Showing papers on "Similarity solution published in 2018"

Similarity solution for cavity expansion in thermoplastic soil

Abstract: Summary This paper presents an analytical solution for cavity expansion in thermoplastic soil considering non-isothermal conditions. The constitutive relationship of thermoplasticity is described by Laloui's advanced and unified constitutive model for environmental geomechanical thermal effect (ACMEG-T), which is based on multi-mechanism plasticity and bounding surface theory. The problem is formulated by incorporating ACMEG-T into the theoretical framework of cavity expansion, yielding a series of partial differential equations (PDEs). Subsequently, the PDEs are transformed into a system of first-order ordinary differential equations (ODEs) using a similarity solution technique. Solutions to the response parameters of cavity expansion (stress, excess pore pressure, and displacement) can then be obtained by solving the ODEs numerically using mathematical software. The results suggest that soil temperature has a significant influence on the pressure-expansion relationships and distributions of stress and excess pore pressure around the cavity wall. The proposed solution quantifies the influence of temperature on cavity expansion for the first time and provides a theoretical framework for predicting thermoplastic soil behavior around the cavity wall. The solution found in this paper can be used as a theoretical tool that can potentially be employed in geotechnical engineering problems, such as thermal cone penetration tests, and nuclear waste disposal problems.

Lie-group similarity solution and analysis for fractional viscoelastic MHD fluid over a stretching sheet

Abstract: Lie-group is introduced for studying boundary layer flow and heat transfer of fractional viscoelastic MHD fluid over a stretching sheet. Fractional boundary layer equations, based on Riemann–Liouville operators, are reduced and solved numerically by Grunwald scheme approximation. Results show that the skin friction and thermal conductivity are strongly affected by magnetic field parameter, fractional derivative and wall stretching exponent. The bigger of the fractional order derivative leads to the faster velocity of viscoelastic fluids near the plate but not to hold near the outer flow. Skin friction increases with increase of magnetic field parameter M , while the heat transfer decreases. For wall stretching exponent parameter β = 1 . 0 , the velocity profile decreases with the increase of similarity variable η . However, for β = − 1 . 5 , the velocity profile increases initially and then decreases afterwards with the biggest velocity at the interior of boundary layer.

Healing capillary films

Abstract: Consider the dynamics of a healing film driven by surface tension, that is, the inward spreading process of a liquid film to fill a hole. The film is modelled using the lubrication (or thin-film) approximation, which results in a fourth-order nonlinear partial differential equation. We obtain a self-similar solution describing the early-time relaxation of an initial step-function condition and a family of self-similar solutions governing the finite-time healing. The similarity exponent of this family of solutions is not determined purely from scaling arguments; instead, the scaling exponent is a function of the finite thickness of the prewetting film, which we determine numerically. Thus, the solutions that govern the finite-time healing are self-similar solutions of the second kind. Laboratory experiments and time-dependent computations of the partial differential equation are also performed. We compare the self-similar profiles and exponents, obtained by matching the estimated prewetting film thickness, with both measurements in experiments and time-dependent computations near the healing time, and we observe good agreement in each case.

Analytical and Numerical Investigations of Unsteady Graphene Oxide Nanofluid Flow Between Two Parallel Plates

Abstract: In this research, different analytical methods were applied to characterize thermal behavior of unsteady graphene oxide–water nanofluid flow between two parallel moving plates. First of all, partial differential equations (PDEs) were transformed to a system of nonlinear ordinary differential equations (ODEs) using similarity solution. Then, collocation method (CM), least square method (LSM) and Galerkin method (GM) were used to solve the system of ODEs and determine velocity and temperature distribution functions. In addition, effects of moving parameter, concentration, Eckert and Prandtl numbers on nanofluid velocity and temperature profiles were examined. Next, using numerical solution of the obtained system of differential equations, the results obtained from the analytical solutions were validated with that of the numerical solution. The validation results indicated high and appropriate accuracy of the analytical solutions compared to the numerical one.

