Critical withdrawal from a two-layer fluid through a line sink

The problem of withdrawing water through a line sink from a region containing two homogenous layers of different density is considered. Assuming steady, irrotational flow of an ideal fluid, a nonlinear integral equation is derived and solved numerically. Confirmation of earlier research is given, and some new results obtained in which the interface between the two layers rises up and then enters the sink vertically from above, even when the sink is located above the undisturbed level of the interface. A diagram is presented which summarises the work on this problem to this time.Part of this work was carried out while the author was at the Centre for Water Research, University of Western Australia.     

Nonlinear double-diffusive convection with local heat and solute sources

Effects of local heat and solute sources on nonlinear double diffusive convection has been investigated. These sources introduce three new parameters. Dependence of the horizontal cells' size, critical solute Rayleigh number, and heat and solute fluxes on these new parameters is determined. In particular, it is found that local sources can significantly reduce the horizontal cells' size. For the case where the strength of the local solute source is insignificant, the preferred horizontal flow cross-section pattern is found to be squares. On the other hand, if the strength of the local solute source is significant, then hexagons become the preferred pattern when flow amplitude ϵ exceeds a critical value. However, both squares and hexagons can be stable for an even higher value of ϵ. The convective motion is upward at the center of the hexagonal cells if a solute sink acts on the flow. When a solute source acts on the flow, downward motion at the cells' center will occur. Furthermore, the local solute source is also found to be able to provide finite amplitude instability.     

Impacts of nonuniform heat source or sink on MHD mixed convection along an exponentially stretching surface

This paper investigates steady MHD mixed convection flow over an exponentially stretching surface in presence of nonuniform heat source or sink. The dimensional nonlinear partial differential equations governing the flow and temperature fields are expressed in nondimensional form using suitable nonsimilar transformations. Then, numerical solutions are obtained by solving these nondimensional equations using implicit finite difference scheme in combination with Quasilinearization technique. Effects of various physical parameters on velocity and temperature profiles are analyzed numerically. Further, numerical results in terms of the skin-friction coefficient and local Nusselt number are presented graphically and also in tabular form.     

Optimal distribution of a heat source within a thermopenetrator

Heat conduction within a heater of an arbitrary shape is investigated. A mathematical model is presented as a mixed boundary-value problem for the Poisson equation converted into a Fredholm boundary integral equation of the first kind which is solved numerically. A closed-form solution for the particular case of a rectangular heater is also found. Provided that the temperature and heat flux on the heater's boundary are given, the problem is treated as an inverse problem where the heat source distribution within the heater is the unknown function. The existence of the unique solution of this inverse problem is proved. Finally, the problem is solved numerically for a one-dimensional heat source.     

Green functions for initial free-surface flows due to 3D impulsive bottom deflections

The immediate impulsive flow of an incompressible fluid due to a concentrated flux through an otherwise impermeable boundary is investigated analytically in three dimensions. The flow is inviscid and irrotational, and obeys the equipotential condition at the free surface, which is initially horizontal. Various elementary bottom geometries are analyzed: rectangular basins, sloping beaches, semi-cylindrical and hemispherical basins. Special attention is paid to the case of impulsive free-surface flows generated on a uniform sloping beach. A general integral solution is presented and compared against a series solution found for a discrete set of angles. The results are relevant for the modeling of tsunami generation due to rapid bottom deflections.     

Influence of radiation on mixed convection over a wedge in non-Darcy porous medium

The influences of radiation on mixed convection flow of an optically dense viscous fluid along an isothermal wedge embedded in non-Darcy porous medium, in the presence of heat source/sink are numerically investigated. The entire mixed convection regime is covered by a single parameter χ from the pure free convection limit (χ=0) to the pure forced convection limit (χ=1). Forchheimer’s extension is employed to describe the fluid flow in the porous medium and the Rosseland diffusion approximation is considered to describe the radiative heat flux in the energy equation. The governing equations, including internal heat source/sink, are first transformed into a dimensionless form by the nonsimilar transformation and then solved by the Keller box method. The effect of the radiation parameter, mixed convection parameter, Forchheimer number and heat source/sink parameter on the velocity and temperature profiles as well as on the local Nusselt number is presented and analyzed. The results are compared with those known from the literature and excellent agreement between the results is obtained.     

Airflow and temperature distribution in two-dimensional drying bins

The design of a drying or cooling store aims to provide an even airflow distribution, when aerated, for preservation purposes. The airflow in some curved bottom bins are studied in this paper. The flow is modelled, using Darcy's law. A generalized Schwarz-Christoffel transformation is employed to reduce the problem of computing streamlines and isobars of airflow to solving a single nonlinear equation for the flow angle along the wall. Corresponding to different bin shapes, a few computed streamlines and isobars of airflow are presented, showing the effect of changing bottom geometries on the air flow. Heat transfer in such bins is also investigated. Based on an analysis of the far field of airflow, finite-height bins are considered. Analytical solutions of the heat conduction equation in terms of streamlines and isobars are obtained.     

