scholarly journals RADIATION HEAT TRANSFER WITH ABLATION IN A FINITE PLATE

2005 ◽  
Vol 4 (2) ◽  
Author(s):  
F. A. A. Gomes ◽  
J. B. C. Silva ◽  
A. J. Diniz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Thermal Protection ◽  
Heated Surface ◽  
Phase Changes ◽  
Solution Technique ◽  
Governing Equations ◽  
Radiation Heat ◽  
Partial Differential ◽  
Non Linear System

The phenomenon of ablation is a process of thermal protection with several applications, mainly, in mechanical and aerospace engineering. Ablative thermal protection is applied using special materials (named ablative materials) externally on the surface of a structure in order to isolate it against thermal effects. The ablative phenomenon is a complex process involving phase changes with partial or total loss of the material. So the position of the boundary is initially unknown. The governing equations of the process form a non-linear system of coupled partial differential equations. The one-dimensional analysis of an ablative process on the plate is performed by using the generalized integral transform technique – GITT for solution of the system of governing equations. By application of this solution technique, the system of partial differential equations is transformed into a system of infinite ordinary differential equations that can be solved after the truncation of that system by numerical techniques codes available. The plate of finite thickness at constant properties is subjected to a time-dependent prescribed radiation heat flux at one face, initially with a uniform temperature T0, and insulated on the other face. After an initial heating period, ablation starts at the heated surface through melting and continuous removal of the plate material. The results of interest are the thickness and the loss rate of the ablative material. The obtained results are compared with available results from other solution techniques in the literature.

scholarly journals RADIATION HEAT TRANSFER WITH ABLATION IN A FINITE PLATE

2005 ◽  
Vol 4 (2) ◽  
pp. 190
Author(s):  
F. A. A. Gomes ◽  
J. B. C. Silva ◽  
A. J. Diniz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Thermal Protection ◽  
Heated Surface ◽  
Phase Changes ◽  
Solution Technique ◽  
Governing Equations ◽  
Radiation Heat ◽  
Partial Differential ◽  
Non Linear System

The phenomenon of ablation is a process of thermal protection with several applications, mainly, in mechanical and aerospace engineering. Ablative thermal protection is applied using special materials (named ablative materials) externally on the surface of a structure in order to isolate it against thermal effects. The ablative phenomenon is a complex process involving phase changes with partial or total loss of the material. So the position of the boundary is initially unknown. The governing equations of the process form a non-linear system of coupled partial differential equations. The one-dimensional analysis of an ablative process on the plate is performed by using the generalized integral transform technique – GITT for solution of the system of governing equations. By application of this solution technique, the system of partial differential equations is transformed into a system of infinite ordinary differential equations that can be solved after the truncation of that system by numerical techniques codes available. The plate of finite thickness at constant properties is subjected to a time-dependent prescribed radiation heat flux at one face, initially with a uniform temperature T0, and insulated on the other face. After an initial heating period, ablation starts at the heated surface through melting and continuous removal of the plate material. The results of interest are the thickness and the loss rate of the ablative material. The obtained results are compared with available results from other solution techniques in the literature.

scholarly journals USE OF ATANGANA–BALEANU FRACTIONAL DERIVATIVE IN HELICAL FLOW OF A CIRCULAR PIPE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040049 ◽  
Author(s):  
KASHIF ALI ABRO ◽  
ILYAS KHAN ◽  
KOTTAKKARAN SOOPPY NISAR
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rheological Parameters ◽  
Shear Stresses ◽  
Fractional Differentiation ◽  
Retardation Time ◽  
Governing Equations ◽  
Partial Differential ◽  
Gamma Functions ◽  
Helical Pipe

