Analytical solution for unsteady adiabatic and isothermal flows behind the shock wave in a rotational axisymmetric mixture of perfect gas and small solid particles

Abstract The approximate analytical solutions are obtained for adiabatic and isothermal flows behind a cylindrical shock wave in a dusty gas. A mixture of perfect gas and micro size small inert solid particles is taken as the dusty gas. The inert solid particles are distributed continuously in the mixture. It is considered that the equilibrium flow conditions are maintained. The flow variables are expanded in power series to obtain the solution of the problem. The analytical solutions are obtained for the first order approximation in both the adiabatic and isothermal cases. Also, the system of ordinary differential equations for second order approximations to the solution is obtained. The influence of an increase in the ratio of the density of the inert solid particles to the initial density of the perfect gas, the rotational parameter and the mass concentration of inert solid particles in the mixture are discussed on the flow variables for first approximation. Our first approximation to the solution corresponds to the Taylor’s solution for the creation of a blast wave by a strong explosion. A comparison is also made between the solutions for isothermal and adiabatic flows. It is investigated that the density and pressure near the line of symmetry in the case of isothermal flow become zero and hence a vacuum is formed at the axis of symmetry when the flow is isothermal. Also, it is found that an increase in the value of rotational parameter or the mass concentration of solid particles in the mixture has a decaying effect on shock wave. The present work may be used to verify the correctness of the solution obtained by self-similarity and numerical methods.

Download Full-text

Focusing of inertial particles after the shock-wave intersection point

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2007.1.016 ◽

2007 ◽

Vol 5 ◽

pp. 145-150

Author(s):

I.V. Golubkina

Keyword(s):

Shock Wave ◽

Mass Concentration ◽

Gas Flow ◽

Intersection Point ◽

Plane Shock ◽

Inertial Particles ◽

Dusty Gas ◽

Focusing Effect ◽

Aerodynamic Focusing ◽

Particle Mass Concentration

The effect of the aerodynamic focusing of inertial particles is investigated in both symmetric and non-symmetric cases of interaction of two plane shock waves in the stationary dusty-gas flow. The particle mass concentration is assumed to be small. Particle trajectories and concentration are calculated numerically with the full Lagrangian approach. A parametric study of the flow is performed in order to find the values of the governing parameters corresponding to the maximum focusing effect.

Download Full-text

Approximate analytical solution for the propagation of shock wave in a mixture of small solid particles and non-ideal gas: isothermal flow

Zeitschrift für Naturforschung A ◽

10.1515/zna-2021-0196 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Gorakh Nath

Keyword(s):

Shock Wave ◽

Analytical Solutions ◽

Order Approximation ◽

Fluid Pressure ◽

Strong Shock ◽

Cylindrical Symmetry ◽

Solid Particles ◽

Approximate Analytical Solutions ◽

Axis Of Symmetry ◽

Physical Variables

Abstract This paper presents the development of mathematical model to obtain the approximate analytical solutions for isothermal flows behind the strong shock (blast) wave in a van der Waals gas and small solid particles mixture. The small solid particles are continuously distributed in the mixture and the equilibrium conditions for flow are maintained. To derive the analytical solutions, the physical variables such as density, pressure, and velocity are expanded using perturbation method in power series. The solutions are derived in analytical form for first approximation, and for second order approximation the set of differential equations are also obtained. The effects of an increase in the problem parameters value on the physical variables are investigated for first order approximation. A comparison is also, made between the solution of cylindrical shock and spherical shock. It is found that the fluid density and fluid pressure become zero near the point or axis of symmetry in spherical or cylindrical symmetry, respectively, and therefore a vacuum is created near the point or axis of symmetry which is in tremendous conformity with the physical condition in laboratory to generate the shock wave.

Download Full-text

Newtonian flow theory for slender bodies in a dusty gas

Journal of Fluid Mechanics ◽

10.1017/s0022112081002048 ◽

1981 ◽

Vol 108 ◽

pp. 147-157 ◽

Cited By ~ 12

Author(s):

R. M. Barron ◽

J. T. Wiley

Keyword(s):

Shock Wave ◽

Gas Flow ◽

The Body ◽

Dusty Gas ◽

Newtonian Theory ◽

Wedge Surface ◽

Slender Bodies ◽

Two Phases ◽

Flow Variables ◽

Thin Wedge

Hypersonic small-disturbance theory is extended to consider the problem of dusty-gas flow past thin two-dimensional bodies. The mass fraction of suspended particles is assumed to be sufficiently large that the two-way interaction between particle phase and gas phase must be considered. The system of eight governing equations is further reduced by considering the Newtonian approximation γ → 1 andM∞→ ∞. The Newtonian theory up to second order is studied and the equations are solved for the case of a thin wedge at zero angle of attack. Expressions for the streamlines, dust-particle paths, shock-wave location and all flow variables are obtained. It is seen that the presence of the dust increases the pressure along the wedge surface and tends to bend the shock wave towards the body surface. Other effects of the interaction of the two phases are also discussed.