Wall jet flows of Glauert type: Heat transfer characteristics and the thermal instabilities in analytic closed forms

Abstract: Heat transfer characteristics of the traditional wall jet flows subject to various thermal boundary conditions including isothermal surface, prescribed temperature, constant heat flux, prescribed heat flux, adiabatic surface and thermally convective surface are documented in analytic closed forms. Heat dissipation has been also included where the similarity energy equation could structurally adjust to this specific term (for the adiabatic case, this term has been necessarily included). In all the studied cases, both the transpiration velocity and moving wall conditions are allowed to exist (where applicable) in such a way, being consistent with the Glauert integral constraint and subject to the context of exponentially decaying wall jet flows. In particular, it is analytically proved (even without having the closed form solutions) that for the Glauert original case, a surface with a prescribed temperature in the form of T w ( x ) = m x − 1 4 + T ∞ serves zero contribution to heat transfer at the wall (the Induced Heat Shield). More precisely, the value − 1 4 as the prescribing parameter is the Surface Heat Transfer Stopping Point and below this point, heat transfer phenomenon falls from a usual physical interpretation, expressing spectrums of thermal instabilities through a hyper geometric function. Furthermore, it is argued that a normalized similarity temperature in the form of θ ( η ) = T − T ∞ m x n − 1 4 results in the appearance of an uninterruptable singularity within the heat transfer phenomenon for the Glauert case subject to a rigorous physical ground; and as an immediate practical consequence, there is no physically-valid similarity solution for an adiabatic surface or more precisely, for energy equation with heat dissipation consideration. It should be mentioned that the concept of Induced Heat Shield is discussed in here for the first time (to our knowledge) which may inspire a concealed fact regarding the restrictions of the thermal similarity solutions. Therefore, it is hopeful that heat transfer phenomenon in the Glauert type wall jet flows and the associated physical representations can be better understood by the present research.

Similarity and numerical analysis of the generalized Levèque problem to predict the thermal boundary layer

Abstract: In the thermal entrance region, a thermal boundary layer develops and also reaches the circular tube center. The fully developed region is the zone in which the flow is both hydrodynamically and thermally developed. The heat flux will be higher near the inlet because the heat transfer coefficient is highest at the tube inlet where the thickness of the thermal boundary layer is zero, and decreases gradually to the fully developed value. In this paper, the assumptions implicit in Leveque’s approximation are re-examined, and the analytical solution of the problem with additional boundary conditions, for the temperature field and the boundary layer thickness through the long tube is presented. From the mathematical side, numerical techniques for solving the problem of fluid–structure interaction with a fully developed laminar incompressible Newtonian flow is described. By defining a similarity variable the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This report gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form, and presents a similarity solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth order (RK4) method. Finally, others important thermal results obtained from this analysis, such as; approximate Nusselt number in the thermal entrance region was discussed in detail.

Study of polymeric liquid between stretching disks with chemical reaction

Abstract: An analysis has been carried out to investigate the heat and mass transfer effects in steady axi-symmetric flow of a polymeric liquid (Maxwell fluid) between two infinite stretching disks in the presence of chemical reaction. Particular attention is to compute similarity solution of the governing nonlinear partial differential equations. The series solution of transformed nonlinear ordinary differential equations are computed via homotopy analysis method. The convergence of the developed series solution is also discussed. The quantities of interest like fluid velocity, fluid pressure, temperature, concentration, skin friction coefficient, local Nusselt and local Sherwood numbers are analyzed for the influence of embedded parameters. It is observed that the skin friction coefficient is an increasing function of Deborah number and Reynolds number. It is also observed that effects of stretching parameter on skin friction coefficient at upper and lower disks are opposite. Moreover, the heat transfer rate at lower disk is increases by increasing Deborah number. It is also observed that the surface mass transfer at both disks decreases by increasing Schmidt number.

