration modes with rotation speed were studied for free planet rings and planet rings with discrete and partially distributed stiffnesses. The influences of several parameters on the vibration properties of planet rings were also investigated. Finally, the forced responses of planet rings resulted from the excitation of rotating and revolving movement were studied. The results show that the revolving movement not only affects the free vibration of planet rings but results in excitation to the rings. Partially distributed stiffness changes the vibration modes heavily compared to the free planet ring. Each vibration mode comprises several nodal diameter components instead of a single component for a free planet ring. The distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. The forced responses caused by revolving movement are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

Sign in / Sign up

Export Citation Format

Share Document