On the evolution of weak discontinuities in radiative magnetogasdynamics

Acta Astronautica 68 (2011) 16–21

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On the evolution of weak discontinuities in radiative magnetogasdynamics L.P. Singh, Akmal Husain n, M. Singh Department of Applied Mathematics, Institute of Technology and DST-CIMS, Banaras Hindu University, Varanasi 221005, India

a r t i c l e i n f o

abstract

Article history: Received 15 May 2010 Received in revised form 25 June 2010 Accepted 29 June 2010 Available online 10 August 2010

A method of wavefront analysis is used to analyze the formation of shock waves in a two-dimensional steady supersonic flow past plane and axisymmetric bodies in radiative magnetogasdynamics. The gas is assumed to be perfectly conducting and to be permeated by a magnetic field orthogonal to the trajectories of the gas particles. The medium is taken to be sufficiently hot for the effects of thermal radiation to be significant, which is treated by the optically thin approximation to the radiative transfer equation. Transport equations, which lead to the determination of the distance at which the first characteristic could intersect with a successive one and also to conditions, which insure that no shock will ever evolve on the wave front, are derived. The effect of upstream flow Mach number and the magnetic field strength on the behavior of the wavefront are examined. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Wavefront analysis Weak discontinuity Magnetogasdynamics Thermal radiation

1. Introduction Discontinuity waves, also known as shock waves, acceleration waves and weak waves are characterized by discontinuity in the normal derivative of the flow variable rather than the variable itself. They form an important class of solution of nonlinear hyperbolic systems. Therefore, for nonlinear systems, the analysis of these waves has been the subject of great interest both from mathematical and physical point of view, due to its application in a variety of fields such as astrophysics, nuclear science, geophysics, plasma physics and interstellar gas masses. The analysis of possible steepening of compression waves leading to shock waves has received considerable attention in the literature with the shock formation distance being used as an important parameter characterizing the relative importance of convective nonlinear steepening and dissipative flattening and setting a limit for the use of certain approximate theories. The pioneering works concerning the problems relating to

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Corresponding author. Tel.: + 91 9452060150. E-mail address: [email protected] (A. Husain).

0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.06.047

wave propagation and some considerations of the nonlinear effects have been studied by a number of investigators [1–4] in various gas dynamic regimes. In classical gasdynamics, the transfer of energy through radiation is usually neglected. However, on account of the high temperature that prevails in many phenomena, it is of interest to consider the effects of thermal radiation in gasdynamics. A further contribution towards the study of thermal radiation effects on the propagation of small disturbances in gaseous flows under steady and unsteady conditions have been discussed previously, in particular, by Pai [5], Penner and Olfe [6], Lick [7], Long and Vincenti [8], Schmitt [9], Singh et al. [10] and Singh and Sharma [11] among others. Also, a number of investigations, yielding experimental and numerical description of deflagration to detonation transition in gas mixtures, have been discussed by several authors such as Oppenheim and Soloukhin [12], Smirnov and Tyurnikov [13] and Smirnov and Panfilov [14]. Smirnov and Tyurnikov [13] used the experimental method of nonintrusive diagnostics of the process to investigate the transition of deflagration to detonation in gaseous mixtures with exothermic chemical reaction, while Smirnov and Panfilov [14] made a numerical

