Showing papers on "Similarity solution published in 2016"

Capillary breakup of a liquid bridge : identifying regimes and transitions

Abstract: Computations of the breakup of a liquid bridge are used to establish the limits of applicability of similarity solutions derived for different breakup regimes. These regimes are based on particular viscous-inertial balances, that is different limits of the Ohnesorge number Oh. To accurately establish the transitions between regimes, the minimum bridge radius is resolved through four orders of magnitude using a purpose-built multiscale finite element method. This allows us to construct a quantitative phase diagram for the breakup phenomenon which includes the appearance of a recently discovered low-Oh viscous regime. The method used to quantify the accuracy of the similarity solutions allows us to identify a number of previously unobserved features of the breakup, most notably an oscillatory convergence towards the viscous-inertial similarity solution. Finally, we discuss how the new findings open up a number of challenges for both theoretical and experimental analysis.

Natural convection boundary layer flow due to gyrotactic microorganisms about a vertical cone in porous media saturated by a nanofluid

Abstract: The similarity solution for Darcy free convection about an isothermal vertical cone with fixed apex half angle, pointing downward in a nanofluid saturated porous medium, has been made. It is assumed that the medium contains gyrotactic microorganisms along with nanoparticles and the cone is subjected to concentration of nanoparticles and density of motile microorganisms. The effects of Brownian motion and thermophoresis are incorporated into the model for nanofluids. The governing partial differential equations are converted into nonlinear ordinary differential equations using unique similarity transformations. The effects of the governing parameters on the dimensionless quantities such as velocity, temperature, nanoparticle concentration, density of motile microorganisms, local Nusselt, local Sherwood and local density numbers for both nanoparticles, and motile microorganism density are explored. A comprehensive numerical computation is carried out for various values of the parameters that describe the flow characteristics.

Oblique stagnation-point flow of a nanofluid past a shrinking sheet

Abstract: Purpose – The laminar two-dimensional stagnation-point flow and heat transfer of a viscous incompressible nanofluid obliquely impinging on a shrinking surface is formulated as a similarity solution of the Navier-Stokes, energy and concentration equations. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The effect of the dimensionless strain rate, shrinking parameter, Brownian motion parameter and thermophoresis parameter on the flow, temperature and nanoparticle volume fraction is investigated in details. The paper aims to discuss these issues. Design/methodology/approach – The transformed system of ordinary differential equations was solved using the function bvp4c from Matlab. The relative tolerance was set to 10−10. Findings – It is found that dimensionless strain rate and shrinking parameter causes a shift in the position of the point of zero skin friction along the stretching sheet. Obliquity of the flow toward the surface increases as the strain rate ...

Semi-analytical solution for the flow of a nanofluid over a permeable stretching/shrinking sheet with velocity slip using Buongiorno's mathematical model

Abstract: In this analysis, the flow and heat transfer characteristics of a nanofluid over a stretching/shrinking surface with suction are investigated. Using a similarity transformation, the nonlinear system of partial differential equations is converted into nonlinear ordinary differential equations. These resulting equations are solved analytically and numerically using a collocation method. Multiple (dual: upper and lower branch) solutions are shown to exist in a range of the governing parameters. In addition, the reduced skin friction coefficient and the reduced heat transfer from the surface of the sheet as well as the velocity, temperature and concentration profiles are analyzed subject to several parameters of interest, namely suction parameter, Brownian motion and thermophoresis parameters, Prandtl number, nanofluid Lewis number and dimensionless slip parameter. The results indicate that the skin friction coefficient and the heat transfer from the surface of the sheet increase with suction effect. It is also observed that suction widens the range of the stretching/shrinking parameter for which the solution exists.

