Showing papers on "Similarity solution published in 2013"

Capillary Flow in an Interior Corner

Abstract: The design of fluids management processes in the low-gravity environment of space requires an accurate model and description of capillarity-controlled flow in containers of irregular geometry. Here we consider the capillary rise of a fluid along an interior corner of a container following a rapid reduction in gravity. The analytical portion of the work presents an asymptotic formulation in the limit of a slender fluid column, slight surface curvature along the corner, small inertia, and low gravity. New similarity solutions are found and a list of closed form expressions is provided for flow rate and column length. In particular, it is found that the flow is proportional to t(exp 1/2) for a constant height boundary condition, t(exp 2/5) for a spreading drop, and t(exp 3/5) for constant flow. In the experimental portion of the work, measurements from a 2.2s drop tower are reported. An extensive data set, collected over a previously unexplored range of flow parameters, includes estimates of repeatability and accuracy, the role of inertia and column slenderness, and the effects of corner angle, container geometry, and fluid properties. Comprehensive comparisons are made which illustrate the applicability of the analytic results to low-g fluid systems design.

Flow of an eyring-powell non-newtonian fluid over a stretching sheet

Abstract: This article is devoted to the study of the boundary layer flow of a non-Newtonian fluid over a stretching sheet. The non-Newtonian behavior of the fluid is characterized by the constitutive equation due to Powell and Eyring (1944). A second-order approximation of the Eyring-Powell model is used to obtain the flow equations. A local similarity solution of the governing problem is obtained numerically using an implicit finite difference scheme known as the Keller box method. The influence of pertinent non-Newtonian fluid parameters M and λ on the velocity and skin-friction coefficient is analyzed through graphical and tabular results.

Flow and heat transfer for three-dimensional flow over an exponentially stretching surface

Abstract: This study analyzes the laminar boundary-layer flow and heat transfer characteristics of a steady, three-dimensional viscous fluid driven by a horizontal surface stretched exponentially in two lateral directions. The simulations in this study assume that the surface temperature is also distributed exponentially and reduce the governing equations to a set of ordinary differential equations using a similarity transformation. This study develops a numerical procedure that combines the Ackroyd method and Runge-Kutta integration scheme to solve the transformed equations. Results show that heat transfer characteristics depend strongly on the stretching ratio, temperature exponent, and Prandtl number.

Analytic solution for magnetohydrodynamic boundary layer flow of Casson fluid over a stretching/shrinking sheet with wall mass transfer

Abstract: In this analysis, the magnetohydrodynamic boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet in the presence of wall mass transfer is studied. Using similarity transformations, the governing equations are converted to an ordinary differential equation and then solved analytically. The introduction of a magnetic field changes the behavior of the entire flow dynamics in the shrinking sheet case and also has a major impact in the stretching sheet case. The similarity solution is always unique in the stretching case, and in the shrinking case the solution shows dual nature for certain values of the parameters. For stronger magnetic field, the similarity solution for the shrinking sheet case becomes unique.

Effects of Exothermic/Endothermic Chemical Reactions with Arrhenius Activation Energy on MHD Free Convection and Mass Transfer Flow in Presence of Thermal Radiation

Abstract: A local similarity solution of unsteady MHD natural convection heat and mass transfer boundary layer flow past a flat porous plate within the presence of thermal radiation is investigated. The effects of exothermic and endothermic chemical reactions with Arrhenius activation energy on the velocity, temperature, and concentration are also studied in this paper. The governing partial differential equations are reduced to ordinary differential equations by introducing locally similarity transformation (Maleque (2010)). Numerical solutions to the reduced nonlinear similarity equations are then obtained by adopting Runge-Kutta and shooting methods using the Nachtsheim-Swigert iteration technique. The results of the numerical solution are obtained for both steady and unsteady cases then presented graphically in the form of velocity, temperature, and concentration profiles. Comparison has been made for steady flow () and shows excellent agreement with Bestman (1990), hence encouragement for the use of the present computations.

An analytical study on entropy generation of nanofluids over a flat plate

Abstract: The steady two-dimensional boundary layer flow of nanofluids over a flat plate is studied analytically to analyze the generated entropy inside the boundary layer at a constant wall temperature. Applying the transformation of the PDE equations of continuity, momentum and energy to ODE ones by similarity variables, a dimensionless equation for entropy generation inside the boundary layer is presented. The most accurate series solution was found by coupling the homotopy-perturbation method (HPM) and the variational iteration method (VIM), which provides an effective technique for solving strongly nonlinear ordinary differential equations. The analytical results indicated that the generated entropy strongly depends on the nanoparticle volume fraction ( ϕ ), Prandtl, Eckert and Reynolds numbers. Based on the series solution, the effects of ϕ on velocity, temperature and entropy generation were explained in details and the related figures are plotted.

