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Self-similar solutions are obtained for one-dimensional unsteady adiabatic flow behind a spherical shock wave propagating in a dusty gas with conductive and radiative heat fluxes under the influence of a gravitational field. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of non-ideal gas and small solid particles, in which solid particles are uniformly distributed. It is assumed that the equilibrium flow-conditions are maintained and variable energy input is continuously supplied by the piston. The heat conduction is expressed in terms of Fourier’s law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity K and the absorption coefficient α R are assumed to vary with temperature and density. The medium is assumed to be under the influence of a gravitational field due to central mass ( m ¯ ) at the origin (Roche Model). It is assumed that the gravitational effect of the mixture itself can be neglected compared with the attraction of the central mass. The initial density of the ambient medium is taken to be always constant. The effects of the variation of the gravitational parameter and nonidealness of the gas in the mixture are investigated. Also, the effects of an increase in (i) the mass concentration of solid particles in the mixture and (ii) the ratio of the density of solid particles to the initial density of the gas on the flow variables are investigated. It is shown that due to an increase in the gravitational parameter the compressibility of the medium at any point in the flow-field behind the shock decreases and all other flow variables and the shock strength are increased. Further, it is found that the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and therefore the distance between the piston and the shock surface is reduced. The shock waves in dusty gas under the influence of a gravitational field can be important for description of shocks in supernova explosions, in the study of central part of star burst galaxies, nuclear explosion, star formation in shocks and shocks in stellar explosion, rupture of a pressurized vessels and explosion in the ionosphere etc. Also, the solution obtained can be used to interpret measurements carried out by spacecraft in the solar wind and in neighborhood of the Earth’s surface.

In the present article, we have obtained the similarity solutions with the help of optimal system of Lie sub-algebras and numerical method for the cylindrical shock wave formed by a moving piston in a non-ideal dusty gas in a rotating medium with the impact of monochromatic radiation. The Lie group theoretic technique is used to obtain the optimal system of Lie sub-algebras for the fundamental equations in the case of 1-D (one-dimensional) flow. By using the Lie group theoretic technique, we are able to drive the similarity and numerical solution for both the power and exponential law shock paths. The similarity solutions with power law shock path in two different cases (i.e., Cases Ia, Ib) are obtained in Case I. Also, in Case II, the similarity solutions exist in two different cases with power law shock path (i.e., Cases IIa, IIc) and in two different cases with exponential law shock path (i.e., Cases IIb, IId) which are discussed in detail. The results presented in Cases Ib, IIa, IIb, IIc, and IId are newly obtained results. The numerical solutions in the power law shock path for Case Ia and the exponential law shock path in Case IIb are performed, and the distribution of the physical flow variables in the flow field region behind the shock front is obtained. The impacts of the several problematic physical parameters on the shock strength and on the physical flow variable are studied. In the present study, it is found that with an increment in the value of the initial ratio of the density of solid particles to the species density of the gas (G1), the shock strength increases, whereas the shock strength decreases with an increment in the gas non-ideality parameter (ω¯). For the lower value of G1 (i.e., for G1=6), the shock strength decreases, whereas for a higher value of G1 (i.e., for G1=20 or 100), the shock strength increases with an increment in the mass concentration of the solid particles in the mixture (μp).

One-dimensional unsteady isothermal and adiabatic flow behind a strong exponential shock wave propagating in a rotational axisymmetric mixture of non-ideal gas and small solid particles, which has variable azimuthal and axial fluid velocities, is analyzed. The shock wave is driven out by a piston moving with time according to exponential law. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to be varying and obeying exponential laws. In the present work, small solid particles are considered as pseudo-fluid with the assumption that the equilibrium flow-conditions are maintained in the flow-field, and the viscous-stress and heat conduction of the mixture are negligible. Solutions are obtained in both the cases, when the flow between the shock and the piston is isothermal or adiabatic by taking into account the components of vorticity vector and compressibility. It is found that the assumption of zero temperature gradient brings a profound change in the density, axial component of vorticity vector and compressibility distributions as compared to that of the adiabatic case. To investigate the behavior of the flow variables and the influence on the shock wave propagation by the parameter of non-idealness of the gas b ¯ in the mixture as well as by the mass concentration of solid particles in the mixture K p and by the ratio of the density of solid particles to the initial density of the gas G 1 are worked out in detail. It is interesting to note that the shock strength increases with an increase in G 1 ; whereas it decreases with an increase in b ¯ . Also, a comparison between the solutions in the cases of isothermal and adiabatic flows is made.

