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In this article, the onset of double-diffusive convection with the effect of viscous dissipation in a horizontal fluid-saturated porous layer is examined. Two impermeable isothermal and isosolutal walls bound the porous layer, and Darcy's law models the flow. The onset of convective instability is studied by two approaches: the linear stability analysis and the nonlinear stability analysis. The nonlinear stability analysis is performed by utilizing the energy method. The literature on the nonlinear stability analysis of onset of convective instability with the viscous dissipation effect is limited. The present article aims to fill this gap. It is observed that, when the fluid is at rest, the effect of viscous dissipation does not influence the critical thermal Rayleigh number corresponding to both the linear and nonlinear stability analyses. Moreover, sub-critical instabilities do not occur when R a S > 0 . But in contrast, the region of sub-critical instabilities increases along the negative R a S direction. [ABSTRACT FROM AUTHOR]

We investigate nonlinear stability in a model for thermal convection in a saturated porous material using Brinkman theory, taking into account viscous dissipation effects. There are (at least) two models for viscous dissipation available, and we include a derivation of one of these by assuming that the flow in the porous medium may be described by a theory for a mixture of an elastic solid and a linearly viscous fluid. A fully nonlinear stability result is provided when either of the viscous dissipation functions is taken into account, and it is shown that from the nonlinear energy stability viewpoint both models are, in a sense, equivalent. [ABSTRACT FROM AUTHOR]

Double-diffusive convective instability of horizontal throughflow in a power-law fluid saturated porous layer is investigated. The boundaries of this horizontal porous layer are impermeable, isothermal and isosolutal. The generalized Darcy's equation for power-law fluid is employed in the momentum balance, along with Oberbeck–Boussinesq approximation for buoyancy effect in the system. It is considered that the viscous dissipation is non-negligible. The basic temperature and concentration profiles for the considered horizontal throughflow are determined analytically, and the linear stability analysis is performed on this basic solution. The eigenvalue problem is solved numerically using bvp4c in MATLAB. The conditions for the onset of instability of base flow in terms of critical Rayleigh number and the corresponding wave number are obtained. It is found that critical values are affected by the flow governing parameters such as the Péclet number (Pe), power-law index (n), viscous dissipation (Ge), angle of inclination (α) , buoyancy ratio (N) and Lewis number (Le). For the considered range of these parameter values, the instability of the basic flow is discussed for the convective rolls. It is observed that instability of any general convective roll lies between that of the longitudinal and transverse rolls. Also, the effect of viscous dissipation on the onset of instability for both the longitudinal and transverse rolls is discussed in detail. This source of heating inside the porous layer increases the thermal instability of base flow. Further, the asymptotic analysis for vanishingly small Péclet number is presented as P e = 0 is a singular point. The boundary between the stationary and the oscillatory instabilities is given using the buoyancy ratio parameter and the Lewis number in this limiting case of P e → 0 . [ABSTRACT FROM AUTHOR]

The onset of double-diffusive convective instability of a horizontal throughflow induced by viscous dissipation in a fluid-saturated porous layer of high permeability is investigated. The porous layer is infinitely long along the horizontal direction and is bounded by two rigid surfaces maintained at constant, but different solute concentrations. The lower surface is thermally insulated, whereas the upper surface is considered to be isothermal. The Darcy-Brinkman model is adopted for deriving the equations governing flow in the medium, and the Soret effect is considered to persist in the flow. The instability in the base flow is considered to be induced by the non-negligible viscous heating. Disturbances in the base flow are assumed in the form of oblique rolls, where the longitudinal and the transverse rolls are at two extreme inclinations. The disturbance functions are assumed to be of O(1). It is considered that Ge≪1 and |Pe|≫1, where Ge is the Gebhart number and Pe is the Péclet number. The eigenvalue problem with coupled ordinary differential equations governing the disturbances in the flow is solved numerically using bvp4c in MATLAB. Results obtained depict that the flow is most stable in the Brinkman regime and the longitudinal rolls are the preferred mode of instability. The solute concentration gradient and the Soret parameter have both stabilizing and destabilizing effect on the flow in the medium, when the values of both are either positive or negative. However, they have either monotonically stabilizing or monotonically destabilizing effect on the flow, when the values of both have opposite signs. [ABSTRACT FROM AUTHOR]

The linear stability of the double-diffusive convection in a horizontal porous layer is studied considering the upper boundary to be open. A horizontal temperature gradient is applied along the upper boundary. It is assumed that the viscous dissipation and Soret effect are significant in the medium. The governing parameters are horizontal Rayleigh number (RaH), solutal Rayleigh number (RaS), Lewis number (Le), Gebhart number (Ge) and Soret parameter (Sr). The Rayleigh number (Ra) corresponding to the applied heat flux at the bottom boundary is considered as the eigenvalue. The influence of the solutal gradient caused due to the thermal diffusion on the double-diffusive instability is investigated by varying the Soret parameter. A horizontal basic flow is induced by the applied horizontal temperature gradient. The stability of this basic flow is analyzed by calculating the critical Rayleigh number (Racr) using the Runge-Kutta scheme accompanied by the Shooting method. The longitudinal rolls are more unstable except for some special cases. The Soret parameter has a significant effect on the stability of the flow when the upper boundary is at constant pressure. The critical Rayleigh number is decreasing in the presence of viscous dissipation except for some positive values of the Soret parameter. How a change in Soret parameter is attributing to the convective rolls is presented. [ABSTRACT FROM AUTHOR]

