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The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

IS there anything magic about the shape of a wing section? Asked to sketch the profile of a wing on the back of an envelope, one would have no difficulty in representing a shape which would probably, for most purposes, be adequate. Assuming this generalization to be true—perhaps it is a rather rash one—one might equally well question the need for an article on aerofoil design, or indeed the need for the long and painstaking research which, over the years, has been conducted on this particular subject. But it is this same research which, in the long run, has resulted in the recognition of certain general rules relating to aerofoil geometry, which are now taken so much for granted that they would probably be embodied in one's preconceived notion of what a wing section should look like. Recently, also, rather complicated theoretical techniques have made possible the design of profiles which, if manufactured faithfully and carefully in each detail, can provide a performance which is considerably better than any more arbitrary shaping to general rules would produce. Finally, of course, one must recognize that there are exceptional conditions where the application of conventional ideas is inadvisable, and where theoretical and experimental research is needed to suggest what is more appropriate. This article will be concerned for the most part with amplifying these remarks; but, by and large, it must be admitted at the outset that we cannot point to any revolutionary discontinuities in the progress of aerofoil design such as have characterized advances in the means of aircraft propulsion, or structural design.

I SHOULD like first to say how much I appreciate the honour of being invited to deliver the Wright Brothers Lecture. To anyone whose work is associated in any way with the aeronautical sciences, it must be a source of pride and gratification to be invited to be a chief participant in one of the greatest occasions in the world of aeronautical research, an occasion designed in honour of the two great pioneers, Wilbur and Orville Wright. In my own case these feelings are shared with a feeling of humility and of my own unworthiness for the task.

In considering the principles of the design of low‐drag aerofoils, we saw that we should like the velocity outside the boundary layer to rise to a position as far back along the wing as possible, but that we are hindered by the danger of turbulent separation if the rate of velocity decrease at the back of the aerofoil is too great; the danger increases, roughly, when the thickness of the aero‐foil increases and when the position of maximum suction is moved further back. If, however, the whole of the pressure recovery, or velocity decrease, is concentrated over a very narrow interval along the chord, over which the boundary layer, or as much of it as necessary, is sucked away to stop separation, all danger of separation is avoided, and we can have, if we wish, a favourable velocity gradient over the whole of the rest of the chord. More specifically, if the upstream boundary layer, or part of it, is sucked in through a slot, there must be a streamline that divides the air crossing the slot from that entering it, and that streamline must meet the surface at a stagnation point (fig. 10). With the stagnation point behind the slot the flow in the boundary layer along the surface is reversed in direction between the stagnation point and the slot, and all danger of separation is avoided with a well‐designed slot if the design of the aerofoil is such that the pressure recovery takes place wholly between the upstream lip of the slot and the stagnation point. Up to now suction aerofoils have actually been designed with a discontinuous drop in velocity; the velocity rises to a given position on the chord, drops discontinuously, and thereafter rises, stays constant, or at any rate docs not decrease rapidly enough to produce any danger of separation, to the trailing edge; the velocity distribution has this character on both the top and bottom surfaces.

Under this heading are published regularly abstracts of all Reports and Memoranda of the Aeronautical Research Council, Reports and Technical Memoranda of the United States National Aeronautics and Space Administration and publications of other similar Research Bodies as issued.

THE Eleventh S.B.A.C. Display, held at Farnborough, September 5 to 10, has marked another year of sustained progress in the British Aircraft Industry. Although the advances since the previous display appeared to be less spectacular than in the preceding years the achievements of the industry, judged by the successful development of both civil and military aircraft which are eminently suited for their tasks, are no less impressive. On the contrary, as the development of an aircraft after its first flight quite often is a more difficult task than the design and construction of the prototype, the impressive display of aircraft types ready to go into service in the near future is an even more convincing proof of the capabilities of the British aircraft industry than any display of spectacular new prototypes could have been.