Numerical Analysis for the Bingham—Papanastasiou Fluid Flow Over a Rotating Disk

Abstract: This study is conducted to investigate the Bingham—Papanastasiou fluid flow driven by a rotating infinite disk. The Bingham—Papanastasiou model is a modification of the Bingham plastic model, which is developed by introducing a continuation parameter to overcome its discontinuity. The von K´arman similarity solution is used to transform the flow equations from ordinary differential equations to a nonlinear system of partial differential equations, which is solved numerically. The effect of the Bingham flow parameters on the radial, tangential, and axial velocities, pressure, and radial and tangential skin friction coefficients is discussed.

Turbulent secondary flows in wall turbulence: vortex forcing, scaling arguments, and similarity solution

Abstract: Spanwise surface heterogeneity beneath high-Reynolds number, fully-rough wall turbulence is known to induce a mean secondary flow in the form of counter-rotating streamwise vortices—this arrangement is prevalent, for example, in open-channel flows relevant to hydraulic engineering. These counter-rotating vortices flank regions of predominant excess(deficit) in mean streamwise velocity and downwelling(upwelling) in mean vertical velocity. The secondary flows have been definitively attributed to the lower surface conditions, and are now known to be a manifestation of Prandtl’s secondary flow of the second kind—driven and sustained by spatial heterogeneity of components of the turbulent (Reynolds averaged) stress tensor (Anderson et al. J Fluid Mech 768:316–347, 2015). The spacing between adjacent surface heterogeneities serves as a control on the spatial extent of the counter-rotating cells, while their intensity is controlled by the spanwise gradient in imposed drag (where larger gradients associated with more dramatic transitions in roughness induce stronger cells). In this work, we have performed an order of magnitude analysis of the mean (Reynolds averaged) transport equation for streamwise vorticity, which has revealed the scaling dependence of streamwise circulation intensity upon characteristics of the problem. The scaling arguments are supported by a recent numerical parametric study on the effect of spacing. Then, we demonstrate that mean streamwise velocity can be predicted a priori via a similarity solution to the mean streamwise vorticity transport equation. A vortex forcing term has been used to represent the effects of spanwise topographic heterogeneity within the flow. Efficacy of the vortex forcing term was established with a series of large-eddy simulation cases wherein vortex forcing model parameters were altered to capture different values of spanwise spacing, all of which demonstrate that the model can impose the effects of spanwise topographic heterogeneity (absent the need to actually model roughness elements); these results also justify use of the vortex forcing model in the similarity solution.

Universality in the nonlinear leveling of capillary films

Abstract: A similarity solution of the lubrication equations is presented for the leveling of an interface after a hole in it closes, in agreement with experimental observations.

MHD boundary layer chemical reacting flow with heat transfer of Eyring–Powell nanofluid past a stretching sheet

Abstract: The problem of laminar nano non-Newtonian fluid through the boundary layer, which results from the stretching of a flat surface, has been investigated. The model of Eyring–Powell is used for the fluid. Constant normal magnetic field, mixed convection, chemical reaction, viscous dissipation, ohmic dissipation, Brownian and thermophoresis effects are considered. The problem is modulated mathematically by a system of partial differential equations, which describe the motion. A similarity solution is presented to transform this system to ordinary non-linear differential equations. The numerical solutions of these equations are obtained as functions of the physical parameters of the problem. Such as, Brownian number Nb, thermophoresis number Nt, Lewis number Le, Prandtl number Pr, Magnetic parameter M and Elastic parameter β. Graphical evaluation is displayed to depict the intrinsic behavior of embedded parameters on velocity, temperature, and nanoparticle concentration profiles.

Analytical solution of convective heat transfer of a quiescent fluid over a nonlinearly stretching surface using Homotopy Analysis Method

Abstract: In this article, an analytical solution of the boundary layer fluid flow and heat transfer of a quiescent viscous fluid over a non-linearly stretching surface is presented. The thermal radiation effects are included in the energy governing equation. Surface velocity and temperature conditions are assumed to be of the power-law form with an exponent of 1/3 for velocity and arbitrary exponent m for surface temperature or heat flux conditions. The system of nonlinear differential equations is solved by Homotopy Analysis Method (HAM) for two cases of Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF). The results of this method appear in the form of series expansions, the convergence of which is analyzed carefully. Graphical results are finally presented in order to investigate the influence of Prandtl number (Pr) and thermal radiation on heat transfer phenomena.