Self-similar mean dynamics in turbulent wall flows

Abstract: This study investigates how and why dynamical self-similarities emerge with increasing Reynolds number within the canonical wall flows beyond the transitional regime. An overarching aim is to advance a mechanistically coherent description of turbulent wall-flow dynamics that is mathematically tractable and grounded in the mean dynamical equations. As revealed by the analysis of Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst. A, vol. 24, 2009, pp. 781–807), the equations that respectively describe the mean dynamics of turbulent channel, pipe and boundary layer flows formally admit invariant forms. These expose an underlying self-similar structure. In all cases, two kinds of dynamical self-similarity are shown to exist on an internal domain that, for all Reynolds numbers, extends from to , where is the kinematic viscosity, is the friction velocity and is the half-channel height, pipe radius, or boundary layer thickness. The simpler of the two self-similarities is operative on a large outer portion of the relevant domain. This self-similarity leads to an explicit analytical closure of the mean momentum equation. This self-similarity also underlies the emergence of a logarithmic mean velocity profile. A more complicated kind a self-similarity emerges asymptotically over a smaller domain closer to the wall. The simpler self-similarity allows the mean dynamical equation to be written as a closed system of nonlinear ordinary differential equations that, like the similarity solution for the laminar flat-plate boundary layer, can be numerically integrated. The resulting similarity solutions are demonstrated to exhibit nearly exact agreement with direct numerical simulations over the solution domain specified by the theory. At the Reynolds numbers investigated, the outer similarity solution is shown to be operative over a domain that encompasses of the overall width of the flow. Other properties predicted by the theory are also shown to be well supported by existing data.

Similarity theory of lubricated Hertzian contacts

Abstract: We consider a heavily loaded, lubricated contact between two elastic bodies at relative speed U, such that there is substantial elastic deformation. As a result of the interplay between hydrodynamics and non-local elasticity, a fluid film develops between the two solids, whose thickness scales as U 3/5. The film profile h is selected by a universal similarity solution along the upstream inlet. Another similarity solution is valid at the outlet, which exhibits a local minimum in the film thickness. The two solutions are connected by a hyperbolic problem underneath the contact. Our asymptotic results for a soft sphere pressed against a hard wall are shown to agree with both experiment and numerical simulations.

Boundary layer stagnation-point flow of casson fluid and heat transfer towards a shrinking/stretching sheet

Abstract: The steady boundary layer stagnation-point flow of Casson fluid and heat transfer towards a shrinking/stretching sheet is studied. Appropriate similarity transformations are employed to transform the governing partial differential equations into the self-similar ordinary differential equations and those are then solved numerically using very efficient shooting method. The numerical computations are carried out for several values of parameters involved (especially, velocity ratio parameter and Casson parameter) to know the possibility of similarity solution for the boundary layer stagnation-point flow. It is found that the range of velocity ratio parameter for which similarity solution exists is unaltered for any change in Casson parameter, though the skin friction changes with Casson parameter. Thus, the possibility of similarity solution for Casson fluid flow is same as that of Newtonian fluid flow.

The Unsteady Flow of a Nanofluid in the Stagnation Point Region of a Time-dependent Rotating Sphere

Abstract: This paper deals with the unsteady boundary layer flow and heat transfer of nanofluid over a time-dependent rotating sphere where the free stream velocity varies continuously with time. The boundary layer equations were normalized via similarity variables and solved numerically. Best accuracy of the results has been obtained for regular fluid with previous studies. The nanofluid is treated as a two-component mixture (base fluid+nanoparticles) that incorporates the effects of Brownian diffusion and thermophoresis simultaneously as the two most important mechanisms of slip velocity in laminar flows. Our outcomes indicated that as

Natural-Convection Flow of Nanofluids Over Vertical Cone Embedded in Non-Darcy Porous Media

Abstract: In this paper, non-Darcy flow and natural convection over a vertical cone embedded in a porous medium saturated with a nanofluid is studied using the Forchheimer-extended Darcy law. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. An analytical technique, similarity solution method, is used to convert the governing equations into a set of ordinary differential equations, and then numerically solved using the finite difference method. The effect of non-Darcy parameter and nanofluid parameters on the velocity, temperature, and nanoparticles volume fraction profiles, as well as on the two important parameters of heat and mass transfer, i.e., reduced Nusselt and Sherwood numbers, are discussed. The results show that an increase in the non-Darcy parameter decreases the velocity profiles, whereas it would increase the temperature and concentration profiles. Simulation results also show that an increase in the non-Darcy parameter would decrease the reduced Nusselt ...