Withdrawal from a fluid of finite depth through a line sink, including surface-tension effects

The steady withdrawal of an inviscid fluid of finite depth into a line sink is considered for the case in which surface tension is acting on the free surface. The problem is solved numerically by use of a boundary-integral-equation method. It is shown that the flow depends on the Froude number, F _B=m(gH ³ _B)^–1/2, and the nondimensional sink depth =H _S/H _B, where m is the sink strength, g the acceleration of gravity, H _B is the total depth upstream, H _S is the height of the sink, and on the surface tension, T. Solutions are obtained in which the free surface has a stagnation point above the sink, and it is found that these exist for almost all Froude numbers less than unity. A train of steady waves is found on the free surface for very small values of the surface tension, while for larger values of surface tension the waves disappear, leaving a waveless free surface. It the sink is a long way off the bottom, the solutions break down at a Froude number which appears to be bounded by a region containing solutions with a cusp in the surface. For certain values of the parameters, two solutions can be obtained.     

Performance evaluation of a two-stage adsorption refrigeration cycle with different mass ratio

The performance of a two-stage adsorption chiller with different mass allocation between upper and bottom beds has been investigated numerically. It is found that the chiller can be driven effectively by the waste heat of temperature 55 °C with the heat sink at environment temperature. Results show that cooling capacity can be improved with the optimum allocation of adsorbent mass to the bottom beds than that to the upper beds. The improvement in Coefficient of Performance (COP) values, however, is less significant. It is also seen that the improvement in cooling capacity is more significant for the relatively higher heat source temperature. It is shown that the cooling capacity can be improved up to 20% if the heat source temperature is 80 °C and the average outlet temperature is fixed at 7 °C.     

Operating Charts for Continuous Sedimentation III: Control of Step Inputs

The main purposes of a clarifier-thickener unit is that it should produce a high underflow concentration and a zero effluent concentration. The main difficulty in the control of the clarification-thickening process (by adjusting a volume flow) is that it is nonlinear with complex relations between concentrations and volume flows via the solution of a PDE – a conservation law with a source term and a space-discontinuous flux function. In order to approach this problem, control objectives for dynamic operation and strategies on how to meet these objectives are presented in the case when the clarifier-thickener unit initially is in steady state in optimal operation and is subjected to step input data. A complete classification of such solutions is given by means of an operating chart (concentration-flux diagram).     

Steady Free Surface Flow of a Fluid-Solid Mixture Down an Inclined Plane

The present work is an extension of the investigations performed by Massoudi and Anand (2001). The free surface flow problem is studied here. Numerical solutions for steady free surface flow of a solid-fluid mixture down an inclined plane are presented. The problem is formulated using the mixture theory framework. The resulting set of three coupled nonlinear differential equations is nondimensionalized. A parametric study is conducted to understand the influence of the dimensionless numbers on the velocity and volume fraction. The maximum fluid velocity is found to decrease with increase in the ratio of the drag force to the viscous forces within the fluid phase (D₁). The fluid phase velocity was found to decrease with increase in the ratio of the drag force to viscous force within the solid component (D₂), and the corresponding solid phase velocity was found to increase.     

Breaking and merging of liquid sheets and filaments

The surface of a sheet of liquid which contracts due to surface tension, breaks, and then pulls apart into two pieces, is calculated. Before breaking, the flow is the self-similar one found by Keller, Milewski and Vanden-Broeck. After breaking, it is the self-similar flow found by Keller and Miksis. A general numerical scheme, which includes the previous ones, is presented and new numerical results are discussed. There is an analogous flow of an axially symmetric liquid filament, but it is not calculated.     

Nonlinear instability of developing streamwise vortices with applications to boundary layer heat transfer intensification through an extended Reynolds analogy

The intent of the present contribution is to explain theoretically the experimentally measured surface heat transfer rates on a slightly concave surface with a thin boundary layer in an otherwise laminar flow. As the flow develops downstream, the measured heat transfer rate deviates from the local laminar value and eventually exceeds the local turbulent value in a non-trivial manner even in the absence of turbulence. While the theory for steady strong nonlinear development of streamwise vortices can bridge the heat transfer from laminar to the local turbulent value, further intensification is attributable to the transport effects of instability of the basic steady streamwise vortex system. The problem of heat transport by steady and fluctuating nonlinear secondary instability is formulated. An extended Reynolds analogy for Prandtl number unity, Pr=1, is developed, showing the similarity between streamwise velocity and the temperature. The role played by the fluctuation-induced heat flux is similar to momentum flux by the Reynolds shear stress. Inferences from the momentum problem indicate that the intensified heat flux developing well beyond the local turbulent value is attributed to the transport effects of the nonlinear secondary instability, which leads to the formation of 'coherent structures' of the flow. The basic underlying pinions of the non-linear hydrodynamic stability problem are the analyses of J. T. Stuart, which uncovered physical mechanisms of nonlinearities that are crucial to the present developing boundary layers supporting streamwise vortices and their efficient scalar transporting mechanisms.     

