Zailan Siri

Stabilization of Steady and Oscillatory Marangoni Instability in Rotating Fluid Layer by Feedback Control Strategy

The effect of feedback control on the onset of steady and oscillatory surface tension-driven (Marangoni) instability in a rotating horizontal fluid layer is considered theoretically using linear stability theory. It is demonstrated that generally the critical Marangoni number for transition from the no-motion (conduction) to the motion state can be drastically increased by the combined effects of feedback control and rotation. Some cases in which increasing the controller gain parameter can be ineffective are also presented. The thresholds and codimension-2 points for the onset of steady and oscillatory convection are determined. We show how the Pr–Ta parameter space is divided into regions in which steady or oscillatory convection is preferred.

A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem.

In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.

Effect of wall inclination on natural convection in a porous trapezoidal cavity.

The present study investigates numerically the effect of wall inclination of a trapezoidal cavity on natural convective flow and heat transfer. The cavity is filled with porous medium. Sinusoidal temperature is applied on the inclined wall and the opposite wall is maintained at a constant temperature. The top and bottom walls are adiabatic. The Darcy model is adopted for porous medium. The governing equations are solved using the finite difference method with various values of wall inclination and Rayleigh number. The heat transfer of the square cavity is found to be higher than that of trapezoidal and triangular cavities.

Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y)

Abstract In this paper three pairs of embedded Runge–Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form y ‴ = f ( x , y ) denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. The second method is the RKD5(4) pair followed by the RKD6(5) pair. The methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates. Variables stepsize codes are developed based on the methods and used to solve a set of special third order problems. Numerical results are compared with the existing embedded Runge–Kutta pairs which require the problems to be reduced into a system of first order ODEs. Numerical results have clearly shown the advantage and the efficiency of the new RKD pairs.

Direct numerical methods for solving a class of third-order partial differential equations

In this paper, three types of third-order partial differential equations (PDEs) are classified to be third-order PDE of type I, II and III. These classes of third-order PDEs usually occur in many subfields of physics and engineering, for example, PDE of type I occurs in the impulsive motion of a flat plate. An efficient numerical method is proposed for PDE of type I. The PDE of type I is converted to a system of third-order ordinary differential equations (ODEs) using the method of lines. The system of ODEs is then solved using direct Runge-Kutta which we derived purposely for solving special third-order ODEs of the form y ? = f ( x , y ) . Simulation results showed that the proposed RKD-based method is more accurate than the existing finite difference method.

Control strategy on the double-diffusive convection in a nanofluid layer with internal heat generation

The influences of feedback control and internal heat source on the onset of Rayleigh–Benard convection in a horizontal nanofluid layer is studied analytically due to Soret and Dufour parameters. The confining boundaries of the nanofluid layer (bottom boundary–top boundary) are assumed to be free–free, rigid–free, and rigid–rigid, with a source of heat from below. Linear stability theory is applied, and the eigenvalue solution is obtained numerically using the Galerkin technique. Focusing on the stationary convection, it is shown that there is a positive thermal resistance in the presence of feedback control on the onset of double-diffusive convection, while there is a positive thermal efficiency in the existence of internal heat generation. The possibilities of suppress or augment of the Rayleigh–Benard convection in a nanofluid layer are also discussed in detail.

Directly Solving Special Second Order Delay Differential Equations Using Runge-Kutta-Nyström Method

Runge-Kutta-Nystrom (RKN) method is adapted for solving the special second order delay differential equations (DDEs). The stability polynomial is obtained when this method is used for solving linear second order delay differential equation. A standard set of test problems is solved using the method together with a cubic interpolation for evaluating the delay terms. The same set of problems is reduced to a system of first order delay differential equations and then solved using the existing Runge-Kutta (RK) method. Numerical results show that the RKN method is more efficient in terms of accuracy and computational time when compared to RK method. The methods are applied to a well-known problem involving delay differential equations, that is, the Mathieu problem. The numerical comparison shows that both methods are in a good agreement.

On a Five-Dimensional Chaotic System Arising from Double-Diffusive Convection in a Fluid Layer

A chaotic system arising from double-diffusive convection in a fluid layer is investigated in this paper based on the theory of dynamical systems. A five-dimensional model of chaotic system is obtained using the Galerkin truncated approximation. The results showed that the transition from steady convection to chaos via a Hopf bifurcation produced a limit cycle which may be associated with a homoclinic explosion at a slightly subcritical value of the Rayleigh number.

Control of Oscillatory of Bénard-Marangoni Convection in Rotating Fluid Layer

The effect of a feedback control on the onset of oscillatory B´enard-Marangoni instability in a rotating horizontal fluid layer is considered theoretically using linear stability theory. It is demonstrated that generally the critical Marangoni number for transition from the no-motion (conduction) to the motion state can be drastically increased by the combined effects of feedback control and rotation.

Effects of chemical reaction on MHD mixed convection stagnation point flow toward a vertical plate in a porous medium with radiation and heat generation

The aim of the present study is to analyze the effects of chemical reaction on MHD mixed convection with the stagnation point flow towards a vertical plate embedded in a porous medium with radiation and internal heat generation. The governing boundary layer equations are transformed into a set of ordinary differential equations using similarity transformations. Then they are solved by shooting technique with Runge-Kutta fourth order iteration. The obtained numerical results are illustrated graphically and the heat and mass transfer rates are given in tabular form. The velocity and temperature profiles overshoot near the plate on increasing the chemical reaction parameter, Richardson number and magnetic field parameter.