Similarity Exponent Research Articles

A perturbation approach to study the shock wave propagation in a non-ideal magnetogasdynamics under isothermal condition

The main goal of this paper is to obtain a global solution to the problem of imploding strong cylindrical shock waves (SWs) collapsing at the axis of symmetry in a non-ideal gas under the influence of an axial magnetic field using the perturbation series technique. This global solution is valid for the entire flow field up to the instant of collapse. Guderley's asymptotic solution, which is only applicable in the neighborhood of the axis of implosion, is properly reproduced by this global solution. Guderley's local self-similar solution allows for the determination of the first dominant similarity exponent only; however, this approach also enables the identification of additional, less dominant similarity exponents along with the corresponding amplitudes. Additionally, the computed values of the similarity exponents have been compared with the results drawn from Guderley's approach. The profiles of fluid variables and shock trajectory are shown graphically for different values of the non-ideal parameter, adiabatic index, and shock Cowling number. The “Mathematica” software has been used to do all numerical computations.

Optimal system of solution using group invariance technique for shock wave in a non-ideal self-gravitating gas in rotating medium in presence of magnetic field

Abstract This work demonstrates the study of the optimal system of solutions for shock wave propagation in a non-ideal self-gravitating gas in rotating medium with magnetic field (axial or azimuthal) for the adiabatic flow in cylindrical geometry by applying the group invariance technique. Using the group invariance technique, we have obtained the one-dimensional (1-D) optimal system of sub-algebra for the basic governing equations. The infinitesimal group optimal classes are obtained and the similarity solution in four possible cases (two cases for perfect gas and two cases for non-ideal gas) with exponential law shock path are discussed. The numerical solution by using the Runge Kutta 4th order method is obtained and the distribution of physical variables are shown via graph. The impact of the rotational parameter, non-idealness parameter, shock Cowling number, similarity exponent and gravitational parameter on the strength of the shock and flow variables are investigated. With an increase in the shock Cowling number, non-idealness and rotational parameters, the shock strength decreases, i.e., they have decaying impact on shock wave; whereas the shock strength increases with gravitational parameter and similarity exponent. Also, the strength of the shock is reduced by considering the magnetic field to be axial instead of azimuthal.

Turbulent flow on a rotating disk

The dissipation theorem is applied to describe fully developed turbulent flow on a rotating disk. A complicated coordinate transformation is proposed to solve the problem. The similarity exponent m becomes a parameter of the problem and can vary gradually with the Reynolds number. This solution may span the entire range from laminar to fully developed turbulent flow, although not accounting for all features observed in the transition region. The profiles of the various quantities are required to be well behaved, and this requirement is used to determine m at different Reynolds numbers. The results are in harmony with the literature and show how different geometries can be treated in a unified manner, with differences arising naturally from those geometric differences but with utilization of similar forms for the decay of the dissipation.

Study of shocks in a nonideal dusty gas using Maslov, Guderley, and CCW methods for shock exponents

Abstract In this work, we consider the system of partial differential equations describing one-dimensional (1D) radially symmetric (i.e., cylindrical or spherical) flow of a nonideal gas with small solid dust particles. We analyze the implosion of cylindrical and spherical symmetric strong shock waves in a mixture of a nonideal gas with small solid dust particles. An evolution equation for the strong cylindrical and spherical shock waves is derived by using the Maslov technique based on the kinematics of 1D motion. The approximate value of the similarity exponent describing the behavior of strong shocks is calculated by applying a first-order truncation approximation. The obtained approximate values of similarity exponent are compared with the values of the similarity exponent obtained from Whitham’s rule and Guderley’s method. All the above computations are performed for the different values of mass fraction of dust particles, relative specific heat, and the ratio of the density of dust particle to the density of the mixture and van der Waals excluded volume.

Solutions of the imploding shock problem in a medium with varying density

We consider the solutions of the Guderley problem, consisting of an imploding strong shock wave in an ideal gas with a power law initial density profile. The self-similar solutions, and, specifically, the similarity exponent that determines the behavior of the accelerating shock, are studied in detail, for cylindrical and spherical symmetries and for a wide range of the adiabatic index and the spatial density exponent. We then demonstrate how the analytic solutions can be reproduced in Lagrangian hydrodynamic codes, thus demonstrating their usefulness as a code validation and verification test problem.

Revisiting the strong shock problem: Converging and diverging shocks in different geometries

Self-similar solutions to converging (implosions) and diverging (explosions) shocks have been studied before, in planar, cylindrical, or spherical symmetry. Here, we offer a unified treatment of these apparently disconnected problems. We study the flow of an ideal gas with adiabatic index γ with initial density ρ∼r−ω, containing a strong shock wave. We characterize the self-similar solutions in the entirety of the parameter space γ,ω and draw the connections between the different geometries. We find that only type II self-similar solutions are valid in converging shocks, and that in some cases, a converging shock might not create a reflected shock after its convergence. Finally, we derive analytical approximations for the similarity exponent in the entirety of parameter space.