The discharge of chromium-contaminated wastewater from industries such as ferrochrome plants and leather manufacturing poses a significant environmental challenge due to the toxic and carcinogenic properties of Chromium [Cr(VI)]. Nanoparticles have emerged as one of the most effective solutions for wastewater treatment due to their high surface area, enhanced reactivity, and ability to target specific contaminants. In recent years, their eco-friendly synthesis, scalability, and efficiency in removing heavy metals and other pollutants have made them vital in addressing environmental challenges, particularly in industrial wastewater management. Their unique properties make them indispensable in modern wastewater treatment technologies. This study explores the application of biogenically synthesized titanium dioxide (TiO₂) for removal of Cr(VI) from synthetic wastewater. Lemon grass leaf extracts has been used as potential precursor in synthesis of TiO₂ nanoparticles from readily available micro size particles of TiO 2 powder. The process was further enhanced by ultrasonic assistance, which promoted the formation of uniformly dispersed nanoparticles with high surface area, improving their adsorption. Experimental techniques, such as X-ray Diffraction, have been utilized to confirm the biogenic synthesis of TiO 2 nanoparticles, demonstrating a size reduction from 10 μm to 35.79 nm. The nanoparticles demonstrated excellent Cr(VI) removal efficiency, achieving 84.55% reduction under optimal conditions. Among the various adsorption isotherm models, the Freundlich model proved to be the best fit, with an R² value exceeding 0.997. This method not only leverages sustainable synthesis processes but also offers potential scalability for industrial applications in waste water treatment.

The similarity solutions using Lie group analysis for shock wave propagation in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the case of isothermal and adiabatic flows are obtained. All possible cases of similarity solutions are discussed using the Lie group analysis for the isothermal and adiabatic flows. The arbitrary constants involved in the generators of local Lie groups bring various possible cases of solutions with exponential law and power law shock paths. Similarity solution for isothermal and adiabatic flows with power law shock path is discussed in detail. The density of ambient medium is taken to be constant. The axial and azimuthal fluid velocities and magnetic field in the ambient medium are assumed to be varying according to power law. The effect on shock wave strength and that on the flow variables due to variation of the Alfven Mach number, adiabatic index of the gas, non-idealness parameter, rotational parameter and initial magnetic field variation exponent are investigated. It is found that these parameters have decaying effects on shock wave. The obtained results in the case of isothermal and adiabatic flows are also compared with each other.

The propagation of ionizing cylindrical magnetogasdynamic shock wave in rotational axisymmetric self-gravitating perfect gas under isothermal flow condition is investigated. Mathematical model for the considered problem using system of PDEs is presented. The density, magnetic pressure, azimuthal fluid velocity and axial fluid velocity are assumed to be varying according to power law with distance from the axis of symmetry in the undisturbed medium. The flow variables are expanded in power series and using that the zeroth and first order approximations are discussed. Solutions for zeroth order approximation are constructed in approximate analytical form. The effect of flow parameters namely: gravitational parameter G 0 , shock Cowling number C 0 , rotational parameter L and ambient density variation index q are studied on the flow variables and total energy of disturbance. Distribution of gasdynamical quantities are discussed. Radial fluid velocity and mass tends to zero near the axis of symmetry in general; but magnetic pressure, axial fluid velocity, non-dimensional components of vorticity vector l θ ( 0 ) and l z ( 0 ) tend to positive infinity near the axis of symmetry. Azimuthal fluid velocity decreases as we move inwards from the shock to the axis of symmetry. Density and pressure vanish near the axis of symmetry thus forming a vacuum there. J 0 decreases with increase in value of ambient density variation index q or gravitational parameter G 0 ; whereas it increases with increase in value of rotational parameter L or shock Cowling number C 0 .

