Journal of Applied Mathematics

Volume 2018, Article ID 9701268, 14 pages

https://doi.org/10.1155/2018/9701268

## Study of Two-Sided Similarity Methods Using a Radiation “Switch on” Imploding Shock in a Magnetic Field

Systems Science Research Group, P.O. Box 4051 Walnut Creek CA., 94596, USA

Correspondence should be addressed to J. R. A. J. NiCastro; ten.knilhtrae@cinjarj

Received 4 December 2017; Revised 29 March 2018; Accepted 28 May 2018; Published 24 June 2018

Academic Editor: Syed Abdul Mohiuddine

Copyright © 2018 J. R. A. J. NiCastro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper explores aspects of two-sided similarity modeling using cylindrical geometry for radiating shock waves embedded in a medium with a magnetic field. Two-sided similarity solution techniques may be used to link states influenced by long range near instantaneous fields that continually modify the pre- and postshock zones. Emergent radiation scaling relations are immediately available from consistent homologies. For both small angle and large angle measurements, an approximate analytic technique in the vicinity of luminous fronts together with the high symmetry implications delineated in Lemma provides direct access to the homology parameters. The parameters obtained using this process can augment the constraint relations and contribute to establishing relevant similarity homologies.

#### 1. Introduction

In nonlinear materials, strong compression or shock phenomenon may result from an initially sufficiently strong impulse, mechanical or otherwise, propagating into the fluid. Similarity techniques have been used to study these problems involving radiation effects. Evidently there has been an interest in radiation driven cylindrical shocks in plasma systems. [1] Vishwakarma and Pandey have studied the propagation of shock waves under the action of monochromatic radiation in nonideal gas using a van deWaals model. [2] Nath and Sahu have studied propagation in an axisymmetric system involving a magnetic field and monochromatic radiation. Further studies have found solutions to spherical plasma systems. [3] NiCastro, in an early work using similarity analysis, studied the possibility of initiating nuclear effects in an electromagnetically driven quasispherical convergent geometry dominated by thermal radiation. [4] Singh has found a solution for the self-similar flow behind a radiative spherical shock wave. [5] Singh and Vishwakarma have studied propagation of spherical waves with radiation heat flux in a dusty gas, a mixture of gas, and small particles and include the effect of an exponential density gradient. [6] D. Mihalas and B. W. Mihalas have prepared a summary bibliography of work on radiation hydrodynamics that spans a history of work and deals with a diversity of problems. [7] Nath, Dutta, and Pathak have obtained an exact solution for propagating shocks including magnetic field, radiative heat flux, and gravitating effects.

When one departs from purely fluid mechanics variables to include radiation (photons and neutrons) and electromagnetic field effects, pre- and postshock regions are coupled by these fields which have long range near instantaneous effects. Both regions will inevitably be modified by these fields during the shock development. Similarity techniques applied to the pre- and postshock regions, and two-sided similarity seems to suggest itself in approaching these problems. More generally, if problems require that the domain be divided by contact surfaces into “n” regions with “p” regions in a constant state, then “n-p” similarity models may be warranted. Some form of conservation equations would be expected to be preserved across the “n-1” contact surfaces.

Similarity equations are based on a symmetrization of the original physics equations and as such imply a strong underlying assumption about the nature of the process. There is no a priori justification for these symmetry assumptions. Assumptions must be justified through an iterative process involving experimental observation and if necessary homology redesign. If the symmetry properties discussed in Lemma 1 are valid, an analytic process, that is an extension of a perspective of [8] Auluck and Tandon, can be used to address this issue using “emergent radiation observables” in the vicinity of a luminous front. The objective of this paper is to develop this process.