L.P. Singh et al. / Acta Astronautica 68 (2011) 16–21

investigation of the process of deflagration to detonation transition in a combustible gas mixture. Since at high temperatures, a gas is likely to be partially ionized, electromagnetic effects may also be significant. A complete analysis of such a problem should therefore consist of the study of the gasdynamic flow and of the electromagnetic and radiation fields simultaneously. Since the formulation of the problem in radiative magnetogasdynamics involves greater complexities, several investigations yielding qualitative description of the flow field have been performed by people with aeronautical interest, using a number of simplifications concerning the gas properties and the boundary walls-producing wave like disturbances. A number of problems relating to wave propagation with radiative effects and their extension to the magnetogasdynamic regime within the context of nonlinear breaking of wave-fronts and its analysis have been carried out by Jeffrey [15], Lustman and Geffen [16], Menon and Sharma [17], Shyam et al. [18], Sharma [19,20] and Sharma et al. [21]. Menon and Sharma [17] analyzed the influence of magnetic field on the process of steepening and flattening of the characteristic wave-fronts. Sharma et al. [21] calculated the shock formation distance in a two-dimensional steady supersonic flow over a concave corner in radiative magnetogasdynamics. In the present study, we employ the method of wavefront analysis to investigate the formation of shock waves in a two-dimensional steady supersonic flow past plane and axisymmetric bodies in radiative magnetogasdynamics. The plasma is assumed to be an ideal gas with infinite electrical conductivity and to be permeated by a magnetic field orthogonal to the trajectories of the gas particles. In case of plane flows, the magnetic field is orthogonal to the plane of the flow, whereas in case of axisymmetric flows, the magnetic lines of force are concentric circles with centers on the axis of symmetry. The medium is taken to be sufficiently hot for the effects of thermal radiation to be significant, which are treated by optically thin approximation to the radiative transfer equation. Transport equations are derived which lead to the determination of the shock formation distance, and also to conditions which insure that no shock will ever evolve on the wave front. The influence of the magnetic field strength and upstream flow Mach number in the presence of thermal radiation on the shock formation distance is assessed. Here, our study concerns, the formation of shock waves when the characteristics begin to coalesce, i.e., at which the first characteristic could intersect the successive one.

17

ruux þ rvuy þðpx þ hx Þ ¼ 0,

ð2Þ

ruvx þ rvvy þðpy þ hy Þ ¼ 0,

ð3Þ

uhx þ vhy þ 2hðux þ vy þ mv=yÞ ¼ 0,

ð4Þ

upx þvpy a2 ðurx þvry Þ þ ðg1ÞQ ¼ 0,

ð5Þ

where r is the gas density, p the pressure, u and v the velocity components directed along the x and y axes, respectively, a the speed of sound given by a2 ¼ gp=r with g as the adiabatic index, h ¼ mH2 =2 is the magnetic pressure with H the transverse magnetic field and m the magnetic permeability. Q is the rate of energy loss by the gas per unit volume through radiation and is given by Q ¼ 4ksðT 4 Tb4 Þ,

ð6Þ

where k is the Planck mean absorption coefficient depending on the density r and temperature T of the gas, s is Stefan–Boltzmann constant and Tb is uniform temperature of the body along which the flow is envisaged. The additional energy gains 4ksTb4 have been added to the energy loss 4ksT4 due to the radiating gas so that infinite optically thin gas has no radiating boundary. The boundary wall with constant temperature Tb has no loss or gain of the energy from the radiating gas. The letter subscripts denote partial differentiation unless stated otherwise. Here, m is a constant which takes values 0 and 1 for plane and axisymmetric flows, respectively. 3. Characteristic formulation Using the summation convention on the repeated indices; the governing Eqs. (1)–(5) can be written as @U i @U i þ Aij þ Fi ¼ 0 @x @y

i, j ¼ 1,2,3,4,5,

ð7Þ

where the column vectors ðU i Þ51 and ðF i Þ51 are defined as 2 3 mruv ðg1ÞQ þ 6 y 7 u 6 7 2 3 6 mv 7 r ð g 1ÞQ 6 7 2  c  6 6u 7 7 6 6 7 7 y r 6 6 7 7 1 6 6 7 7, U ¼ 6 v 7, F ¼ 2 2 6 0 7 ðu c Þ 6 mruv 6 7 7 ð g 1ÞQ 2 2 2 6 4p 5 7 a ðu þ b Þ 6 y 7 u 6 7 h 6 7 4 mruv 2 ðg1ÞQ 2 5 b þ b y u ð8Þ

2. Basic equations

ij

and the non-zero components of the matrix (A )5  5 are as follows:

The governing equations for two-dimensional steady supersonic flow in radiative magnetogasdynamics, where the effects of thermal radiation are treated by the optically thin approximation to the radiative transfer equation and applied magnetic field is orthogonal to the trajectories of the gas particles with infinite electrical conductivity can be written down in familiar form [18,20] urx þ vry þ rðux þ vy þmv=yÞ ¼ 0,

ð1Þ

rv , ðu2 c2 Þ ru v , A13 ¼ 2 2 , A14 ¼ A15 ¼ ðu c Þ uðu2 c2 Þ 2 uv c A22 ¼ 2 2 , A23 ¼  2 2 , ðu c Þ ðu c Þ v 1 , , A34 ¼ A35 ¼ A24 ¼ A25 ¼  ru rðu2 c2 Þ