Flow driven by an externally imposed pressure gradient in a vertical porous channel is analysed. The combined effects of viscous dissipation and thermal buoyancy are taken into account. These effects yield a basic mixed convection regime given by dual flow branches. Duality of flow emerges for a given vertical pressure gradient. In the case of downward pressure gradient, i.e. upward mean flow, dual solutions coincide when the intensity of the downward pressure gradient attains a maximum. Above this maximum no stationary and parallel flow solution exists. A nonlinear stability analysis of the dual solution branches is carried out limited to parallel flow perturbations. This analysis is sufficient to prove that one of the dual solution branches is unstable. The evolution in time of a solution in the unstable branch is also studied by a direct numerical solution of the governing equation. [ABSTRACT FROM AUTHOR]

This paper reports a detailed analytical and numerical investigation on free convection flow of viscous fluid through a porous medium due to the combined effects of thermal and mass diffusion. Effect of temperature-dependent viscosity and thermal conductivity is investigated in the presence of first-order chemical reaction. The non-Darcy model is applied to define the porous matrix. The effects of viscous and Darcy dissipations are taken into account. The governing equations of continuity, momentum, energy, and concentration which are coupled nonlinear ordinary differential equations are solved analytically using regular perturbation method and numerically using Runge-Kutta shooting method. Brinkman number and variable thermal conductivity parameters are used as perturbation parameters. The velocity, temperature, and concentration distributions are discussed numerically and plotted in graphs. The effects of variable viscosity parameter, variable thermal conductivity parameter, thermal Grashof number, mass Grashof number, Brinkman number, wall temperature ratio, and first-order chemical reaction parameter on the flow fields are explored. The effects of physical parameters such as skin friction and Nusselt number at both the plates are derived and discussed, and the numerical values for various values of physical parameters are presented in tables. The solutions obtained using Runge-Kutta shooting method and using perturbation method are compared and the solutions agree very well in the absence of perturbation parameter. [ABSTRACT FROM AUTHOR]

It is known that even entirely symmetric boundary value problems can admit solutions in which the inherent symmetry of the governing equations gets spontaneously broken. When this happens, two non-symmetric 'twin solutions' occur so that the symmetry of the boundary value problem broken in the two individual solutions becomes restored by the twins. The present paper shows that, as a combined effect of buoyancy, viscous dissipation and pressure work, in a mixed convection flow in a vertical porous channel with isothermal walls, in addition to symmetric solutions, precisely this kind of twin solutions with broken symmetry can occur, although the walls are kept at the same temperature. The existence of another remarkable solution branch which describes symmetric adiabatic flows corresponding to nonlinear eigenvalues of the problem is also reported. The heat-work balance of all these steady-flow states is discussed in detail. The analysis is supported by an intuitive point-mechanical analogy. The equivalence of this mechanical energy analysis to the standard phase space method is also discussed shortly. [ABSTRACT FROM AUTHOR]

The paper presents an investigation of the influence of thermal radiation and viscous dissipation on the mixed convective flow due to a vertical plate immersed in a non-Darcy porous medium saturated with a Newtonian fluid. The physical properties of the fluid are assumed to be constant. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The governing partial differential equations are transformed into a system of ordinary differential equations and solved numerically using a shooting method. The results are analyzed for the effects of various physical parameters such as viscous dissipation, thermal radiation, mixed convection parameters, and the modified Reynolds number on dynamics. The heat transfer coefficient is also tabulated for different values of the physical parameters. [ABSTRACT FROM AUTHOR]

Numerical solutions for the free convection heat transfer in a viscous fluid at a permeable surface embedded in a saturated porous medium, in the presence of viscous dissipation with temperature-dependent variable fluid properties, are obtained. The governing equations for the problem are derived using the Darcy model and the Boussinesq approximation (with nonlinear density temperature variation in the buoyancy force term). The coupled non-linearities arising from the temperature-dependent density, viscosity, thermal conductivity, and viscous dissipation are included. The partial differential equations of the model are reduced to ordinary differential equations by a similarity transformation and the resulting coupled, nonlinear ordinary differential equations are solved numerically by a second order finite difference scheme for several sets of values of the parameters. Also, asymptotic results are obtained for large values of | f|. Moreover, the numerical results for the velocity, the temperature, and the wall-temperature gradient are presented through graphs and tables, and are discussed. It is observed that by increasing the fluid variable viscosity parameter, one could reduce the velocity and thermal boundary layer thickness. However, quite the opposite is true with the non-linear density temperature variation parameter. [ABSTRACT FROM AUTHOR]

