Steady free convection boundary layer over a vertical flat plate embedded in a porous medium filled with water at 4 °C

This paper reports a theoretical investigation of the boundary layer flow over a vertical flat plate embedded in a porous medium filled with water near the vicinity of its density maximum associated with the temperature of 3.98 °C at atmospheric pressure. The study aims at determining similarity solutions of the governing boundary layer equations for a class of problems where the variable wall temperature (VWT), variable heat flux (VHF), or variable heat transfer coefficient (VHTC), vary as power functions of the distance from the leading edge of the plate. The existence and uniqueness of the solutions are considered and studied. The analytical and numerical solutions of the similarity form of the boundary layer equations yield velocity and temperature profiles as well as values of the stream function at the edge of the boundary layer, the heat transfer coefficient and the temperature on the plate.     

Free-Forced Convection from a Heated Longitudinal Horizontal Cylinder

Abstract

Finite-difference solutions of a longitudinal three-dimensional boundary layer along a heated horizontal cylinder are presented for Pr = 1 and 10. The numerical results are compared with earlier asymptotic solutions. The comparison shows that the asymptotic solution is valid only for a narrow region close to the leading edge. Downstream from this narrow region, the asymptotic solution overpredicts the buoyancy effect along the bottom of the cylinder and underpredicts it along the top. The numerical solutions indicate that the flow becomes free-convection dominant far downstream from the leading edge even when Gr/Re² is small.     

Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface

The present study deals with new similarity solution of steady mixed convection boundary layer flow over a permeable surface for convective boundary condition. It has been shown that a self similar solution is possible when the mass transfer velocity at the surface of the plate varies like x^−1/2, where x is the distance from the leading edge of the solid surface. Two point boundary value problem governed by non-linear coupled ordinary differential equations have been solved numerically using implicit finite difference scheme in combination with the quasi-linearization technique. It is interesting to note that dual solutions exist for buoyancy assisting flow, besides that usually reported in literature for buoyancy opposing flow. Further, the buoyancy assisting force causes considerable overshoot in the velocity profile and the Prandtl number strongly affects the thermal boundary layer thickness including the surface heat transfer rate.     

Heat transfer with contact resistanceTransfert thermique avec resistance de contactWärmeübergang mit kontaktwiderstandTeплoпereнos пrи кoнтaктнoм soпroтивлeнии

The thermal contact resistance of two solids touching each another was analyzed with particular reference to the shape of a single heat channel. This channel was assumed to have a cylindrical contour whose radius near the interface decreases gradually to the contact area forming a truncated cone.The contact resistance of the interface was found as a function of the cone angle, the ratio of the radii of the truncated cone and the properties of the materials involved.From a numerical solution it was found that the contact resistance can be described by the properties of the materials, the number of contact areas and a single function of the radii ratio and that, for small values of cot γ, the contact resistance is almost insensitive to the cone angle.     

Double-diffusive natural convection along a vertical wavy truncated cone in non-Newtonian fluid saturated porous media with thermal and mass stratification

This paper studies the double-diffusive natural convection near a vertical wavy truncated cone in a non-Newtonian fluid saturated porous medium with thermal and mass stratification. The surface of the truncated cone is kept at constant wall temperature and concentration. A coordinate transformation is employed to transform the complex wavy surface to a smooth surface, and the obtained boundary-layer equations are then solved by the cubic spline collocation method. Effects of thermal and concentration stratification parameters, Lewis number, buoyancy ratio, power-law index, and wavy geometry on the heat and mass transfer characteristics are studied. Results show that the streamwise distributions of the local Nusselt number and the local Sherwood number are harmonic curves with a wave number twice the wave number of the surface of the vertical wavy truncated cone. An increase in the power-law index leads to a smaller fluctuation of the local Nusselt and Sherwood numbers. Moreover, increasing the thermal and concentration stratification parameter decreases the buoyancy force and retards the flow, thus decreasing the heat and mass transfer rates between the fluid and the wavy surface of the vertical truncated cone.     

井下换热器周围对流型地热储传热特性

采用Darcy自然对流传换模型分析了同轴管式井下换热器（DCHE）在对流型地热储中的换热物性，在假设井下换热器外壁温度变化与其高度成幂函数关系的基础上，建立了数学模型并得出了相似妥，为其进一步工程应用奠定了较好的理论基础。     

COMPUTATION OF BOUNDARY LAYER TRANSITION USING LOW-REYNOLDS-NUMBER TURBULENCE MODELS

A literature review shows that, for the commonly used turbulence models, the predicted location of boundary layer transition is very sensitive to the initial profiles of turbulence quantities and starting location of calculation. To eliminate these effects, two independent solution approaches are proposed: (1) to solve the boundary layer equations over a flat plate with the starting location of calculation very close to the leading edge of the plate, and (2) to solve the elliptic Navier-Stokes equations over the whole plate, including the leading edge and some region upstream of it. Computations show both approaches lead to identical results. Three well-known low-Reynolds-number (LRN) turbulence models are evaluated with respect to the transition on a flat plate. None of the models are able to predict the quantitative aspects of transition correctly without an ad hoc adjustment. A satisfactory new turbulence model is presented in a companion paper.     

