Rheol. Acta 14, 619-625 (1975) Department of Mathematics, Indian Institute of Technology, Powai, Bombay 400076 ( India ) Stability of plane Poiseuille flow of viscoelastic fluids with uniform cross flow R. K. Bhatnagar and Om Prakash Sharma With 3 figures (Received October 3, 1973) lntroduction The stability of plane Poiseuille flow of viscoelastic fluids has been investigated in several papers, notably in those by Chan Man Fon9 and Walters (1), Chun and Schwartz(2), Bhatnagar and Giesekus(3), Mook and Graebel (4) and Porteous and Denn (5). They examined the problem by considering infinitesimal disturbances such that the equations may be linearized. The visco- elastic fluids considered were described either by the constitutive equation of a second order liquid proposed by Coleman and Noll (6) or by an equation of integral form proposed by Oldroyd (7). Since Giesekus (8) pointed out that viscoelasticity gives rise to cellular type of instabilities even in the absence of the inertial forces if second normal stress-difference is chosen as positive i.e. inertia only modifies the critical value of the characteristic number associated with neutral stability, therefore, Bhatnagar and Giesekus(3) employed the cellular type of disturbances in their study whereas other workers employed the wavy disturbances. The problem was solved numerically by Chun and Schwartz (2) and Porteous and Denn (5) and asymptotically by Chan Man Fong and Walters (1) and Mook and Graebel (4). Bhatnagar and Giesekus (3), however, obtained the solution of the disturbance equation by an approximate method (Galerkin method) and by direct numerical integration. The result of these papers being that viscoelasticity is found to have a destabilizing effect in the Poiseuille flow (Similar effect has been observed in Couette flow). Further, Bhatna9ar and Giesekus (3, 9) pointed out that two types of disturbances with dif- ferent cell widths may exist simultaneously provided the parameter representing the ratio of inertial to elastic forces lies in a certain range. While discussing the overstability of plane Couette flow (10) they found that, in general, the overstable mode is higher than the stationary mode but both can come close to each other if certain conditions are satisfied. If the walls are porous and a uniform cross-flow is superposed by injecting a certain amount of fluid at one wall and removing an equal amount at the other wall, it is well known that the symmetry of the flow is destroyed and the introduction of cross-flow results in a nonparallel flow with curved streamlines. For classical viscous fluids the stability of flow between porous plane walls has been studied by Hains (11) and Sheppard (12), and for annular flows (i. e. flows between two concentric porous cylinders) by Gerstin9 and Jankowski (13) and Gerstin9 (14). For the flows between 168 plane porous walls the cross-flow is found to be stabilizing whereas for annular flows it is found to have both the stabilizing and destabilizing effect for the appropriate values of the pertinent parameter. Recently, Bhatnagar and Sharma (15) investigated the stability of plane Couette flow of a viscoelastic fluid with uniform cross-flow employing the approximate method due to GaIerkin. To the authors' knowledge this is the only work reported so ~far in this direction and they found that cross-flow fnay haue both the stabilizing and destabilizing effect in a certain range of the parameter representing the ratio of inertial to elastic forces. This encouraged us to investigate some other types of flows of viscoelastic fluids which may be interesting from the point of view of practical applica- tions. Therefore, the present study has been undertaken and the cross-flow effects are found similar to those for the Couette flow. The method of solution is essen- tially the one used in reference (15). Formulation of the problem and solution of the base flow We choose the constitutive equation of a viscoelastic fluid, applying the usual approxi= mation of slow motions, in the form S = -pI + 2t/o Il°)+ K(2)f(2)q - K(11)f(1)2], [1] where f")=½(v~ + ~v), ~ =l(vv-vV), [2] f(2)= Of(l) & + v-Vf°)+ ¢9.f(1)-f(1).¢9, [3] mean respectively the usual rate of strain tensor, the vorticity tensor and the second corotational kinematic tensor; ~/o representing a viscosity parameter, ~c (11) and ~c (2) parameters having the dimensions of time and characterizing the elasticity of the fluid. Further in [1] S represents the stress tensor and p the undetermined pres- sure. For the details of the eqs. [1]-[3] cf. Giesekus (24).