A similarity solution of time dependent MHD liquid film flow over stretching sheet with variable physical properties

Abstract: An analysis is performed for the fluid dynamics incorporating the variation of viscosity and thermal conductivity on an unsteady two-dimensional free surface flow of a viscous incompressible conducting fluid taking into account the effect of a magnetic field. Surface tension quadratically vary with temperature while fluid viscosity and thermal conductivity are assumed to vary as a linear function of temperature. The boundary layer partial differential equations in cartesian coordinates are transformed into a system of nonlinear ordinary differential equations (ODEs) by similarity transformation. The developed nonlinear equations are solved analytically by Homotopy Analysis Method (HAM) while numerically by using the shooting method. The Effects of natural parameters such as the variable viscosity parameter A, variable thermal conductivity parameter N, Hartmann number Ma, film Thickness, unsteadiness parameter S, Thermocapillary number M and Prandtl number Pr on the velocity and temperature profiles are investigated. The results for the surface skin friction coefficient f ″ ( 0 ) , Nusselt number (heat flux) - θ ′ ( 0 ) and free surface temperature θ ( 1 ) are presented graphically and in tabular form.

Resistively-limited current sheet implosions in planar anti-parallel (1D) and null-point containing (2D) magnetic field geometries

Abstract: Implosive formation of current sheets is a fundamental plasma process. Previous studies focused on the early time evolution, while here our primary aim is to explore the longer-term evolution, which may be critical for determining the efficiency of energy release. To address this problem we investigate two closely-related problems, namely: (i) 1D, pinched anti-parallel magnetic fields and (ii) 2D, null point containing fields which are locally imbalanced ('null-collapse' or 'X-point collapse'). Within the framework of resistive MHD, we simulate the full nonlinear evolution through three distinct phases: the initial implosion, its eventual halting mechanism, and subsequent evolution post-halting. In a parameter study, we find the scaling with resistivity of current sheet properties at the halting time is in good agreement - in both geometries - with that inferred from a known 1D similarity solution. We find that the halting of the implosions occurs rapidly after reaching the diffusion scale by sudden Ohmic heating of the dense plasma within the current sheet, which provides a pressure gradient sufficient to oppose further collapse and decelerate the converging flow. This back-pressure grows to exceed that required for force balance and so the post-implosion evolution is characterised by the consequences of the current sheet `bouncing' outwards. These are: (i) the launching of propagating fast MHD waves (shocks) outwards and (ii) the width-wise expansion of the current sheet itself. The expansion is only observed to stall in the 2D case, where the pressurisation is relieved by outflow in the reconnection jets. In the 2D case, we quantify the maximum amount of current sheet expansion as it scales with resistivity, and analyse the structure of the reconnection region which forms post-expansion, replete with Petschek-type slow shocks and fast termination shocks.

Explicit solution for the one-phase stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face

Abstract: An explicit solution of a similarity type is obtained for a onephase Stefan problem in a semi-infinite material using Kummer functions. It is considered a phase-change problem with a latent heat defined as a power function of the position with a non-negative real exponent and a convective boundary condition at the fixed face x = 0. Existence and uniqueness of the solution is proved. Relationship between this problem and the problems with temperature and flux boundary condition is also analysed. Furthermore it is studied the limit behaviour of the solution when the coefficient which characterizes the heat transfer at the fixed boundary tends to infinity. Computing this limit allows to demonstrate that the problem proposed in this paper with a convective boundary condition generalizes the problem with Dirichlet boundary condition. Numerical computation of the solution is done over certain examples, with a view to comparing this results with those obtained by general algorithms that solve Stefan problems. AMS Subject Classification: 35R35, 80A22, 35C05