Natural convection of nanofluid over vertical plate embedded in porous medium: prescribed surface heat flux

Abstract: The aim of the present paper is to analyze the natural convection heat and mass transfer of nanofluids over a vertical plate embedded in a saturated Darcy porous medium subjected to surface heat and nanoparticle fluxes. To carry out the numerical solution, two steps are performed. The governing partial differential equations are firstly simplified into a set of highly coupled nonlinear ordinary differential equations by appropriate similarity variables, and then numerically solved by the finite difference method. The obtained similarity solution depends on four non-dimensional parameters, i.e., the Brownian motion parameter (N b), the Buoyancy ratio (N r), the thermophoresis parameter (N t), and the Lewis number (Le). The variations of the reduced Nusselt number and the reduced Sherwood number with N b and N t for various values of Le and N r are discussed in detail. Simulation results depict that the increase in N b, N t, or N r decreases the reduced Nusselt number. An increase in the Lewis number increases both of the reduced Nusselt number and the Sherwood number. The results also reveal that the nanoparticle concentration boundary layer thickness is much thinner than those of the thermal and hydrodynamic boundary layers.

Stability of dual solutions in stagnation-point flow and heat transfer over a porous shrinking sheet with thermal radiation

Abstract: An analysis is carried out to study the steady two-dimensional stagnation-point flow and heat transfer of an incompressible viscous fluid over a porous shrinking sheet in the presence of thermal radiation. A set of similarity transformations reduce the boundary layer equations to a set of non-linear ordinary differential equations which are solved numerically using fourth order Runge-Kutta method with shooting technique. The analysis of the result obtained shows that as the porosity parameter β increases, the range of region of existence of similarity solution increases. It is also observed that multiple solutions exist for a certain range of the ratio of the shrinking velocity to the free stream velocity (i.e., α) which again depends on β. We then discuss the stability of the unsteady solutions about each steady solution, showing that one steady state solution corresponds to a stable solution whereas the other corresponds to an unstable solution. The stable solution corresponds to the physically relevant solution. Further we obtain numerical results for each solution, which enable us to discuss the features of the respective solutions.

Swirling flow of Bingham fluids above a rotating disk: An exact solution

Abstract: An exact solution is found for swirling flow of Bingham fluids above a rotating disk. The similarity parameter, first introduced by von Karman for Newtonian fluids, is found to be applicable for Bingham fluids effectively reducing the governing PDEs into a set of coupled ODEs. The system of nonlinear ordinary differential equations so obtained is solved numerically. Numerical results are presented addressing the effect of a fluid’s yield stress (as represented by the Bingham number) on the velocity profiles, wall shear stress distribution, and volumetric flow rate. Unlike the radial and axial velocity component, the tangential velocity is predicted to increase by an increase in the yield stress. On the other hand, the wall shear stress and the volumetric flow rate are both predicted to decrease when the fluid’s yield stress is increased.

Entraining gravity currents

Abstract: Entrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of the bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as t 0:447 . For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as t 4=5 . In both cases, entrainment is most significant close to the current front.

Thermodynamic optimization of fluid flow over an isothermal moving plate

Abstract: In this paper, entropy generation minimization (EGM) was employed in order to achieve a thermodynamic optimization of fluid flow and heat transfer over a flat plate. The basic boundary layer equations including continuity, momentum, energy, and entropy generation have been reduced to a two-point boundary value problem via similarity variables and solved numerically via Runge–Kutta–Fehlberg scheme. The novelty of this study was to consider the effects of velocity ratio λ – which represents the ratio of the wall velocity to the free stream fluid velocity – in a thermodynamic system. Focusing on the velocity ratio as a pivotal parameter, in view of minimizing the entropy generation, the optimum value of λ = λ o was achieved. Moreover, considering Bejan number, it was shown that the region, in which the maximum entropy generates, gets closer to the plate as λ increases.

