Natural convection from a semi-infinite flat plate inclined at a small angle to the horizontal in a saturated porous medium

Summary The free convection boundary layer on a semi-infinite heated flat plate which is inclined at a small angle to the horizontal in a saturated porous medium is considered. When the plate is inclined upwards, series solutions, one valid near the leading edge and the other valid at large distances from it, are obtained. Sufficient terms in the leading edge solution are taken so that the two asymptotic solutions may be joined. When the plate is inclined downwards the series solution valid near the leading edge is again obtained and sufficient terms are again employed to investigate the nature of the boundary layer separation. There is no evidence of a singularity at the separation point and a mathematical explanation of the behaviour at separation is presented.With 10 Figures     

The effects of MHD on free-convection flow past a semi-infinite isothermal inclined plate

An analysis of the effects of MHD on a two-dimensional free-convective flow of a viscous incompressible fluid past a semi-infinite isothermal inclined plate is carried out. The effects of viscous dissipation and the induced magnetic field are assumed to be negligible. A more accurate, unconditionally stable, and fast converging implicit finite difference scheme is used to solve the dimensionless governing equations. The effects of MHD on velocity, temperature, local and average skin friction, and local and average Nusselt number are studied. It is observed that the magnetic field parameter has a retarding effect on the velocity. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 81, No. 4, pp. 697–704, July–August, 2008.     

Singularities of an inclined crack terminating at the bi-material interface in a Reissner plate

On the basis of Reissner’s plate theory, the stress singularities at the tip of an arbitrarily inclined semiinfinite crack terminating at the interface of two dissimilar materials are investigated in the present paper. Using the eigenfunction expansion method, the eigenequation of the corresponding problem is derived explicitly by directly solving the governing equations of Reissner’s plate theory in terms of three generalized displacement components. In this paper, the focus is on the calculation of the singularity order as a fundamental quantity in fracture mechanics. The singularity orders of the moments and shear force at the crack tip are determined by the dominant eigenvalues whose real parts lie between 0 and 1. The influences of the bi-material parameters and the crack inclination angle on the moment and shear force singularity orders are discussed in detail. Specifically, the variations of the shear force singularity order with the bi-material parameter and the crack inclination angle are examined in detail. It is proved that the shear force singularity order is a completely monotonic function of the bi-material parameter and the inclination angle. Some numerical results are given in order to prove the validity of the present study.     

Motion of a nonlinearly viscoplastic fluid jet along a plate moving at an angle to the horizon

Stationary stabilized flow of a plane fluid jet along a moving inclined plane is considered.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 52, No. 4, pp. 597–605, April, 1987.     

Potential flow of a film down an inclined plate

Summary The flow of a liquid film along a semi-infinite flat plate due to gravity is considered, the fluid being assumed inviscid and incompressible. When the Froude numberFr, based on the initial film thickness and velocity, is large compared to unity, solutions can be found by the method of matched asymptotic expansions. The fluid speed and deflection, and the pressure gradient are found toO(Fr ^–2). Hydraulic theory enters as the first term in the outer expansion, which is valid far downstream from the leading edge. When the liquid falls vertically, the motion represents half of a freely falling jet.     

Free convection of gas mixture above a flat horizontal plate in constant-velocity flow

The problem of mixed convection for a mixture of viscous heat-conducting gases above a horizontal plate is solved by using the method of integral relations.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 31, No. 3, pp. 523–530, September, 1976.     

Free convective heat transfer at a vertical semiinfinite plate

The effect of the boundary layer at the leading edge on heat transfer near a vertical semiinfinite heated plate is determined by means of matched asymptotic expansions. The criterial relation for air is in good agreement with existing experimental data.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 1, pp. 32–39, July, 1977.     

Generalized analytical solution to wave interaction with submerged multi-layer horizontal porous plate breakwaters

This study deals with the flow induced by a rigid flat plate of finite length, initially touching a horizontal water surface, when it starts to move upwards with constant acceleration. In the present model, negative hydrodynamic pressures on the lower (wetted) surface of the plate are allowed, and thus, the water follows the plate due to the resulting suction force. The acceleration of the plate and the plate length are such that gravity, surface tension and viscous effects can be neglected during the early stages of the motion. Under these assumptions, the initial two-dimensional, potential flow caused by the plate lifting is obtained by using the small-time expansion of the velocity potential. This small-time solution is not valid close to the plate edges, as it predicts there singular flow velocities and unbounded displacements of the water-free surface. It is shown that close to the plate edges the flow is nonlinear and self-similar to leading order. This nonlinear flow is computed by the boundary-element method combined with a time-marching scheme. The numerical time-dependent solution approaches the self-similar local solution with time.     

