Assuming the occurrence of a boundary shock wave as postulated by Martin, this paper studies its structure in conducting gas in presence of a magnetic field

Prasad, Baleshwar

Verma, B. G.

English

Defence Science Journal

STRUCTURE OF A PLANE BOUNDARY SHOCK WAVE B. G. VERMA & BALESHWAR P ASAD , U.~ioersity of Gorakhpur, Gorakhpur (Received 20 December 1970 ; revised 25 June 1971) , Assunling the occurrence of a bormdzry shock wave as postulated by Martin, this paper studies its structure in conducting gas in the presence of a magnetic field. Martin1 postulated the concept of a boundary shock wave, a thin viscous region dominated by viscous compressive stress (rather than viscous shear) and the associated heat conduction in the gas adjacent to the body surface. He discussed the occurrence of boundary shock waves, developed the general theory for one dimensional laminar steady flow through a boundary shock in a perfect gas with longitudina* Prltndtl number equal to unity. He' also obtained explicit expressions for the jump conditions, the heat conduction co-efficient at the wall and the structure or variation of quantities through the boundary shosk. From arguments similar to those given by Martin, Verma & Prasad2, presumed the occurrence of boundary shock wave in a conducting gas in the presence of a magnetic field and extended the remilts of Martin to obtain the analogues of the well known Rankine Hugoniot jump relations, Pralzdtl numbep as well as other properties of the boundary shock. In this note, our purpose is to obtain the structure of a boun- dary shock wave in a conducting gas. B A S I C  E Q U A T I O N S  The equations for conservation of mass, momentum, field and energy in one-dimensional steady flow in a non-accelerating GO-ordinate system are ua and where x is the distance to the right of the boundary, e is the internal energy per unit mass, H is the intensity of the magnetic field, and f is the sum of the surface forces in the x-direction on an element of mass and is - given by, where p is the shear viscosity coefficient and p, is the magnetic permeability. Also, k being the coefficient of thermal conductivity. The boundary conditions at x = xbl= Ot , are, and at x+ 00, On integration the equations (1) to (4) give p u = Constant. ' and u H = Constant. p u (e + & us) + q - u f = Constant. S T R U C T U R E  O F A B O U N D A R Y S H O C K  The differential equatians (1) to (4) and the b a u n d q  canditions (8) and (9) may be put in terms of two variables u rwzd T. Then it is convenient to make the equations dimensianless by defining and substituting the dependent variables, and a new independent variable . dzc 5 = PbU) j-7 With the assumption, Fr = constant, i; is proportional to b, since cp is constant. Then from (11) a d  (13), we have . and the boundary canditions, d T  5--3-00: - dli d5 - - t o ,  - d l  --+ 0. where in deriving (17) we have assumed that In the special case where we have considered 7, = 1, it is caavenient to define, so that rn VERMA & P B A ~ ~ D  : Plane Boundary Shook Wave the equatian (17) can be written as, d Q  where by dehitian, Q = 0, at 6 = 0 and fmm the boundary ccmditians (18), -J t  -t 0 as6 -too. The only passible salution is, obviously, Hence (20) to (22) give q = UT (25) Equatian (16) then becomes, dii - R S d t  = p i i - Q + ~ + ~ .  u where by definition, 1 Y - 1  1 R = -- y - 1  Mba 1 +   --- 2 M ~ H ~  and Far integrating (26) we put it in the fallowing fm S iia dii (+ el) ( a  - e,) (o - e,) = +Conr;tant. if the roots of - are all r e d  In oase one root of (34) ia real and other two me coqlex, (26) can be written as iia d 4  $ ( G + a r + p ) ( ~ - ~ )  = P d + Constant. DEB. SOI. J., VOL. 22, APRIL 1972 TABLE 1 VALUER OF P -, ;ii, 5 COBRESPONDING TO REPRESENTATIVE VALUES OF ; - - I u P P F 1.0 1.0 1.0 1.0 Fig. 1-Variation of pressure density, temperature and velo- city within s boundary shock for y = 513, 1, C h , = t ;  M H ~ =  l o ;  p,= 1. The form (35) is however not admissible in the physical problem we are encountered withbkcauae the value of a cannot be complex, it being the ratio between the velocities in front and behind the boundary shook. Then, integrating (33) in the usual manner and applying the condition (27) we have In order to have a qualitative if theL(%),-we take and substitute them in the relations (29) to (32) to obtain-values of P, Q, R and S. Then (34) becomes Solving the equation numerically, its roots come out to be approximately .6,1.7 and -406. The equation (36) then gives ( y 4 . 6  )-'(4 ( C + 6 3 ( GI -1; 7 = e'85 - 1.006 (38) Evidently when a = 1 ; 5 + 0 and when 12 = '6, 5 + m. For values of u' between 1 and -6, 4 takes steadily large values. There is no need of finding the values of 5 corresponding to ti greater than 1 and negative values of a since they are inconsistant with the physioal problem under consideration. Keeping in mind that the corresponding values of j , T and 3 are given in the Table 1 and the variations of these quantities with 6 are plotted in the Fig. 1. R E F E R E N C E S  1. MARTIN, E. D., J .  Fluid Mech., 28 (1967), 337. 2. VERMA, B. G.  & PRASAD, B.. Pure Appl. Ueophys., 87 (1971), 207. 

Structure of a Plane Boundary Shock Wave

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Structure of a Plane Boundary Shock Wave

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