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The uniqueness of the solution can be proved as follows: Suppose that 4>i(x) and $z(x) are two solutions such that *i(x) = f(x) + ("" K(x, sfaWds,

Abstract.The primary objective of this study has been to prepare a chart for computing the growth of Taylor-Gortler vortices in laminar flow along walls of both high and low concave curvature. Taylor-Gortler vortices are streamwise vortices having alternate right-and left-hand rotation that may develop in the laminar boundary layer along a concave surface.The equations of motion are derived anew and re-examined with regard to the importance of the various terms. The final equations used in preparation of the chart are found to be valid for radii of curvature as small as 30 times the boundary layer thickness. Furthermore, it is shown that the equations are not restricted in validity to cases of constant wall curvature, constant free stream velocities, or to boundary layers of constant thickness. Whereas the previous analyses by Taylor and Gortler assumed the vortex to grow exponentially as a function of time, the present study recasts the growth into a more convenient form in which the vortex grows as a function of distance.The solution is an eigenvalue problem, which in the present study has been solved mainly by Galerkin's method-a variational method. Both the eigenvalues and the eigenfunctions are presented, the former in the aforementioned chart. It is possible to compare the solutions for neutral stability with those given by Gortler. The two solutions are in approximate agreement.A second method of solution also is described. This method is believed to offer considerable improvement, provided a high-speed digital computer is available. In the one case checked by both methods agreement was within 2%.Finally, the stability chart was applied to all the known experimental data concerning the effect of concave curvature on the transition point. The well known parameter Rs(d/r)1/2 is shown to be inadequate as an indicator of the transition point. Instead, the experimental data indicate that an apparent amplification factor, exp / /3 d.r, is a much better measure. Available results show that transition of this type will occur when / /3 dx reaches a value of about ten.2. The flow past a concave plate. A considerable body of indirect evidence indicates that a laminar flow along a concave wall does not remain two-dimensional. Instead, it rolls up into alternate right-and left-hand vortices as indicated in Fig. 1. To obtain some insight into the forces that cause the formation of these vortices, consider the streaming of an incompressible, viscous fluid past a concave wall, Fig. 2. If the Reynolds number is not extremely low, a boundary layer will develop. At some arbitrary height, y1 , within the boundary layer, the velocity is . At some other height y2 = yx + Ay the velocity is u2 = ux + (du/dy)Ay. At the height yt , the fluid is under a pressure . By the usual boundary layer equations for two-dimensional

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The theory of two-dimensional instability of laminar flow of water over solid surfaces is extended to include the effects of heat transfer. The equation that governs the stability of these flows to Tollmien-Schlichting disturbances is the Orr-Sommerfeld equation “modified” to include the effect of viscosity variation with temperature. Numerical solutions to this equation at high Reynolds numbers are obtained using a new method of integration. The method makes use of the Gram-Schmidt orthogonalization technique to obtain linearly independent solutions upon numerically integrating the “modified Orr-Sommerfeld” equation using single precision arithmetic. The method leads to satisfactory answers for Reynolds numbers as high as Rδ* = 100,000. The analysis is applied to the case of flow over both heated and cooled flat plates. The results indicate that heating and cooling of the wall have a large influence on the stability of boundary-layer flow in water. At a free-stream temperature of 60 deg F and wall temperatures of 60, 90, 120, 135, 150, 200, and 300deg F, the critical Reynolds numbers Rδ* are 520, 7200, 15200, 15600, 14800, 10250, and 4600, respectively. At a free-stream temperature of 200F and wall temperature of 60 deg F (cooled case), the critical Reynolds number is 151. Therefore, it is evident that a heated wall has a stabilizing effect, whereas a cooled wall has a destabilizing effect. These stability calculations show that heating increases the critical Reynolds number to a maximum value (Rδ* max = 15,700 at a temperature of TW = 130 deg F) but that further heating decreases the critical Reynolds number. In order to determine the influence of the viscosity derivatives upon the results, the critical Reynolds number for the heated case of T∞ = 40 and TW = 130 deg F was determined using (a) the Orr-Sommerfeld equation and (b) the present governing equation. The resulting critical Reynolds numbers are Rδ* = 140,000 and 16,200, respectively. Therefore, it is concluded that the terms pertaining to the first and second derivatives of the viscosity have a considerable destabilizing influence.