This paper has three main objectives. First, it aims to show that basic general conservation principles for viscous flow can be formulated in terms of diffusion and convection. Secondly, it aims to show that three scalar conservation principles suffice to provide a method for characterizing swirling axisymmetric flows in terms of axial and boundary production of the conserved quantities. Thirdly, it aims to exemplify these two objectives by giving a complete specification of the axial causes for swirl-free conically similar flow in otherwise free space.This series of papers, overall, is concerned with the analysis and characterization of swirling conically similar flows in terms of the singularities that generate the conserved quantities. In conically similar flows there is no natural lengthscale, and the sole parameters governing the flow are provided by the strengths of the singularities that cause the flow. These are required to have the same dimensions as a power of the kinematic viscosity v. The axisymmetric flow generated by uniform production of swirl angular momentum per unit mass along a half-axis at a constant rate provides a simple example.In conically similar flow the three conservation principles for axisymmetric flow provide a sixth-order non-autonomous system of two ordinary differential equations governing the flow. Here, in Part 1, these equations are derived for the general case of swirling flow, and are shown to reduce to a fourth-order system when swirl is absent. The two scalar conservation principles describing swirl-free flow are used to classify the basic axial causes for this system.Part 2 analyses these basic exact one-parameter swirl-free families of solutions, and Part 3 extends the analysis to the remaining one-parameter family of swirling flows associated with uniform swirl angular-momentum production on a half-axis. Each of the families is characterized by a single independent cause, and two of them provide new non-trivial solutions of the Navier–Stokes equations. The effects of nonlinear coupling of these basic one-parameter causes and of conically similar distributions over conical boundaries will be examined in later papers.
In Part 1 of this series conservation principles for ring circulation and kinematic swirl angular momentum were developed for general axisymmetric incompressible viscous flow. These principles were then used to classify the four independent axial causes of swirl-free conically similar viscous flow. Part 2 provided a detailed analysis of the one-parameter swirl-free flows that are generated by each one of the axial singularities acting alone. The present paper extends, to swirling flow, the description of the axial singularities that drive axisymmetric viscous flow. In the special case of conically similar viscous flow, two independent half-line sources of swirl angular momentum suffice to complete the set of axial singularities that can generate such swirling flows. The individual strengths of the six independent axial causes provide a complete characterization of all conically similar viscous flows that can be generated in this way. This Part 3 completes the task of analysing in detail the independent one-parameter flows generated by axial causes by studying the flow caused by uniform production of kinematic swirl angular momentum on a half-axis. This flow demonstrates how swirl may induce an axial half-plane flow. For large swirl circulation strengths, swirl angular momentum diffuses and convects so as to fill slightly more than half the space with an almost constant density of swirl angular momentum. A well-developed internal boundary layer, in the form of an outward radial jet, then separates this region from one in which the flow is almost irrotational. The jet entrains two impinging convection fields. The angular location of the jet is determined by relating the axial component of moment of whirl produced at the origin to the strength of the swirling circulation singularity on the axis.
It was shown in Part 1 of this series that in swirl-free flow there are three different types of axial causes of steady conically similar viscous flow. The three corresponding swirl-free one-parameter families of exact solutions to the Navier–Stokes equations are presented and analysed here in terms of the basic conservation principles for volume and ring circulation. The simplest is the irrotational flow generated by a uniform distribution of volume sources along a half-axis. A second, independent, one-parameter family of solutions is provided by Landau's (1943) solution, where the second moment of ring circulation about the axis is produced at the origin at a finite constant rate. Fresh insight into the nature of this flow is gained by separating and comparing the roles of the diffusion and convection terms in the flux vector for ring circulation. A similar analysis is applied to the remaining independent one-parameter family caused by an antisymmetric (about the origin) conically similar axial distribution of the singularity in Landau's solution. This simple new family of exact solutions is characterized by opposed jets neighbouring the axis of symmetry. When the axial jets are directed inwards, they always erupt into an emergent axisymmetric jet normal to the axis of symmetry. Solutions fail to exist, however, for sufficiently strong axial jets directed outwards.
In this paper the problem solved is that of unsteady flow of a viscous incompressible fluid between two parallel infinite disks, which are performing torsional oscillations about a common axis.The solution is restricted to high Reynolds numbers, and thus extends an earlier solution by Rosenblat for low Reynolds numbers.The solution is obtained by the method of matched asymptotic expansions. In the main body of the fluid the flow is inviscid, but may be rotational, and in the boundary layers adjacent to the disks the non-linear convection terms are small. These two regions do not overlap, and it is found that in order to match the solutions a third region is required in which viscous diffusion is balanced by steady convection. The angular velocity is found to be non-zero only in the boundary layers adjacent to the disks.
In many compressible fluid flow problems the classical solution breaks down completely owing to the formation of regions of infinite acceleration in the flow field. The actual behaviour of the fluid in such cases does not seem to have been investigated mathematically and this is largely due to the difficulties which enter with non-uniform shocks. It is with these difficulties that this paper is principally concerned.The way in which discontinuities may arise mathematically in a flow field is first discussed. The equations governing the one-dimensional motion of a gas due to an accelerating piston are then set up. It is shown that when allowance is made for varying entropy conditions due to the presence of non-uniform shocks these differential equations reduce (outside the shock wave) to three first order quasilinear ones.The initial solution breaks down when a point of infinite acceleration occurs in the flow field. From this point onwards a shock wave grows in the fluid and behind it three different sets of characteristics are required to describe the flow. By working in the plane of two quantities that are constant along two different sets of characteristics, we can use the shock-jump conditions to determine the equation of the shock-line in this plane and to reduce the equations of the characteristics to three differential equations, which would be linear if the relation between the entropy and other flow variables were known.In the case of a constantly accelerating piston a first approximation is found by neglecting reflexions and entropy variations behind the shock. Using this as a basis we then find a second approximation for the entropy in the neighbourhood of the initial portion of the shock-line and show that the problem reduces to the solution of a second order linear partial differential equation. The introduction of a Riemann function and the satisfaction of the boundary conditions at the shock lead to an integral equation whose solution enables us to determine the position of the shock as a function of the time. The solution is in the form of a power series and is valid provided the shock wave does not become too strong.Finally, it is shown that if the piston is given a constant terminal velocity a reflected wave from the shock is reflected again from the piston and eventually overtakes the shock and reduces its velocity to a final steady value which is in agreement with the value arising from an impulsive start.
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