A numerical study of a mildly non-linear fartiai differential equation

2008 ◽  
pp. 112-119
Author(s):  
L. M. Hocking
Keyword(s):  
Differential Equation ◽  
Numerical Study ◽  
Non Linear

Un-Shocked High Amplitude Standing Waves in Wave-Shaped Resonators

2012 ◽  
Author(s):  
Dion Savio Antao ◽  
Bakhtier Farouk
Keyword(s):  
Numerical Study ◽  
Fluid Dynamic ◽  
Ambient Pressure ◽  
Standing Waves ◽  
High Amplitude ◽  
Prime Mover ◽  
Non Linear ◽  
Resonator System ◽  
Cylindrical Resonators ◽  
Linear Nature

A numerical study of non-linear, high amplitude standing waves in non-cylindrical circular resonators is reported here. These waves are shock-less and can generate peak acoustic overpressures that can exceed the ambient pressure by three/four times its nominal value. A high fidelity compressible computational fluid dynamic model is used to simulate the phenomena in cylindrical and arbitrarily shaped axisymmetric resonators. A right circular cylinder and frustum of cone are the two geometries studied. The model is validated using past numerical and experimental results of standing waves in cylindrical resonators. The non-linear nature of the harmonic response of the frustum of cone resonator system is investigated for two different working fluids (carbon dioxide and argon) operating at various values of piston amplitude. The high amplitude non-linear oscillations demonstrated can be used as a prime mover in a variety of applications including thermoacoustic cryocooling.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

scholarly journals A numerical study of glacier advance over deforming till

2010 ◽  
Vol 4 (3) ◽  
pp. 359-372 ◽  
Author(s):  
G. J.-M. C. Leysinger Vieli ◽  
G. H. Gudmundsson
Keyword(s):  
Effective Viscosity ◽  
Numerical Study ◽  
Thickness Distribution ◽  
Plug Flow ◽  
Linear Wave ◽  
Sediment Layer ◽  
Mixed Flow ◽  
Model Experiments ◽  
Non Linear ◽  
Glacier Advance

Abstract. The advance of a glacier over a deforming sediment layer is analysed numerically. We treat this problem as a contact problem involving two slowly-deforming viscous bodies. The surface evolution of the two bodies, and of the contact interface between them, is followed through time. Using various different non-linear till rheologies, we show how the mode of advance depends on the relative effective viscosities of ice and till. Three modes of advances are observed: (1) overriding, where the glacier advances through ice deformation only and without deforming the sediment; (2) plug-flow, where the sediment is strongly deformed, the ice moves forward as a block and a bulge is built in front of the glacier; and (3) mixed-flow, where the glacier advances through both ice and sediment deformation. For the cases of both overriding and mixed-flow, an inverse depth-age relationship within the ice is obtained. A series of model experiments show the contrast in effective viscosity between ice and till to be the single most important model parameter defining the mode of advance and the resulting thickness distribution of the till. Our model experiments indicate that the thickness of the deforming till layer is greatest close to the glacier front. Measurements of till thickness taken in such locations may not be representative of deforming till thickness elsewhere. Given sufficiently large contrast in effective viscosity between ice and till, a sediment bulge is formed in front of the glacier. During glacier advance, the bulge quickly reaches a steady state form strongly resembling single-crested push moraines. Inspection of particle paths within the sediment bulge, shows that particles within the till travel at a different speed from the bulge itself, and the push moraine to advance as a form-conserving non-linear wave.

Efficient Finite Element for Evaluation of Strain Concentrations in Concrete Coated Pipelines

2008 ◽  
Author(s):  
Svein Sævik ◽  
Martin Storheim ◽  
Erik Levold
Keyword(s):  
Finite Element ◽  
Numerical Study ◽  
Sandwich Beam ◽  
Beam Theory ◽  
Material Model ◽  
Body Motion ◽  
Theoretical Formulation ◽  
Concrete Material ◽  
Non Linear ◽  
Full Scale Tests

MARINTEK has developed software for detailed analysis of pipelines during installation and operation. As part of the software development a new coating finite element was developed in cooperation with StatoilHydro enabling efficient analysis of field joint strain concentrations of long concrete coated pipeline sections. The element was formulated based on sandwich beam theory and application of the Principle of Potential Energy. Large deformations and non-linear geometry effects were handled by a Co-rotated “ghost” reference description where elimination of rigid body motion was taken care of by referring to relative displacements in the strain energy term. The non-linearity related to shear interaction and concrete material behaviour was handled by applying non-linear springs and a purpose made concrete material model. The paper describes the theoretical formulation and numerical studies carried out to verify the model. The numerical study included comparison between model and full-scale tests as well as between model and other commercial software. At last a 3000 m long pipeline was analysed to demonstrate the strain concentration behaviour of a concrete coated pipeline exposed to high temperature snaking on the seabed.

