Narrow Results

The application of nanofluids has exploded in recent decades to improve the local number, mean Nusselt number, and rate of heat transfer. However, boundary layer equations of nanofluid across a flat plate with radiation have not been studied, and therefore this paper studies them mathematically for the first time. For water-based copper and aluminum oxide nanofluids, a similarity solution is presented in this study, and the subsequent system of the ordinary differential equation (ODE) is numerically solved by the Runge–Kutta method in MATLAB. Two different hydraulic boundary conditions are used in the simulations. In the first, the flow across a moving plate and the direction of the flow are analyzed, while in the second, the flow over a nonlinearly moving plate in a still fluid is investigated. The nanoparticle’s boundary layer thickness is found less than the thermal and hydraulics boundary layers. The local Nusselt number and friction factor of both the nanofluids are calculated and compared with the base fluid. The results demonstrate that the friction coefficient is high and the Nusselt number is low for nanoparticles with a high volume fraction. It also revealed that the friction factor for water–aluminum oxide is 16% greater than that for the water–CuO whereas the local Nusselt number for water–aluminum oxide is only 5% more than that for the water–CuO.

In this paper the two-dimensional Euler equations, with a simple chemical reaction model, are used as the governing equations for the detonation problem. The spatial derivatives are evaluated using the fifth-order WENO scheme, and the third-order TVD Runge-Kutta method is employed for the temporal derivative. The characteristics of the two-dimensional detonation in an argon-diluted mixture of hydrogen and oxygen are investigated using Adaptive Mesh Refinement (AMR) method. From computational accuracy point of view, AMR enables the detonation front to be clearer than the method with basic meshes. From the other point of computational time, AMR also saves about half the time as compared with the case of refining the entire field. It is obvious that AMR not only increases the resolution of local field, but also improves the efficiency of numerical simulation.

Electrical circuits based on linear and nonlinear modelling principles have difficulties to meet demands caused by a large amount of data generated and processed. The aim is to examine the existing models from bigeometric calculus point of view to obtain accuracy on the results. This work is an application of bigeometric Runge–Kutta (BRK4) method aiming to solve differential equations with nonzero initial condition. This type of work arises from applications where the systems are defined by ordinary differential equations such as noise, filter, audio, chaotic circuits, etc. Solutions to these types of equations are not always easy. The improvement in this work is obtained by introducing bigeometric calculus in the process of seeking a solution to differential equations. Different classes of input signals are applied as input to the system and processed to determine the accuracy of the output. The applicability is tested against the classical method called Runge–Kutta (RK4). Simulation results confirm the application of BRK4 method in electrical circuit analysis. The new method also provides better results for all types of input signals, i.e., linear, nonlinear, constant or Gaussian.

This work focuses to solve any order of scalar differential equation involved in analog circuit representation. These types of mathematical representations have many applications in analysis and processing such as noise, filter, audio, RLC distributed interconnection (nodes) and transmission lines. Such systems are represented with scalar type differential equations and use numerical method to find a solution. One of the most successful methods is the fourth-order Runge–Kutta. This study introduced a multiplicative version of Runge–Kutta (MRK4) method. The performance analysis of the MRK4 is examined based on the error analysis method. The MRK4 method is applied to solve equations representing the linear and the nonlinear type systems. Results indicate the MRK4 to be superior with respect to the RK4 method.

Finite difference approximations for the convection equation are developed, which exhibit enhanced stability limits for explicit Runge–Kutta integration. Stability limits are increased by adding artificial dissipation terms, which are optimized to yield greatest stable time steps. For the artificial dissipation terms, symmetric finite difference approximations of even-order derivatives are used with differencing stencils equal to the convective stencils. The spatial discretization inclusive of the added dissipation term is shown to be consistent with a first derivative. The formal order of accuracy in space is decreased by one order, while the order of time integration is not affected. As a result, the time step limits of originally stable Runge–Kutta integration is increased, for some combinations of spatial discretization and time integration by a factor of two. Algorithms, which are unstable without damping are stabilized. The dispersion properties of the algorithms are not influenced by the proposed damping terms. Spectral analysis of the algorithms show very low dissipation error for dimensionless wave numbers k Δ x < 0.5. Stability conditions based on von Neumann stability analysis are given for the proposed schemes for explicit Runge–Kutta time integration of orders up to ten.

The following sections are included:

• Introduction
• Formulating Classical Mechanics
• Numerical Methods for Differential Equations
  • The Størmer-Verlet Centered-Difference Algorithm
  • Runge-Kutta Integration
  • Compiler and Graphics Software
• Thermomechanical Equilibrium from Gibbs’ Entropy
• Development of Newtonian Molecular Dynamics
• “Corresponding States” At and Not At Equilibrium
  • van der Waals’ Equation and Corresponding States
• Gauss’ and Nosé-Hoover Thermal Mechanics
  • Example Problem in Gaussian Isokinetic Mechanics
  • “Stiff” Equations and the Nosé-Hoover Oscillator
  • Adaptive Integration Solves Nosé’s Equations
  • Is the Restriction $\mathcal{H}$ = 0 Signi?cant?
• Lattice Dynamics of One-Dimensional Chains
• A Classical ManyBody Example Problem
• Simulating Hydrodynamic Manybody Properties
  • The Virial Theorem for the Pressure Tensor P
  • The Pressure Tensor is Necessarily Symmetric
  • The Heat Theorem for the Heat-Flux Vector Q
• Summary of Lecture 1
• References