Natural convection about a vertical plate embedded in a bidisperse porous medium

The classical Cheng–Minkowycz study of convection past a vertical plate embedded in a porous medium has been extended to the case of a bidisperse porous medium (BDPM). The boundary layer analysis leads to expressions for the velocity and temperature fields in terms of a geometrical parameter, an inter-phase momentum transfer parameter, a thermal diffusivity ratio, a permeability ratio, a thermal conductivity ratio, and an inter-phase heat transfer parameter. For the leading edge region, and for an inner layer, a similarity solution is obtained numerically. This involves the first four parameters, each of which is a characteristic of the BDPM.     

Natural convection of a micropolar fluid from a vertical truncated cone with power-law variation in surface temperature

This work presents a boundary-layer analysis about the natural convection heat transfer near a vertical truncated cone with power-law variation in surface temperature in a micropolar fluid. The transformed boundary layer governing equations are solved by the cubic spline collocation method. Results for local Nusselt numbers are presented as functions of vortex viscosity parameter, the surface temperature exponent, and the Prandtl number. The heat transfer rates of the truncated cones with higher surface temperature exponents are higher than those with lower surface temperature exponents. Moreover, the heat transfer rate from a vertical truncated cone in Newtonian fluids is higher than that in micropolar fluids.     

The Cheng–Minkowycz problem for natural convection about a vertical plate embedded in a tridisperse porous medium

The classical Cheng–Minkowycz study of convection past a vertical plate embedded in a porous medium has been extended to the case of a tridisperse porous medium (TDPM). The boundary-layer analysis leads to expressions for the velocity and temperature fields in terms of two geometrical parameters, two inter-phase momentum transfer parameters, two thermal diffusivity ratios, two permeability ratios, two thermal conductivity ratios, and two inter-phase heat transfer parameters. For the leading edge region, and for an inner layer, a similarity solution is obtained. This involves the first eight parameters, each of which is a characteristic of the TDPM.     

Double diffusion from a vertical truncated cone in a non-Newtonian fluid saturated porous medium with variable heat and mass fluxes

This paper studies the double diffusion flow over a vertical truncated cone with variable heat and mass fluxes in a porous medium saturated with non-Newtonian power-law fluids. A coordinate transformation is used to obtain the nonsimilar governing equations, and the transformed boundary layer equations are then solved by the cubic spline collocation method. Results for local surface temperature and concentration are presented as functions of power-law indexes, exponents for variable heat and mass fluxes, buoyancy ratios, and Lewis numbers. The local surface temperature and concentration of the truncated cone decrease as the exponents for variable heat and mass fluxes are increased. Moreover, a decrease in the power-law index of fluids tends to decrease the local surface temperature and concentration of the truncated cone.     

Heat transfer from a rotating cone or disk to fluids of any prandtl number

A new similarity variable is proposed for the analysis of laminar boundary-layer heat transfer from a rotating cone or disk to fluids of any Prandtl number. Very accurate similarity solutions for both the isothermal and uniform-flux cases are obtained over a very wide range of Prandtl number from 0.001 to infinity. Dimensionless temperature profiles in this range of Prandtl number are very similar. A heat transfer correlation equation of high accuracy for all Prandtl numbers is given.     

Irreversibility analysis for Casson thermo-fluidics inside a cone: Cattaneo–Christov heat flux

In this communication, thermodynamic irreversibility arising in dissipative Casson fluid flow inside a cone is investigated. The boundary–layer flow is considered wherein the motion is caused due to a point sink at the cone's vertex and the movement of the wall of the cone. The wall of the cone is subjected to mass transpiration that alters the flow and thermal regime. The cone having fluid-saturated porous medium experiences Cattaneo–Christov heat flux. The configuration admits a similarity transformation that yields a boundary value problem (BVP) comprising an ordinary differential equation. The BVP is treated by the fourth-order R-K method along with the shooting algorithm. The system yields a dual solution for momentum and energy, which gives rise to a dual regime for entropy distribution. Numerical computations provide quantities of interest viz. velocity and temperature distributions, skin friction coefficient, Nusselt number, and entropy distribution. Phenomena exhibited through profiles/tables for velocity, temperature, entropy, streamlines, and other quantities of interest reveal interesting results.     

The effect of longitudinal surface waves on free convection from vertical surfaces in porous media

In this paper we consider the effect of longitudinal surface waves on the thermal boundary layer flow induced by a vertically aligned heated surface embedded in a porous medium. The full governing equations are considered and the boundary layer equations are derived in a systematic way. It is found that, for a wide range of values of x, the distance from the leading edge, the boundary layer equations for the three—dimensional flowfield are satisfied by a two-dimensional similarity solution.     