Additionally, radiation phenomenon while not initially dominant may become so as the shock onset goes through its development phase and becomes a developed shock front. The initial conditions are masked by what is essentially a radiation switch on (RSO) phenomenon. A radiative “precursor” and a “postcursor” detach the radiating shock from information (initial conditions) about its formation. Indeed it may easily be the case that many different initial conditions lead to the same or indistinguishably similar RSO results! While it is convenient to use assigned values on boundaries to solve numerical problems, the results are academic if the boundary values do not evolve reflecting the energy released in the pre- and postshock zones (fissions and fusion events or even exothermic chemical explosives). Also it would be suspect to define the solution by initial conditions when the dominant phenomenology determining the homology does not develop until the shock is formed. Consider the example case of a tamper plate (contact surface) moving into a latent explosive material. A compression wave may build into a shock and a detonation front is formed. The fluid and thermodynamic parameters at the plate and the movement of the tamper must reflect the evolutionary release of energy occurring behind the detonation front.

These comments suggest in many cases the shock may have to be considered as a standalone event. Complex problems may require dividing the solution domain into more than two regions partitioned by appropriate contact surfaces. In these cases models should be built from the conservation laws and shock jump conditions meshed with measurements and observations. Radiation observables at luminous fronts may play an important role in this process.

Historically the analysis of various aspects of the nature of a shock front has undergone considerable discussion in the literature. It has been suggested by [9] Germain that, to some degree, there appears to have been two avenues of shock wave studies.

In the first case, the continuum physics equations are used to describe the asymptotic character of the shock transition which is represented as a discontinuity across which the conservation laws must apply. Rankine and Hugoniot (R-H) produced the classic result formulating the shock jump conditions from the conservation laws in the shock moving frame which are summarized in [10] Friedrich and Courant. Implosions studied by [11] Guderly and explosions studied by [12] Taylor have used similarity techniques in the strong shock limit. Similarity solutions in the strong shock limit of a magneto hydrodynamic shock have also received some attention by [13] Cole and Greifinger and [14] Greenspan. It would appear that these early works and others might have been motivated by the study of large weapons systems. The “Strong Shock” model assumption, acceptable in modeling strong explosions, belies the issue of coupling the two regions using any two-sided similarity methods.

In the second case, attention is concentrated on the shock transition. Important are the effects of radiation and the dissipative mean free paths, among others, on the character of the transition. Numerous research papers have developed these issues. Some work by [15] Heaslet and Baldwin and [16] Clarke suggests that radiation may smooth the shock structure under certain flow conditions. [17] Mitchner and Vinokur suggest the presence of a magnetic field might to some extent countervail this smoothing effects. [18] Imshennik has discussed shock wave structure in the case the radiation energy and pressure are significant and may exceed the kinetic pressure and energy. These conditions are likely to prevail in stars. One might expect they could also prevail in the early stages of a fusion explosion.

#### 2. Problem

The predicate for using two-sided similarity techniques is that the pre- and postshock regions are tied together by some form of radiation exchange. In this paper, we develop the discussion with thermal radiation but neutron transport and electromagnetic effects are also issues. For example, a neutron “switch on” wave is likely to be accompanied by significant thermal radiation. A neutron “switch on” wave may be characterized as a wave initiated in a medium that has the potentiality of a self-supporting process of fission or fusion or both leading to the emission of neutron radiation. When the wave steepens into a shock or compression wave and either the number of MFP or the “*ρ*x”product of the fluid reaches a critical value, the process may become self-sustaining or may simply be a dissipating pulse. The neutron and thermal radiation fronts are likely to coexist and require new physical considerations. These adjunct processes distinguish these problems from purely mechanical shock problems.

In this paper we use imploding radiating shock in a magnetic field as a vehicle to study some aspects of two-sided similarity problem formulations. The same ideas used in this model can be used in more complex problems, if the solution space is divided by “(n-1)” contact surfaces into “n” zones and “p” zones are a constant states.