Similarity Solutions for Imploding Shocks in a Non-ideal Magnetogasdynamics

In the present paper, we use a Lie group of transformations to obtain a class of similarity solutions to a problem of a cylindrically symmetric shock wave propagating through one-dimensional, unsteady flow of a non-ideal gas. We assume that the gas has infinite electrical conductivity under the influence of axial component of the magnetic field, with isothermal flow condition. The density of the ambient medium is taken to be uniform ahead of the shock. The generators of the Lie group of transformations involve arbitrary constants which yield two different cases of possible solutions with a power law shock path and exponential shock path. We present a detailed investigation for the case of power law shock path. Numerical computations have been carried out to find out the flow patterns in the flow-field behind the shock and the values of similarity exponent. Further, we compare the obtained similarity exponent with that obtained by Guderley’s (Luftfahrtforschung 19:302–312, 1942) method. The effects of the non-ideal parameter and magnetic field strength on the similarity exponent and the flow patterns have also been studied. All computations have been done using the software package MATHEMATICA.

Similarity Solutions of Strong Shock Waves for Isothermal Flow in an Ideal Gas

Self-similar solutions of the system of non-linear partial differential equations are obtained using the Lie group of invariance technique. The system of equations governs the one dimensional and unsteady motion for the isothermal flow of an ideal gas. The medium has been taken the uniform. From the expressions of infinitesimal generators involving arbitrary constants, different cases arise as per the choice of the arbitrary constants. In this paper, the case of a collapse of an implosion of a cylindrical shock wave is shown in detail along with the comparison between the similarity exponent obtained by Guderley's method and by Crammer's rule. Also, the effects of the adiabatic index and the ambient density exponent on the flow variables are illustrated through the figures. The flow variables are computed behind the leading shock and are shown graphically.

Healing capillary films

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.

Self-similar solutions and converging shocks at the stellar surfaces

In this paper, Lie group of transformations method is used to investigate the self-similar solutions for the system of one-dimensional hydrodynamic equations describing the post-shock flow at the stellar surface. Necessary conditions for the existence of similarity solutions for strong shocks are obtained. The similarity exponent using Lie group of transformations and characteristic rule are obtained. Collapses of imploding cylindrical and spherical shocks are worked out in detail.

Non-Newtonian nanofluid free convection flow subject to mixed thermal boundary conditions about a vertical cone

A similarity analysis for natural convection past a vertical cone in Darcy porous medium saturated with a non-Newtonian nanofluid is considered assuming that the cone is subjected to a mixed thermal boundary condition. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The generalized governing equations derived in this work can be applied to the cases of prescribed surface temperature and prescribed heat flux. The effects of the power-law viscosity index and the similarity exponent on the heat transfer characteristics under mixed thermal boundary conditions have been studied. Moreover, numerical solutions of the boundary layer equations are obtained for several values of the nanofluid parameters (Brownian motion number Nb, thermophoresis parameter Nt, buoyancy ratio number Nr). A comparison with previously published work on a special case of the general problem was performed and the results were found to be in excellent agreement.

Existence of self-similar solutions in reacting gases

In this paper, the Lie group theoretical method is used to establish the entire class of self-similar solutions to the problem of shock wave propagation through reacting polytropic gases with the same $$\gamma $$ -law. The system consists of two species, burnt gas and unburnt gas. Necessary conditions for the existence of similarity solutions for shocks of arbitrary strength as well as for strong shocks are obtained. The arbitrary constants, occurring in the expressions of the infinitesimals of Lie group of transformations give rise to different cases of possible solutions with a power law, exponential and logarithmic shock paths. For the existence of self-similar solutions, the forms of reaction rate are found out in different cases. A particular solution is considered to study the effect of reaction rate on the similarity exponent.

The influence of magnetic field upon the collapse of a cylindrical shock wave

In the present paper, the problem of propagation of collapsing cylindrical shock wave in an ideal gas permeated by a transverse magnetic field with infinite electrical conductivity is investigated. Here it is assumed that the medium ahead of the shock front is uniform and at rest. Also, its counter pressure concerning the motion of the wave front is neglected. This problem admits a self similar solution of second kind. The similarity exponent has been computed by solving a nonlinear eigenvalue problem and integrating numerically the self-similar equations for various values of adiabatic heat exponent and Cowling number. Numerical computations have been performed to determine the flow field behind the shock wave. The influence of magnetic field strength and adiabatic heat exponent on the flow parameters for various cases is presented.