Operation of thermosyphons at small angles of inclination to the horizontal

Thermosyphons with internal coaxial inserts in the heating zone are investigated. Their operation at small angles of inclination to the horizontal is analyzed. The optimum design of the inserts that ensures operation of the thermosyphons at an angle of inclination of 5° is determined.Institute of Technical Thermophysics, Academy of Sciences of the Ukraine, Kiev. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 67, Nos. 3–4, pp. 258–260, September–October, 1994.     

Flow past an accelerated horizontal plate in a rotating fluid

Summary A semi-infinite mass of an incompressible viscous fluid bounded by an infinite flat plate is initially rotating with uniform angular velocity about an axis normal to the plate. An analysis is presented for the subsequent flow when the plate started impulsively from rest relative to the rotating fluid moves with uniform acceleration in its own plane. It is found that when 0, the velocity profiles for varying times are nonsimilar in contrast to the velocity profiles which are similar in the absence of rotation (=0). At a given instant, the velocity component along the direction of motion of the plate decreases with an increase in rotation but the transverse velocity component (induced by the Coriolis force) increases with increasing rotation. Due to the gradual thinning of the boundary layer with rotation, both the skin-friction components along and transverse to the direction of motion of the plate increase with increasing rotation. A study of the asymptotic behavior of the velocity field for large time reveals a novel feature of the flow; it develops inertial oscillations with frequency 2, which grow with time. This behavior has not been reported in the absence of rotation.     

Free convective flow in an enclosure with a cooled inclined upper surface

Natural convective air flow in an enclosure with a horizontal lower wall, vertical side-walls and a straight inclined top wall has been numerically studied. The lower wall is at a uniform high temperature while the top wall is cooled to a uniform lower temperature. The temperature of the side-walls varies in a prescribed way between the temperatures of the bottom and top walls. This flow situation is related to that occurring in a proposed system for use in developing countries for drying crops such as corn and rice. It has been assumed that the flow is steady, laminar, and two-dimensional and that the fluid properties are constant except for the density change with temperature that gives rise to the buoyancy forces. The governing equations have been expressed in terms of stream function and vorticity and written in dimensionless form. The finite element method has been used to obtain the solution to these dimensionless equations. Results have been obtained for enclosures with aspect ratios of between 0.25 and 1 for top surface angles of inclination of between 0 and 45° and for Rayleigh numbers, based on the enclosure width, of between 1000 and 10⁷. The results have been used to study the effect of changes in the governing parameters on the flow pattern in the enclosure and on the mean heat transfer rate to the upper surface.List of symbols A aspect ratio, h/w - h mean height of cavity - k thermal conductivity - Nu local Nusselt number based on w - mean Nusselt number for top surface based on w - n n/w - n coordinate measured normal to boundary surface considered - p pressure - Pr Prandtl number - q local dimensionless heat transfer rate - mean heat transfer rate to top surface - Ra Rayleigh number based on w - T dimensionless temperature - T temperature - T _H temperature of bottom surface - T _C temperature of top surface - u velocity component in x direction - v velocity component in y direction - w width of cavity - x dimensionless x coordinate - x horizontal coordinate position - y dimensionless y coordinate - y vertical coordinate position - thermal diffusivity This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.     

Natural convection flow from an isothermal horizontal circular cylinder in presence of heat generation

Natural convection laminar boundary layer flow from a horizontal circular cylinder with a uniform surface temperature at presence of heat generation has been investigated. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear systems of partial differential equations are solved numerically applying two distinct methods namely (i) implicit finite difference method together with the Keller box scheme and (ii) series solution technique. The results of the surface shear stress in terms of the local skin friction and the surface rate of heat transfer in terms of the local Nusselt number for a selection of the heat generation parameter γ (= 0.0, 0.2, 0.5, 0.8, 1.0) are obtained and presented in both tabular and graphical formats. Without effect of the internal heat generation inside the fluid domain for which we take γ = 0.0, the present numerical results show an excellent agreement with those of Merkin [J.H. Merkin, Free convection boundary layer on an isothermal horizontal circular cylinders, in: ASME/AIChE, Heat Transfer Conference, St. Louis, MO, August 9–11, 1976]. The effects of γ on the fluid velocity, temperature distribution, streamlines and isotherms are examined.     