n analytical study of viscous dissipation effect on the fully developed forced convection Couette flow through a parallel plate channel partially filled with porous medium is presented. A uniform heat flux is imposed at the moving plate while the fixed plate is insulated. In the fluid-only region the flow field is governed by Navier-Stokes equation while the Brinkman-extended Darcy law relationship is considered in the fully saturated porous medium. The interface conditions are formulated with an empirical constant β due to the stress jump boundary condition. Fluid properties are assumed to be constant and the longitudinal heat conduction is neglected. A closed-form solution for the velocity and temperature distributions and also the Nusselt number in the channel are obtained and the viscous dissipation effect on these profiles is briefly investigated. [ABSTRACT FROM AUTHOR]

The convective instability activated by the sole effect of viscous dissipation in a fluid saturated porous layer is studied. The basic parallel flow in a highly permeable porous medium is analysed by considering the viscous heating contribution in the local energy balance, by assuming a thermally insulated lower boundary and an isothermal upper boundary. The Brinkman model of momentum transfer is adopted. Arbitrarily oriented oblique roll disturbances are considered in the linear stability analysis. Among them, the longitudinal rolls, having axis parallel to the basic flow direction, are shown to be the preferred mode of instability. Some considerations on the reliability of the Brinkman model, when the viscous dissipation contribution is not negligible and when the flow conditions are close to the limiting case of a clear fluid, are finally expressed. [ABSTRACT FROM AUTHOR]

buoyancy-induced stationary flow with viscous dissipation in a horizontal porous layer is investigated. The lower boundary surface is impermeable and subject to a uniform heat flux. The upper open boundary has a prescribed, linearly varying, temperature distribution. The buoyancy-induced basic velocity profile is parallel and non-uniform. The linear stability of this basic solution is analysed numerically by solving the disturbance equations for oblique rolls arbitrarily oriented with respect to the basic velocity field. The onset conditions of thermal instability are governed by the Rayleigh number associated with the prescribed wall heat flux at the lower boundary, by the horizontal Rayleigh number associated with the imposed temperature gradient on the upper open boundary, and by the Gebhart number associated with the effect of viscous dissipation. The critical value of the Rayleigh number for the onset of the thermal instability is evaluated as a function of the horizontal Rayleigh number and of the Gebhart number. It is shown that the longitudinal rolls, having axis parallel to the basic velocity, are the most unstable in all the cases examined. Moreover, the imposed horizontal temperature gradient tends to stabilise the basic flow, while the viscous dissipation turns out to have a destabilising effect. [ABSTRACT FROM AUTHOR]

The onset of convective rolls instability in a horizontal porous layer subject to a basic temperature gradient inclined with respect to gravity is investigated. The basic velocity has a linear profile with a non-vanishing mass flow rate, i.e., it is the superposition of a Hadley-type flow and a uniform flow. The influence of the viscous heating contribution on the critical conditions for the onset of the instability is assessed. There are four governing parameters: a horizontal Rayleigh number and a vertical Rayleigh number defining the intensity of the inclined temperature gradient, a Péclet number associated with the basic horizontal flow rate, and a Gebhart number associated with the viscous dissipation effect. The critical wave number and the critical vertical Rayleigh number are evaluated for assigned values of the horizontal Rayleigh number, of the Péclet number, and of the Gebhart number. The linear stability analysis is performed with reference either to transverse or to longitudinal roll disturbances. It is shown that generally the longitudinal rolls represent the preferred mode of instability. [ABSTRACT FROM AUTHOR]

The modelling of viscous dissipation in a porous medium saturated by an incompressible fluid is discussed, for the case of Darcy, Forchheimer and Brinkman models. An apparent paradox relating to the effect of inertial effects on viscous dissipation is resolved, and some wider aspects of resistance to flow (concerning quadratic drag and cubic drag) in a porous medium are discussed. Criteria are given for the importance or otherwise of viscous dissipation in various situations. [ABSTRACT FROM AUTHOR]

A new analytical solution is introduced for the effect of viscous dissipation on mixed convection flow and heat transfer about an isothermal vertical wall embedded in Darcy and non-Darcy porous media with uniform free stream velocity. The effect of viscous dissipation on mixed convection in both regimes has been analyzed for both the aiding and opposing flows using Gebhart number, Ge x = gβ x/cp. The governing parameters are Re, Ra, Pe and Ge x. The case of Re = 0 corresponds to Darcy mixed convection region and Re/Pe is identified as the mixed convection governing parameter, Ra = 0 leading to pure forced convection. A good agreement was found between the numerical and analytical solutions. It was found from the Nusselt number results that viscous dissipation lowers the heat transfer rate in both Darcy and Forchheimer flow regimes for aiding as well as opposing flows. [ABSTRACT FROM AUTHOR]