How Quickly Can a Submarine Dive?

2019 ◽  
Vol 01 (04) ◽  
pp. 1950013
Author(s):  
Carl E. Mungan ◽  
Trevor C. Lipscombe
Keyword(s):  
Fluid Dynamics ◽  
Differential Equations ◽  
Numerical Integration ◽  
Equation Model ◽  
Order Equation ◽  
Intermediate Level ◽  
Second Law ◽  
Basic Fluid ◽  
Ballast Tanks ◽  
Spatial Derivatives

A two-equation model is developed to describe the descent of a submarine as it floods its ballast tanks. The flow into the tanks is assumed to be inviscid, and the drag on the vertical sinking motion of the craft is neglected. The two coupled differential equations are the generalized form of Newton’s second law and the Bernoulli relation. Time derivatives are converted to spatial derivatives to decouple the equations, and the resulting second-order equation is solved using the Euler–Cromer algorithm. The theory and the method of numerical integration are suitable for an intermediate-level undergraduate course in mechanics that includes some basic fluid dynamics.

Fundamentals of fluid dynamics with an introduction to the importance of interfaces

2017 ◽  
Author(s):  
H. A. Stone
Keyword(s):  
Fluid Dynamics ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Scaling Laws ◽  
Fluid Flows ◽  
Governing Equations ◽  
Soft Interfaces ◽  
Physical Intuition ◽  
Basic Fluid ◽  
Wide Range

The topics discussed are all related to basic fluid mechanics. In these introductory notes I highlight some of the main features of fluid flows and their mathematical characterization. There is much physical intuition encapsulated in the differential equations, and one of our goals is to gain more experience (i) understanding the governing equations and various related principles of kinematics, (ii) developing intuition with approximating the equations, (iii) applying the principles to a wide range of problems, which includes (iv) being able to rationalize scaling laws and quantitative trends, often without having a detailed solution in hand. Where possible we provide examples of the ideas with ‘soft interfaces’ in mind.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

scholarly journals New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1159
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Mihai Postolache
Keyword(s):  
Fluid Dynamics ◽  
Neural Networks ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Delay Differential

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

scholarly journals Non-Linear Neutral Differential Equations with Damping: Oscillation of Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 285
Author(s):  
Saad Althobati ◽  
Jehad Alzabut ◽  
Omar Bazighifan
Keyword(s):  
Differential Equations ◽  
Oscillatory Behavior ◽  
Order Equation ◽  
Second Order ◽  
Riccati Technique ◽  
Neutral Equations ◽  
Neutral Differential Equations ◽  
Main Equation ◽  
Non Linear ◽  
Even Order

The oscillation of non-linear neutral equations contributes to many applications, such as torsional oscillations, which have been observed during earthquakes. These oscillations are generally caused by the asymmetry of the structures. The objective of this work is to establish new oscillation criteria for a class of nonlinear even-order differential equations with damping. We employ different approach based on using Riccati technique to reduce the main equation into a second order equation and then comparing with a second order equation whose oscillatory behavior is known. The new conditions complement several results in the literature. Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

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