A11 ¼ A33 ¼

v , u

A12 ¼ 

18

L.P. Singh et al. / Acta Astronautica 68 (2011) 16–21

rva2 rua2 , A43 ¼ 2 2 , ðu2 c2 Þ ðu c Þ a2 v vðu2 b2 Þ A45 ¼ , A44 ¼ , 2 2 uðu c Þ uðu2 c2 Þ rvb2 rub2 A52 ¼  2 2 , A53 ¼ 2 2 , ðu c Þ ðu c Þ b2 v vðu2 a2 Þ A54 ¼ , A55 ¼ : 2 2 uðu c Þ uðu2 c2 Þ A42 ¼ 

ð9Þ

The matrix Aij has the eigenvalues

l

ð1,2Þ

uv 7c2 ½ðM 2 =eÞ11=2 ¼ , u2 c2

ð3,4,5Þ

l

lð1Þ lðiÞ lð1Þ x LðiÞ Uy0 þ ð1Þ ðiÞ xx LðiÞ F ¼ 0, ð1Þ ðiÞ x l l l l

LðiÞ Ux þ

v ¼ , u

Lð3Þ ¼ ½1, 0, 0, ð1=a2 Þ, 0, Lð4Þ ¼ ½0, 1, v=u, e=ru, 0, Lð5Þ ¼ ½0, 0, 0, ð1eÞ, 0:

where xx ¼ 1=xx is the Jacobian of the transformation. Across the wavefront x = 0, U and Uy0 are continuous and have their subscripts-0 values, whilst Ux and xx are discontinuous. On using Eq. (11) and the flow conditions ahead, Eq. (13) at the rear side of x = 0 for i= 2, 3, 4 and 5 becomes

rx ¼ ð10Þ

with the corresponding left eigenvectors h i Lð1Þ ¼ 0, 1, u=v, ½ðM2 =eÞ11=2 =rv, 0 , h i Lð2Þ ¼ 0, 1, u=v, ½ðM2 =eÞ11=2 =rv, 0 ,

1 p , a0 2 x

vx ¼

Here, M= q/a with q ¼ ðu2 þ v2 Þ1=2 , is the upstream flow Mach number, e ¼ 1 þðb2 =a2 Þ is the Alfve´n number and c ¼ ða2 þb2 Þ1=2 is the magnetosonic speed with b ¼ ð2h=rÞ1=2 as the Alfve´n speed. It is obvious from Eq. (11) that the system (7) possesses, except along the stream line on which l ¼ v=u, two families of characterð1,2Þ istics along which dy=dx ¼ l ; these characteristics represent the propagation of waves in opposite directions with characteristic speeds l(1,2). It is also evident that these characteristic velocities are real if and only if, the flow is supersonic, with M 4 ðeÞ1=2 ; for M o ðeÞ1=2 , the speed of propagation of the disturbance becomes imaginary and the wave propagation phenomenon ceases to exist.

4. Evolution of transport equations for weak discontinuities Let us assume that l(1) describes the initial wavefront trace, x(x, y)= 0, which passes through the point (x0, y0). The medium ahead of the initial wavefront is assumed to have a uniform temperature T0 = Tb and a uniform velocity u0 in the x direction with v0 =0. In the foregoing analysis, we shall use the suffix 0 to denote a quantity in the region ahead of the wavefront x =0. Now, we derive the transport equations for the jump discontinuities in U as they move along the wavefront x = 0. We introduce new curvilinear co-ordinates x, y0 defined as (see Jeffrey [2]) ) xx þ lð1Þ xy ¼ 0 ð12Þ xðx,y0 Þ ¼ xx0, and y= y0 . Then x has the required coordinate property that x is positive (negative) behind (ahead of) the leading characteristic on which x = 0. In terms of these new co-ordinates, Eq. (7) on premultiplying by L(i) reduces to the form

1

ð14Þ

e 0 px ,

ð15Þ

½ðM0 2 =e0 Þ11=2 px , r0 u0

ð16Þ

ux ¼ 

ð11Þ

ð13Þ

r0 u0

hx ¼ ðe0 1Þpx :