Assume a cylindrical column of gas is permeated by a background magnetic field. The cylinder implodes. Conventional explosives would be one of several means of accomplishing this. In laboratory plasmas, external pulsed fields initiate snowplow and shock wave processes. A compression wave is initiated and may build into an isothermal shock. The gas is ionized in front of and behind the shock front by the radiation cooling of the shock zone. The resulting high conductivity can lead to what is referred to as “frozen in magnetic field.” The fluid, radiation, and magnetic field sensitively depend on the homologies and the initial values of the dependent variables. It is of importance to exploit the high symmetry characteristics, to be discussed in Lemma 1, of a presumed similarity solution to obtain measurements that would validate or guide the values of homology parameters.

#### 3. Basic Model Equations

The descriptive equations can be tailored depending on the specific engineering problem. In this discussion we will use basic conservation laws. In the text by [19] Sutton and Sherman, these equations are sometimes referred to as the MHD approximation. It is frequently useful to gain insight into the nature of the physics with a less detailed model, especially if there are measurements available to guide the model development. In cylindrical coordinates the equations take the form Equations (1)-(4) are the continuity, momentum, and energy and magnetic diffusion, respectively. The symbols *ρ*, v, P, B, j, F, and E are the density, velocity, kinetic pressure, magnetic field, current, radiation flux, and internal energy. In general, the radiation flux may be of any kind, photons or neutrons, for example. Also, the pressure may generally include hydrodynamic, magnetic, and radiation contributions. We have excluded Ohmic heating, assuming the conductivity will be sufficiently large in the region of interest to justify this. It is unlikely this will be the case in all phases of an experiment or phenomenon under study. If thermal gradients are large, then (4) needs to be reassessed.

Generally, P=P(*ρ*,S). Since F and ΔF≠0, ΔS≠0. We will use the ideal gas equation of state functional form together with an internal energy proportional to the temperature. So P=R*ρ*T and E=c_{v}T serve in this analysis. This is called the polytropic model. We view these equations and the specific heat ratio as an heuristic. If S=constant, the pressure would be proportional to the density to a constant power *γ*, the specific heat ratio. Also, the ideal gas form of the constitutive equation of state does not always represent the best modeling of high temperature processes.

In some analysis, different values of *γ* are used to capture a range of behavior due to different degrees of ionization, collision frequencies, and magnetic field strength. In a “frozen in field” B/*ρ* is constant. One might be inclined to discuss a *γ*_{⊥} and *γ*_{∥} for the fluid due to the field in a collisionless or weakly collisional gas, as might occur in some laboratory or an astrophysical problem. This might be a fruitful modeling approach in such cases, but in this analysis, we are assuming a collision dominated fluid. Additionally, we will not consider contributions from the thermal conductivity or viscosity. It is important to model the detailed constitutive properties of the fluid, if one does a complete computer calculation.

These basic model equations are usually augmented by Faraday and Ampere equation (5) and generalized Ohm's Law. In this example the magnetic field is axial, the bulk fluid velocity is radial, and the induction electric field and consequent currents are azimuthal. The displacement field term in Ampere equation is ignored.Because of the geometry of the B field and its gradient, jxB in (2) can be represented as in We note that the diffusion equation (4) for the magnetic field is derived by using the generalized Ohm's Law in Ampere's equation (5), then by taking the curl of (5).While the high conductivity approximation is acceptable in a high temperature zone, ignoring gradients in the conductivity is suspect if there are strong thermal gradients. In this case (4) would have to be reassessed. Faraday's equation is necessary if the relative gradient in conductivity cannot be ignored and if flux slippage is something to be modeled. Ignoring the tensor character of the conductivity, some researchers have used a constant value or the Spitzer Harm (SH) power law approximation to capture aspects of the temperature dependence of conductivity. =*χ*T^{ζ}; *χ* ≈1.5x10^{−2}/; *ζ*=3/2. Note that depending on the structure of the plasma, ≈6-23. This covers a significant range effect.

Within a given problem it is unlikely that just one radiation functional structure will represent the entire domain of the problem. This fact was noted by F.C. Auluck and J.N. Tandon, hereafter referred to as AT, and their perspective will form the basis of the approximate calculations to follow. The Flux “F,” (3), is radiation energy transfer internal to the plasma and is likely due to many different processes. The mean free path is of significance, for example, in determining whether the plasma is optically thick, thin, or something intermediate. There are a number of recognized expressions for emergent radiation loss mechanisms that would constitute the “external observables” of the experiment or phenomenon depending on the properties of the plasma.