Hypersonic rarefied flow past an insulated flat plate with suction/injection

Von Kármán-Pohlhausen method has been used to study the rarefied hypersonic flow past an insulated flat plate with slip velocity and suction or injection on the surface. Analytical solutions are obtained for the leading edge and strong interaction regions. The governing equations are, then numerically integrated. Numerical results agree with the analytical results in the leading edge and strong interaction regions and provide solution in the intervening zone. Effects of slip and suction or injection on various flow quantities are discussed.     

Natural convection heat and mass transfer from a vertical truncated cone in a porous medium saturated with a non-Newtonian fluid with variable wall temperature and concentration

This work presents a boundary-layer analysis about the natural convection heat and mass transfer near a vertical truncated cone with variable wall temperature and concentration in a porous medium saturated with non-Newtonian power-law fluids. A coordinate transform is used to obtain the nonsimilar governing equations, and the transformed boundary-layer equations are solved by the cubic spline collocation method. Results for local Nusselt numbers are presented as functions of power-law indexes, surface temperature and concentration exponents, buoyancy ratios, and Lewis numbers. The heat and mass transfer rates of the truncated cones with higher surface temperature and concentration exponents are higher than those with lower exponents. Moreover, an increase in the power-law index of fluids tends to decrease the heat and mass transfer from a vertical truncated cone in a porous medium saturated with non-Newtonian power-law fluids.     

A New Multistep Technique Based on the Nonuniform Rational Basis Spline Curves for Nonlinear Transient Heat Transfer Analysis of Functionally Graded Truncated Cone

A novel multistep method based on the nonuniform rational basis spline curves is developed for the solution of a system of nonlinear differential equations. Efficiency of the presented method in terms of convergence and accuracy is demonstrated through the two-dimensional nonlinear transient heat transfer analysis of multilayered functionally graded truncated cone subjected to an internal thermal load with radiative–convective boundary condition at outer surface. In order to discretize the governing equations in the spatial domain, a layerwise-differential quadrature method is implemented. This transforms the partial differential equations into a system of nonlinear ordinary differential equations in the temporal domain, which is then solved using the presented technique. Comparative studies between predictions of the presented multistep technique and various existing numerical methods demonstrate superior computational efficiency of the method with low computational cost. High efficiency of this method indicated that the presented multistep approach can be successfully employed in various nonlinear problems. Finally, the effects of different parameters on the transient temperature of the truncated cone are studied. It is expected that the presented multistep method will be applied on various practical engineering problems in future studies.     

Soret and Dufour effects on heat and mass transfer by natural convection from a vertical truncated cone in a fluid-saturated porous medium with variable wall temperature and concentration

This work studies the Soret and Dufour effects on the natural convection heat and mass transfer near a vertical truncated cone with variable wall temperature and concentration in a fluid-saturated porous medium. A coordinate transform is used to obtain the nonsimilar governing equations, and the transformed boundary layer equations are solved by the cubic spline collocation method. Results for local Nusselt number and the local Sherwood number are presented as functions of Soret parameters, Dufour parameters, surface temperature and concentration exponents, buoyancy ratios, and Lewis numbers. Results show that increasing the Dufour parameter tends to decrease the local Nusselt number, while it tends to increase the local Sherwood number. An increase in the Soret number leads to an increase in the Nusselt number and a decrease in the Sherwood number from a vertical truncated cone in a fluid-saturated porous medium. The local Nusselt number and the local Sherwood number of the truncated cones with higher surface temperature and concentration exponents are higher than those with lower exponents.     

AN EIGENFUNCTION APPROACH TO SINGULAR THERMAL STRESSES IN BONDED STRIPS

Solutions for thermal stress singularities infinite bonded strips are sought by using an eigenfunction expansion in the neighborhood of the singularity. The coefficients in the resulting series are determined by satisfying the boundary conditions on surfaces far removed from the singularity either pointwise or in an integrated sense. The latter of these techniques is found to be more reliable. The accuracy of the solution is checked by comparing it to a semianalytical solution for thermal stresses in bonded quarter planes, which is derived by using the Mellin transformation. It is shown that the eigenfunction approach provides accurate solutions for the leading term in the series, thus capturing the essence of the thermal stress fields near the edge of the interface. The far-field solutions, however, are found to feature excessive inaccuracies, which are attributed to numerical errors     

Thermal convection over flat plates possessing an irregular leading edge

Boundary layer similarity reductions for thermal convection adjacent to inclined heated plates with an irregular leading edge are presented. The theory is outlined using three examples: Newtonian convection over an isothermal plate, Darcian convection over a nonuniformly heated plate, and mixed convection of Newtonian shear flow over a nonuniformly heated plate. These reductions represent a natural extension of results obtained by the author for Blasius flow over a flat plate with an irregular leading edge.