ð17Þ

We set i=1 in Eq. (13), differentiate the resulting equation with respect to x, and then evaluating it at the rear of x =0, we obtain c0 2 ½ðM02 =e0 Þ11=2 pxy0 þ r0 u0 c0 2 vxy0 þ

mr0 u0 2 a0 vx yu

2

þ

ðg1Þðu0 2 b0 Þ Qx ¼ 0: u0

ð18Þ

In view of equation of state p= rRT, Eq. (6) on differentiating with respect to x and evaluating it at the rear of x =0 yields Qx ¼

16ðg1ÞksTb 4 px : r0 a0 2

ð19Þ

Plugging in Eqs. (16) and (19), Eq. (18) can be reduced to the form   m þ LY0 e0 1 px ¼ 0, ð20Þ pxy0 þ 0 2y 40 is a measure of importance of thermal where L ¼ 8ðg1Þk d radiation with d ¼ r0 a0 3 =ðg1ÞsTb 4 , the Boltzmann number, representing the rate of convective energy flux and Y0 ¼ ðM0 2 e0 þ1Þ=M0 ½ðM02 =e0 Þ11=2 is constant. Also, along x = constant, we obtain xy0 ¼

u2 c2 uv þ c2 ½ðM0 2 =e0 Þ11=2

:

ð21Þ

Differentiating Eq. (21) with respect to x and evaluated at the rear of wavefront x =0, yields xxyu ¼ 

  ½f1 þ ðg=e0 ÞgM0 2 þ 2ð1e0 Þ y0 m=2e0 yu 2r0 a0 c0 ðM0 2 e0 Þ1=2

expfLz0 ðy0 yuÞgpx0 ,

ð22Þ

where z0 ¼ Y0 =e0 . Eqs. (20) and (22) are required transport equations for the discontinuities px and xx, which shall be used to study the behavior of waves in the disturbed region.

L.P. Singh et al. / Acta Astronautica 68 (2011) 16–21

5. Nonlinear steepening of waves

19

y Wavefront ξ=0

0

Integrating Eq. (20), with respect to y , we obtain  m=2e0   y0 exp Lz0 ðy0 yuÞ px0 , ð23Þ px ¼ y0

ξ <0 ξ >0

where px0 ¼ lim px , taken along x = 0. 0 y -y0

On substituting Eq. (23) into Eq. (22) and then integrating we get xx ¼ 1

½fðg=e0 Þ þ1gM02 þ 2ð1e0 Þ 2r0 a0 c0 ðM02 e0 Þ1=2 Zy

ðy0 Þm=2e0 eLz0 y0 px0

wm=2e0 expfLz0 wgdw,



x0

M0 > 1

ð24Þ

y0

where we have used the fact that xx0 =xx9x = 0  =xx9x = 0 + =1; this follows from Eq. (12). Let y= Y(x) be the equation of the body contour with tangent, being parallel to the velocity of the stream line, at the leading body edge. We, thus, have dy=dx ¼ v=u, which on differentiating with respect to x and evaluating it at the rear of x =0 yields 00

vx0 ¼ u0 Y0 ,

ð25Þ

y0

x

0

Fig. 1. Flow field and convergence of characteristic for a plane and axisymmetric supersonic flow.

00

where Y0 is the body curvature at the tip. With the help of Eqs. (16) and (25), Eq. (24) may be cast in the following form xx ¼ 1

½fðg=e0 Þ þ1gM0 2 þ 2ð1e0 ÞM0 2 2

2ðM0 e0 Þ Zy



1=2

wm=2e0 expfLz0 wgdw:

ðy0 Þm=2e0 eLz0 y0 Y000

ð26Þ

y0

The left hand side of Eq. (26) is the Jacobian of coordinate transformation in the region immediately behind x = 0, therefore for some y =yw this Jacobian vanishes, the neighboring characteristics of the family x = constant must intersect on the wavefront x = 0 and a strong discontinuity known as shock wave then occurs in the solution vector U. This will be the case if Ux is finite at y= yw as xx =0, for then, just behind the wavefront x = 0, Ux ¼ Ux =xx becomes infinite; this describes the phenomenon of the steepening of the wavefront. A detailed analysis of result Eq. (26) for plane (m= 0) and axisymmetric (m= 1) flow configuration is presented in the following section. 6. Results and discussion In this section, we shall deal with a supersonic flow past a plane beak (m= 0) and a sharp edged ring (m= 1). The described phenomenon is sketched in Fig. 1. (i) Plane beak case (m =0) In case of a plane beak with body contour y= Yb(x), the initial disturbance is released by a sharp edge of the contour with a vanishing small initial tangent (beak) and therefore Eq. (26) reduces to xx ¼ 1