By “external observables” the signature a measuring instrument external to the phenomenon would record is meant. The measurements might be a superposition of the functions in (7). The “external observables” might have to be decomposed into individual contributions from different sources before using (45). This might require deconvolving the frequency distribution of the measured radiation signature.

Equations (7) are a listing of some these radiation processes along with their dependence on thermodynamic and fluid variables. It is likely that a measurement would contain a superposition of the various sources indicated in (7). In order to condense the presentation, modeling details are condensed in the scaling coefficients. The values of the coefficients depend on the particular model calculation and plasma composition. For example, for Bremsstrahlung, for a pure deuterium plasma *ρ*^{2} would be proportional to n_{d}^{2} and for a mixed deuterium-tritium plasma it would be proportional to n_{d}n_{t}. For Cyclotron radiation *ρ* is proportional to the electron number density. In both cases T is proportional to T_{e} the electron temperature. For a Black body, T is the equilibrium temperature.RF emissions, which are the result of the bulk compression (or movement) of any “embedded” axial magnetic field and the azimuthal induction field at the shock wave interface, are included. RF emission is present in a complete treatment of the energy equation and might serve as a useful diagnostic signature.

Black Body radiation flux diffusion is characterized by the gradient of the radiative energy and is a model considered in many papers. It is not the focus in this paper, but we will indicate the transformation constraints for that radiative model. The frequency averaged mass absorption or Rosseland opacity is *κ*. The mean free path, *μ*=1/* ρκ*, is frequently modeled as a product of a power of the density times a separate power of the temperature. This is not done but recognize it allows modeling flexibility and will change the homology representation of the fluid's opacity. It may be possible to model the pre- and postshock material with different mean free path functions, when using the more general representation of the mean free path, as long as the homologies cancel across the shock. The quantity ac/4 is the Stephan Boltzmann constant . This is summarized inIt is unlikely that the electron and ion kinetic temperatures and the radiation field are in thermal equilibrium in most cases. A shape factor equal to the ratio of the systems dimensions to the radiation MFP has been used in modeling radiation diffusion to compensate for this property. In addition to Bremsstrahlung, a significant contribution of the radiation in the presence of a magnetic field would likely come from Cyclotron effects. Unless one is analyzing extreme radiation environments, radiation pressure and energy density can usually be ignored in the energy and momentum equations. In those cases a more complete description of the radiative process is required.

#### 4. Solution

##### 4.1. The Overall Approach

To use (45), a solution (model) on both sides of the vicinity of the shock front needs to be obtained. This calculation is an example of the iterative process of calculation and measurement to help define relevant homologies consistent with measured radiation spectral structure. The solution involves solving or constructing a preshock state and joining the solution across the shock front to the postshock state using the shock jump conditions. We choose to illustrate the technique using analytic approximation but a detailed computer calculation is equally acceptable. However the neighborhood of the luminous front is the only region of interest to perform this analysis. The homologies of (17a), , , (17c), (17d), (17e), and (17f) reveal important aspects of the flow, if narrow angle observations about a luminous front, shock front, or contact surface can be made. The approximate analytic or computer solutions about the shock front can be used when the experimental observations about a line of constant *η* require more than a very narrow subtended observation angle.

To obtain the analytic approximation, this paper extends the suggestion and perspective of AT. In their particular calculation, they work under the assumption that the details of the radiation field are difficult to postulate a priori. This is well suited to the process that involves incorporating experimental data of the luminous front to determine the character of the radiation instead of stipulating it as an assumption. This analysis asserts that if the radiation loss is not a dominant mechanism, the homology should not determine the dynamics. Their analysis centers on the jump equations. We expand their point of view by developing simple approximate two-sided similarity representations in the vicinity of the shock front. The complex dependence on homology parameters, which is frequently lost with detailed discreet computer calculations, can be retained. Then as an operational alternative to postulating the homology of the radiation, the homologies of the external emergent radiation functions can be evaluated and compared with observation ((44a)-(46)). There are at least three approaches to consider.

