Natural Convection on a Horizontal Cone in a Porous Medium with Non-Uniform Wall Temperature/Concentration or Heat/Mass Flux and Suction/Injection

The steady natural convection flow on a horizontal cone embedded in a saturated porous medium with non-uniform wall temperature/concentration or heat/mass flux and suction/injection has been investigated. Non-similar solutions have been obtained. The nonlinear coupled differential equations under boundary layer approximations governing the flow have been numerically solved. The Nusselt and Sherwood numbers are found to depend on the buoyancy forces, suction/injection rates, variation of wall temperature/concentration or heat/mass flux, Lewis number and the non-Darcy parameter.     

Non-darcy natural convection on a vertical cylinder in a saturated porous medium

The steady free convection boundary layer flow of non-Darcy fluid along an isothermal vertical cylinder embedded in a saturated porous medium using the Ergun model has been studied. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme developed by Keller. It is found that the heat transfer is strongly affected by the modified Grashof number which characterizes the non-Darcy fluid, and the curvature parameter. Also the heat transfer is found to be more than that of the flat plate.     

Natural convection from a vertical cylinder in a thermally stratified porous medium

The problem of non-Darcy natural convection adjacent to a vertical cylinder embedded in a thermally stratified porous medium has been analyzed. Nonsimilarity solutions are obtained for the case that the ambient temperature increases linearly with height of the cylinder. A generalized flow model was used in the present study to include the effects of the macroscopic viscous term and the microscopic inertial force. Also, the thermal dispersion effect is considered in the energy equation. Thus, the main aim of this work is to examine the effects of thermal stratification and non-Darcy flow phenomena on the free convection flow and heat transfer characteristics. It was found that the present problem depends on six parameters, namely, the local thermal stratification parameter ξ, the boundary effect parameter Bp, the modified Grashof number Gr^*, wall temperature exponent m, the curvature parameter ω, and the modified Rayleigh number based on pore diameter Ra _d . The impacts of these governing parameters on the local heat transfer parameter are discussed in great detail. Also, representative velocity and temperature profiles are presented at selected values of the thermal stratification parameter. In general, the local heat transfer parameter is increased with increasing the values of m, ω, and Ra _d ; while it is decreased with increasing the values of ξ, Bp, and Gr^*. Received on 19 May 1998     

The Effect of Double Dispersion on Natural Convection Heat and Mass Transfer in a Non-Newtonian Fluid Saturated Non-Darcy Porous Medium

The effect of power law index parameter of the non-Newtonian fluid on free convection heat and mass transfer from a vertical wall is analyzed by considering double dispersion in a non-Darcy porous medium with constant wall temperature and concentration conditions. The Ostwald–de Waele power law model is used to characterize the non-Newtonian fluid behavior. In this case a similarity solution is possible. The variation of heat and mass transfer coefficients with the governing parameters such as power law index, thermal and solutal dispersion parameters, inertia parameter, buoyancy ratio, and the Lewis number is discussed for a wide range of values of these parameters.     

Non-isothermal Poiseuille flow and its stability in a vertical annulus filled with porous medium

In this article, the non-isothermal Poiseuille flow and its stability in a vertical annulus filled with porous medium are investigated. The flow is induced by external pressure gradient and buoyancy force due to linearly varying inner wall temperature. The non-Darcy model along with Boussinesq approximation has been used. The Chebyshev spectral-collocation method has been adopted to solve the governing equations related to basic flow as well as its stability. Special attention is given to understand the effect of curvature parameter of the annular geometry on the flow, heat transfer rate and stability of the stably stratified flow. A comprehensive numerical experiment indicates that reducing gap between two concentric cylinders decreases the heat transfer rate as well as the maximum magnitude of the flow velocity. It stabilizes the flow which has been shown through stability analysis. Furthermore, appropriateness of the Forchheimer term in the momentum equation has been examined by investigating the flow regime as well as its stability in the presence and absence of Forchheimer term. Finally, it has been found from the energy analysis at critical point that the thermal-buoyant instability is the only mode of instability for the considered range of different parameters.     

An integral treatment for non-Darcy free convection over a vertical flat plate and cone embedded in a fluid-saturated porous medium

An integral treatment was proposed for analysis of non-Darcy free convection over a vertical flat plate and cone within a fluid-saturated porous medium. A flexible one-parameter family of third order polynomials was employed to cope with vast changes in the velocity and temperature profiles encountered in the Darcy flow limit through to the Forchheimer flow limit. Zero curvature requirement for the temperature profile at the wall was exploited as an auxiliary relation to determine the shape parameter. Comparison of the approximate results with the exact solution reveals a high performance of the present integral procedure for heat transfer rat prediction.     

