Showing papers on "Similarity solution published in 2017"

Thermophoresis and Brownian motion effects on unsteady MHD nanofluid flow over a slendering stretching surface with slip effects

Abstract: In the current exploration, a similarity solution is bestowed for hydromagnetic motion of a nanofluid over a slendering stretching sheet with the existence of thermophoresis and Brownian moment. We first altered the time dependent governing mathematical equations into coupled dimensionless differential equations by making use of apposite transmutations. Numerical solution for these equations is procured deploying R.K.-Fehlberg numerical methodology. Further, the impacts of sundry non-dimensional parameters on the flow field along with shear stress, reduced heat and mass transfer have been tackled with the succor of plots and tabular forms. It is worth to divulge that an upturn in the magnitude of thermophoretic and Brownian motion parameters amplifies the fluid temperature, whereas the concentration profiles get depreciated with a hike in thermophoretic parameter.

Similarity Solution for the Synchronous Grouting of Shield Tunnel Under the Vertical Non-Axisymmetric Displacement Boundary Condition

Abstract: Similarity solution is investigated for the synchronous grouting of shield tunnel under the vertical non-axisymmetric displacement boundary condition in the paper. The synchronous grouting process of shield tunnel was simplified as the cylindrical expansion problem, which was based on the mechanism between the slurry and stratum of the synchronous grouting. The stress harmonic function on the horizontal and vertical ground surfaces is improved. Based on the virtual image technique, stress function solutions and Boussinesq's solution, elastic solution under the vertical non-axisymmetric displacement boundary condition on the vertical surface was proposed for synchronous grouting problems of shield tunnel. In addition, the maximum grouting pressure was also obtained to control the vertical displacement of horizontal ground surface. The validity of the proposed approach was proved by the numerical method. It can be known from the parameter analysis that larger vertical displacement of the horizontal ground surface was induced by smaller tunnel depth, smaller tunnel excavation radius, shorter limb distance, larger expansion pressure and smaller elastic modulus of soils.

New similarity solution of micropolar fluid flow problem over an uhspr in the presence of quartic kind of autocatalytic chemical reaction

Abstract: The motion of air (i.e fluid) in which tiny particle rotates past a pointed surface of a rocket (as in space science), over a bonnet of a car and past a pointed surface of an aircraft is of important to experts in all these fields. Geometrically, all the domains of fluid flow in all these cases can be referred to as the upper horizontal surface of a paraboloid of revolution (uhspr). Meanwhile, the solution of the corresponding partial differential equation is an open question due to unavailability of suitable similarity variable to non-dimensionalize the angular momentum equation. This article unravels the nature of skin friction coefficient, heat transfer rate, velocity, temperature, concentration of homogeneous bulk fluid and heterogeneous catalyst which exists on a stretchable surface which is neither a perfect horizontal/vertical nor inclined/cone. Theory of similarity solution was adopted to obtain the similarity variable suitable to scale the proposed angular momentum equation. These equations along with the boundary conditions are solved numerically using Runge-Kutta technique along with shooting method. The similarity variable successfully nondimensionalized and parameterized the angular momentum for boundary layer flow past uhspr. Temperature dependent dynamic viscosity parameter increases vertical velocity near a free stream but reduces micro-rotation near uhspr. Effect of thermal radiation parameter on temperature profile and heat transfer rate can be greatly influenced by thickness parameter.

Nanofluid flow on the stagnation point of a permeable non-linearly stretching/shrinking sheet

Abstract: This paper studies a steady two-dimensional stagnation-point flow of nanofluids over a nonlinearly stretching/shrinking sheet in the presence of blowing/suction. The effects of different nanoparticle materials, namely copper, alumina and titania on the flow and heat transfer rate are investigated. Employing similarity variables, the governing partial differential equations including continuity, momentum and energy have been reduced to ordinary equations and are solved numerically via Runge-Kutta-Fehlberg scheme. It is shown that two solutions exist for shrinking sheets in both blowing and impermeable cases, while an additional solution appears in the case of suction (there are three solutions). Moreover, the effects of nonlinearly parameter β , blowing/suction S , and solid volume fraction ϕ on the heat and fluid flow characteristics are investigated in details.