The inverse problem in creeping film flows

We consider the inverse problem for gravity-driven free surface flows at vanishing Reynolds numbers. In contrast to the direct problem, where information about the underlying topographic structure is given and the steady free surface shape and the flow field are unknown, the inverse problem deals with the flow along unknown topographies. The bottom shape and the corresponding flow field are reconstructed from information at the steady free surface only. We discuss two different configurations for the inverse problem. In the first case, we assume a given free surface shape, and by simplifying the field equations, we find an analytical solution for the corresponding bottom topography, velocity field, wall shear stress, and pressure distribution. The analytical results are successfully compared with experimental data from the literature and with numerical data of the Navier–Stokes equations. In the second inverse problem, we prescribe a free surface velocity and then solve numerically for the full flow domain, i.e. the free surface shape, the topography and simultaneously the wall shear stress and the pressure field. The results are validated with the numerical solution of the corresponding direct problem.     

Axisymmetric Flow in an Oil Reservoir of Finite Depth Caused by a Point sink above an Oil-Water Interface

The flow of a stratified fluid (e.g., oil/water) withdrawn from a vertically confined porous medium through a point sink is considered. The withdrawal tends to cause the oil-water interface to move upwards. So long as the interface is below the well, the less dense fluid (oil) is pumped into the well without the denser fluid (water) until a critical flow rate is reached. The flow is considered to be axisymmetric, and involves a nonlinear boundary condition along the free surface. A boundary-integral equation method (BIEM) is used to find the interface position for different pumping rates. For small flow rates, a small-parameter expansion is derived and the results are compared with numerical solutions to the problem. There exists a critical withdrawal rate beneath which the water does not break through into the sink, this rate depending on the sink location and bottom geometry.     

Cusp flows due to an extended sink in two dimensions

The steady two-dimensioanl potential flow of a finite-depth fluid into an extended or distributed sink, in which the free surface dips to form a cusp above the centre of the sink, is examined. The extended sink is a region where the vertical outflow velocity V is constant and uniform. Numerical solutions for the free-surface profiles are obtained by use of a boundary-integral technique. Solutions are only found for the supercritical case where the Froude numbers are greater than one. In the limiting case where the extended sink width tends to zero, the problem reduces to that of a line sink beneath the free surface, and comparisons are made to existing results for this type of flow.     

Unsteady free convection flow over a continuous moving vertical surface

Summary The problem of heat transfer in the unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid has been investigated. Both constant surface temperature and constant surface heat flux conditions have been considered. The nonlinear coupled partial differential equations governing the flow have been solved numerically using the Keller box method and the Nakamura method which both give closely similar solutions. The results indicate that the cooling rate of the sheet can be enhanced by increasing the buoancy parameter or the velocity of the sheet. It is found that a better cooling performance could be achieved by using a liquid as a cooling medium rather than a gas. The overshoot in the velocity occurs near the surface when the buoyancy parameter exceeds a certain critical value.     

Free convection in hydromagnetic flows in a vertical wavy channel

A study is made of the free convection in hydromagnetic flows in a vertical wavy channel in the presence of heat source or sink. The governing equations for the hydromagnetic fluid flow and the heat transfer are solved subject to the relevant boundary conditions with the assumption that the solution consists of a mean part and a perturbed part. The zeroth-order, the first order and the total solution of the problem are numerically evaluated for various values of the magnetic parameter, heat source/sink parameter, wall-waviness parameter, and free convection parameter. The velocity and the temperature profiles are graphically represented for these parameters. The qualitative features of the hydromagnetic solution are discussed. A comparison is made between the hydromagnetic and the hydrodynamic solutions. The numerical values of the skin friction and the Nusselt number are tabulated for various parameters involved in the analysis. Special attention is given on the characteristic features of the flow, heat transfer, skin friction and the Nusselt numbers at the walls.     

The generalized Riemann problem method for the shallow water equations with bottom topography

This paper extends the generalized Riemann problem method (GRP) to the system of shallow water equations with bottom topography. The main contribution is that the generalized Riemann problem method (J. Comput. Phys. 1984; 55 (1):1–32) is used to evaluate the midpoint values of solutions at each cell interface so that the bottom topography effect is included in numerical fluxes, and at the same step the source term is discretized with an interface method in which only mid‐point values are plugged in. This scheme is well balanced between the flux gradient and bottom topography when incorporating the surface gradient method (SGM) (J. Comput. Phys. 2001; 168 (1):1–25) into data reconstruction step, and it is also suitable for both steady and unsteady flow simulations. We illustrate the accuracy of this scheme by several 1‐D and 2‐D numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.