Simulation and scaling analysis of a spherical particle-laden blast wave

Abstract: A spherical particle-laden blast wave, generated by a sudden release of a sphere of compressed gas–particle mixture, is investigated by numerical simulation. The present problem is a multiphase extension of the classic finite-source spherical blast-wave problem. The gas–particle flow can be fully determined by the initial radius of the spherical mixture and the properties of gas and particles. In many applications, the key dimensionless parameters, such as the initial pressure and density ratios between the compressed gas and the ambient air, can vary over a wide range. Parametric studies are thus performed to investigate the effects of these parameters on the characteristic time and spatial scales of the particle-laden blast wave, such as the maximum radius the contact discontinuity can reach and the time when the particle front crosses the contact discontinuity. A scaling analysis is conducted to establish a scaling relation between the characteristic scales and the controlling parameters. A length scale that incorporates the initial pressure ratio is proposed, which is able to approximately collapse the simulation results for the gas flow for a wide range of initial pressure ratios. This indicates that an approximate similarity solution for a spherical blast wave exists, which is independent of the initial pressure ratio. The approximate scaling is also valid for the particle front if the particles are small and closely follow the surrounding gas.

Exact Solution for a Magnetogasdynamical Cylindrical Shock Wave in a Self-Gravitating Rotating Perfect Gas with Radiation Heat Flux and Variable Density

Abstract: An exact similarity solution for a magnetoradiative cylindrical shock wave in a self-gravitating rotating perfect gas is obtained. The density, azimuthal velocity, and magnetic field strength are assumed to vary in an undisturbed medium. It is shown that the flow variables, namely, the radial velocity, pressure, magnetic field strength, azimuthal velocity, mass, and the radiation flux, decrease from the highest values at the shock front to zero; however, the density tends to infinity as the symmetry axis is approached. The effects of variation in the magnetic field strength, gravitational parameter, rotational parameter, and in the adiabatic exponent on the flow variables and shock strength are discussed. The solutions obtained for self-gravitating and nongravitating media are compared. The total energy of the shock wave is shown to be not constant.

Similarity solution and Runge Kutta method to a thermal boundary layer model at the entrance region of a circular tube

Abstract: Purpose The purpose of this paper is to re-examine the assumptions implicit in Leveque’s approximation, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth-order (RK4) method. Finally, other important thermal results obtained from this analysis, such as approximate Nusselt number in the thermal entrance region, was discussed in detail. After that, the analytical results of the present paper are validated with certain previous investigations which were found in the specialized literature. Design/methodology/approach By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This paper gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form and presents a similarity solution. The calculation methodology for numerical resolution is based on the RK4 technique. Findings The profiles of the solutions are provided from which the authors infer that the numerical and exact solutions agreed very well. Another result that the authors obtained from this paper is the number of Nusselt in the thermal entrance region for which a parametric study was carried out and discussed well for the impact of scientific contribution. Originality/value The novelty of this paper is the application of the RK4 with a step size control, as a sequential numerical method of a ODEs system compared with the exact similarity solution of the thermal boundary layer problem.

Similarity solution and Runge Kutta method to a thermal boundary layer model at the entrance region of a circular tube: The Lévêque Approximation

Abstract: In the thermal entrance region, a thermal boundary layer develops and also reaches the circular tube center. The fully developed region is the zone in which the flow is both hydrodynamically and thermally developed. The heat flux will be higher near the inlet because the heat transfer coefficient is highest at the tube inlet where the thickness of the thermal boundary layer is zero and decreases gradually to the fully developed value. In this paper, the assumptions implicit in Leveque's approximation are re-examined, and the analytical solution of the problem with additional boundary conditions, for the temperature field and the boundary layer thickness through the long tube is presented. By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This report gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form, and presents a similarity solution. The analytical solutions are then checked against numerical solution programming by Fortran code obtained via using Runge-Kutta fourth order (RK4) method. Finally, others important thermal results obtained from this analysis, such as; approximate Nusselt number in the thermal entrance region was discussed in detail.