Thermophoresis and Brownian effects on the Blasius flow of a nanofluid due to a curved surface with thermal radiation

Abstract: In this analysis, we have discussed the Blasius flow of a nanofluid over a curved surface coiled in a circle of radius R . The physical situation is formulated in a mathematical model using a curvilinear coordinates system. The model is considered for the nanofluid including the effects of Brownian motion and thermophoresis in the presence of thermal radiation. A similarity solution of the developed ordinary differential equations is obtained numerically using the shooting method. The influence of the various involved parameters on the flow phenomena are analyzed through graphs and tables.

Capillary Breakup of a Liquid Bridge: Identifying Regimes and Transitions

Abstract: Computations of the breakup of a liquid bridge are used to establish the limits of applicability of similarity solutions derived for different breakup regimes. These regimes are based on particular viscous-inertial balances, that is different limits of the Ohnesorge number $Oh$. To accurately establish the transitions between regimes, the minimum bridge radius is resolved through four orders of magnitude using a purpose-built multiscale finite element method. This allows us to construct a quantitative phase diagram for the breakup phenomenon which includes the appearance of a recently discovered low-$Oh$ viscous regime. The method used to quantify the accuracy of the similarity solutions allows us to identify a number of previously unobserved features of the breakup, most notably an oscillatory convergence towards the viscous-inertial similarity solution. Finally, we discuss how the new findings open up a number of challenges for both theoretical and experimental analysis.

Thermophysical effects of water driven copper nanoparticles on MHD axisymmetric permeable shrinking sheet: Dual-nature study

Abstract: The present study is dedicated to analyze the dual-nature solutions of the axisymmetric flow of a magneto-hydrodynamics (MHD) nanofluid over a permeable shrinking sheet. In those phenomena where the fluid flow is due to the shrinking surface, some reverse behaviors of the flow arise because of vorticity effects. Despite of heat transfer analysis, the main purpose of the present study is to attain the solutions of the complex nature problem that appear in reverse flow phenomena. Thermophysical properties of both base fluid (water) and nanoparticles (copper) are also taken into account. By means of similarity transformation, partial differential equations are converted into a system of coupled nonlinear ordinary differential equations and then solved via the Runge-Kutta method. These results are divided separately into two cases: the first one is the unidirectional shrinking along the surface (m = 1) and the other one is for axisymmetric shrinking phenomena (m = 2) . To enhance the thermal conductivity of base fluid, nanoparticle volume fractions (0≤φ ≤ 0.2)) are incorporated within the base fluid. The numerical investigation explores the condition of existence, non-existence and the duality of similarity solution depends upon the range of suction parameter (S) and Hartmann number (M). The reduced skin friction coefficient and local Nusselt number are plotted to analyze the fluid flow and heat transfer at the surface of the shrinking sheet. Streamlines and isotherms are also plotted against the engineering control parameters to analyze the flow behavior and heat transfer within the whole domain. Throughout this analysis it is found that both nanoparticle volume fraction and Hartmann number are increasing functions of both skin friction coefficient and Nusselt number.

Linear disturbances in the rotating-disk flow: A comparison between results from simulations, experiments and theory

Abstract: The incompressible Navier-Stokes equations have an exact similarity solution for the flow over an infinite rotating disk giving a laminar boundary layer of constant thickness, also known as the von ...

Unsteady turbulent buoyant plumes

Abstract: We model the unsteady evolution of turbulent buoyant plumes following temporal changes to the source conditions. The integral model is derived from radial integration of the governing equations expressing the evolution of mass, axial momentum and buoyancy in the plume. The non-uniform radial profiles of the axial velocity and density deficit in the plume are explicitly captured by shape factors in the integral equations; the commonly assumed top-hat profiles lead to shape factors equal to unity. The resultant model for unsteady plumes is hyperbolic when the momentum shape factor, determined from the radial profile of the mean axial velocity in the plume, differs from unity. The solutions of the model when source conditions are maintained at constant values are shown to retain the form of the well-established steady plume solutions. We demonstrate through a linear stability analysis of these steady solutions that the inclusion of a momentum shape factor in the governing equations that differs from unity leads to a well-posed integral model. Therefore, our model does not exhibit the mathematical pathologies that appear in previously proposed unsteady integral models of turbulent plumes. A stability threshold for the value of the shape factor is also identified, resulting in a range of its values where the amplitudes of small perturbations to the steady solutions decay with distance from the source. The hyperbolic character of the system of equations allows the formation of discontinuities in the fields describing the plume properties during the unsteady evolution, and we compute numerical solutions to illustrate the transient development of a plume following an abrupt change in the source conditions. The adjustment of the plume to the new source conditions occurs through the propagation of a pulse of fluid through the plume. The dynamics of this pulse is described by a similarity solution and, through the construction of this new similarity solution, we identify three regimes in which the evolution of the transient pulse following adjustment of the source qualitatively differs.