PurposeThis article aims to find the similarity solutions for the one-dimensional motion of spherical symmetric shock wave in non-ideal gas influenced by the azimuthal magnetic field and monochromatic radiation in the presence or absence of gravitational field. This paper also aims to study the effects of physical parameters on the strength of shock wave, and on the flow variables in the flow-field region behind the shock front.Design/methodology/approachThe Roche model is used to describe the gravitational field effects due to a massive nucleus at the point of symmetry. To derive the similarity solutions, the Lie group symmetry method has been used. Also, the numerical solutions to the present problem are obtained by using Rung–Kutta method of the fourth order with the use of Mathematica software. The effects of variation in the parameter of non-idealness of the gas, the gravitation parameter, the strength of the ambient magnetic field and the adiabatic index of the gas on the shock wave, and on the flow variables is discussed. A comparative study between with and without gravitational field is also, made.FindingsFor different choices of the arbitrary constants that appeared in the solution of infinitesimal generators, we have obtained seven distinct cases of similarity solutions. In the absence of the gravitational field, the similarity solution exists to the power and exponential law shock paths, but in the presence of gravitational field, the similarity solution exists to the power law shock path case only. In the absence of gravitational field, the shock strength is enhanced in the exponential law shock path case in comparison to the power law shock path case. It is found that the shock wave decays with an increase in the value of the adiabatic exponent, the strength of magnetic field, non-idealness of the gas or gravitational parameter.Research limitations/implicationsThe consideration of medium under the influence of gravitational field due to a heavy nucleus at the center and presence of magnetic field decrease the shock strength. This result may be helpful in designing space vehicle and jet engine.Practical implicationsThe result of the present study may be used in the analysis of data from the measurements by space craft in the solar wind and in neighborhood of the Earth’s magnetosphere.Social implicationsThe obtained results may be used for mankind.Originality/valueThe study of spherical shock wave propagation influenced by monochromatic radiation and azimuthal magnetic field in a non-ideal gas with or without gravitational field has yet to be discussed by any authors by using the Lie group symmetry method. In this article, we have discussed all possible cases of similarity solutions using the Lie group symmetry method, which is not studied by anyone as known to us.

Propagation of cylindrical shock wave in a self-gravitating perfect gas under the influence of axial magnetic field using Lie group of transformation method is investigated. The flow is considered to be isothermal. Density and magnetic field are assumed to be varying in the undisturbed medium. Two different cases of solutions are brought out by the arbitrary constants appearing in the expressions of infinitesimals of local Lie group of transformations. One is with a power law shock path and the other one is with an exponential law shock path. Numerical solutions are obtained for both the cases of power law and exponential law shock paths. The effects of variation in Alfven-Mach number, gravitational parameter and ambient density variation index for power law shock path and effects of variation in Alfven-Mach number, gravitational parameter and ambient magnetic field variation index on the flow variables in the case of exponential law shock path are studied. Also the effects of increase in value of gravitational parameter and in the strength of ambient magnetic field on the shock strength are investigated. The increase in value of Alfven-Mach number leads to the increase in the density ratio which infers to the decrease in shock strength.

Using the characteristics of the governing quasi-linear system as the referencing coordinate system in the presence of a transverse magnetic field, the evolution of acceleration waves in a non-ideal relaxing gas has been examined along its characteristic path. It is demonstrated that a linear solution in the characteristic plane can behave non-linearly in the physical plane. We have determined the critical amplitude of the initial disturbance; if the initial amplitude of the compressive disturbance is greater than the critical value, the disturbance must culminate into a shock wave, while if it is less than this value, the disturbance will decay, and no shock formation will happen. We establish the criteria for shock generation and the transport equation that governs the development of weak shock waves. Acceleration waves having planar and cylindrical symmetry are analyzed as their steepening, or flattening is investigated as a function of the non-idealness parameter, relaxation parameter, adiabatic index, and magnetic field strength parameter. In both the planar and cylindrical symmetries, the shock formation process is slowed by increasing the relaxation parameter as well as the magnetic field parameter but accelerated by non-idealness and the adiabatic index. In the ideal gas case with adiabatic exponent γ = 2 , the magnetic field has no effect on the steepening or flattening of the wavefront in both the planar and cylindrical symmetries.

The propagation of a cylindrical shock wave in rotating ideal gas under adiabatic flow condition is investigated using Lie group of transformation method. Both the cases of shock without magnetic field and under the influence of axial magnetic field are considered separately. The density and azimuthal fluid velocity in the case of shock without magnetic field and density, azimuthal fluid velocity and axial magnetic field in the case of shock under the influence of magnetic field are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the local Lie group of transformations bring about two different cases of solutions, i.e. with a power law and exponential law shock paths. Exact solutions are obtained in the case of power law shock path for both the cases of cylindrical shock with and without magnetic field. It is not possible to obtain the exact solution in the case of exponential law shock path. In this case, the numerical solutions can be obtained by using the respective boundary conditions. Distribution of gasdynamical and magnetogasdynamical flow quantities are illustrated through figures.