Yong-Hong Wu

Analysis of flux flow and the formation of oscillation marks in the continuous caster

In the industrial process of continuous steel casting, flux added at the top of the casting mould melts and forms a lubricating layer in the gap between the steel and the oscillating mould walls. The flow of flux in the gap plays an essential role in smoothing the casting operation. The aim of the present work is to better understand the mechanics of flux flow, with an emphasis on such problems as how the flux actually moves down the mould, the physical parameters governing the consumption rate of the flux and the geometry of the lubricating layer. The problem considered is a coupled problem of liquid flow and multi-phase heat transfer. In the first part of the paper, the formation of the lubricating layer is analysed and a set of equations to describe the flux flow is derived. Then, based on an analysis of the heat transfer from the molten steel through the lubricating layer to the mould wall, a system of equations correlating the temperature field in the steel and flux with the geometry of the lubricating layer is derived. Subsequently, the equations for the flux flow are coupled with those arising from heat-transfer analysis and then a numerical scheme for the calculation of the consumption rate of flux, the geometry of the lubricating layer and the solidification surface of the steel is presented.

Plastic flows of granular materials of shear index n-I. Yield functions

Granular materials fail due to frictional slip between particles when the shear component of stress τ attains a critical value which depends on the normal component of stress σ. A number of authors have investigated the so-called Warren Spring equation (τc)n = 1 − (σt) where c, t and n are positive constants which are referred to as the cohesion, tensile strength and shear index respectively and known numerical values of the shear index indicate that for certain materials n lies between the values 1 and 2. Here, the yield function in terms of principal stresses corresponding to the Warren Spring equation is derived and bounding external and internal yield cones are deduced. Other than the Coulomb-Mohr yield function arising from n = 1, the yield function corresponding to the shear index n = 2 turns out to be the simplest for n lying in the range of physical interest and in Part II of the paper simple plane and axially symmetric flows are obtained for this special case. This means that the well known Coulomb-Mohr yield function (n = 1) and that for n = 2 provide idealized limiting behaviour for many real granular materials. Finally, some mathematical details which relate to the general yield function of the Warren Spring equation are noted in the Appendix.

Some axially symmetric flows of Mohr–Coulomb compressible granular materials

In this paper we consider a number of axially symmetric flows of compressible granular materials obeying the Coulomb–Mohr yield condition and the associated flow rule. We pay particular attention to those plastic régimes and flows not included in the seminal work of Cox, Eason & Hopkins (1961). For certain plastic régimes, the velocity equations uncouple from the stress equations and the flow is said to be kinematically determined. We present a number of kinematically determined flows and the development given follows the known solutions applicable to the so-called ‘double-shearing’ model of granular materials which assumes incompressibility and for which the governing equations are almost the same. Similarly, for certain other plastic régimes the stresses may be completely determined without reference to the velocity equations and these are referred to as statically determined flows. In the latter sections of the paper we examine statically determined flows arising from the assumption that the shear stress in either cylindrical or spherical polar coordinates is zero. In the final section we present a numerical solution, which incorporates gravitational effects, for the flow of a granular material in a converging hopper. In addition, we examine the Butterfield & Harkness (1972) modification of the double-shearing model of granular materials which formally includes both the double-shearing theory and the Coulomb–Mohr flow rule theory as special cases. Moreover, for kinematically determined régimes, the velocity equations are the same apart from a different constant, while for statically determined régimes the governing velocity equations are slightly more complicated, involving another constant which is a different combination of the basic physical parameters. Thus some of the solutions presented here can be immediately extended to this alternative theory of granular material behaviour and therefore the prospect arises of devising experiments which might validate or otherwise one theory or the other.

