On the Propagation of Weak Discontinuities Along Bicharacteristics in a Radiating Gas

1981 ◽  
Vol 48 (4) ◽  
pp. 737-742
Author(s):  
R. S. Singh ◽  
V. D. Sharma
Keyword(s):  
Mach Number ◽  
Differential Equations ◽  
Velocity Gradient ◽  
Spherical Waves ◽  
Characteristic Manifold ◽  
Geometric Factors ◽  
Weak Discontinuities ◽  
Upstream Flow ◽  
Special Case ◽  
Explicit Criteria

The propagation of weak discontinuities along bicharacteristic curves in the characteristic manifold of the differential equations governing the flow of a radiating gas near the optically thin limit has been discussed. Some explicit criteria for the growth and decay of weak discontinuities along bicharacteristics are given. As a special case, when the discontinuity surface is adjacent to a region of uniform flow, the solution for the velocity gradient at the wave head is specialized to the plane, cylindrical, and spherical waves. For expandng waves, the attenuation induced by geometric factors and the radiative flux, and the growth induced by the upstream flow Mach number are discussed. It is shown that a compressive disturbance steepens into a shock only if the initial disturbance is sufficiently strong.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

ON TAYLOR'S SCRAPING PROBLEM AND FLOW OF A SISKO FLUID

2009 ◽  
Vol 14 (4) ◽  
pp. 515-529 ◽  
Author(s):  
Abdul M. Siddiqui ◽  
Ali R. Ansari ◽  
Ahmed Ahmad ◽  
N. Ahmad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Viscous Fluid ◽  
Normal Stress ◽  
Partial Differential ◽  
Sisko Fluid ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Special Case

The aim of the present investigation is to study the properties of a Sisko fluid flowing between two intersecting planes. The problem is similar to Taylor's scraping problem for a viscous fluid. We determine the solution of the complicated set of non‐linear partial differential equations describing the flow analytically. The analysis is carried out in detail reflecting the effects of varying the angle of the scraper on the flow. In addition, the tangential and normal stress are also computed. We also show the well known Taylor scraper problem as a special case.

A Mechanical Model of Helical Elements in Textured Yarns

1976 ◽  
Vol 46 (4) ◽  
pp. 278-283 ◽  
Author(s):  
M. Konopasek
Keyword(s):  
Differential Equations ◽  
Mechanical Model ◽  
Mechanical Characteristics ◽  
Functional Relationship ◽  
Three Dimensional ◽  
Infinite Length ◽  
Fiber Properties ◽  
Helical Angle ◽  
Limiting Case ◽  
Special Case

The helical model of the spontaneously collapsed filaments in twist-textured yarns is defined as reflecting the limiting case of a free-filament segment with infinite length (or number of coils) between two reversal points. The fundamental relationships linking fiber properties and parameters of the texturing process with geometrical and mechanical characteristics of the helices are derived directly from the differential equations of the three-dimensional elastica. Bicomponent and similar fibers are interpreted as a special case of twist-textured filaments with original (permanently set) helical angle equal to π/2; for this case an explicit functional relationship between contraction and stretching force is obtained.

On the St. Venant Problems for Inhomogeneous Circular Bars

1999 ◽  
Vol 66 (1) ◽  
pp. 32-40 ◽  
Author(s):  
F. Rooney ◽  
M. Ferrari
Keyword(s):  
Shear Modulus ◽  
Differential Equations ◽  
Uniform Distribution ◽  
Elastic Moduli ◽  
Transverse Force ◽  
Radial Coordinate ◽  
Effective Moduli ◽  
Pure Bending ◽  
Simple Tension ◽  
Special Case

The classical St. Venant problems, i.e., simple tension, pure bending, and flexure by a transverse force, are considered for circular bars with elastic moduli that vary as a function of the radial coordinate. The problems are reduced to second-order ordinary differential equations, which are solved for a particular choice of elastic moduli. The special case of a bar with a constant shear modulus and the Poisson’s ratio varying is also considered and for this situation the problems are solved completely. The solutions are then used to obtain homogeneous effective moduli for inhomogeneous cylinders. Material inhomogeneities associated with spatially variable distributions of the reinforcing phase in a composite are considered. It is demonstrated that uniform distribution of the reinforcement leads to a minimum of the Young’s modulus in the class of spatial variations in the concentration considered.

scholarly journals Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
T. E. Govindan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Special Case ◽  
Partial Functional Differential Equation

This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.

scholarly journals On the solution of two-sided fractional ordinary differential equations of Caputo type

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Ma. Elena Hernández-Hernández ◽  
Vassili N. Kolokoltsov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Well Posedness ◽  
Solutions To Equations ◽  
Fractional Differential ◽  
The Right ◽  
Stochastic Representations ◽  
Exit Problems ◽  
Special Case

AbstractThis paper provides well-posedness results and stochastic representations for the solutions to equations involving both the right- and the left-sided generalized operators of Caputo type. As a special case, these results show the interplay between two-sided fractional differential equations and two-sided exit problems for certain Lévy processes.

Lagrangian investigations of velocity gradients in compressible turbulence: lifetime of flow-field topologies

2019 ◽  
Vol 872 ◽  
pp. 492-514 ◽  
Author(s):  
Nishant Parashar ◽  
Sawan Suman Sinha ◽  
Balaji Srinivasan
Keyword(s):  
Mach Number ◽  
Velocity Gradient ◽  
Direct Numerical Simulations ◽  
Compressible Turbulence ◽  
Lagrangian Particle ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Conditional Mean ◽  
Velocity Gradients ◽  
Decaying Turbulence

We perform Lagrangian investigations of the dynamics of velocity gradients in compressible decaying turbulence. Specifically, we examine the evolution of the invariants of the velocity-gradient tensor. We employ well-resolved direct numerical simulations over a range of Mach number along with a Lagrangian particle tracker to examine trajectories of fluid particles in the space of the invariants of the velocity gradient tensor. This allows us to accurately measure the lifetimes of major topologies of compressible turbulence and provide an explanation of why some selective topologies tend to exist longer than the others. Further, the influence of dilatation on the lifetime of various topologies is examined. Finally, we explain why the so-called conditional mean trajectories (CMT) used previously by several researchers fail to predict the lifetime of topologies accurately.

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