There is no denying fact that helically moving pipe/cylinder has versatile utilization in industries; as it has multi-purposes, such as foundation helical piers, drilling of rigs, hydraulic simultaneous lift system, foundation helical brackets and many others. This paper incorporates the new analysis based on modern fractional differentiation on infinite helically moving pipe. The mathematical modeling of infinite helically moving pipe results in governing equations involving partial differential equations of integer order. In order to highlight the effects of fractional differentiation, namely, Atangana–Baleanu on the governing partial differential equations, the Laplace and Hankel transforms are invoked for finding the angular and oscillating velocities corresponding to applied shear stresses. Our investigated general solutions involve the gamma functions of linear expressions. For eliminating the gamma functions of linear expressions, the solutions of angular and oscillating velocities corresponding to applied shear stresses are communicated in terms of Fox- H function. At last, various embedded rheological parameters such as friction and viscous factor, curvature diameter of the helical pipe, dynamic analogies of relaxation and retardation time and comparison of viscoelastic fluid models (Burger, Oldroyd-B, Maxwell and Newtonian) have significant discrepancies and semblances based on helically moving pipe.

scholarly journals Analysis Bending Solutions of Clamped Rectangular Thick Plate

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Yang Zhong ◽  
Qian Xu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Shear Deformation ◽  
Thick Plate ◽  
Plate Theory ◽  
Analytic Solutions ◽  
Higher Order ◽  
Numerical Comparison ◽  
Governing Equations ◽  
Partial Differential

The bending solutions of rectangular thick plate with all edges clamped and supported were investigated in this study. The basic governing equations used for analysis are based on Mindlin’s higher-order shear deformation plate theory. Using a new function, the three coupled governing equations have been modified to independent partial differential equations that can be solved separately. These equations are coded in terms of deflection of the plate and the mentioned functions. By solving these decoupled equations, the analytic solutions of rectangular thick plate with all edges clamped and supported have been derived. The proposed method eliminates the complicated derivation for calculating coefficients and addresses the solution to problems directly. Moreover, numerical comparison shows the correctness and accuracy of the results.

Modeling and Control of Flexible Transporter System With Arbitrarily Time-Varying Cable Lengths

2003 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil K. Agrawal ◽  
Peter Hagedorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ritz Method ◽  
Nonlinear Partial Differential Equations ◽  
Time Varying ◽  
Governing Equations ◽  
Partial Differential ◽  
Modeling And Control ◽  
Lyapunov Controller ◽  
And Control

A systematic procedure for deriving the system model of a cable transporter system with arbitrarily time-varying lengths is presented. Two different approaches are used to develop the model, namely, Newton’s Law and Hamilton’s Principle. The derived governing equations are nonlinear partial differential equations. The same results are obtained using the two methods. The Rayleigh-Ritz method is used to obtain an approximate numerical solution of the governing equations by transforming the infinite order partial differential equations into a finite order discretized system. A Lyapunov controller which can both dissipate the vibratory energy and assure the attainment of the desired goal is derived. The validity of the proposed controller is verified by numerical simulation.

Longitudinal Vibration Modeling and Control of a Flexible Transporter System with Arbitrarily Varying Cable Lengths

2005 ◽  
Vol 11 (3) ◽  
pp. 431-456 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil K. Agrawal ◽  
Peter Hagedorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Vibration ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Governing Equations ◽  
Partial Differential ◽  
Modeling And Control ◽  
Assumed Mode Method ◽  
Mode Method

We present a systematic procedure for deriving the model of a cable transporter system with arbitrarily varying cable lengths. The Hamilton principle is applied to derive the governing equations of motion. The derived governing equations are nonlinear partial differential equations. The results are verified using the Newton law. The assumed mode method is used to obtain an approximate numerical solution of the governing equations by transforming the infinite-dimensional partial differential equations into a finite-dimensional discretized system. We propose a Lyapunov controller, based directly on the governing partial differential equations, which can both dissipate the vibratory energy during the motion of the transporter and guarantee the attainment of the desired goal point. The validity of the proposed controller is verified by numerical simulation.