The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free-stream

Abstract: In this study, a new centrifugal instability mode, which dominates within the boundary-layer flow over a slender rotating cone in still fluid, is used for the first time to model the problem within an enforced oncoming axial flow. The resulting problem necessitates an updated similarity solution to represent the basic flow more accurately than previous studies in the literature. The new mean flow field is subsequently perturbed, leading to disturbance equations that are solved via numerical and short-wavelength asymptotic approaches, yielding favourable comparisons with existing experiments. Essentially, the boundary-layer flow undergoes competition between the streamwise flow component, due to the oncoming flow, and the rotational flow component, due to effect of the spinning cone surface, which can be described mathematically in terms of a control parameter, namely the ratio of streamwise to axial flow. For a slender cone rotating in a sufficiently strong axial flow, the instability mode breaks down into Gortler-type counter-rotating spiral vortices, governed by an underlying centrifugal mechanism, which is consistent with experimental and theoretical studies for a slender rotating cone in otherwise still fluid.

Entropy generation analysis for film boiling: A simple model of quenching

Abstract: In this paper, quenching in high-temperature materials processing is modeled as a superheated isothermal flat plate. In these phenomena, a liquid flows over the highly superheated surfaces for cooling. So the surface and the liquid are separated by the vapor layer that is formed because of the liquid which is in contact with the superheated surface. This is named forced film boiling. As an objective, the distribution of the entropy generation in the laminar forced film boiling is obtained by similarity solution for the first time in the quenching processes. The PDE governing differential equations of the laminar film boiling including continuity, momentum, and energy are reduced to ODE ones, and a dimensionless equation for entropy generation inside the liquid boundary and vapor layer is obtained. Then the ODEs are solved by applying the 4th-order Runge-Kutta method with a shooting procedure. Moreover, the Bejan number is used as a design criterion parameter for a qualitative study about the rate of cooling and the effects of plate speed are studied in the quenching processes. It is observed that for high speed of the plate the rate of cooling (heat transfer) is more.

Three-Dimensional Stagnation-Point Flow and Heat Transfer of a Dusty Fluid Toward a Stretching Sheet

Abstract: The steady three-dimensional stagnation-point flow and heat transfer of a dusty fluid toward a stretching sheet is investigated by using similarity solution approach The freestream along z-direction impinges on the stretching sheet to produce a flow with different velocity components The governing equations are transformed into ordinary differential equations by introducing appropriate similarity variables and an exact solution is obtained The nonlinear ordinary differential equations are solved numerically using Runge–Kutta fourth-order method The effects of the physical parameters like velocity ratio, fluid and thermal particle interaction parameter, ratio of freestream velocity parameter to stretching sheet velocity parameter, Prandtl number, and Eckert number on the flow field and heat transfer characteristics are obtained, illustrated graphically, and discussed Also, a comparison of the obtained numerical results is made with two-dimensional cases existing in the literature and good agreement is approved Moreover, it is found that the heat transfer coefficient and shear stress on the surface for axisymmetric case are larger than nonaxisymmetric case Also, for stationary flat plat case, a similarity solution is presented and a comparison of the obtained results is made with previously published results and full agreement is reported

