We consider planar flow involving two viscous fluids in a porous medium. One fluid is injected through a line source at the origin and moves radially outwards, pushing the second, ambient fluid outwards. There is an interface between the two fluids and if the inner injected fluid is of lower viscosity, the interface is unstable to small disturbances and radially directed unstable Saffman–Taylor fingers are produced. A linearized theory is presented and is compared with nonlinear results obtained using a numerical spectral method. An additional theory is also discussed, in which the sharp interface is replaced with a narrow diffuse interfacial region. We show that the nonlinear results are in close agreement with the linearized theory for small-amplitude disturbances at early times, but that large-amplitude fingers develop at later times and can even detach completely from the initial injection region.

Numerical analysis was investigated for steady two-dimensional double diffusive mixed convection boundary layer flow over a semi-infinite vertical plate embedded non-Darcy porous medium filled with nanofluid, in presence of thermal dispersion and under convective boundary conditions. The Buongiorno nanofluid model is used, while the porous medium is described by the Darcy-Forchheimer extension. The governing partial differential equations are transformed into four coupled nonlinear ordinary differential equations using an appropriate similarity transformations and the resulting system of equations is then solved numerically by the finite-difference method. Numerical results are presented to illustrate how the physical parameters affect the flow field, temperature, concentration and solid volume fraction profiles. In addition, the variation of heat, mass and nanoparticle transfer rates at the plate are exhibited graphically for different values of pertinent parameters.

Free convective flow and heat transfer of nanofluid close to the inclined plate immersed in the porous medium under the effects of uniform magnetic field and solar radiation has been studied. Boundary-layer approach, Boussinesq approximation and two-phase nanofluid model have been used for a formulation of the governing equations taking into account convective-radiative heat exchange with an environment. The local similarity method has been adopted for the analysis of the considered phenomenon. The obtained equations have been solved numerically using MATLAB software. The effects of control characteristics on profiles of velocity, temperature and nanoparticles volume fraction as well as Nusselt number have been studied in detail.

The present analysis has been developed to investigate the heat transfer phenomenon in peristaltic flow of Carreau fluid in a curved channel with rhythmic contraction and expansion of waves along the walls (similar to blood flow in tubes). Magnetic field is imposed in radial direction. The heat transfer aspect is further studied with viscous dissipation effect. The curved channel walls are influenced by flow and thermal partial slip. In addition the flow stream comprised porous medium. The system of relevant non-linear PDEs have been reduced to ODEs by utilizing the long wavelength approximation. The striking features of flow and temperature characteristics under the involved parameters are examined by plotting graphs. The generation of fluid temperature and velocity due to viscous dissipation and gravitational efforts are recorded respectively. Moreover indicated results signify activation of velocity, temperature and heat transfer rate with Darcy number.

We derive a macroscopic model for biofilm formation in a porous medium reactor to investigate the role of suspended bacteria on reactor performance. The starting point is the mesoscopic one-dimensional Wanner–Gujer biofilm model. The following processes are included: hydrodynamics and transport of substrate in the reactor, biofilm and suspended bacteria growth in the pore space, attachment of suspended cells to the biofilm, and detachment of biofilm cells. The mesoscopic equations are up-scaled from the biofilm scale to the reactor scale, yielding a stiff system of balance laws, which we study numerically. We find that suspended bacteria and attachment can have a significant effect on biofilm reactor performance.

We consider a non-stationary incompressible non-Newtonian Stokes system in a porous medium with characteristic size of the pores ϵ and containing a thin fissure of width ηϵ. The viscosity is supposed to obey the power law with flow index $\frac{5}{3}\leq q\leq 2$. The limit when size of the pores tends to zero gives the homogenized behaviour of the flow. We obtain three different models depending on the magnitude ηϵ with respect to ϵ: if ηϵ ≪ $\varepsilon^{q\over 2q-1}$ the homogenized fluid flow is governed by a time-dependent non-linear Darcy law, while if ηϵ ≫ $\varepsilon^{q\over 2q-1}$ is governed by a time-dependent non-linear Reynolds problem. In the critical case, ηϵ ≈ $\varepsilon^{q\over 2q-1}$, the flow is described by a time-dependent non-linear Darcy law coupled with a time-dependent non-linear Reynolds problem.

This paper presents the feasibility of torsional surface wave propagation in an anisotropic layer sandwiched between two anisotropic inhomogeneous media. The anisotropy considered in the upper layer and the lower half-space is of transversely isotropic kind while the sandwiched anisotropic layer is a porous layer. The directional rigidities and density have been considered linearly and exponentially varying in the half-space and in the upper layer respectively, while it is taken as a variable in the sandwiched layer. The compact form of dispersion equation governing the propagation of the torsional surface wave has been derived by using the Whittaker function under appropriate boundary conditions. The dispersion of the torsional wave and the effects of inhomogeneity parameters, initial stress and poroelastic constant have been calculated numerically and demonstrated through graphs.