(1) Assume a strong shock condition where the shock assumes limiting values and the preshock state is not strongly influential in the process. This eliminates the need for structuring the preshock state explicitly but has limited applicability. It is well known that there are substantial photon and neutron precursors that strongly influence the preshock regions in nuclear weapons effects into which the blast wave eventually propagates. Nonetheless the strong blast approach produces useful insights into the mechanical nature of the blast waves and has been successfully utilized in a number of past studies.

(2) The relevant equation can be solved with boundary point conditions that are physically reasonable, such as bounded physical variable values at a well-defined contact surface (piston) for the functions and if necessary their first derivatives. Assigning initial values at select points to obtain solutions is sometimes a necessity but it is in reality at best a convenience since discreet contact surfaces seldom exist in real situations for all the dependent variables. The solutions are more useful if the initial conditions are tied to experimental measurements. There are various techniques for measuring the plasma properties, as described by [20] Griem. Temperature and densities are among them.

(3) The preshock state being strongly influenced by near instantaneous long range field effects can be postulated (calculated), be a prepared laboratory experimental state, or be the observed result of a natural process. The state can be mathematically structured enabling wide modeling applicability. The strong blast assumption is a limiting case.

#### 5. The Transformation

Introduce the homology representation for the dependent and independent variables equations (10a)-(10b). One of the first applications of this was by [21] Boltzman in his treatment of the heat diffusion problem. Before WWII, [22] Weizacker studied similarity solutions presumable as part of the fission implosion program. While (10c) is a composite variable, the quantities, *ρ*_{λ}, T_{λ}, B_{λ}, F_{λ}, r_{λ}, and t_{λ} are, respectively, the scale density, temperature and magnetic field and radiation flux, scale length, and time. The functional structures of (10a)-(11) are not unique and have been used by many authors. The similarity transformation, while a simple mathematical process, implies a strong underlying symmetry assumption about the dependent variables. Equations (12a) are the similarity structures and the scaling relations of the composite emergent radiation functions in (7). The terms in brackets are the respective homologies.The approximate analysis in subsequent sections provides a representation for the dependent invariants in (12a), (12b), and (12c). Equations 12(a) are the radiation observables, among others, that can be used to test the similarity hypothesis.

Lemma 1. *If the problem is self-similar and if any set of functions such as B( η),T(η),υ(η), and ρ(η) are dependent invariants, then any function Ψ, (12b), of dependent invariants is a dependent invariant and a constant for each η. The physical function's explicit time and space dependence is determined solely by the composite homology.This would apply to (12a), the quantities in brackets in (12a) being the composite homology. Furthermore Γ, (12c), is a three-parameter family of functions of physical variables that is invariant for every consistent solution of (13)-(16). This follows since there are four-theta parameters and one δ constraint.There are strong rationals for physical symmetry transformation such as the Lorentz transformation. There is no a priori physical law that requires sets of physical variables to be mapped (evolve) by these automorphism symmetries. Collapsing space and time into a single independent variable η has induced these symmetries. The symmetry is a modeling assumption. If the similarity hypothesis is true for the particular problem under study, it makes possible comparisons between calculated emergent radiation characteristics and small angle measurements about a line of constant η to test the similarity hypothesis. This includes the trajectories of contact surfaces, shocks and plasma sheaths, and density time profiles. No differential equations have to be solved at this point to accomplish these comparisons, if Lemma 1 is found to apply. If large angle measurements need to be made in the vicinity of the shock line, an approximation procedure based on an extension of the AT concept could be used. This is developed further on.*

*Comment on Lemma 1*. It is likely that in real physical problems many cases would involve a weakly broken automorphism symmetry. The measured homology parameters might be close too but not precisely the calculated values from (17a), , , (17c), (17d), (17e), and (17f). This might suggest consideration of a calculation that is a perturbation approach to the similarity equations. Depending upon measurements, this could involve a perturbation to the shock path or any of the dependent invariants. The perturbation would likely break the strong internal symmetries of (12b) and (12c).

