This paper carries out a linear stability analysis of a plane Couette flow in a porous layer underlying a fluid layer where the porous layer is anisotropic and inhomogeneous. The plane Couette flow is induced due to the uniform movement of the upper plate and convection arises due to the maintenance of the temperature difference between the upper plate and the lower plate. The fluid considered is Newtonian and incompressible. Darcy model is used to narrate the flow in the porous layer and at the interface, the Beavers-Joseph condition is used. The Chebyshev collocation method is used to solve the generalized eigenvalue problem. Here, the effect of anisotropy and inhomogeneity of the porous medium, along with the ratio of the thickness of fluid to porous layer, i.e., depth ratio $(\hat{d})$, Reynolds number $(Re)$ and Darcy number $(δ)$ are studied. The analysis is carried out majorly for water; however, the impact of anisotropy and inhomogeneity on different fluids by varying Prandtl number ($Pr$) is also studied. Depending on the value of parameters, the unimodal (porous mode or fluid mode), bimodal (porous mode and fluid mode) and also trimodal (porous mode, fluid mode and porous mode) nature of the neutral curve is obtained. The increasing value of the inhomogeneity parameter, depth ratio or decreasing value of the anisotropy parameter, Reynolds number, Prandtl number and Darcy number raises the system instability. For $δ=0.002$, $Pr=6.9$ and $Re=10$, dominating nature of porous mode is always observed for $\hat{d}<0.07$, and fluid mode for $\hat{d}>0.21$ irrespective of anisotropy and inhomogeneity parameter. With the help of energy budget analysis, the types of instability are categorized and also the types of mode obtained from linear stability analysis are verified. Secondary flow patterns are also visualized to understand the flow dynamics.
We study the stability of the Couette-Taylor flow between porous cylinders with radial throughflow. It had been shown earlier that this flow can be unstable with respect to non-axisymmetric (azimuthal or helical) waves provided that the radial Reynolds number, $R$ (constructed using the radial velocity at the inner cylinder and its radius), is high. In this paper, we present a very detailed and, in many respects, novel chart of critical curves in a region of moderate values of $R$, and we show that, starting from values of $R$, as low as $10$, the critical modes inherited from the inviscid ins…
The long-wave dynamics of liquid films on isothermal substrates show a dynamic competition between various physical mechanisms. If the destabilizing effect of thermocapillarity overcomes the stabilizing effect of surface tension and gravity, the liquid film ruptures in finite time, through the formation of primary and secondary thermocapillary finger structures. The long-wave evolution dynamics are compared for two different substrate types: isothermal non-porous and isothermal porous for small Biot number in a zero gravity environment. The multi-time-scale dynamics is revealed through time sc…
We develop constitutive equations for multi-component, multi-phase, macro-scale flow in a porous medium exposed to temperature-, composition-, and pressure -gradients. The porous medium is non-deformable. We define the pressure and the composition of the representative elementary volume (REV) in terms of the volume and surface averaged pressure and the saturation, and the respective driving forces from these variables. New contributions due to varying porosity or surface tension offer explanations for non-Darcy behavior. The interaction of a thermal and mechanical driving forces give thermal o…
We derive the entropy production for transport of multi-phase fluids in a non-deformable, porous medium exposed to differences in pressure, temperature, and chemical potentials. Thermodynamic extensive variables on the macro-scale are obtained by integrating over a representative elementary volume (REV). Using Euler homogeneity of the first order, we obtain the Gibbs equation for the REV. From this we define the intensive variables, the temperature, pressure and chemical potentials and, using the balance equations, derive the entropy production for the REV. The entropy production defines sets …
Thin liquid films on isothermal substrates, where film is flat and parallel to the substrate, succumb to thermocapillary instabilities and rupture forming local hot-spots. These instabilities are called long wavelength instabilities and are specific to aspect ratios where the liquid film mean thickness is several orders of magnitude less than the substrate characteristic dimension. In the absence of stabilizing gravitational acceleration, the growth rate of thermocapillary instabilities is further intensified, driving the film to rupture at an even earlier time. Numerical simulations of zero g…
In the above paper by Bechtel, Cai, Rooney and Wang, Physics of Fluids, 2004, 16, 3955-3974 six different theories of a Newtonian viscous fluid are investigated and compared, namely, the theory of a compressible Newtonian fluid, and five constitutive limits of this theory: the incompressible theory, the limit where density changes only due to changes in temperature, the limit where density changes only with changes in entropy, the limit where pressure is a function only of temperature, and the limit of pressure a function only of entropy. The six theories are compared through their ability to …
Intelligent backpropagated neural networks application on Couette-Poiseuille flow of variable viscosity in a composite porous channel filled with an anisotropic porous layer
The study investigates the problem of magnetohydrodynamic (MHD) free convection over a non-isothermal axisymmetric body under the action of transverse magnetic field. The body is embedded in a porous medium saturated with electrically conducting non-Newtonian power law fluid. In order to obtain similarity solution, it is assumed that the viscosity of the fluid decays exponentially with temperature. The qualitative results are illustrated for a vertical flat plate, horizontal cylinder and sphere.