On Hele–Shaw flow of power-law fluids

1992 ◽  
Vol 3 (4) ◽  
pp. 343-366 ◽  
Author(s):  
Gunnar Aronsson ◽  
Ulf Janfalk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Stream Function ◽  
Power Law ◽  
Representation Theorem ◽  
Governing Equations ◽  
Power Law Fluid ◽  
Partial Differential ◽  
Power Law Fluids

This paper reviews the governing equations for a plane Hele–Shaw flow of a power-law fluid. We find two closely related partial differential equations, one for the pressure and one for the stream function. Some mathematical results for these equations are presented, in particular some exact solutions and a representation theorem. The results are applied to Hele–Shaw flow. It is then possible to determine the flow near an arbitrary corner for any power-law fluid. Other examples are also given.

scholarly journals Numerical Investigations of the Effect of Nonlinear Quadratic Pressure Gradient Term on a Moving Boundary Problem of Radial Flow in Low-Permeable Reservoirs with Threshold Pressure Gradient

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Wenchao Liu ◽  
Jun Yao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Pressure Gradient ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Radial Flow ◽  
Moving Boundary Problem ◽  
Strong Nonlinearity ◽  
Governing Equations ◽  
Partial Differential

The existence of a TPG can generate a relatively high pressure gradient in the process of fluid flow in porous media in low-permeable reservoirs, and neglecting the QPGTs in the governing equations, by assuming a small pressure gradient for such a problem, can cause a significant error in predicting the formation pressure. Based on these concerns, in consideration of the QPGT, a moving boundary model of radial flow in low-permeable reservoirs with the TPG for the case of a constant flow rate at the inner boundary is constructed. Due to strong nonlinearity of the mathematical model, a numerical method is presented: the system of partial differential equations for the moving boundary problem is first transformed equivalently into a closed system of partial differential equations with fixed boundary conditions by a spatial coordinate transformation method; and then a stable, fully implicit finite difference method is used to obtain its numerical solution. Numerical result analysis shows that the mathematical models of radial flow in low-permeable reservoirs with TPG must take the QPGT into account in their governing equations, which is more important than those of Darcy’s flow; the sensitive effects of the QPGT for the radial flow model do not change with an increase of the dimensionless TPG.

Analytical/Numerical Hybrid Solution for One-Dimensional Ablation Problem

2003 ◽  
Author(s):  
Fa´bio Yukio Kurokawa ◽  
Antonio Joa˜o Diniz ◽  
Joa˜o Batista Campos-Silva
Keyword(s):  
Differential Equations ◽  
Integral Transform ◽  
Thermal Protection ◽  
Linear Partial Differential Equation ◽  
Equation System ◽  
Partial Differential ◽  
One Dimensional ◽  
Plane Plate ◽  
Non Linear ◽  
Non Linear System

Ablation is a thermal protection process with several applications in engineering, mainly in the field of airspace industry. The use of conventional materials must be quite restricted, because they would suffer catastrophic flaws due to thermal degradation of their structures. However, the same materials can be quite suitable once being protected by well-known ablative materials. The process that involves the ablative phenomena is complex, could involve the whole or partial loss of material that is sacrificed for absorption of energy. The analysis of the ablative process in a blunt body with revolution geometry will be made on the stagnation point area that can be simplified as a one-dimensional plane plate problem. In this work the Generalized Integral Transform Technique (GITT) is employed for the solution of the non-linear system of coupled partial differential equations that model the phenomena. The solution of the problem is obtained by transforming the non-linear partial differential equation system to a system of coupled first order ordinary differential equations and then solving it by using well-established numerical routines. The results of interest such as the temperature field, the depth and the rate of removal of the ablative material are presented and compared with those ones available in the open literature.

scholarly journals A Novel Method for Analytical Solutions of Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Mehmet Ali Akinlar ◽  
Muhammet Kurulay
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Solution Technique ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Vector Type ◽  
Novel Method ◽  
Fractional Differential ◽  
Finite Sum

A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.

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