Yb00 ð0Þ 

c

  1exp Lz0 ðyy0 Þ ,

ð27Þ

where c ¼ 2Lz0 ðM0 2 e0 Þ

hn

g e0

i1  o þ 1 M02 þ 2ð1e0 Þ M0 2 4 0,

and Yb00 (0) is the radius of curvature of the body shape at the tip, where the body contour begins to bend. As discussed earlier, the formation of shock is characterized by the vanishing of the Jacobian xx, i.e., when the characteristics begin to coalesce. Since L 40, it is evident from Eq. (27) that the Jacobian will vanish on the leading wavefront for y4y0 only when Yb00 (0)4 0 (which corresponds to the situation when the body shape has a compressive corner at x = 0) with Yb00 (0) 4 c. For Yb00 (0)r c, the Jacobian remains positive for finite y4y0 and consequently a shock will not form on the leading wavefront. Thus, the parameter c represents a critical level such that when this level is exceeded by the radius of curvature Yb00 (0) at the body tip, a shock will form at a finite distance away from the body. This is in contrast with the corresponding case of an ideal gas in the absence of the thermal radiation, where one always finds a shock after a finite length of run, no matter how small an initial body curvature may be. It may be recalled that at the wavehead x = 0, vx and vx are related according to vx ¼ vx =xx ; it is therefore immaterial whether we seek an expression for vx or vx at the wavehead. Since vx has a slightly more direct physical interpretation, we shall opt to work in terms of that quantity and note from Eqs. (14), (19) and (27) that for the plane beak (m =0) case vx ¼

a0 M0 Yb00 ð0Þ½expfLz0 ðyy0 Þg : 1fYb00 ð0Þ=cg½1expfLz0 ðyy0 Þg

ð28Þ

The condition for the growth and decay behavior of the wave can be derived from Eq. (28). It is evident from Eq. (28) that for Yb00 (0) is positive and 9Yb00 (0)94 c, a shock will form and the corresponding shock formation distance,

20

L.P. Singh et al. / Acta Astronautica 68 (2011) 16–21

1.8 2.2 ε 0 = 1.4

yω 1.6

2.0

2

M0 = 2.5

yω 1.8

1.4

2

M0 = 1.5

1.6

ε 0 = 1.2

2

M0 = 3 2 M0

=2

1.4 1.2 1.2 1.0

1.0 1.5

2.0

2.5

3.0

3.5

4.0

4.5

1.0

5.0

1.5

2.0

M0

2.5 ε0

3.0

Fig. 2. Effect of upstream flow Mach number (M0) on shock formation distance (yo) for g = 1.67 (solid line) and g =1.4 (dotted line).

Fig. 3. Effect of magnetic field strength (e0) on shock formation distance (yo) for g = 1.67 (solid line) and g = 1.4 (dotted line).

y= yw, is given by ( 00 ) Yb ð0Þ 1 , log yw ¼ y0 þ Lz0 Yb00 ð0Þc

corresponds to nonmagnetic case, the analysis presented in [10] is recovered. Physically, we can say that the dissipative mechanism due to presence of magnetic field causes to slow down the formation of shock wave as compared to what it would be in the non-magnetic case. Further, an increase in the value of g causes to increase the shock formation distance in a radiative magnetogasdynamic flow. Also, when Yb00 (0)o0, which corresponds to the situation when the body shape has an expansive corner at x =0, then for 9Yb00 (0)9b L, Eq. (28) given that 