The effect of the horizontal vibrations on natural heat transfer from an isothermal array of cylinders

A numerical investigation is carried out to study an unsteady laminar natural convection heat transfer caused by an array of isothermal oscillating circular cylinders. Under oscillating conditions, flow and thermal fields are categorized into a class of moving boundary problems. In this study, the moving interfaces between the fluid and cylinders have been considered. The numerical model used in the present paper, is based on a 2D Navier–Stokes momentum and energy equations for an incompressible flow solver on an unstructured grid. Discretization of the governing equations including continuity, momentum and energy equations is achieved through a finite element scheme based on characteristic based split algorithm using the arbitrary Lagrangian–Eulerian approach to satisfy boundary movement. Besides a dual time stepping method is employed to capture unsteady flow and thermal characteristics. The working fluid is designated a Prandtl number of 0.71(air) and assumed to be incompressible with constant physical properties. The radiation, viscous dissipation and pressure work are also assumed to be negligible throughout this investigation. Fluid flow and heat transfer characteristics are examined in the domain of the Rayleigh number, cylinders spacing, amplitude, and frequency of oscillations such that: 10³ ≤ Ra ≤ 10⁵, 2 ≤ s/d ≤ 4, 0.5 ≤ l ≤ 2, and 0.1 ≤ f ≤ 0.4. The obtained results reveal that increment of Rayleigh number and cylinders’ spacing augment the average Nusselt of each cylinder as well as higher oscillation amplitude and frequency. Moreover, it was found that horizontal vibration makes vortices appear in the left and right area of the cylinders. These vortices reduce heat transfer from two upper cylinders.     

Free convection on a horizontal plate in a saturated porous medium with prescribed heat transfer coefficient

Summary The free convection on a horizontal plate embedded in a saturated porous medium is considered assuming that the plate is subjected to a prescribed temperature or a prescribed heat flux or a prescribed heat transfer coefficient. By similarity transformation the governing equations are reduced to identical coupled equations for all the three cases with three common boundary conditions and one boundary condition depending on the thermal boundary condition imposed. It is shown that there is no need to solve the three boundary value problems independently and that the solution for one case can be used to obtain the solution for any other case by a simple algebraic method.Notation A transition parameter used in Eq. (19) - C function ofx defined in Eq. (11.1) - f dimensionless stream function - g acceleration due to gravity - k thermal conductivity - K permeability of the porous medium - N heat transfer coefficient - p fluid pressure - p _e ambient pressure - PT prescribed temperature - PHF prescribed heat flux - PHTC prescribed heat transfer coefficient - q _w surface heat flux - R function ofx defined in Eq. (11.2) - T temperature - T _w plate temperature - T _e ambient temperature - T _w temperature difference=T _w-T _e - u velocity in thex-direction - v velocity in they-direction - x coordinate along the plate in the upward direction - y coordinate normal to the plate - equivalent thermal diffusivity - coefficient of thermal expansion - dimensionless similarity variable - dimensionless temperature - exponent inC - viscosity - kinematic viscosity - fluid density - _e ambient density - stream function     

Stress solution for a crack at an arbitrary angle to an interface

Two-dimensional elasticity solution and the stress intensity factors are determined for a finite crack in one of the materials of a bimaterial composite. The crack has an arbitrary orientation and distance from the straight interface. The solution for general stress boundary conditions on the crack surface is presented in the form of coupled Fredholm integral equations of the second kind. Numerical values of the stress intensity factors are computed for various crack orientations, distances from the interface, and different combinations of material properties when the boundary conditions are uniform pressure and uniform shear stress.     

Fracture angle and strain-energy-density-factor of a crack at hole at an arbitrary angle

For both the maximum stress criterion and strain-energy-density-factor (S) theory, fracture angle (the initial angle of crack growth) − θ_o is predicted by using opening and sliding mate stress intensity factors, k₁ and k₂. These theoretical predictions are consistent with experimental fracture angles.

For the S theory, the crack spreads in the negative θ_o-direction in a plane for which S is a minimum, S_min. This quantity was obtained analytically. The experimental data of the critical S (S_cr) on plexiglass fracture specimens remains essentially constant.