A new form of Boussinesq equations for long waves in water of non-uniform depth

2012 ◽  
Vol 60 (3) ◽  
pp. 631-643
Author(s):  
J.K. Szmidt ◽  
B. Hedzielski
Keyword(s):  
Differential Equations ◽  
Order Approximation ◽  
Fluid Velocity ◽  
Power Series Expansion ◽  
Periodic Problem ◽  
Long Waves ◽  
Variable Depth ◽  
Governing Equations ◽  
Non Linear ◽  
New Form

Abstract The paper describes the non-linear transformation of long waves in shallow water of variable depth. Governing equations of the problem are derived under the assumption that the non-viscous fluid is incompressible and the fluid flow is a rotation free. A new form of Boussinesq-type equations is derived employing a power series expansion of the fluid velocity components with respect to the water depth. These non-linear partial differential equations correspond to the conservation of mass and momentum. In order to find the dispersion characteristic of the description, a linear approximation of these equations is derived. A second order approximation of the governing equations is applied to study a time dependent transformation of waves in a rectangular basin of water of variable depth. Such a case corresponds to a spatially periodic problem of sea waves approaching a near-shore zone. In order to overcome difficulties in integrating these equations, the finite difference method is applied to transform them into a set of non-linear ordinary differential equations with respect to the time variable. This final set of these equations is integrated numerically by employing the fourth order Runge - Kutta method.

Geometrical and Material Non-Linear Formulation for Bend Stiffeners

2004 ◽  
Author(s):  
Murilo Augusto Vaz ◽  
Carlos Alberto Duarte de Lemos
Keyword(s):  
Differential Equations ◽  
Constitutive Relations ◽  
Mathematical Formulation ◽  
Power Series Expansion ◽  
Constitutive Models ◽  
Bending Moment ◽  
Accurate Analysis ◽  
Linear Ordinary Differential Equations ◽  
Linear Elastic ◽  
Non Linear

A mathematical formulation and a numerical solution for the geometrical and material non-linear analysis of bend stiffeners — employed to protect the upper terminations of flexible risers and subsea umbilical cables — are presented in this paper. The differential equations governing the problem result from geometrical compatibility, equilibrium of forces and moments and material constitutive relations, which can be linear elastic symmetric or non-linear elastic asymmetric. In this latter case, the bending moment versus curvature for each cross-section is calculated and then expressed by a polynomial power series expansion. Hence, a set of four first order non-linear ordinary differential equations is written and boundary conditions are defined at both ends. A one-parameter shooting method is employed and results are presented for a case study where linear elastic symmetric and non-linear elastic asymmetric constitutive models are compared and discussed. It is shown that an accurate analysis of bend stiffeners depends on a precise assessment of the material constitutive property.

MHD Boundary Layer Slip Flow and Heat Transfer of Nanofluid Past a Vertical Stretching Sheet with Non-Uniform Heat Generation/Absorption

2014 ◽  
Vol 13 (03) ◽  
pp. 1450019 ◽  
Author(s):  
S. Das ◽  
R. N. Jana ◽  
O. D. Makinde
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Heat Generation ◽  
Stretching Sheet ◽  
Fluid Velocity ◽  
Slip Flow ◽  
Uniform Heat ◽  
Mhd Boundary Layer ◽  
Non Linear

An investigation of the magnetohydrodynamics (MHD) boundary layer slip flow over a vertical stretching sheet in nanofluid with non-uniform heat generation/absorbtion in the presence of a uniform transverse magnetic field has been carried out. The governing non-linear partial differential equations are transformed into a system of coupled non-linear ordinary differential equations using similarity transformations and then solved numerically using the Runge–Kutta fourth order method with shooting technique. Numerical results are obtained for the fluid velocity, temperature as well as the shear stress and the rate of heat transfer at the surface of the sheet. The results show that there are significant effects of various pertinent parameters on velocity and temperature profiles.

Stokes’ first problem for MHD flow of Casson nanofluid

2017 ◽  
Vol 13 (1) ◽  
pp. 2-10
Author(s):  
Syed Tauseef Mohyud-din ◽  
Umar Khan ◽  
Naveed Ahmed ◽  
M.M. Rashidi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Mhd Flow ◽  
Governing Equations ◽  
Content Type ◽  
Buongiorno’S Model ◽  
Linear Ordinary Differential Equations ◽  
Flux Condition ◽  
Non Linear ◽  
Flow Heat

Purpose The purpose of this paper is to present investigation of the flow, heat and mass transfer of a nanofluid over a suddenly moved flat plate using Buongiorno’s model. This study is different from some of the previous studies as the effects of Brownian motion and thermophoresis on nanoparticle fraction are passively controlled on the boundary rather than actively. Design/methodology/approach The partial differential equations governing the flow are reduced to a system of non-linear ordinary differential equations. Viable similarity transforms are used for this purpose. A well-known numerical scheme called Runge-Kutta-Fehlberg method coupled with shooting procedure has been used to find the solution of resulting system of equations. Discussions on the effects of different emerging parameters are provided using graphical aid. A table is also given that provides the results of different parameters on local Nusselt and Sherwood numbers. Findings A revised model for Stokes’ first problem in nanofluids is presented in this paper. This model considers a zero flux condition at the boundary. Governing equations after implementing the similarity transforms get converted into a system of non-linear ordinary differential equations. Numerical solution using RK-Fehlberg method is also carried out. Emerging parameters are analyzed graphically. Figures indicate a quite significant change in concentration profile due to zero flux condition at the wall. Originality/value This work can be extended for other problems involving nanofluids for the better understanding of different properties of nanofluids.

