Lie group analysis of a non-Newtonian fluid flow over a porous surface (original) (raw)
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Paper: LIE GROUP ANALYSIS OF A NON-NEWTONIAN FLUID FLOW OVER A POROUS SURFACE
Two-dimensional, unsteady squeezed flow over a porous surface for a power-law non-Newtonian fluid is considered. Continuity, momentum and energy equations are written and cast into a non-dimensional form. Boundary conditions are selected in a general form. Lie Group theory is applied to the equations. Then, a partial differential system with three independent variables is converted into an ordinary differential system, via application of two successive symmetry generators. The ordinary differential equations are solved numerically. Effects of flow behavior index, Prandtl number, squeezing parameter, surface velocity parameter and suction/injection parameter on the flow are outlined in the figures.
The passing hydrodynamics actions of the non-Newtonian fluid flow in horizontal parallel-plate channels without porous medium are searched numerically and Newtonian fluid flow is investigated analytically and numerically. We have organized solutions of a boundary-value problem for a partial differential equation arising in the research of the flow of Newtonian and non-Newtonian fluids without a porous medium. The problem of Newtonian and non-Newtonian fluids flow is solved by means of the Lie-group analysis approach. The one-parameter group transformation reduces the number of independent variables by one and the governing partial differential equation with the boundary conditions reduces to an ordinary differential equation with the suitable boundary conditions. After solving the ordinary differential equation, we are obtained results of the problem.
Lie Group Analysis of Unsteady Flow and Heat Transfer over a Porous Surface for a Viscous Fluid
Journal of Applied Mathematics, 2012
The problem of a two-dimensional, unsteady flow and a heat transfer of a viscous fluid past a surface in the presence of variable suction/injection is analyzed. The unsteadiness is due to the time dependent free stream flow. The governing equations are derived with the usual boundary layer approximation. Using Lie group theory, a group classification of the equations with respect to the variable free stream flow and suction/injection velocity is performed. Restrictions imposed by the boundary conditions on the symmetries are discussed. Adopting the obtained symmetry groups, governing partial differential equations are converted into ordinary differential equations and then solved numerically. Effects of the dimensionless problem parameters on the velocity and temperature profiles are outlined in the figures.
A Semi-Analytical Solution for a Porous Channel Flow of a Non-Newtonian Fluid
Journal of Applied Fluid Mechanics, 2016
A theoretical study of steady laminar two-dimensional flow of a non-Newtonian fluid in a parallel porous channel with variable permeable walls is carried out. Solution by Differential Transform Method (DTM) is obtained and the flow behavior is studied. The non-Newtonian fluid considered for the study is couple stress fluid. Thus, in addition to the effects of inertia and permeabilities on the flow, the couple stress effects are also analyzed. Results are presented and comparisons are made between the behaviour of Newtonian and non-Newtonian fluids.
Flow of non-newtonian fluids in porous media
Journal of Polymer Science Part B: Polymer Physics, 2010
The study of flow of non-Newtonian fluids in porous media is very important and serves a wide variety of practical applications in processes such as enhanced oil recovery from underground reservoirs, filtration of polymer solutions and soil remediation through the removal of liquid pollutants. These fluids occur in diverse natural and synthetic forms and can be regarded as the rule rather than the exception. They show very complex strain and time dependent behavior and may have initial yieldstress. Their common feature is that they do not obey the simple Newtonian relation of proportionality between stress and rate of deformation. Non-Newtonian fluids are generally classified into three main categories: time-independent whose strain rate solely depends on the instantaneous stress, time-dependent whose strain rate is a function of both magnitude and duration of the applied stress and viscoelastic which shows partial elastic recovery on removal of the deforming stress and usually demonstrates both time and strain dependency. In this article, the key aspects of these fluids are reviewed with particular emphasis on singlephase flow through porous media. The four main approaches for describing the flow in porous media are examined and assessed. These are: continuum models, bundle of tubes models, numerical methods and pore-scale network modeling.
An analysis is carried out to study the momentum and mass transfer characteristics in a visco-elastic fluid flow over a porous stretching sheet in the presence of a transverse magnetic field. The flow is generated solely due to the linear stretching of the sheet. The symmetry groups obtained using a special form of Lie group transformations viz. Scaling group of transformations, reduce the momentum equation and the concentration conservation equation into fourth order and second order ordinary differential equations respectively. Closed form analytical solutions have been derived for non-dimensional concentration and mass flux profiles in the form of confluent hyper geometric (Kum-mer's) functions, for two different cases of the boundary conditions, namely (1) Prescribed Sheet Concentration (PSC) and (2) Prescribed Mass Flux (PMF). The main emphasis of this paper is to derive the final equations using the scaling group of transformations and to study the effects of the visco-elastic parameter, suction/blowing parameter, magnetic parameter, concentration and mass flux parameters and Schmidt number on the mass transfer characteristics. It has been observed that, for the case of suction and for the values of the parameters considered, an ideal combination to obtain a reduced concentration boundary layer thickness would be to choose smaller values of the visco-elastic and magnetic parameters and relatively larger value for the Schmidt number. This is seen to be more significant in the PMF case. An increase in the concentration and mass flux parameters has shown a steep decrease in the concentration boundary layer thickness.
