54 



Bouthier (1973), Nayfeh, Saris, and Mook (1974), 

 Gaster (1974), and Saric and Nayfeh (1975, 1977). 



The formulation of the problem and method of 

 solution is taken in the next section, the solution 

 of the first-order problem is given in Section 3, 

 the solution of the second-order problem is given 

 in Section 4, the mean flow is discussed in Section 

 5, and the numerical results and their comparison 

 with the experimental data of Strazisar et al. (1975, 

 1977) is given in Section 6. 



2. PROBLEM FORMULATION AND METHOD OF SOLUTION 



The present study is concerned with the two- 

 dimensional, nonparallel stability of two-dimensional, 

 viscous , heat conducting liquid boundary layers to 

 small amplitude disturbances. The analysis takes 

 into accoxmt variations in the fluid properties but 

 neglects buoyancy, dissipation, and expansion ener- 

 gies. All fluid properties are assumed to be known 

 functions of the temperature alone. 



Dimensionless quantities are introduced by using 

 a suitable reference length L* and the freestream 

 values as reference quantities, where the star 

 denotes dimensional quantities. 



To study the linear stability of a mean boundary- 

 layer flow, we superpose a small time-dependent dis- 

 turbance on each mean flow, thermodynamic, and 

 trasport quantity. Thus, we let 



q(x,y,t) 



QQ(x,y) + q(x,y,t) 



(1) 



where Qo(x,y) is a mean steady quantity and q(x,y,t) 

 is an unsteady disturbance quantity. Here, q stands 

 for the streamwise and transverse velocity compo- 

 nents u and V, the temperature T, the pressure p, 

 the density p, the specific heat Cp, the viscosity 

 y , and the thermal conductivity <. Substituting 

 Eq. (1) into the Navier-Stokes and energy equations, 

 subtracting the mean quantities and linearizing the 

 resulting equations in the q's, we obtain the fol- 

 lowing disturbance equation: 



8p 3 , 



at 



3x 



37 *'0" ^ 



pvo) 



(2) 



3y R 



/3u + 3v 



^ 3 r / 3v ^ 3u 



+ y r 



PO 



3Vn 

 3y 



3T 



8Un 



3 

 3T 



(4) 



3T ^ 3To , ._ 



3^ ^ " sT ^ "0 il ^ " 3y 



+ V, 



3T 

 3y 



Po- 



+ P 



Po 



3T 



3x 



+ V 



RPr c 

 e Pn 



P, V, K, C 



3_ 

 3x 



3T 

 ^0 3ir* ^ 



3To 



3y 



3To \ 



3x / 



3T 3Tf 

 3y V^° 3y "^ '^ 3y 



functions (T) 



(5) 



(6) 



Here, Cp is the liquid specific heat at constant 



pressure, R = p|U|L*/vi| is the Reynolds number and 

 Moreover , 



Pr = CpgiJ|/K* is the freestream Prandtl number 



(e + 2) , 

 (1 + 2e) 



!'• 



1), 



A = - y(e-l) 



(7) 



where e is the ratio of the second to the first 

 viscosity coefficients (e = is the Stokes assump- 

 tion) . 



The problem is completed by the specification of 

 the boundary conditions ; they are 



=v=T=0=aty=0 

 u,v,T ->■ as y ^- " 



(8) 

 (9) 



"(II * ". ij - 1?^ -. t - If ; 

 *»(°»if*'»if) 



3P 

 3x 



3 f du 3v\ 



_3^ L'°v ^ ^ " i^j 



3v 



'h 



— + 



''[-^ " ^. 



^\W 



' 3Uo + 



/3v 



3V^^ 

 3x J_ 



(3) 



, 3v 3Vo ^ ,, 3v , 3Vo\ 



'0 3^^ "i^+^0 37^^ aTJ 



3Vo\ 



* <"« I? * V. |<1 



We restrict our analysis to mean flows which are 

 slightly nonparallel; that is, the transverse ve- 

 locity component is small compared with the stream- 

 wise velocity component. This condition demands all 

 mean flow variables to be weak functions of the 

 streamwise position. These assumptions are expressed 

 mathematically by writing the mean flow variables in 

 the form 



U 



Uo(xi ,y) , Vg + eVo(xi,y) , 

 Pq = Po(xi) , To= To{xi,y) 

 Po = Po'^l 'V 



Po 



c (xi,y) 



Po 



\iQ = yo'^^l'Y' ' *^0 = Ko'^1 -y' 



(10) 



where xj = ex with e being a small dimensionless 

 parameter characterizing the nonparallelism of the 

 mean flow. In what follows, we drop the caret from 

 Vq. 