Closed form analytic solutions to heat and mass transfer characteristics of wall jet flow of nanofluids

Abstract: In the present work, an interesting and simple analytic solution to wall jet flow of nanofluids is presented. The concept of exponentially decaying wall jet flows (those the integral constraint proposed by Glauert is retained) is targeted. A general scheme for momentum similarity equation is considered in which the effects of suction together with moving wall are allowed. A parametric two-phase modeling framework is brought into account to study the wall jet flow of nanofluids. The only simplification is with respect to thermophoresis effect elimination which in particular reveals some facts regarding the effectiveness of boundary conditions for particle transport equation. In this regard, a brief discussion is initially provided as may be of help to a better understanding of the nanoparticles behavior at the wall. Finally, closed form analytic expressions are obtained for heat and mass transfer characteristics of wall jet flow of nanofluids. It is hopeful that the rate of heat and mass transfer as well as temperature and concentration distributions can be better understood by means of the presented formulae.

Flow of Reiner–Philippoff fluid over a stretching sheet with variable thickness

Abstract: The characteristics of flow of Reiner–Philippoff fluid over a nonlinearly stretching sheet with variable thickness are investigated in this article. The similarity solution of associated boundary layer equation is obtained to examine the effects of non-uniform surface on the flow of Reiner–Philippoff fluid. It is worth mentioning that the skin friction is decreasing with non-uniform surface parameter. Furthermore, skin friction is an increasing function of Bingham number for dilatant fluid, a decreasing function for pseudo-plastic, and a constant function for viscous fluid.

Thermodynamics Analysis of Unsteady MHD Mixed Convection with Slip and Thermal Radiation over a Permeable Surface

Abstract: This paper discusses the thermodynamics irreversibility in an unsteady hydromagnetic mixed convective flow of an electrically conducting optically dense fluid over a permeable vertical surface under the combined influence of thermal radiation, velocity slip, temperature jump, buoyancy force, viscous dissipation, Joule heating and magnetic field. The governing partial differential equations are reduced to ordinary differential equations by using similarity variable. A local similarity solution is obtained numerically using shooting technique coupled with Runge-Kutta Fehlberg integration method. The influence of various thermophysical parameters on velocity and temperature profiles, skin friction, Nusselt number, entropy generation rate and Bejan number are presented graphically and discussed quantitatively. It is found that velocity slip, surface injection and temperature jump can successfully reduce entropy generation rate in the presence of an applied magnetic field. A comparison of numerical solution is made with the exact solution under a special case scenario and excellent agreement is found.

Mixed convection non-axisymmetric Homann stagnation-point flow

Abstract: The steady mixed convection non-axisymmetric (Homann, Z. Angew. Math. Mech., vol. 16, 1936, pp. 153–164) stagnation-point flow over a vertical flat wall placed in a viscous and incompressible fluid is considered. A similarity solution is derived which involves the dimensionless parameters , representing the shear-to-strain-rate ratio, and , a mixed convection parameter. Forced convection, , is treated first where solutions additional to those given previously by Weidman (J. Fluid Mech., vol. 702, 2012, pp. 460–469) are found arising from singularities as . Numerical solutions are obtained for representative values of both and . Critical values of are seen in opposing flow and these are treated in detail. Asymptotic results for large and are derived.

Internal shear layers from librating objects

Abstract: In this work, we analyse the internal shear layer structures generated by the libration of an axisymmetric object in an unbounded fluid rotating at a rotation rate Ω * using direct numerical simulation and small Ekman number asymptotic analysis. We consider weak libration amplitude and libration frequency ω * within the inertial wave interval (0, 2Ω *) such that the fluid dynamics is mainly described by a linear axisymmetric harmonic solution. The internal shear layer structures appear along the characteristic cones of angle θ c = acos(ω * /(2Ω *)) which are tangent to the librating object at so-called critical latitudes. These layers correspond to thin viscous regions where the singulari-ties of the inviscid solution are smoothed. We assume that the velocity field in these layers is described by the class of similarity solutions introduced by Moore & Saffman [Phil. Trans. R. Soc. A 264, 597-634 (1969)]. These solutions are characterised by two parameters only: a real parameter m, which measures the strength of the underlying singularity, and a complex amplitude coefficient C 0. We first analyse the case of a disk for which a general asymptotic solution for small Ekman numbers is known when the disk is in a plane. We demonstrate that the numerical solutions obtained for a free disk and for a disk in a plane are both well-described by the asymptotic solution and by its similarity form within the internal shear layers. For the disk, we obtain a parameter m = 1 corresponding to a Dirac source at the edge of the disk and a coefficient C 0 ∝ E 1/6 where E is the Ekman number. The case of a smoothed librating object such as a spheroid is found to be different. By asymptotically matching the boundary layer solution to similarity solutions close to a critical latitude on the surface, we show that the adequate parameter m for the similarity solution is m = 5/4, leading to a coefficient C 0 ∝ E 1/12 , that is larger than for the case of a disk for small Ekman numbers. A simple general expression for C 0 valid for any axisymmetric object is obtained as a function of the local curvature radius at the critical latitude in agreement with this change of scaling. This result is tested and validated against direct numerical simulations.