Non-Darcy double-diffusive natural convection in axisymmetric fluid saturated porous cavities

Double-diffusive natural convection in a fluid saturated porous medium has been investigated using the finite element method. A generalised porous medium model is used to study both Darcy and non-Darcy flow regimes in an axisymmetric cavity. Results indicate that the Darcy number should be a separate parameter to understand flow characteristics in non-Darcy regime. The influence of porosity on heat and mass transfer is significant and the transport rates may differ by 25% or more, at higher Darcy and Rayleigh numbers. When compared with the Darcy and other specialised models of Brinkman and Forchheimer, the present generalised model predicts the least heat and mass transfer rates. It is also observed that an increase in radius ratio leads to higher Nusselt and Sherwood numbers along the inner wall.     

Double-Diffusive Free Convection Flow Past an Inclined Plate Embedded in a Non-Darcy Porous Medium Saturated with a Nanofluid

In this investigation, we intend to present the influence of the prominent Soret effect on double-diffusive free convection heat and mass transfer in the boundary layer region of a semi-infinite inclined flat plate in a nanofluid saturated non-Darcy porous medium. The transformed boundary layer ordinary differential equations are solved numerically using the shooting and matching technique. Consideration of the nanofluid and the coupled convective process enhanced the number of non-dimensional parameters considerably thereby increasing the complexity of the present problem. A wide range of parameter values are chosen to bring out the effect of Soret parameter on the free convection process with varying angle of inclinations making the wall geometry from vertical to horizontal plate. The effects of angle of inclination and Soret parameter on the flow, heat and mass transfer coefficients are analyzed. The numerical results obtained for the velocity, temperature, volume fraction, and concentration profiles, local wall temperature, local nanoparticle concentration, and local wall concentration reveal interesting phenomenon, and some of these qualitative results are presented through the plots.     

Unsteady flow with heat and mass transfer due to impingement of slot jet on an inclined plate

The unsteady laminar incompressible boundary layer flow due to a two-dimensional slot jet on a flat plate at an angle of attack has been studied. The unsteadiness in the flow field is due to the free stream velocity distribution or wall temperature (concentration) which varies with time. The governing partial differential equations in primitive variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The effect of the variation of the free stream velocity distribution with time is found to be more pronounced on the skin friction than on the heat or mass transfer. The Prandtl number and the variation of the wall temperature with time strongly affect the heat transfer. Similarly, the Schmidt number and the variation of the concentration at the wall with time strongly affect the mass transfer. Beyond a certain critical value of the viscous dissipation parameter, the plate gets heated instead of being cooled.     

Effect of thermal dispersion on free convection in a fluid saturated porous medium

The present article considers a numerical study of thermal dispersion effect on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. Forchheimer extension is considered in the flow equations. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The non-dimensional governing equations are solved by the finite element method (FEM) with a Newton–Raphson solver. Numerical results for the details of the stream function, velocity and temperature contours and profiles as well as heat transfer rates in terms of Nusselt number are obtained. The study shows that the increase in thermal dispersion coefficient of the porous medium results in more heat energy to disperse away in the normal direction to the wall. This induces more fluid to flow along the wall, enhancing the heat transfer coefficient particularly near the wall.     

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.     

Thermal dispersion effects on non-Darcy natural convection with lateral mass flux

The method of similarity solution is used to study the influence of lateral mass flux and thermal dispersion on non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. Forchheimer extension is considered in the flow equations and the coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The suction/injection velocity distribution has been assumed to have power function form Ax ^l , where x is the distance from the leading edge and the wall temperature distribution is assumed to be uniform. When l=−1/2, similarity solution is possible, and the results indicate that the boundary layer thickness decreases where as the heat transfer rate increases as the mass flux parameter passes from injection domain to the suction domain. The increase in the thermal dispersion parameter is observed to enhance the heat transfer. The combined effect of thermal dispersion and fluid suction/injection on the heat transfer rate is discussed. Received on 9 September 1996     

Effect of variable viscosity on non-Darcy free or mixed convection flow on a vertical surface in a fluid saturated porous medium

This paper analyzes the variable viscosity effects on non-Darcy free or mixed convection flow on a vertical surface in a fluid saturated porous medium. The viscosity of the fluid is assumed to be a inverse linear function of temperature. Velocity and heat transfer are found to be significantly affected by the variable viscosity parameter, Ergun number, Peclet number or Rayleigh number.     