This paper analyses the effects of small injection/suction Reynolds number, Hartmann number, permeability parameter and wave number on a viscous incompressilbe electrically conduction fluid flow in a parallel porous channel. The plates of the channel with small constant injection/suction, have constant temperature. The upper plate is allowed to mouve in flow direction and the lower plate is kept at rest. A magnetic field of uniform strength is also applied normally to the plates what are parallel. The originality of the paper is to study the effect of the above parameter in temporal linear sta…
In petroleum reservoir simulation, the industry standard preconditioner, the constrained pressure residual method (CPR), is a two-stage process which involves solving a restricted pressure system with Algebraic Multigrid (AMG). Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not have a specific treatment of the additional energy conservation equation and temperature variable. We seek to develop preconditioners which better capture thermal effects such as heat diffusion. In order to study the effects of both pressure and temperature on…
For both isothermal and thermal petroleum reservoir simulation, the Constrained Pressure Residual (CPR) method is the industry-standard preconditioner. This method is a two-stage process involving the solution of a restricted pressure system. While initially designed for the isothermal case, CPR is also the standard for thermal cases. However, its treatment of the energy conservation equation does not incorporate heat diffusion, which is often dominant in thermal cases. In this paper, we present an extension of CPR: the Constrained Pressure-Temperature Residual (CPTR) method, where a restricte…
Modelling Material Injection Into Porous Structures Under Non-isothermal Conditions
University of Stuttgart（斯图加特大学）；Institute for Modelling and Simulation of Biomechanical Systems, University of Stuttgart（斯图加特大学生物力学系统建模与仿真研究所）；Institute of Structural Mechanics and Dynamics of Aerospace Structures, University of Stuttgart（斯图加特大学航空航天结构结构力学与动力学研究所）；Ansbach University of Applied Sciences（安斯巴赫应用科学大学）；Stuttgart Center for Simulation Science (SC SimTech), University of Stuttgart（斯图加特大学斯图加特模拟科学中心）
Darcy's law for porous media transport is given a new local thermodynamic basis in terms of the grand potential of confined fluids. The local effective pressure gradient is determined using non-equilibrium molecular dynamics, and the hydraulic conductivity and permeability are investigated. The transport coefficients are determined for single-phase flow in face-centered cubic lattices of solid spheres. The porosity changed from that in the closest packing of spheres to near unity in a pure fluid, while the fluid mass density varied from that of a dilute gas to a dense liquid. The permeability …
The buoyancy-induced parallel flow in a vertical cylindrical porous layer is analysed. A radial thermal gradient caused by a uniformly distributed heat source is assumed to induce the buoyant flow. The layer boundaries are modelled as isothermal and permeable to an external fluid reservoir. The onset of the convective instability is analysed by linearising the governing equations for the perturbations. The governing parameters driving the instability are the heat-source Rayleigh number and the ratio between the internal radius and the external radius. Neutral stability curves and the critical …
We develop a two-fluid model (TFM) for heat transfer in dense non-Brownian suspensions. Specifically, we propose closure relations for the inter-phase heat transfer coefficient and the thermal diffusivity of the particle phase based on calibration against experimental data. The model is then employed to simulate non-isothermal flow in an annular Couette cell. We find that, when the shear rate is controlled by the rotation of the inner cylinder, both the shear and thermal gradients are responsible for particle migration. Within the TFM framework, we identify the origin and functional form of a …
Mesh-free methods have significant potential for simulations of flows in complex geometries, with the difficulties of domain discretisation greatly reduced. However, many mesh-free methods are limited to low order accuracy. In order to compete with conventional mesh-based methods, high order accuracy is essential. The Local Anisotropic Basis Function Method (LABFM) is a mesh-free method introduced in King et al., J. Comput. Phys. 415:109549 (2020), which enables the construction of highly accurate difference operators on disordered node discretisations. Here, we introduce a number of developme…
The temporal stability of an inviscid flow through cylindrical geometries with a porous wall subjected to non-axisymmetric perturbations is investigated in the present work using an unsteady Darcy equation for the porous layer. An expression for the perturbation energy exchange term between the fluid and porous layers is derived by integrating the perturbation equation for the porous layer which is then used to prove the propositions. The necessary and sufficient condition for the existence of the instability in flows through cylindrical geometries is shown to be $\frac{d}{dr} \left( \frac{r \…
We have investigated the effects of permeable walls, modeled by linear acoustic impedance with zero reactance, on compressible channel flow via linear stability analysis (LSA). Base flow profiles are taken from impermeable isothermal-wall laminar and turbulent channel flow simulations at bulk Reynolds number, $Re_b$= 6900 and Mach numbers, $M_b$ = 0.2, 0.5, 0.85. For a sufficiently high value of permeability, Two dominant modes are made unstable: a bulk pressure mode, causing symmetric expulsion and suction of mass from the porous walls (Mode 0); a standing-wave-like mode, with a pressure node…