Using the transformation equations (10a), (10b), (10c), and (10d), a symmetrization of (1)-(4) is being modeled. Furthermore in reducing the PDEs to ODEs we are not modeling the full physical equations subject to boundary values that might lead to a unique solution in a strict mathematical sense. Furthermore, the process predicate (switch on waves) has detached the shock from initial condition of its formation. The notion of constructing a solution to the ODEs that satisfies some arbitrary initial value problem becomes at best moot unless there is a well-defined contact surface or line of constant *η* on which there has been measured values or observations.

With these caveats in mind, the PDEs (1)-(4) are transformed into the ODEs (13)-(16). We note that for explosion “t”” increases and implosions “t” decreases but it always remains positive. Since the fundamental equations are time translation invariant one could introduce a new time variable *τ*=-t, t < for implosions where is the implosion time.where

The continuity equation does not generate a constraint. The constraints from the momentum, energy, and magnetic diffusion are indicated below, respectively. Equation is an alternate and more restrictive form than . Equation (17f) replaces (17c) in the case one uses (8) as a radiation model. Note that *ζ*=3/2 and *ζ*=0 correspond, respectively, to SH and a constant conductivity. Equations (13)-(16) are the transformed continuity, momentum, energy, and field diffusion, respectively. Equations (17a), , , (17c), (17d), (17e), and (17f) are the constraints. Since we must match similarity structures across the shock front along a line of constant *η* further constraints may be imposed on the homologies.

Corollary 2. *For the particular class of transformations characterized by equations (10a), (10b), (10c), and (10d), the origin is a singular point. By Lemma 1, it will be the case that dependent invariant functions are multivalued at an intersection point of η lines. Hence they are not uniquely defined at this point. For this reason, the solution space should exclude a small region centered about the origin (r=0,t=0). Depending on the circumstances, it may be possible to generate similarity transformations that do not have this problem.*

#### 6. Homology Analysis

It is necessary to find a consistent set of parameters (n, m, *δ*, *ζ*, l,w,k) to satisfy the conditions of equations (17a), , , (17c), (17d), (17e), and (17f). There is no unique way of doing this. The goal is to obtain (13)-(16) solely a function of *η*. Powers of *η* to the qth power will necessarily appear in these equations. In obtaining an homology solution to transform the system, it tacitly assumes the governing constraint properties are valid throughout the entire domain. Plasmas are unlikely to be that consistent. This situation is not reflected in global similarity solution applications. This study is focused on locally approximate solutions in the vicinity of luminous contact surfaces. We consider several cases.

In both Cases 1(a) and 1(b), the conductivity is assumed to be very large so explicit solution of the entire diffusion equation can be ignored but still characterized by the conductivity homology. The parameters *ζ*,k and n are free.

*Case **1(a)*. For *ζ*=3/2, *σ* corresponds to the SH law. The parameter “m” is determined from (17a) and (17d). Using (17a) determines *δ*. From , use k+m=2w and n=2l which makes the kinetic and magnetic pressure terms completely homogeneous in *η*, i.e., = ; q_{1}=q_{2}. Flow chart (18) follows:

*Case **1(b)*. For *ζ* =0, *σ* is a constant. Then from (17d), necessarily *δ*=1/2. Equation (17a) defines “m” and as before use for l and w. Flow chart (19) followsSince the radiation is not a priori specified, for any values of k, m, n, and *δ*, a relationship exists between a and b from (17c). The flowcharts are not unique and the parameter space has many solutions. The solutions generated by the transformation of equation (10a), (10b), (10c), and (10d) are a symmetry approximation and are used to build an approximate model. Some solutions using (18) and (19) are indicated in Table 1.