Complex Bifurcation from Real Paths

A new bifurcation phenomenon, called complex bifurcation, is studied. The basic idea is simply that real solution paths of real analytic problems frequently have complex paths bifurcating from them. It is shown that this phenomenon occurs at fold points, at pitchfork bifurcation points, and at isola centers. It is also shown that perturbed bifurcations can yield two disjoint real solution branches that are connected by complex paths bifurcating from the perturbed solution paths. This may be useful in finding new real solutions. A discussion of how existing codes for computing real solution paths may be trivially modified to compute complex paths is included, and examples of numerically computed complex solution paths for a nonlinear two point boundary value problem, and a problem from fluid mechanics are given

Liquid Rocket Engine Turbopump Rotating-shaft Seals

A monograph is organized and presents, for effective use in design, the significant experience and knowledge accumulated in development and operational programs to date. It reviews and assesses current practices, and from them establishes firm guidance for achieving greater consistency in design, increased reliability in the end product, and greater efficiency in the design effort. The monograph is divided into two major sections: state of the art and design criteria

Solid rocket motor internal insulation

Internal insulation in a solid rocket motor is defined as a layer of heat barrier material placed between the internal surface of the case propellant. The primary purpose is to prevent the case from reaching temperatures that endanger its structural integrity. Secondary functions of the insulation are listed and guidelines for avoiding critical problems in the development of internal insulation for rocket motors are presented

Backward Free Convection Boundary Layers in Porous Media

The well known steady free convection forward boundary layer (FBL) flows ascending over a heated upwards projecting semi-infinite flat plate embedded in a fluid saturated porous medium are compared in this paper to their less well known backward (BBL) counterparts descending over a cooled (also upwards projecting!) semi-infinite flat plate. The circumstance that the definite edge of the plate (x = 0) in the former case is a leading edge and in the latter one a trailing edge, leads to substantially different mathematical and physical features of the FBL and BBL flows, respectively. The paper considers under this aspect the case of similar flows corresponding to surface temperature distributions which are power-law functions of the distance x from the definite edge. For permeable plates the effect of an adequate lateral suction and injection of the fluid is also taken into account. The detailed investigation, however, is restricted to the particular values m = +1 and m = −1/3 of the power-law exponent m, where both FBL and BBL solutions are available in exact analytic form. For each of these values, both exponentially and algebraically decaying BBL solutions were found. In addition, the existence of an exact algebraic BBL solution valid for any value of m is reporte

Buoyancy Sustained by Viscous Dissipation

The quasi-parallel regime of a Darcy-Boussinesq boundary-layer flow over a permeable vertical flat plate adjacent to a fluid saturated porous medium is considered. ‘Quasi-parallel' means here a plane flow with a constant transversal velocity v=−v 0 directed perpendicularly towards the vertical surface, where a lateral suction with the same velocity −v 0 is applied. The plate is held at a constant temperature T w which coincides with the ambient temperature T ∞ of the fluid. The heat released by viscous dissipation induces a density gradient in the fluid. Thus, although T w=T ∞, a thermal convection occurs. The steady regime of this ‘self-sustaining buoyant flow' has been examined in detail. Wall jet-like profiles with a continuous but finite spectrum of the momentum flow have been found. These self-sustaining buoyant jets show a universal behavior, that is, there exist certain length, velocity and temperature scales such that the flow characteristics become independent of the (constant) material properties of the fluid and the porous medium as wel

The Free Convection Boundary-Layer Flow Induced in a Fluid Saturated Porous Medium by a Non-Isothermal Vertical Cylinder Approaches the Shape of Schlichting's Round Jet as the Cylinder Radius Tends to Zero

The title statement is proven for a circular cylinder whose surface temperature (above that of the ambient fluid) varies inversely proportional with the axial distance from the leading edg

Heat transfer characteristics of boundary-layer flows induced by continuous surfaces stretched with prescribed skin friction

A continuous surface stretched with velocity u w=u w (x) and having the temperature distribution T w=T w (x) interacts with the viscous fluid in which it is immersed both mechanically and thermally. The thermal interaction is characterized by the surface heat flux q w=q w (x) and the mechanical one by the skin friction τ w=τ w (x). In the whole previous theoretical research concerned with such processes, either (u w and T w) or (u w and q w) have been prescribed as known boundary conditions. The goal of the present paper is to initiate the investigation of the boundary layer flows induced by stretching processes for which either (τ w and T w ) or (τ w and q w) are the prescribed quantities. The case of an isothermal surface stretched with constant skin friction, (τ w=const., T w=const. ≠ T ∞) is worked out in detail. The corresponding flow and heat transfer characteristics are compared to those obtained for the (well known) case of a uniformly moving isothermal surface (u w=const., T w=const. ≠ T ∞