2013
A B S T R A C T In this paper general group symmetry analysis so-called deductive group-theoretical method is applied to analyze the boundary layer flow of electrically conducting viscous fluid over with heat transfer over a non-linear surface. The symmetry groups admitted by the corresponding boundary value problem are obtained. With the use of the entailed similarity transformations the governing equations reduce to a set of non-linear ordinary differential equations. The system of equations is solved numerically using MATLAB coding. The effect of various flow parameters is studied for both Newtonian and Non-Newtonian power-law fluids in case of stretching and shrinking sheet.
Physica Scripta, 2013
A numerical investigation of magnetoconvective boundary layer slip flow along a nonisothermal continuously moving permeable nonlinear radiating plate in Darcian porous media is reported. The concentration dependent mass diffusivity, viscous dissipation, Joule heating, and chemical reaction are taken into account. A Lie group of transformation is applied to the governing transport equations and boundary condition to find the corresponding similarity equations. Furthermore, the similarity equations with the relevant boundary conditions are solved numerically using the Runge-Kutta-Fehlberg fourth-fifth order numerical method. Numerical results for the dimensionless velocity, temperature, and concentration distributions as well as friction factor, local Nusselt, and local Sherwood numbers are discussed for various controlling parameters. It is found that that the dimensionless concentration increases whilst the rate of mass transfer decreases with the mass diffusivity parameter. An excellent correlation is found between the present results and published results. The study finds applications in the polymer industry and metallurgy.
On non-Newtonian fluid flow in ducts and porous media
Chemical Engineering Science, 1998
In this communication a three-shape-factor approach is developed to characterize both the flow of non-Newtonian fluids, in particular generalized Newtonian fluids, in an arbitrarily shaped duct and the flow over an isolated sphere. The flow of Herschel-Bulkley fluid and Meter fluid is studied. While the detailed solution for a Herschel-Bulkley fluid can be used to deduce the values of the shape factors, the simplified solution for a Meter fluid is also provided to enable for easy application. The interaction of macromolecules with fine capillary duct walls, i.e. the discontinuity in flow, is modeled using the Meter fluid model. At the low shear limit (or very low flow rate), the solutions of Chauveteau (1982, J. Rheol. 26, 111-142) and Kozicki et al. (1987, Chem. Engng Commun. 59, 137-160) are recovered. For a high flow rate, the present study predicts an increased pressure drop for surface-adhesive capillary and a pressure drop reduction for a wall macromolecule depleted/aligned system. The success in modeling flow in arbitrary shaped ducts and flow over isolated spheres using the same approach suggests that the present approach is also applicable to flow in porous media. When a volume averaging technique is employed to arrive at the continuum governing equations for flow of a generalized Newtonian fluid in porous media, the macroscopic viscosity is obtained. The macroscopic viscosity is the single quantity in the governing equations that must be determined for a non-Newtonian fluid flow in addition to the parameters already known for a Newtonian fluid flow. In the Newtonian limit, the macroscopic viscosity becomes identical to the Newtonian viscosity. Otherwise, the macroscopic viscosity is the apparent viscosity of the fluid under the average microscopic shear conditions in porous media. The expressions for the macroscopic viscosity for various fluids: namely, Herschel-Bulkley fluid, Meter fluid and Cross fluid, are derived in this communication. Porous medium matrix-macromolecule interactions due to flow discontinuity are studied using two phenomenological models: macromolecule retention and macromolecule alignment/depletion. In particular, the Meter fluid model is used to derive the effects of the porous medium matrix-macromolecule interactions. Predictions based on this study agree well with experimental data for flow of non-Newtonian fluids in packed beds and consolidated sandstones.
PLOS ONE, 2015
The aim of this article is to model and analyze an unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates passing through porous medium channel with slip boundary condition. A single fourth order nonlinear ordinary differential equation is obtained using similarity transformation. The resulting boundary value problem is solved using Homotopy Perturbation Method (HPM) and fourth order Explicit Runge Kutta Method (RK4). Convergence of HPM solution is verified by obtaining various order approximate solutions along with absolute residuals. Validity of HPM solution is confirmed by comparing analytical and numerical solutions. Furthermore, the effects of various dimensionless parameters on the longitudinal and normal velocity profiles are studied graphically.