Slip effects on unsteady stagnation-point flow and heat transfer over a shrinking sheet

Abstract: This paper investigates the unsteady boundary layer stagnation-point flow and heat transfer over a linearly shrinking sheet in the presence of velocity and thermal slips. Similarity solutions for the transformed governing equations are obtained and the reduced equations are then solved numerically using fourth order Runge-Kutta method with shooting technique. The numerical results show that multiple solutions exist for certain range of the ratio of shrinking velocity to the free stream velocity (i.e., α) which again depend on the unsteadiness parameter β and the velocity slip parameter (i.e., δ). An enhancement of the velocity slip parameter δ causes more increment in the existence range of similarity solution. Fluid velocity at a point increases increases with the increase in the value of the velocity slip parameter δ, resulting in a decrease in the temperature field. The effects of the velocity and thermal slip parameters, unsteadiness parameter (β) and the velocity ratio parameter (α) on the velocity and temperature distributions are computed, analyzed and discussed. The reported results are in good agreement with the available published results in the literature.

Fluxes through steady chimneys in a mushy layer during binary alloy solidification

Abstract: Solute transport within solidifying binary alloys occurs predominantly by convection from narrow liquid chimneys within a porous mushy layer. We develop a simple model that elucidates the dominant structure and driving forces of the flow, which could be applied to modelling brine fluxes from sea ice, where a cheaply implementable approach is essential. A horizontal density gradient within the mushy layer in the vicinity of the chimneys leads to baroclinic torque which sustains the convective flow. In the bulk of the mushy layer, the isotherms are essentially horizontal. In this region, we impose a vertically linear temperature field and immediately find that the flow field is a simple corner flow. We determine the strength of this flow by finding a similarity solution to the governing mushy-layer equations in an active region near the chimney. We also determine the corresponding shape of the chimney, the vertical structure of the solid fraction and the interstitial flow field. We apply this model first to a periodic, planar array of chimneys and show analytically that the solute flux through the chimneys is proportional to a mush Rayleigh number. Secondly we extend the model to three dimensions and find that an array of chimneys can be characterized by the average drainage area alone. Therefore we solve the model in an axisymmetric geometry and find new, sometimes nonlinear, relationships between the solute flux, the Rayleigh number and the other dimensionless parameters of the system.

Gravity-Driven Thin Film Flow of an Ellis Fluid

Abstract: The thin film lubrication approximation has been studied extensively for moving contact lines of Newtonian fluids. However, many industrial and biological applications of the thin film equation involve shear-thinning fluids, which often also exhibit a Newtonian plateau at low shear. This study presents new numerical simulations of the three-dimensional (i.e. two-dimensional spreading), constant-volume, gravity-driven, free surface flow of an Ellis fluid. The numerical solution was validated with a new similarity solution, compared to previous experiments, and then used in a parametric study. The parametric study centered around rheological data for an example biological application of thin film flow: topical drug delivery of anti-HIV microbicide formulations, e.g. hydroxyethylcellulose (HEC) polymer solutions. The parametric study evaluated how spreading length and front velocity saturation depend on Ellis parameters. A lower concentration polymer solution with smaller zero shear viscosity (η0), τ1/2, and λ values spread further. However, when comparing any two fluids with any possible combinations of Ellis parameters, the impact of changing one parameter on spreading length depends on the direction and magnitude of changes in the other two parameters. In addition, the isolated effect of the shear-thinning parameter, λ, on the front velocity saturation depended on τ1/2. This study highlighted the relative effects of the individual Ellis parameters, and showed that the shear rates in this flow were in both the shear-thinning and plateau regions of rheological behavior, emphasizing the importance of characterizing the full range of shear-rates in rheological measurements. The validated numerical model and parametric study provides a useful tool for future steps to optimize flow of a fluid with rheological behavior well-described by the Ellis constitutive model, in a range of industrial and biological applications.

Effect of variable permeability on the propagation of thin gravity currents in porous media

Abstract: A new formulation is proposed to study the influence of deterministic heterogeneity on the propagation of thin two-dimensional gravity currents in a porous medium above a horizontal impervious boundary. Heterogeneity is conceptualized as a monotonic power-law variation of medium permeability transverse or parallel to the direction of propagation. Considering the injection of a constant or time-variable volume of fluid, the nonlinear differential problem admits a similarity solution which describes the shape and rate of propagation of the current. The bounds on parameters necessary to respect model assumptions are derived asymptotically and for finite time, to clarify the range of applicability of the proposed models. An application to the migration of a contaminant gravity current in the subsurface is then discussed, showing the impact of permeability variations on extension and shape of the intrusion.

