Common bearing theory cannot be applied to a Rayleigh bearing which we place at 0<y<h1forx<0,0<y<h2forx>0,h1>h2,(1) near x = 0. This paper is concerned with a more exact theory, providing (partial) differential equations whose solution can handle even the region near x = 0. In section 1, after the ordinary, approximate solution is reviewed, more exact equations are obtained for the flow, under the assumptions that the flow is nonturbulent, that the fluid is incompressible, and that its viscosity μ is constant. It is shown that the streamline function ψ is biharmonic, that it is a solution of the repeated Laplace equation ∇4ψ=∇2(∇2ψ)=0,(2) and proper boundary conditions are obtained for it. In section 2 it is recalled that equation (2) also occurs in certain plate problems, such as the normal deflection of a constant thickness plate given by equations (1), free from distributed normal load, and subject to proper deflections and slopes at its boundary. Again, equation (1) is also satisfied by Airy function H of the plate (1) subjected to normal and shearing tractions over its boundary. Based on the above, analogies are outlined between the fluid flow and the two plate problems, with ψ corresponding to ω and to H. These analogies can be used to obtain experimental solutions of the bearing flow problem. Section 3 is devoted to graphical and numerical solutions. The numerical methods are based on covering the area (1) with a square mesh, and approximating to the differential operators by finite difference quotients of values at the mesh points, yielding to a set of linear equations. These are solved either on a computer, or by assuming a solution and improving it successively. The graphical method involves conformal mapping of (1) onto a plane with a simpler boundary, such as an infinite strip. This can be carried out analytically, or graphically by means of two sets of orthogonal curves cutting (1) into small squates. Section 4 utilizes separate expansions of ψ in product biharmonics to each side of x = 0, and joining them up so that the integrated errors over x = 0 are minimized. Finally in Section 5 the biharmonic ψ is expressed in the form ψ=F+yG(3) where F and G are harmonic, and from the boundary conditions on ψ are obtained boundary values of F, G and/or their conjugate harmonics. The methods used involve Green’s functions and therefore are carried out best in the plane of the infinite strip.
New Analytical Solutions for Time-Fractional Kolmogorov–Petrovsky–Piskunov Equation with Variety of Initial Boundary Conditions