Download Full-text

Propagation of strong cylindrical shock wave in a self-gravitating rotational axisymmetric mixture of small solid particles and perfect gas with density varying exponentially

Acta Astronautica ◽

10.1016/j.actaastro.2019.06.016 ◽

2019 ◽

Vol 162 ◽

pp. 447-460 ◽

Cited By ~ 2

Author(s):

G. Nath

Keyword(s):

Shock Wave ◽

Solid Particles ◽

Perfect Gas ◽

Cylindrical Shock Wave

Download Full-text

Dusty Shock Waves

Applied Mechanics Reviews ◽

10.1115/1.3151872 ◽

1988 ◽

Vol 41 (11) ◽

pp. 379-437 ◽

Cited By ~ 47

Author(s):

O. Igra ◽

G. Ben-Dor

Keyword(s):

Shock Wave ◽

Shock Waves ◽

Flow Field ◽

Gaseous Medium ◽

Present Review ◽

Normal Shock ◽

Dusty Gas ◽

Perfect Gas ◽

Post Shock ◽

Dusty Shock Waves

The flow field developed behind shock waves in a pure gaseous medium is well known and documented in all gasdynamics textbooks. This is not the case when the gaseous medium is seeded with small solid particles. The present review treats various cases of shock waves propagation into a gas-dust suspension (dusty shock waves). It starts (chapter 1) with basic definitions of two-phase (gas-dust) suspensions and presents a general form of the conservation equations which govern dusty shock wave flows. In chapter two, the simple case of a steady flow of a suspension consisting of an inert dust and a perfect gas through a normal shock wave is studied. The effect of the dust presence, and of changes in its physical parameters, on the post-shock wave flow are discussed. Obviously, these discussions are limited to relatively weak shock waves (perfect gas). For stronger normal shock waves, the assumption of a perfect gas no longer holds. Therefore, in chapter three, real gas effects (ionization or dissociation) are taken into account when calculating the post-shock flow field. In chapter four, the dust chemistry is included and its effects on the post-shock flow is studied. In order to emphasize the role played by the dust chemistry, a comparison between a reactive and a similar inert suspension is presented. The case of an oblique shock wave in a dusty gas is discussed in chapter five. In all cases treated in chapters two to five the flow is steady; however, in many engineering applications this is not the case. In reality, even for the simplest case of a one-dimensional flow (normal shock wave propagation into quiescent suspension—the dusty shock tube) the shock wave attenuates and the flow field behind it is not steady. This case is treated in chapter six. The cases treated in chapters two to six deal with planar shock waves. However, all explosion generated shock waves in the atmosphere are spherical. Due to the engineering importance of this case, the post-shock flow for spherical shock waves in a dusty gas is studied, in detail, in chapter seven. It is shown in the present review that the dust presence has significant effects on the post-shock flow field. In all cases studied, a relaxation zone is developed behind the shock wave front. Throughout this zone momentum and energy exchange between the two phases of the suspension takes place. Through these interactions a new state of equilibrium is reached. The extent of the relaxation zone depends upon the dust loading ratio, the dust particle diameter, its specific heat capacity, and the dust spatial density. Due to the complexity of conducting experimental investigations with dusty shock waves, the number of published experimental results is very limited. As a result most of the present review contains numerical studies. However, in the few cases where experimental data are available, (e.g. dusty shock tube flow; see chapter six) a comparison between the numerical and experimental results is given.

Download Full-text

Propagation of Waves in a Nonideal Magnetogasdynamics with Dust Particles

Zeitschrift für Naturforschung A ◽

10.1515/zna-2019-0255 ◽

2020 ◽

Vol 75 (3) ◽

pp. 193-200 ◽

Cited By ~ 3

Author(s):

Kajal Sharma ◽

Rajan Arora ◽

Astha Chauhan ◽

Ashish Tiwari

Keyword(s):

Shock Wave ◽

Mass Concentration ◽

Numerical Approach ◽

Solid Particles ◽

Dust Particles ◽

Compatibility Conditions ◽

One Dimensional ◽

Specific Heats ◽

The One ◽

Propagation Of Waves

AbstractIn this article, we use the surface theory and compatibility conditions to describe the behaviour of wave propagation and their culmination into a shock wave in nonideal reacting gas with dust particles. The one-dimensional steepening of waves has been considered. A Bernoulli-type transport equation for the velocity gradient has been obtained. A numerical approach is used to explain the effects of van der Waals excluded volume of the medium, the ratio of specific heats, and the mass concentration of the solid particles on the shock wave.

Download Full-text

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

Zeitschrift für Naturforschung A ◽

10.1515/zna-2020-0248 ◽

2021 ◽

Vol 76 (3) ◽

pp. 265-283

Author(s):

G. Nath

Keyword(s):

Magnetic Field ◽

Shock Wave ◽

Analytical Solution ◽

Axial Magnetic Field ◽

Order Approximation ◽

Ideal Gas ◽

Field Region ◽

First Order ◽

First Order Approximation ◽

Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

Download Full-text