The Guderley problem revisited

The self-similar converging-diverging shock wave problem introduced by Guderley in 1942 has been the source of considerable mathematical and physical interest. We investigate a novel application of the Guderley solution as a unique and challenging code verification test problem for compressible flow algorithms; this effort requires a unified understanding of the problem's mathematical and computational subtleties. Hence, we review the simplifications and group invariance properties that reduce the compressible flow equations for a polytropic gas to two coupled nonlinear eigenvalue problems: the first for the similarity exponent in the converging regime, and the second for a trajectory multiplier in the diverging regime. The information we provide, together with previously published material, gives a complete description of the computational steps required to construct a semi-analytic Guderley solution. We employ the problem in a quantitative code verification analysis of a cell-centred, finite volume, Eulerian compressible flow algorithm. Lastly, in appended material, we introduce a new approximation for the similarity exponent, which may prove useful in the future construction of certain semi-analytic Guderley solutions.

Landau—Stanyukovich Rule and the Similarity Parameter of Converging Shock Waves in Magnetogasdynamics

Second-kind self-similar solutions to a problem of converging cylindrical shock waves in magnetogasdynamics are investigated. Two trial functions suggested by Chisnell and the shooting method of Landau—Stanyukovich are used to determine the similarity exponent for different values of specific heat ratio γ and the parameter k, where k ∊ (0, 1]. Detailed analyses of flow patterns for different values of adiabatic heat exponent and magnetic field strength are carried out. It is observed that the general behavior of the velocity and density profiles is not affected in a magnetogasdynamics regime whereas there is an increase in the absolute value of the flow parameters with an increase in the magnetic field strength. However, the pressure profiles are greatly affected by the magnetic field interaction.

The BTW mechanism on a self-similar image of a square: A path to unexpected exponents

We define the BTW mechanism on a two-dimensional heterogeneous self-similar lattice. Our model exhibits the power distribution of avalanches with the exponent τ=2−2/ν, where ν is the similarity exponent of the lattice. The inequality τ<1, for the first time detected in this paper inside a broad class of sand-piles, is ensured by random loading uniformly distributed over the lattice.

Analytical Solution of Converging Shock Wave in Magnetogasdynamics

In this paper, a similarity solution to a problem in magnetogasdynamics with strong converging cylindrical shock wave is examined. An analytical description of converging shock waves is presented by replacing the previous approach for numerical solutions of the ordinary differential equations with a theoretical study of the singular points of the differential equations. A study of singular points of the system of differential equations leads to an analytic description of the flowfield and a determination of the similarity exponent. The influences of adiabatic heat exponent and magnetic field strength on the flow pattern for various cases are assessed. The general behavior of the velocity and density distribution remains unaffected. However, the pressure profiles are greatly affected by the magnetic field interaction.

Similarity solutions for imploding shocks in non-ideal magnetogasdynamics

In the present article, similarity solutions of second kind to a problem of imploding cylindrical shock wave in non-ideal magnetogasdynamics are investigated. The equation of state of the medium is assumed to be in the form of the Mie-Gruneisen type. A numerical study of singular points of the differential equations leads to determination of the similarity exponent. Numerical description of the flow field has been presented in non-ideal magnetogasdynamics. The results obtained are compared with the numerical solution obtained by using CCW approximation method. Detailed studies are carried out for two different physically meaningful non-ideal medium in the presence of magnetic field. Also, the effect on magnetic field of flow variables such as density, velocity, pressure and magnetic pressure behind the wave front is illustrated through figures.

Thin-film rupture for large slip

The rupture of thin liquid films on hydrophobic substrates, assuming large slip at the liquid-solid interface, is studied using a recently developed strong slip lubrication model, it is shown that the rupture passes through up to three self-similar regimes with different dominant balances and different scaling exponents. For one of these regimes the similarity is of second kind, and the similarity exponent is determined by solving a boundary-value problem for a nonlinear ODE. Furthermore, finite-time rupture is proved for this regime.

Natural convection heat transfer of non-Newtonian fluids in porous media from a vertical cone under mixed thermal boundary conditions

This work studies the problem of the steady natural convection boundary layer flow over a downward-pointing vertical cone in porous media saturated with non-Newtonian power-law fluids under mixed thermal boundary conditions. A similarity analysis is performed, and the obtained similar equations are solved by cubic spline collocation method. The effects of the power-law viscosity index and the similarity exponent on the heat transfer characteristics under mixed thermal boundary conditions have been studied. Under mixed thermal boundary conditions, both the surface heat flux and the surface temperature are found to decrease when the power-law viscosity index of the non-Newtonian power-law fluid in porous media is increased. Moreover, an increase in the similarity exponent tends to increase the boundary layer thickness and thus decreases the surface heat flux under mixed thermal conditions. The generalized governing equations derived in this work can be applied to the cases of prescribed surface temperature and prescribed heat flux.