The Energy Parameter B for Strong Blast Waves

Abstract: : The energy parameter B used in the strong blast wave equations is calculated for monatomic and diatomic gases. Three geometries, spherical, cylindrical, and plane are considered. Comparisons are made with previously published values of B. Tables and curves of the distribution functions are given for each case. The equations of the blast waves, in the similarity solution, are compiled for the six cases. An application of the analysis of a cylindrical blast wave from an exploding wire is given.

Two-phase gravity currents resulting from the release of a fixed volume of fluid in a porous medium

Abstract: We consider the instantaneous release of a finite volume of fluid in a porous medium saturated with a second, immiscible fluid of different density. The resulting two-phase gravity current exhibits a rich array of behaviours due to both the residual trapping of fluid as the current recedes and the differing effects of surface tension between advancing and receding regions of the current. We develop a framework for the evolution of such a current with particular focus on the large-scale implications of the form of the constitutive relation between residual trapping and initial saturation. Pore-scale hysteresis within the current is represented by families of scanning curves relating capillary pressure and relative permeability to saturation. In the resulting vertically integrated model, all capillary effects are incorporated within specially defined saturation and flux functions specific to each region. In the long-time limit, when the height of the current and the saturations within it are low, the saturation and flux functions can be approximated by mathematically convenient power laws. If the trapping model is approximately linear at low saturations, the equations admit a similarity solution for the propagation rate and height profile of the late-time gravity current. We also solve the governing partial differential equation numerically for the nonlinear Land’s trapping model, which is commonly used in studies of two-phase flows. Our investigation suggests that for trapping relations for which the proportion of trapped to initial fluid saturation increases and tends to unity as the initial saturation decreases, both of which are properties of Land’s model, a gravity current slows and eventually stops. This trapping behaviour has important applications, for example to the ultimate distance contaminants or stored carbon dioxide may travel in the subsurface.

Numerical tackling for viscoelastic fluid flow in rotating frame considering homogeneous-heterogeneous reactions

Abstract: This study provides a numerical treatment for rotating flow of viscoelastic (Maxwell) fluid bounded by a linearly deforming elastic surface. Mass transfer analysis is carried out in the existence of homogeneous-heterogeneous reactions. By means of usual transformation, the governing equations are changed into global similarity equations which have been tackled by an expedient shooting approach. A contemporary numerical routine bvp4c of software MATLAB is also opted to develop numerical approximations. Both methods of solution are found in complete agreement in all the cases. Velocity and concentration profiles are computed and elucidated for certain range of viscoelastic fluid parameter. The solutions contain a rotation-strength parameter λ that has a considerable impact on the flow fields. For sufficiently large value of λ , the velocity fields are oscillatory decaying function of the non-dimensional vertical distance. Concentration distribution at the surface is found to decrease upon increasing the strengths of chemical reactions. A comparison of present computations is made with those of already published ones and such comparison appears convincing.

A simple method for solving unidirectional methane gas flow in coal seam based on similarity solution

Abstract: The equation used to model the unidirectional flow of methane gas in coal seams is usually formulated as a nonlinear partial differential equation, which needs to be solved numerically with a computer program. Nevertheless, for people without access to the computer program, the conventional numerical method may be inconvenient. Thus, the objective here is to seek some method simpler than the conventional one for solving the flow problem. A commonly used model of the unidirectional methane gas flow is considered, where the methane adsorption is described by the Langmuir isotherm and the free gas is treated as real gas. By introducing the similarity solution, a simple method for solving the flow model is proposed, which can be done on a hand calculator. It is shown by two examples that the gas pressure profile obtained by the proposed method agrees well with the direct numerical solution of the flow model.

Asymptotically based self-similarity solution of the Navier-Stokes equations for a porous tube with a non-circular cross-section

Abstract: This work introduces a similarity solution to the problem of a viscous, incompressible and rotational fluid in a right-cylindrical chamber with uniformly porous walls and a non-circular cross-section. The attendant idealization may be used to model the non-reactive internal flow field of a solid rocket motor with a star-shaped grain configuration. By mapping the radial domain to a circular pipe flow, the Navier–Stokes equations are converted to a fourth-order differential equation that is reminiscent of Berman’s classic expression. Then assuming a small radial deviation from a fixed chamber radius, asymptotic expansions of the three-component velocity and pressure fields are systematically pursued to the second order in the radial deviation amplitude. This enables us to derive a set of ordinary differential relations that can be readily solved for the mean flow variables. In the process of characterizing the ensuing flow motion, the axial, radial and tangential velocities are compared and shown to agree favourably with the simulation results of a finite-volume Navier–Stokes solver at different cross-flow Reynolds numbers, deviation amplitudes and circular wavenumbers.