Electrical Unsteady MHD Natural Convection Flow of Nanofluid with Thermal Stratification and Heat Generation/Absorption

Abstract: In this paper, we analyzed the effects of thermal radiation, chemical reaction, heat generation/absorption, magnetic and electric fields on unsteady natural convection flow and heat transfer due to nanofluid over a permeable stretching sheet. The transport equations used passively controlled boundary condition rather than actively. A similarity solution is employed to transformed the governing equations from nonlinear partial differential equations to a set of ordinary differential equations, and then solve using Keller box method. It was found that the temperature is a decreasing function with the thermal stratification due to the fact the density of the fluid in the lower vicinity is much higher compared to the upper region, whereas the thermal radiation, viscous dissipation and heat generation enhanced the nanofluid temperature and thermal layer thickness.

Natural convection of third-grade non-Newtonian fluid flow in a porous medium with heat source: Analytical solution

Abstract: Heat transfer by natural convection occurs in many physical and engineering applications. Governing equations of these problems are non-linear and need special methods for being solved. This paper aims to conduct an analytical analysis on the natural convection of a non-Newtonian fluid flow between two infinite vertical flat plates within a porous medium considering a variable heat source. The addition of porous medium and heat source marks the difference between the present work and previous researches. In the first phase of the current analysis, the governing equations including partial differential equations (PDE) are turned into ordinary differential equations (ODE) utilizing the similarity solution. Afterwards, a system of differential equations is solved using the Least Square Method (LSM), and reliable functions for the temperature and velocity distributions are presented. In order to investigate the accuracy of this method, the governing equations are also solved using numerical solutions. Proper agreement is observed between the analytical and numerical results. Regarding the very small errors observed in the results yielded by the LSM method, it can be concluded that this method is an efficient and reliable approach for solving non-linear ordinary differential equations. Finally, the effects of two main parameters, namely porosity and heat source parameters, are meticulously discussed. It is shown that as the value of the heat source parameter increases, the values of velocity and temperature decrease. Also, the results indicate that by enhancing the porosity parameter the flow velocity decreases, whereas there is no change in temperature. The results could be helpful for systems such as geothermal systems, heat exchangers, petroleum reservoirs and nuclear waste repositories in which natural convection is important.

The effects of fluid transport on the creation of a dense cluster of activated fractures in a porous medium

Abstract: The formation of a cluster of activated fractures when fluid is injected in a low permeability rock is analysed. A fractured rock is modelled as a dual porosity medium that consists of a growing cluster of activated fractures and the rock’s intrinsic porosity. An integro-differential equation for fluid pressure in the developing cluster of fractures is introduced to account for the pressure-driven flow through the cluster, the loss of fluid into the porous matrix and the evolution of the cluster’s permeability and porosity as the fractures are activated. Conditions under which the dependence of the permeability and porosity on the fluid pressure can be derived from percolation theory are discussed. It is shown that the integro-differential equation admits a similarity solution for the fluid pressure and that the cluster radius grows as a power law of time in two regimes: (i) a short-time regime, when many fractures are activated but pressure-driven flow in the network still dominates over fluid loss; and (ii) a long-time regime, when fluid loss dominates. The power law exponents in the two regimes are functions of the Euclidean dimension of the cluster, percolation universal exponents and the injection protocol. The model predicts that the effects of the fluid properties on the morphology of the formed network are different in the two similarity regimes. For example, increasing the injection rate with time, in the flow dominant regime, produces a smaller cluster of activated fractures than that formed by injecting the fluid at a constant rate. In the fluid loss dominated regime, however, ramping up the injection rate produces a larger cluster.

Similarity solution for a two-phase one-dimensional Stefan problem with a convective boundary condition and a mushy zone model

Abstract: A two-phase solidification process for a one-dimensional semi-infinite material is considered. It is assumed that it is ensued from a constant bulk temperature present in the vicinity of the fixed boundary, which it is modelled through a convective condition (Robin condition). The interface between the two phases is idealized as a mushy region and it is represented following the model of Solomon, Wilson, and Alexiades. An exact similarity solution is obtained when a restriction on data is verified, and it is analysed the relation between the problem considered here and the problem with a temperature condition at the fixed boundary. Moreover, it is proved that the solution to the problem with the convective boundary condition converges to the solution to a problem with a temperature condition when the heat transfer coefficient at the fixed boundary goes to infinity, and it is given an estimation of the difference between these two solutions. Results in this article complete and improve the ones obtained in Tarzia (Comput Appl Math 9:201–211, 1990).