An Enthalpy Control Volume Method for Transient Mass and Heat Transport with Solidification

A single domain enthalpy control volume method is developed for solving the coupled fluid flow and heat transfer with solidification problem arising from the continuous casting process. The governing equations consist of the continuity equation, the Navier–Stokes equations and the convection–diffusion equation. The formulation of the method is cast into the framework of the Petrov–Galerkin finite element method with a step test function across the control volume and locally constant approximation to the fluxes of heat and fluid. The use of the step test function and the constant flux approximation leads to the derivation of the exponential interpolating functions for the velocity and temperature fields within each control volume. The exponential fitting makes it possible to capture the sharp boundary layers around the solidification front. The method is then applied to investigate the effect of various casting parameters on the solidification profile and flow pattern of fluids in the casting process.

Mathematical analysis of long-wave breaking on open channels with bottom friction

In this paper we study the breaking of long waves propagating along an open channel with linear friction on the bottom. The equations governing the wave propagation consist of a pair of first-order nonlinear hyperbolic partial differential equations (PDEs). We first transformed the PDEs into a pair of ordinary differential equations (ODEs) along the characteristic directions by means of a pair of Riemann invariants. By analyzing the ODEs, we found that the breaking of waves can be identified by the singularity of the derivative of the Riemann invariants. Thus, we derived an analytical solution for the derivative of the Riemann invariants. Then, a breaking criterion and an analytical formula for the estimation of breaking time were developed and validated through numerical experiments. It is also shown in the paper that the present model includes the previous model neglecting bottom friction as a special case.

Plastic flows of granular materials of shear index n—II. Plane and axially symmetric problems for n = 2

In part I the general three-dimensional yield function is formulated in terms of principal stresses for the so-called Warren Spring equation (τc)n = 1 − (σt) which for a granular material at yield, relates the shear component of stress τ with the normal component σ (assumed positive in tension) where c, t and n are various positive constants such that 1 ⩽ n ⩽ 2. For n > 1 the yield function for n = 2 turns out to be the simplest and in Part II simple plane and axially symmetric flows are examined for this special case. In general, for the Warren Spring equation with n ≠ 1, the stress and velocity equations do not uncouple and accordingly even the determination of the simplest solutions is a highly non-trivial matter, for which ultimately a numerical scheme must be adopted. For both plane and axially symmetric flows the basic governing equations are formulated and a number of special solutions are attempted which are subsequently solved numerically using a modified Euler scheme and a number of the solutions presented are illustrated graphically.

Probability distribution of random wave forces in weakly nonlinear random waves

Abstract Based on the second-order random wave theory, the joint statistical distribution of the horizontal velocity and acceleration is derived using the characteristic function expansion method. From the joint distribution and the Morison equation, the theoretical distributions of drag forces, inertia forces and total random wave forces are determined. The distribution of inertia forces is Gaussian as that derived using the linear wave model, whereas the distributions of drag forces and total random forces deviate slightly from those derived utilizing the linear wave model. It is found that the distribution of wave forces depends solely on the frequency spectrum of sea waves associated with the first order approximation and the second order wave–wave interaction.

Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework

This paper investigates the asset liability management problem for an ordinary insurance system incorporating the standard concept of proportional reinsurance coverage in a stochastic interest rate and stochastic volatility framework. The goal of the insurer is to maximize the expectation of the constant relative risk aversion (CRRA) of the terminal value of the wealth, while the goal of the reinsurer is to maximize the expected exponential utility (CARA) of the terminal wealth held by the reinsurer. We assume that the financial market consists of risk-free assets and risky assets, and both the insurer and the reinsurer invest on one risk-free asset and one risky asset. By using the stochastic optimal control method, analytical expressions are derived for the optimal reinsurance control strategy and the optimal investment strategies for both the insurer and the reinsurer in terms of the solutions to the underlying Hamilton-Jacobi-Bellman equations and stochastic differential equations for the wealths. Subsequently, a semi-analytical method has been developed to solve the Hamilton-Jacobi-Bellman equation. Finally, we present numerical examples to illustrate the theoretical results obtained in this paper, followed by sensitivity tests to investigate the impact of reinsurance, risk aversion, and the key parameters on the optimal strategies.