Similarity and singularity in adhesive elastohydrodynamic touchdown

Abstract: We consider the dynamics of an elasticsheet as it starts to adhere to a wall, a process that is limited by the viscous squeeze flow of the intervening liquid. Elastohydrodynamic lubrication theory allows us to derive a partial differential equation coupling the elastic deformation of the sheet, the microscopic van der Waals adhesion, and viscousthin film flow. We use a combination of numerical simulations of the governing equation and a scaling analysis to describe the self-similar touchdown of the sheet as it approaches the wall. An analysis of the equation in terms of similarity variables in the vicinity of the touchdown event shows that only the fundamental similarity solution is observed in the time-dependent numerical simulations, consistent with the fact that it alone is stable. Our analysis generalizes similar approaches for rupture in capillary thin film hydrodynamics and suggests experimentally verifiable predictions for a new class of singular flows linking elasticity, hydrodynamics, and adhesion.

Nonlinear Stefan problem with convective boundary condition in Storm’s materials

Abstract: We consider a nonlinear one-dimensional Stefan problem for a semi-infinite material x > 0, with phase change temperature T f . We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition, and we assume a convective boundary condition at the fixed face x = 0. A unique explicit solution of similarity type is obtained. Moreover, asymptotic behavior of the solution when $${h\rightarrow + \infty}$$ is studied.

The Equatorial Ekman Layer

Abstract: The steady incompressible viscous flow in the wide gap between spheres rotating about a common axis at slightly different rates (small Ekman number E) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the limit E<<1, was undertaken by Stewartson (J. Fluid Mech. 1966, vol. 26, pp. 131-144). The mainstream flow, exterior to the E^{1/2} Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder C, is geostrophic. Stewartson identified a complicated nested layer structure on C, which comprises relatively thick quasi-geostrophic E^{2/7} (inside C) and E^{1/4} (outside C) layers. They embed a thinner E^{1/3} ageostrophic shear layer (on C), which merges with the inner sphere Ekman layer to form the E^{2/5} Equatorial Ekman layer of axial length E^{1/5}. Under appropriate scaling, this $E^{2/5}$--layer problem may be formulated, correct to leading order, independent of E. Accordingly, the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial co-ordinate z) solution. We present a numerical solution, which uses a non-local integral boundary condition at finite $z$ to account for the far-field behaviour. Adopting z^{-1} as a small parameter we extend Stewartson's similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.

Numerical simulation of the stagnation point flow past a permeable stretching/shrinking sheet with convective boundary condition and heat generation

Abstract: Purpose – The purpose of this paper is the stagnation-point flow driven by a permeable stretching/shrinking surface with convective boundary condition and heat generation. Design/methodology/approach – It is known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, for the present case it is demonstrated that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is constant. Findings – The governing boundary layer equations are transformed to self-similar nonlinear ordinary differential equations using similarity transformations. Numerical results of the resulting equations are obtained using the function bvp4c from Matlab for different values of the governing parameters. In addition an analytical solution has been obtained for the energy equation when heat generation is absent. The streamlines for the upper branch solution show ...

On the equatorial Ekman layer

Abstract: The steady incompressible viscous flow in the wide gap between spheres rotating rapidly about a common axis at slightly different rates (small Rossby number) has a long and celebrated history. The problem is relevant to the dynamics of geophysical and planetary core flows, for which, in the case of electrically conducting fluids, the possible operation of a dynamo is of considerable interest. A comprehensive asymptotic study, in the small Ekman number limit , was undertaken by Stewartson (J. Fluid Mech., vol. 26, 1966, pp. 131–144). The mainstream flow, exterior to the Ekman layers on the inner/outer boundaries and the shear layer on the inner sphere tangent cylinder , is geostrophic. Stewartson identified a complicated nested layer structure on , which comprises relatively thick quasigeostrophic - (inside ) and - (outside ) layers. They embed a thinner ageostrophic shear layer (on ), which merges with the inner sphere Ekman layer to form the -equatorial Ekman layer of axial length . Under appropriate scaling, this -layer problem may be formulated, correct to leading order, independent of . Then the Ekman boundary layer and ageostrophic shear layer become features of the far-field (as identified by the large value of the scaled axial coordinate ) solution. We present a numerical solution of the previously unsolved equatorial Ekman layer problem using a non-local integral boundary condition at finite to account for the far-field behaviour. Adopting as a small parameter we extend Stewartson’s similarity solution for the ageostrophic shear layer to higher orders. This far-field solution agrees well with that obtained from our numerical model.