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.     

Effect of longitudinal surface curvature on heat transfer

The effect of longitudinal surface curvature on heat transfer has been analysed for laminar forced convection by the method of matched asymptotic expansions. Utilizing the classical solution of boundary layer equations as the first order approximations, the second order perturbation for the velocity and temperature field has been calculated by a similarity analysis. The analysis permits the wall temperature to vary as a power function of distance from the stagnation point. Numerical solutions have been obtained for the resulting coupled ordinary differential equations. The results for the variation in the second order temperature profile and the second order wall temperature gradient due to surface curvature parameter, Prandtl number, wall temperature distribution parameter, and pressure gradient parameters are presented graphically. The variation in a typical temperature profile due to curvature, and percentage variations from the first order theory due to longitudinal surface curvature are also presented graphically.     

Flow and heat transfer due to impulsive motion of a cone in a viscous fluid

Numerical solution has been obtained for the development of the flow over a cone which is impulsively set into motion. Initially the flow is described by the solution of Rayleigh and then it tends to the ultimate steady state solution of Falkner-Skan equation. But due to the leading edge effect the semi-similar equation describing the transient flow changes its character after certain time and the solution depends also on the ultimate steady state solution of the Falkner-Skan equation. A second-order upwind difference scheme has been used for discretisation. The temperature distribution and heat transfer has also been obtained for constant wall temperature as well as for constant heat flux at the wall. With the increase ofm, Falkner-Skan parameter, the magnitude of skin friction and wall heat transfer increases. It has been found that form≥?0.275 flow separation does not occur.     

Transient free convection flow between long vertical parallel plates with ramped wall temperature at one boundary in the presence of thermal radiation and constant mass diffusion

The unsteady laminar free convection flow between two long vertical parallel plates with ramped wall temperature at one boundary has been investigated in the presence of thermal radiation and chemical species concentration. The exact solutions of the momentum, energy and concentration equations have been obtained using the Laplace transform technique. The velocity and temperature profiles, skin-friction and Nusselt number variations are shown graphically and the numerical values of the volume flow rate, the total heat rate and species rate added to the fluid are presented in a table. The influence of different system parameters such as the radiation parameter (R), buoyancy ratio parameter (N), Schmidt number (Sc) and time (t) has been analyzed carefully. A critical analysis of the coupled heat and mass transfer phenomena is provided. The free convective flow due to ramped wall temperature has also been compared with the baseline case of flow due to constant wall temperature.     

Darcian mixed convection plumes along vertical adiabatic surfaces in a saturated porous medium

The mixed convection flow due to a line thermal source embedded at the leading edge of an adiabatic vertical plane surface immersed in a saturated porous medium has been studied. Both weakly and strongly buoyant plume regimes have been considered. The cases of buoyancy assisting and buoyancy opposing flow conditions have been incorporated in the analysis. The results are presented for the entire range of buoyancy parameter from the pure forced convection (ξ=0) to the pure free convection (ξ → ∞@#@) regimes. For buoyancy-assisting flow, the wall temperature and the velocity at the wall increase as the plume strength increases. However, they all decrease as the free-stream velocity increases. For buoyancyopposing flow, the temperature at the wall increases as the strength of the plume increases but velocity at the wall decreases.     

A Criterion for Non-Darcy Flow in Porous Media

Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-Darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-Darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-Darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid–solid interactions to that by viscous resistance. It is directly related to the non-Darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-Darcy flow is expressed in terms of the critical non-Darcy effect. Considering a 10% non-Darcy effect, the critical Forchheimer number would be 0.11.     

Numerical Simulation of Single-Phase and Multiphase Non-Darcy Flow in Porous and Fractured Reservoirs

A numerical method as well as a theoretical study of non-Darcy fluid flow through porous and fractured reservoirs is described. The non-Darcy behavior is handled in a three-dimensional, multiphase flow reservoir simulator, while the model formulation incorporates the Forchheimer equation for describing single-phase or multiphase non-Darcy flow and displacement. The non-Darcy flow through a fractured reservoir is handled using a general dual-continuum approach. The numerical scheme has been verified by comparing its results against those of analytical methods. Numerical solutions are used to obtain some insight into the physics of non-Darcy flow and displacement in reservoirs. In addition, several type curves are provided for well-test analyses of non-Darcy flow to demonstrate a methodology for modeling this type of flow in porous and fractured rocks, including flow in petroleum and geothermal reservoirs.