Numerical analysis of heat and mass transfer along a stretching wedge surface

Abstract: In this work, the effects of dimensionless parameters on the velocity field, thermal field and nanoparticle concentration have been analyzed. In this respect, the magnetohydrodynamic (MHD) boundary layer nanofluid flow along a moving wedge is considered. Therefore, a similarity solution has been derived like Falkner – Skan solution and identified the point of inflexion. So the governing partial differential equations transform into ordinary differential equations by using the similarity transformation. These ordinary differential equations are numerically solved using fourth order Runge–Kutta method along with shooting technique. The present results have been shown graphically and in tabular form. From the graph, the results indicate that the velocity increases with increasing values of pressure gradient, magnetic induction and velocity ratio. The temperature decreases for velocity ratio, Brownian motion and Prandtl number but opposite result arises for increasing values of thermophoresis. The nanoparticle concentration decreases with an increase in pressure gradient, Brownian motion and Lewis number, but increases for thermophoresis. Besides, the solution of nanoparticle concentration exists in the case of Brownian motion is less than 0.2, thermophoresis is less than 0.14 and lewis number is greater than 1.0. Finally, for validity and accuracy the present results have been compared with previous work and found to be in good agreement.

Similarity Solution of Axisymmetric Stagnation-Point Flow and Heat Transfer of a Nanofluid on a Stationary Cylinder with Constant Wall Temperature

Abstract: The steady state, viscous flow and heat transfer of a nanofluid in the vicinity of an axisymmetric stagnation point of a stationary cylinder is investigated. Exact solution of the Navier–Stokes equations and energy equation are derived in this problem. For all Reynolds numbers, as the particle fraction increases, the depth of diffusion of the thermal boundary layer and shear stresses decreases. It’s clear that by adding nanoparticles to the base fluid there is a significant enhancement in convective heat transfer coefficient and convective heat flux.

MHD flow and heat transfer of a magnetite–water nanofluid in porous medium under the effects of chemical reaction

Abstract: Purpose Due to the extensive industrial applications of stagnation flow problems, the present work aims to investigate the magnetohydrodynamics (MHD) flow and heat transfer of a magnetite nanofluid (here Fe3O4–water nanofluid) impinging a flat porous plate under the effects of a non-uniform magnetic field and chemical reaction with variable reaction rate. Design/methodology/approach Similarity transformations are applied to reduce the governing partial differential equations with boundary conditions into a system of ordinary differential equations over a semi-infinite domain. The modified fourth-order Runge–Kutta method with the shooting technique which is developed for unbounded domains is conducted to give approximate solutions of the problem, which are then verified by results of other researchers, showing very good agreements. Findings The effects of the volume fraction of nanoparticles, permeability, magnetic field, chemical reaction and Schmidt number on velocity, temperature and concentration fields are examined and graphically illustrated. It was found that fluid velocity and temperature fields are affected strongly by the types of nanoparticles. Moreover, magnetic field and radiation have strong effects on velocity and temperature fields, fluid velocity increases and thickness of the velocity boundary layer decreases as magnetic parameter M increases. The results also showed that the thickness of the concentration boundary layer decreases with an increase in the Schmidt number, as well as an increase in the chemical reaction coefficient. Research limitations/implications The thermophysical properties of the magnetite nanofluid (Fe3O4–water nanofluid) in different conditions should be checked. Practical implications Stagnation flow of viscous fluid is important due to its vast industrial applications, such as the flows over the tips of rockets, aircrafts, submarines and oil ships. Moreover, nanofluid, a liquid containing a dispersion of sub-micronic solid particles (nanoparticles) with typical length of the order of 1-50 nm, showed abnormal convective heat transfer enhancement, which is remarkable. Originality/value The major novelty of the present work corresponds to utilization of a magnetite nanofluid (Fe3O4–water nanofluid) in a stagnation flow influenced by chemical reaction and magnetic field. It should be noted that in addition to a variable chemical reaction, the permeability is non-uniform, while the imposed magnetic field also varies along the sheet. These, all, make the present work rather original.