NUMERICAL STUDY OF MIXED BIOCONVECTION IN POROUS MEDIA SATURATED WITH NANOFLUID CONTAINING OXYTACTIC MICROORGANISMS

2013 ◽  
Vol 13 (04) ◽  
pp. 1350067 ◽  
Author(s):  
O. ANWAR BÉG ◽  
V. R. PRASAD ◽  
B. VASU
Keyword(s):  
Boundary Layer ◽  
Peclet Number ◽  
Mechanical Design ◽  
Numerical Study ◽  
Lewis Number ◽  
Nanoparticle Concentration ◽  
Péclet Number ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Oxytactic Microorganisms

A mathematical model has been developed for steady-state boundary layer flow of a nanofluid past an impermeable vertical flat wall in a porous medium saturated with a water-based dilute nanofluid containing oxytactic microorganisms. The nanoparticles were distributed sufficiently to permit bioconvection. The product of chemotaxis constant and maximum cell swimming speed was assumed invariant. Using appropriate transformations, the partial differential conservation equations were non-dimensionalised to yield a quartet of coupled, non-linear ordinary differential equations for momentum, energy, nanoparticle concentration and dimensionless motile microorganism density, with appropriate boundary conditions. The dominant parameters emerging in the normalised model included the bioconvection Lewis number, bioconvection Peclet number, Lewis number, buoyancy ratio parameter, Brownian motion parameter, thermophoresis parameter, local Darcy-Rayleigh number and the local Peclet number. An implicit numerical solution to the well-posed two-point non-linear boundary value problem is developed using the well-tested and highly efficient Keller box method. Computations are validated with the Nakamura tridiagonal implicit finite difference method, demonstrating excellent agreement. Nanoparticle concentration and temperature were found to be generally enhanced through the boundary layer with increasing bioconvection Lewis number, whereas dimensionless motile microorganism density was only increased closer to the wall. Temperature, nanoparticle concentration and dimensionless motile microorganism density were all greatly increased with a rise in Peclet number. Temperature and dimensionless motile microorganism density were reduced with increasing buoyancy parameter, whereas nanoparticle concentration was increased. The present study found applications in the fluid mechanical design of microbial fuel cell and bioconvection nanotechnological devices.

scholarly journals A Numerical Study of Chemical Reaction in a Nanofluid Flow Due to Rotating Disk in the Presence of Magnetic Field

2021 ◽  
Author(s):  
Muhammad Ramzan ◽  
Poom Kumam ◽  
Kottakkaran Sooppy Nisar ◽  
Ilyas Khan ◽  
Wasim Jamshed
Keyword(s):  
Differential Equation ◽  
Radial Velocity ◽  
Chemical Reaction ◽  
Axial Velocity ◽  
Numerical Study ◽  
Tangential Velocity ◽  
Rotating Disk ◽  
Partial Differential ◽  
Suction Parameter ◽  
Matlab Programming

Abstract In this paper, a numerical study of MHD steady flow due to the rotating disk with chemical reaction was explored. Effect of different parameters such as Schmidt number, chemical reaction parameter, Prandtl number, Suction parameter, heat absorption/generation parameter, Nano-particle concentration, Reynold number, Magnetic parameter, skin friction, shear stress, temperature distribution, Nusselt number, mass transfer rate, radial velocity, axial velocity, and tangential velocity was analyzed and discussed. For the simplification of non-linear partial differential equations (PDEs) into the nonlinear ordinary differential equation (ODEs), the method of Similarity transformation was employed, and the resulting partial differential equation was solved by using finite difference method through MATLAB programming. This work's remarkable finding is that with the expansion of nanoparticle concentration radial velocity, tangential velocity and temperature of the fluid was enhanced but reverse reaction for axial velocity. Furthermore, the present results are found to be in excellent agreement with previously published work.

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