In this paper, the problem of magnetohydrodynamics (MHD) boundary layer flow of nanofluid with heat and mass transfer through a porous media in the presence of thermal radiation, viscous dissipation and chemical reaction is studied. Three types of nanofluids, namely Copper (Cu)-water, Alumina (Al 2 O 3)-water and Titanium Oxide (TiO 2)-water are considered. The governing set of partial differential equations of the problem is reduced into the coupled nonlinear system of ordinary differential equations (ODEs) by means of similarity transformations. Finite element solution of the resulting system of nonlinear differential equations is obtained using continuous Galerkin-Petrov discretization together with the well-known shooting technique. The obtained results are validated using MATLAB “bvp4c” function and with the existing results in the literature. Numerical results for the dimensionless velocity, temperature and concentration profiles are obtained and the impact of various physical parameters such as the magnetic parameter M, solid volume fraction of nanoparticles 𝜙 and type of nanofluid on the flow is discussed. The results obtained in this study confirm the idea that the finite element method (FEM) is a powerful mathematical technique which can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

A numerical analysis was performed to study the effects of combined magnetohydrodynamic and thermal radiation under convective boundary condition over a semi infinite vertical plate embedded in a non-Darcy porous medium. Coupled heat and mass transfer of free convective boundary layer with viscous nanofluid are considered. The model used for the nanofluid includes the effects of Brownian motion and thermophoresis mechanisms, while the Darcy-Forchheimer model is used for the porous medium. The governing partial differential equations are transformed into the ordinary differential equations using the similarity transformations. The accuracy of the method is observed by a comparison with other works reduced to a common case. Many results are tabulated and representative set is displayed graphically to illustrate the influence of the various parameters of interest on different profiles. Extensive numerical investigations show that the flow field, temperature, concentration and nanoparticle volume fraction shapes are significantly influenced by magnetic parameter, regular Lewis number, Brownian motion parameter, thermophoresis parameter, regular buoyancy ratio parameter and Biot number. Heat and mass transfer rates are significantly affected by the level of the applied magnetic field and the convective heat coefficient.

This paper concerns with the unsteady MHD flow of a second grade fluid between two parallel walls through porous media induced by rectified sine pulses shear stress. The analytical expressions for the velocity field and the adequate shear stress are determined by means of the Laplace transform technique and Fourier cosine and sine transforms and are written as a sum of steady state and transient solutions. The influence of side walls on the fluid motion, the distance between walls for which the velocity of the fluid in the middle of the channel is negligible, and the required time to reach the steady state are presented by graphical illustrations. As the second grade fluid parameter → 0 the problem reduces to the Newtonian fluids performing the same motion.

The unsteady MHD free convection and mass transfer boundary layer flow of an incompressible electrically conducting fluid past an accelerated infinite vertical flat plate embedded in porous medium with ramped wall temperature is considered here. It is assumed that the plate accelerates in its own plane in the presence of thermal radiation incorporating first order chemical reaction. The governing equations are solved analytically using the Laplace transformation technique. The flow phenomenon has been characterized with the help of flow parameters such as permeability parameter, Hartmann number, phenomenon has been characterized with the help of flow parameters such as permeability parameter, Hartmann number, thermal radiation parameter etc. The influences of these parameters on the velocity, temperature field and concentration distribution have been studied and the results are presented graphically and discussed quantitatively. Also, the effects of the various parameters on the skin friction coefficient, the rate of heat and mass transfer at the surface are discussed.

A non-linear analysis has been made to study the unsteady hydromagnetic boundary layer flow and heat transfer of a micropolar fluid over a stretching sheet embedded in a porous medium. The effects of thermal radiation in the boundary layer flow over a stretching sheet have also been investigated. The system of governing partial differential equations in the boundary layer have reduced to a system of non-linear ordinary differential equations using a suitable similarity transformation. The resulting non-linear coupled ordinary differential equations are solved numerically by using an implicit finite difference scheme. The numerical results concern with the axial velocity, micro-rotation component and temperature profiles as well as local skin-friction coefficient and the rate of heat transfer at the sheet. The study reveals that the unsteady parameter S has an increasing effect on the flow and heat transfer characteristics.