Self similar flow behind an exponential shock wave in a self-gravitating, rotating, axisymmetric dusty gas with heat conduction and radiation heat flux

Abstract: One-dimensional unsteady adiabatic flow behind an exponential shock wave propagating in a self-gravitating, rotating, axisymmetric dusty gas with heat conduction and radiation heat flux, which has exponentially varying azimuthal and axial fluid velocities, is investigated. The shock wave is driven out by a piston moving with time according to an exponential law. The dusty gas is taken to be a mixture of a non-ideal gas and small solid particles. The density of the ambient medium is assumed to be constant. The equilibrium flow conditions are maintained and energy is varying exponentially, which is continuously supplied by the piston. The heat conduction is expressed in the terms of Fourier’s law, and the radiation is assumed of diffusion type for an optically thick grey gas model. The thermal conductivity and the absorption coefficient are assumed to vary with temperature and density according to a power law. The effects of the variation of heat transfer parameters, gravitation parameter and dusty gas parameters on the shock strength, the distance between the piston and the shock front, and on the flow variables are studied out in detail. It is interesting to note that the similarity solution exists under the constant initial angular velocity, and the shock strength is independent from the self gravitation, heat conduction and radiation heat flux.

Asymptotic scaling laws and semi-similarity solutions for a finite-source spherical blast wave

Abstract: A spherical blast wave generated by a sudden release of a sphere of compressed gas is an important model problem to understand blast phenomena such as volcanic eruptions and explosive detonations. The resulting explosion flow physics, such as the instability at the gas contact discontinuity and the interaction between the shock wave and the gas contact, are dictated by the initial pressure and sound-speed ratios between the compressed gas and the ambience. Since the initial pressure and sound-speed ratios vary over a wide range in practical applications, it is of interest to investigate the scaling laws and similarity solutions for the spherical symmetric explosion flow. In the present study, numerical simulation of a spherical blast wave is performed. A long-term length scale that incorporates the initial charge radius and the initial pressure ratio is introduced. The trajectories of the main shock normalized by the long-term length scale for a wide range of parameters collapse after a short transition time, indicating an asymptotic similarity solution exists for the far field in the long term. With the assistance of this similarity solution, the full evolution of the main shock can be obtained semi-analytically. For near-field features, i.e. the gas contact and the secondary shock wave, only semi-similarity solutions are observed, which depend on the initial sound-speed ratio but not the initial pressure ratio. The gas contact and the secondary shock share the same scaling relations. Asymptotic analysis is performed to obtain the short-term dynamics of the gas contact, including the gas contact acceleration and the Atwood number, which are the key parameters determining the Rayleigh–Taylor instability development at the gas contact. The asymptotic contact radius as is also obtained, which is found to be well represented by the long-term length scale and thus only depends on the initial pressure ratio. A simple model of an oscillating bubble is employed to explain the scaling relation of the asymptotic gas contact radius.

Three-Dimensional Boundary Layer Flow and Heat Transfer of a Dusty Fluid Towards a Stretching Sheet with Convective Boundary Conditions

Abstract: The steady three-dimensional boundary layer flow and heat transfer of a dusty fluid towards a stretching sheet with convective boundary conditions is investigated by using similarity solution approach. The free stream along z-direction impinges on the stretching sheet to produce a flow with different velocity components. The governing equations are reduced into ordinary differential equations by using appropriate similarity variables. Reduced nonlinear ordinary differential equations subjected to the associated boundary conditions are solved numerically by using Runge–Kutta fourth-fifth order method along with Shooting technique. The effects of the physical parameters like magnetic parameter, velocity ratio, fluid and thermal particle interaction parameter, Prandtl number, Eckert number and Biot number on flow and heat characteristics are examined, illustrated graphically, and discussed in detail. The results indicate that the fluid phase velocity is always greater than that of the particle phase and temperature profiles of fluid and dust phases increases with the increase of the Eckert number.