We study the effect of spatially-correlated heterogeneity on isothermal drying of porous media. We combine a minimal pore-scale model with microfluidic experiments with the same pore geometry. Our simulated drying behavior compare favorably with experiments, considering the large sensitivity of the emergent behavior to the uncertainty associated with even small manufacturing errors. We show that increasing the correlation length in particle sizes promotes preferential drying of clusters of large pores, prolonging liquid connectivity and surface wetness and thus higher drying rates for longer p…
The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface model which is based on van der Waals theory of capillarity. Diffuse interface methods have gained much interest to model two-phase flow in porous media. However, for the numerical solution of the NSK equations two major challenges have to be faced. First, an extended numerical stencil is required due to a third-order term in the linear momentum and the total energy equations. In addition, the dispersive contribution in the linear momentum equations prevents the straightforward use of contact angle boundary conditions. Se…
The linear stability of buoyant parallel flow in a vertical porous layer with an annular cross-section is investigated. The vertical cylindrical boundaries are kept at different uniform temperatures and they are assumed to be impermeable. The emergence of linear instability by convection cells is excluded on the basis of a numerical solution of the linearised governing equations. This result extends to the annular geometry the well-known Gill's theorem regarding the impossibility of convective instability in a vertical porous plane slab whose boundaries are impermeable and isothermal with diff…
The basic stationary buoyant flow in a vertical annular porous passage induced by a boundary temperature difference is investigated. The vertical cylindrical boundaries are considered both isothermal and permeable to external fluid reservoirs. There exists a stationary parallel velocity field with a zero flow rate and pure conduction heat transfer. Its linear stability is analysed with normal mode perturbations of the pressure and temperature fields. The transition to convective instability is caused by the basic horizontal temperature gradient. Hence, its nature differs from that of the usual…
Perfectly incompressible materials do not exist in nature but are a useful approximation of several media which can be deformed in non-isothermal processes but undergo very small volume variation. In this paper the linear analysis of the Darcy-Bénard problem is performed in the class of extended-quasi-thermal-incompressible fluids, introducing a factor $β$ which describes the compressibility of the fluid and plays an essential role in the instability results. In particular, in the Oberbeck-Boussinesq approximation, a more realistic constitutive equation for the fluid density is employed in ord…
The Navier-Stokes-Fourier model for a 3D thermoconducting viscous fluid, where the evolution equation for the temperature T contains a term proportional to the rate of energy dissipation, is investigated analitically at the light of the rotational invariance property. Two cases are considered: the Couette flow and a flow with a radial velocity between two rotating impermeable and porous coaxial cylinders, respectively. In both cases, we show the existence of a maximum value of T, T_max, when the difference of temperature Delta T=T_2-T_1 on the surfaces of the cylinders is assigned. The role of…
We consider the stability of two-dimensional viscous flows in an annulus with permeable boundary. In the basic flow, the velocity has nonzero azimuthal and radial components, and the direction of the radial flow can be from the inner cylinder to the outer one or vice versa. In most earlier studies, all components of the velocity were assumed to be given on the entire boundary of the flow domain. Our aim is to study the effect of different boundary conditions on the stability of such flows. We focus on the following boundary conditions: at the inflow part if the boundary (which may be either in…
We study the convective patterns that arise in a nearly semi-cylindrical cavity fed in with hot fluid at the upper boundary, bounded by a cold, porous semi-circular boundary at the bottom, and infinitely extended in the third direction. While this configuration is relevant to continuous casting processes that are significantly more complex, we focus on the flow patterns associated with the particular form of mixed convection that arises in it. Linear stability analysis and direct numerical simulations (DNS) are conducted, using the spectral element method to identify observable states. The nat…
We have developed a fully sparse, compact-scheme based biglobal stability analysis numerical solver applied, for the scope of the current paper, to the investigation of the effects of impedance boundary conditions (IBCs) on the structure of a fully developed compressible turbulent channel flow. A sixth-order compact finite difference scheme is used to discretize the linearized Navier-Stokes equations leading to a Generalized Eigenvalue Problem (GEVP). Sparsity is retained by explicitly introducing derivatives of the perturbation as additional unknowns, increasing the overall problem size (numb…