Self-similar equilibration of strongly interacting systems from holography

Abstract: We study the equilibration of a class of far-from-equilibrium strongly interacting systems using gauge-gravity duality. The systems we analyze are 2+1 dimensional and have a four-dimensional gravitational dual. A prototype example of a system we analyze is the equilibration of a two-dimensional fluid which is translational invariant in one direction and is attached to two different heat baths with different temperatures at infinity in the other direction. We realize such setup in gauge-gravity duality by joining two semi-infinite asymptotically anti-de Sitter (AdS) black branes of different temperatures, which subsequently evolve towards equilibrium by emitting gravitational radiation towards the boundary of AdS. At sufficiently late times the solution converges to a similarity solution, which is only sensitive to the left and right equilibrium states and not to the details of the initial conditions. This attractor solution not only incorporates the growing region of equilibrated plasma but also the outwardly propagating transition regions, and can be constructed by solving a single ordinary differential equation.

The solutions for the flow of micropolar fluid through an expanding or contracting channel with porous walls

Abstract: The unsteady, two-dimensional laminar flow of an incompressible micropolar fluid in a channel with expanding or contracting porous walls is investigated. The governing equations are transformed into a coupled nonlinear two-points boundary value problem by a suitable similarity transformation. Unlike the classic Berman problem (Berman in J. Appl. Phys. 24:1232-1235, 1953), three new solutions (totally six solutions) and no-solution interval, which is one of important characteristics for the laminar flow through porous pipe with stationary wall (Terrill and Thomas in Appl. Sci. Res. 21:37-67, 1969), are found numerically for the first time. The multiplicity of the solutions is strictly dependent on the expansion ratio. Furthermore, the asymptotic solutions are constructed by the Lighthill method, which eliminates the singularity of the similarity solution, for large injection and by the matching theorem for the suction Reynolds number, respectively. The analytical solutions also are compared with the numerical ones and the results agree well.

MHD flow and heat transfer of a power-law non-Newtonian nanofluid (Cu–H 2 O) over a vertical stretching sheet

Abstract: A steady-state mixed convection boundary layer flow of an electrically conducting nanofluid (Cu–H2O) obeying a power-law model in the presence of an alternating magnetic field due to a stretching vertical heated sheet is investigated numerically through the use of Wolfram Mathematica. The surface stretching velocity and the surface temperature are assumed to vary as linear functions of the distance from the origin. A similarity solution is presented, which depends on the nanoparticle volume fraction, power-law parameter, magnetic field parameter, buoyancy convection parameter, and modified Prandtl number.

Similarity solutions of compressible flow over a rotating cone with surface suction

Abstract: We present solutions of the laminar compressible boundary-layer flows over the family of rotating cones subject to surface mass flux. The work is a generalization of previous studies of the compressible rotating-disk flow and incompressible rotating-cone flow without surface mass flux. Transformations are used which lead to a system of generalized von Karman equations with boundary conditions parameterized by half-angle and a mass-flux parameter. Results are discussed in terms of wall temperature and local Mach number in the particular case of air, although the formulation is readily extended to other fluids. It is suggested that suction acts a stabilizing mechanism, whereas increased wall temperature and local Mach number have destabilizing influences.