To finite-difference model elastic wave propagation in a combined structure with solid, fluid and porous subregions, a set of modified Biot’s equations are used, which can be reduced to the governing equations in solids, fluids as well as fluid-saturated porous media. Based on the modified Biot’s equations, the field quantities are finite-difference discretized into unified forms in the whole structure, including those on any interface between the solid, fluid and porous subregions. For the discrete equations on interfaces, however, the harmonic mean of shear modulus and the arithmetic mean of the other parameters on both sides of the interfaces are used. These parameter averaging equations are validated by deriving from the continuity conditions on the interfaces. As an example of using the parameter averaging technique, a 2-D finite-difference scheme with a velocity-stress staggered grid in cylindrical coordinates is implemented to simulate the acoustic logs in porous formations. The finite-difference simulations of the acoustic logging in a homogeneous formation agree well with those obtained by the analytical method. The acoustic logs with mud cakes clinging to the borehole well are simulated for investigating the effect of mud cake on the acoustic logs. The acoustic logs with a varying radius borehole embedded in a horizontally stratified formation are also simulated by using the proposed finite-difference scheme.

In recent decades, problems related to the squeeze of fluid films in the presence of aporous medium draw attention of researchers and are the subject of many applied studiesfor industry and biomechanics. Our concerns in this paper are the numerical simulation ofthe viscous shear stresses effects on the fluid film characteristics between two discswith one porous. This study is based on the coupling, at the fluid film-porous discinterface, of the Darcy-Brinkman equations in the porous medium and the modified Reynoldsequation describing the flow in the fluid film. The system of equations obtained isdiscretized by the means of finite differences method and solved numerically using thetechnique of Successive Over-Relaxation (SOR). The results show that the viscous sheareffects increase the radial and the axial fluid film velocities as well as the squeezefilm velocity but decrease the response time. Moreover, these effects are enlarged forsmaller viscous shear parameter and for smaller fluid film thickness.

The paper is devoted to the computation of two-phase flows in a porous mediumwhen applying the two-fluid approach. The basic formulation is presented first, together with the main properties of the model. A few basic analytic solutions are then provided, some of them correspondingto solutions of the one-dimensional Riemann problem. Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme,are shown to give wrong approximations in some cases involving sharp porous profiles.The third one, which is an extension of a scheme proposed by Kröner and Thanh [SIAM J. Numer. Anal. 43 (2006) 796–824]for the computation of single phase flows in varying cross section ducts,provides fair results in all situations. Properties of schemes and numerical results are presented. Analytic tests enable to compute the L 1 norm of the error.

The turbulent flow in a channel with periodic porous ribs on one wall is numerically studied. The numerical model utilizes the Reynolds averaged Navier-Stokes (RANS) equations with a k−ε turbulent model for turbulence closure. Computational results show good agreements with experimental data in flows over a porous rib. The parameter effects, including the pitch ratio PR (1 ∼ 9) and porosity γ (0.4 ∼ 0.6), on flow fields are further examined in detail. Systematic variations of streamline, streamwise and vertical velocities, and turbulent kinetic energy are clearly identified. As to the PR effect, the interaction between outer flow and flow within the cavity is promoted by arranging ribs due to the penetration of the outer flow. Increasing porosity can reduce the downward outer flow by strong flows passing through the porous ribs. The numerical calculations suggest that the flow characteristics for porous ribs are not only a function of the rib geometry, i.e. pitch ratio, but also the porous property, i.e. porosity.

We consider a non-local filtration equation of the form

and a porous medium equation, in this case K(u) = um, with some boundary and initial data u0, where 0 < p < 1 and f, f′, f″ > 0. We prove blow-up of solutions for sufficiently large values of the parameter λ > 0 and for any u0 > 0, or for sufficiently large values of u0 > 0 and for any λ λ 0.

The aim of this paper is to study the effect of vibrations on convective instability ofreaction fronts in porous media. The model contains reaction-diffusion equations coupledwith the Darcy equation. Linear stability analysis is carried out and the convectiveinstability boundary is found. The results are compared with direct numericalsimulations.

This study theoretically and experimentally investigates the velocity distributions of the interface boundary layer region in a porous medium. By combining a quadratic non-Darcy law and a practical eddy viscosity model, an analytical solution is derived and presented. Three additional parameters, i.e., the depth of the interface boundary layer region, the slip velocity, and proportionality constant, are contained in the analytical solution. The measured experimental data show the depth of the interface boundary layer region only depends on the characteristics of the porous medium rather than the relative flow depth, bed slop, or Reynolds number. The values of the slip velocity are found to increase with increasing relative mean clear fluid velocities. The proportionality constant plays an important role in modeling the penetration of turbulence and the associated momentum transfer. The measured experimental velocity distribution is used to evaluate the accuracy of the analytical predicted profile. The analytical results obtained in this study are in agreement with the measured experimental results.