  1 g þ 1 M0 2 þ 2ð1e0 Þ M0 2 vx ¼ 2u0 Lz0 ðM0 2 e0 Þ

ð29Þ

and the denominator of Eq. (28) becomes zero, whereas numerator remains finite, i.e., the velocity gradient at the wavehead becomes unbounded at y= yw, thus signifying the steepening of the wave into a shock wave; the coincidence of this behavior with the vanishing of the Jacobian xx is clear from Eq. (29). For the case Yb00 (0) r c, the wave is still compressive, but the steepening of the velocity gradient does not occur. On the contrary vx either diminishes out along the wavehead or it remains stationary according as Yb00 (0)o c or Yb00 (0) = c, respectively, and no shock will ever form on the leading wavehead x = 0. The shock formation distance specified by Eq. (29) is computed for various values of e0, the parameter of magnetic field strength; g the specific heat ratio; and M0 the upstream flow Mach number. For computational purpose y0 is taken as 1. The typical values of parameters involved in computation are taken as [21]

e0 ¼ 1:2,1:4, g ¼ 1:4,1:67, M02 ¼ 1:5,2:0,2:5,3:0, Yb00 ð0Þ ¼ 0:4: Also from Eq. (28) and Figs. (2) and (3), it is remarked that for M0  e0, the shock formation distance yw, is given by   4 M 1 0 , yw  y0  00  2 e0 Yb ð0Þ e0 þðg2Þe0 þ2 whereas for M0 b e0   1 2 g yw  00 þ 1 M0 2 þ 2ð1e0 Þ , Yb ð0Þ e0 this is a decreasing function of M0, which exhibits that an increase in the value of M0 causes an early shock formation; the corresponding situation is illustrated through Figs. 2 and 3. It is also evident that an increase in the value of e0, the parameter of magnetic field strength, in the presence of thermal radiation, enhances the shock formation distance. However, for e0 = 1, which

e0



expfLz0 ðyy0 Þg , ½1expfLz0 ðyy0 Þg

which is an expression for the velocity gradient at the head of the Prandtl–Meyer expansion flow. (ii) Axisymmetric case (m= 1) Here, we consider a ring shaped body y= Yw(x) with a sharp edged inlet releasing the initial disturbance, which runs both inwards and outwards along characteristic lines. Now, putting m= 1, Eq. (26) can be written as h i fðg=e0 Þ þ1gM0 2 þ 2ð1e0 Þ M0 2 xx ¼ 1 ðy0 Þ1=2e0 eLz0 y0 Yw00 f, 2ðM0 2 e0 Þ1=2 ð30Þ where

f¼

Zy

w1=2e0 expfLz0 wgdw:

ð31Þ

y0

From Eq. (30), it is evident that for y 4y0 the entity within the square bracket is always positive and less than unity. Therefore, the left hand side of Eq. (30) will vanish, leading to the formation of shock, provided Yw00 (0) is positive and exceeds the critical value f  1; when 1 Yw00 ð0Þ r f the Jacobian xx is always positive, and hence no shock wave will ever form on the leading wavehead. From Eqs. (29) and (30), it is obvious that the shock

L.P. Singh et al. / Acta Astronautica 68 (2011) 16–21

formation distance yw depends upon the upstream flow Mach number initial body curvature Z being 1 M0, the either Yb00 ð0Þ or ½Yw00 ð0Þ1 , the parameter of magnetic field strength e0 and the parameter L, which represents the importance of thermal radiation. Since @yw =@Z, @yw =@L and @yw =@e0 are positive, which means that an increase in body curvature or e0, parameter of magnetic field strength enhances the shock formation distance, while decreasing values of Boltzmann number causes to increase the shock formation distance in the presence of magnetic field. It is also evident from Eqs. (29) and (30) that the presence of magnetic field causes an early shock formation in an axisymmetric case as compared to plane flow under the same initial conditions. 7. Concluding remarks A problem of propagation of weak shock wave in a two-dimensional steady supersonic flow past plane and axisymmetric bodies in radiative magnetogasdynamics is theoretically investigated. The effect of thermal radiation is treated by the optically thin approximation. Using the method of wavefront analysis, transport equations are derived which lead to the determination of the distance at which the first characteristic could intersect with a successive one and also to conditions which insure that no shock will ever evolve on the wave front. The effect of an increase in the upstream flow Mach number and the magnetic field strength is to enhance the shock formation distance in the presence of radiative heat transfer, which is illustrated through Figs. 2 and 3.

Acknowledgements The authors are thankful to the referees for making certain points more explicit. The second author acknowledges the financial support from DST-CIMS, Banaras Hindu University, Varanasi, India under the JRF scheme.

21

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