Spherical accretion: the influence of inner boundary and quasi-periodic oscillations

Abstract: Bondi accretion assumes that there is a sink of mass at the centre - which in the case of a black hole (BH) corresponds to the advection of matter across the event horizon. Other stars, such as a neutron star (NS), have surfaces and hence the infalling matter has to slow down at the surface. We study the initial value problem in which the matter distribution is uniform and at rest at t = 0. We consider different inner boundary conditions for BHs and NSs: outflow boundary condition (mimicking mass sink at the centre) valid for BHs; and reflective and steady-shock (allowing gas to cross the inner boundary at subsonic speeds) boundary conditions for NSs. We also obtain a similarity solution for cold accretion on to BHs and NSs. 1D simulations show the formation of an outward-propagating and a standing shock in NSs for reflective and steady-shock boundary conditions, respectively. Entropy is the highest at the bottom of the subsonic region for reflective boundary conditions. In 2D this profile is convectively unstable. Using steady-shock inner boundary conditions, the flow is unstable to the standing accretion shock instability in 2D, which leads to global shock oscillations and may be responsible for quasi-periodic oscillations seen in the light curves of accreting systems. For steady accretion in the quiescent state, spherical accretion rate on to an NS can be suppressed by orders of magnitude compared to that on to a BH.

Water entry of an expanding wedge/plate with flow detachment

Abstract: A general similarity solution for water-entry problems of a wedge with its inner angle fixed and its sides in expansion is obtained with flow detachment, in which the speed of expansion is a free parameter. The known solutions for a wedge of a fixed length at the initial stage of water entry without flow detachment and at the final stage corresponding to Helmholtz flow are obtained as two special cases, at some finite and zero expansion speeds, respectively. An expanding horizontal plate impacting a flat free surface is considered as the special case of the general solution for a wedge inner angle equal to π. An initial impulse solution for a plate of a fixed length is obtained as the special case of the present formulation. The general solution is obtained in the form of integral equations using the integral hodograph method. The results are presented in terms of free-surface shapes, streamlines and pressure distributions.

A similarity solution for reaction front propagation in a fracture-matrix system

Abstract: We propose a new composite similarity variable, based on which a similarity solution is derived for reaction front propagation in fracture-matrix systems. The similarity solution neglects diffusion/dispersion within the fracture and assumes the existence of a sharp reaction front in the rock matrix. The reaction front location in the rock matrix is shown to follow a linear decrease with distance along the fracture. The reaction front propagation along the fracture is shown to scale like diffusion (i.e. as the square root of time). The similarity solution using the composite similarity variable appears to be applicable to a broad class of reactive transport problems involving mineral reactions in fracture-matrix systems. It also reproduces the solutions for non-reactive solute and heat transport when diffusion/dispersion/conduction are neglected in the fracture. We compared our similarity solution against numerical simulations for nonlinear reactive transport of an aqueous species with a mineral in the rock matrix. The similarity solutions agree very well with the numerical solutions, especially at later times when diffusion limitations are more pronounced.This article is part of the themed issue 'Energy and the subsurface'.

The motion induced between a rotating plate and a radially stretching or shrinking plate

Abstract: The flow induced between parallel plates separated by a distance h executing different types of in-plane motion is investigated. The upper plate radially stretches at strain rate a and the lower plate rotates at angular velocity Ω about a common axis. A similarity solution form reduces the Navier-Stokes equations to a coupled pair of ordinary differential equations in two parameters: R = |a| h2/ν and σ = Ω/|a|, where ν is the kinematic viscosity of the fluid. Solutions are obtained for both stretching and shrinking upper plates and numerical results for pressure gradient and wall shear stress parameters are found and compared with low-R series solutions and large-R asymptotic behaviors. Sample radial and azimuthal velocity profiles reveal regions of zero radial and azimuthal wall shear stress which are studied in detail. This work represents the first study of the flow induced between parallel plates for which each plate executes a different type of in-plane motion.