Mathematical Model of a Railway Pneumatic Brake System With Varying Cylinder Capacity Effects

1990 ◽  
Vol 112 (3) ◽  
pp. 456-462 ◽  
Author(s):  
S. Bharath ◽  
B. C. Nakra ◽  
K. N. Gupta
Keyword(s):  
Differential Equations ◽  
Volume Effect ◽  
Brake System ◽  
Governing Equations ◽  
Linear Ordinary Differential Equations ◽  
Linear Partial Differential Equations ◽  
Reservoir Volume ◽  
State Approximation ◽  
Piston Displacement ◽  
Non Linear

Governing equations for the analysis of pressure transient are derived from the principle of conservation of mass and momentum for a pneumatic brake system, which consists of a train pipe connected to a number of linear actuators (brake cylinders with piston displacement). The governing one-dimensional non-linear partial differential equations for the train pipe, non-linear ordinary differential equations for the brake cylinders, and second-order differential equation of motion for piston displacement are solved to determine the pressure transients in the brake system for a step change in pressure at the inlet. The governing equations are nondimensionalized and reduced to a set of ordinary nonlinear differential difference equations and integrated by standard numerical methods. The flow is considered isothermal, and the friction effects for turbulent and laminar flow are evaluated by quasi-steady state approximation. The auxiliary reservoir volume effect is also included. The results are compared with the experimental data obtained on a brake test rig.

The solitary wave in water of variable depth

1970 ◽  
Vol 42 (3) ◽  
pp. 639-656 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Solitary Wave ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Wave Amplitude ◽  
Finite Amplitude ◽  
Long Waves ◽  
Two Dimensional ◽  
Variable Depth ◽  
Constant Depth

Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.

scholarly journals DTM-PADE APPROACH TO MHD SLIP FLOW AND HEAT TRANSFER OVER A RADIALLY STRETCHING SHEET WITH THERMAL RADIATION

2020 ◽  
Vol 50 (3) ◽  
pp. 175-184
Author(s):  
Shoeb Rashid Sayyed ◽  
Brijbhan Singh ◽  
Oluwole Makinde ◽  
Nasreen Bano Muslim Shaikh
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Fluid Velocity ◽  
Slip Flow ◽  
Fluid Temperature ◽  
Linear Ordinary Differential Equations ◽  
Flow And Heat Transfer ◽  
Non Linear

The present paper deals with the boundary layer flow and heat transfer of an electrically conducting magnetohydrodynamic viscous fluid over a radially stretching power-law sheet with suction/injection within a porous medium by considering the effects of momentum and thermal slips and thermal radiation. The governing non-linear partial boundary layer differential equations are reduced to a  system of coupled non-linear ordinary differential equations (ODEs) with the aid of appropriate similarity transformations. These transformed ODEs are then solved by employing a semi-analytic technique known as differential transformation method (DTM) in combination with Pade approximation. The numerical values of skin friction and local Nusselt number are tabulated and validated by comparing them with the corresponding values available in the literature. Our results have been found in precise agreement  with the results published earlier. The effects of different governing parameters on velocity and temperature distributions are analyzed graphically. It has been found that with an increase in the velocity slip parameter the fluid velocity decreases whereas the temperature of the fluid increases. Also, the fluid temperature gets enhanced with an increase in the radiation parameter, but it decreases with an increase in thermal slip.

Mathematical Solution on MHD Stagnation Point Flow of Ferrofluid

2020 ◽  
Vol 399 ◽  
pp. 38-54
Author(s):  
Siti Hanani Mat Yasin ◽  
Muhammad Khairul Anuar Mohamed ◽  
Zulkhibri Ismail ◽  
Mohd Zuki Salleh
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Volume Fraction ◽  
Stagnation Point Flow ◽  
Governing Equations ◽  
Linear Ordinary Differential Equations ◽  
Linear Partial Differential Equations ◽  
Keller Box Method ◽  
Non Linear

This study presents a numerical investigation on the magnetohydrodynamic (MHD) stagnation point flow of a ferrofluid with Newtonian heating. The black oxide of iron, magnetite (Fe3O4) which acts as magnetic materials and water as a base fluid are considered. The two dimensional stagnation point flow of cold ferrofluid against a hot wall under the influence of the uniform magnetic field of strength is located some distance behind the stagnation point. The effect of magnetic and volume fraction on the velocity and temperature boundary layer profiles are obtained through the formulated governing equations. The governing equations which are in the form of dimensional non-linear partial differential equations are reduced to dimensionless non-linear ordinary differential equations by using appropriate similarity transformation. Then, they are solved numerically by using the Keller-box method which is programmed in the Matlab software. It is found that the cold fluid moves towards the magnetic source that is close to the hot wall. Hence, leads to the better cooling rate and enhances the heat transfer rate. Meanwhile, an increase of the magnetite nanoparticles volume fraction, increases the ferrofluid capabilities in thermal conductivity and consequently enhances the heat transfer.

Glycemia monitoring

1998 ◽  
Vol 2 ◽  
pp. 23-30
Author(s):  
Igor Basov ◽  
Donatas Švitra
Keyword(s):  
Diabetes Mellitus ◽  
Mathematical Model ◽  
Numerical Methods ◽  
Differential Equations ◽  
Blood Sugar ◽  
Self Regulation ◽  
Physiological System ◽  
Non Linear ◽  
The Mathematical Model

Here a system of two non-linear difference-differential equations, which is mathematical model of self-regulation of the sugar level in blood, is investigated. The analysis carried out by qualitative and numerical methods allows us to conclude that the mathematical model explains the functioning of the physiological system "insulin-blood sugar" in both normal and pathological cases, i.e. diabetes mellitus and hyperinsulinism.

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