Reducing Entropy Generation in MHD Fluid Flow over Open Parallel Microchannels Embedded in a Micropatterned Permeable Surface

Abstract: The present study examines embedded open parallel microchannels within a micropatterned permeable surface for reducing entropy generation in MHD fluid flow in microscale systems. A local similarity solution for the transformed governing equations is obtained. The governing partial differential equations along with the boundary conditions are first cast into a dimensionless form and then the reduced ordinary differential equations are solved numerically via the Dormand-Prince pair and shooting method. The dimensionless entropy generation number is formulated by an integral of the local rate of entropy generation along the width of the surface based on an equal number of microchannels and no-slip gaps interspersed between those microchannels. Finally, the entropy generation numbers, as well as the Bejan number, are investigated. It is seen that surface-embedded microchannels can successfully reduce entropy generation in the presence of an applied magnetic field.

Soret and Dufour Effects on Hydro Magnetic Heat and Mass Transfer over a Vertical Plate with a Convective Surface Boundary Condition and Chemical Reaction

Abstract: This paper examined the hydro magnetic boundary layer flow with heat and mass transfer over a vertical plate in the presence of magnetic field with Soret and Dufour effects, chemical reaction and a convective heat exchange at the surface with the surrounding has been studied. The similarity solution is used to transform the system of partial differential equations and an efficient numerical technique is implemented to solve the reduced system by using the Runge-Kutta fourth order method with shooting technique. A comparison study with the previous results shows a very good agreement. The results are presented graphically and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters.

Slip Effects on Boundary Layer Flow and Heat Transfer Along a Stretching Cylinder

Abstract: An axi-symmetric laminar boundary layer flow of a viscous incompressible fluid and heat transfer towards a stretching cylinder is presented. Velocity slip is considered instead of the no-slip condition at the boundary. Similarity transformations are used to convert the partial differential equations corresponding to the momentum and heat equations into non-linear ordinary differential equations. Numerical solutions of these equations are obtained by the shooting method. It is found that the velocity decreases with increasing the slip parameter. The skin friction as well as the heat transfer rate at the surface is larger for a cylinder compared to those for a flat plate.

Self-similar vortex dynamics in superfluid 4He under the Cartesian representation of the Hall-Vinen model including superfluid friction

Abstract: We determine conditions under which the fully nonlinear form of the local induction approximation (LIA) governing the motion of a vortex filament in superfluid 4He (that is, the Hall-Vinen model) in the Cartesian frame of reference permits the existence of self-similar solutions, even in the presence of superfluid friction parameters. Writing the Cartesian Hall-Vinen LIA in potential form for the motion of a vortex filament, we find that a necessary condition for self-similarity is that the normal-fluid component vanishes (which makes sense in the low temperature limit), and we reduce the potential form of the Hall-Vinen LIA to a complex nonlinear ordinary differential equation governing the behavior of a similarity solution. In the limit where superfluid friction parameters are negligible, we provide some analytical and asymptotic results for various regimes. While such analytical results are useful for determining the qualitative behavior of the vortex filament in the limit where superfluid friction par...

Approximate solution for the temperature field of 1-D soil freezing process in a semi-infinite region

Abstract: Unfrozen liquid water always exists in the humid soil system below freezing point, and the amount of the unfrozen water decreases continuously with the temperature decreasing. This phenomenon is a special characteristic for the freezing of the humid soil system. The temperature field of 1-D soil freezing process in a semi-infinite region has been studied. The problem is a Stefan-like problem. After the continuous phase change process of soil water is divided into a finite number of substeps, the Stefan problem of a multi-phase material is obtained. A similarity solution is found and determined. In order to get the right solution of the nonlinear equations, a variable substitution technique is introduced. The approximate solution is verified by the numerical results of the continuous phase change model of soil freezing process. Finally, for practical purpose, the advancing factor of the freezing front and the mean squared error of the temperature caused by the measurement errors are defined. Computational examples concerning the effect of different parameters on the advancing factor of the freezing front and the effect of the measurement errors on the accuracy of the solution are presented and discussed.

An exact non‐linear Navier–Stokes compressible‐flow solution for CFD code verification

Abstract: SUMMARY An exact similarity solution of the compressible-flow Navier–Stokes equations is presented, which embeds supersonic, transonic, and subsonic regions. Describing the viscous and heat-conducting high-gradient flow in a shock wave, the solution accommodates non-linear temperature-dependent viscosity as well as heat-conduction coefficients and provides the variation of all the flow variables and their derivatives. Also presented are methods to obtain time-dependent and/or multi-dimensional solutions as well as verification benchmarks of increasing severity. Comparisons between the developed analytical solution and CFD solutions of the Navier–Stokes equations, with determination of convergence rates and orders of accuracy of these solutions, illustrate the utility of the developed exact solution for verification purposes. Copyright © 2012 John Wiley & Sons, Ltd.