Use Of Conformal Mapping Research Articles

On the departure of a sphere from contact with a permeable membrane

A sphere in contact with a porous membrane can depart with non-zero velocity under the influence of a finite force. The flow field and the viscous force resisting the motion are evaluated by reducing the Stokes equations to a fourth-order ordinary differential equation through the use of conformal mapping. Explicit results are given for high and low membrane permeabilities by using regular and singular perturbation techniques respectively.

Iterative methods for solving the exterior Dirichlet problem for the Helmholtz equation with applications to the inverse scattering problem for low frequency acoustic waves

Abstract : A previous paper presented an iterative method for solving the exterior Dirichlet problem for the Helmholtz equation defined in the plane and used this result to provide a constructive approach for solving the low frequency inverse scattering problem for a cylinder. These results were based on the use of conformal mapping and the fact that the integral of the normal derivative of the total field over the boundary of the obstacle vanishes, neither of which is valid in the three dimensional case. The present paper shows how the previous analysis can be modified in order to extend these results to the case of the exterior Dirichlet problem for the Helmholtz equation in R3. Our results are based on choosing an appropriate fundamental solution such that the integral equation associated with the exterior Dirichlet problem can be solved by iteration for sufficiently small values of the wave number.

On the slow viscous rolling of a sphere in contact with a permeable surface

An exact solution for the velocity field induced by the slow rolling motion of a sphere in contact with a permeable surface is derived. The porous solid and the viscous fluid each occupies a semi-infinite space. It is shown that, by the use of conformal mapping, the problem is reduced to a fourth order ordinary linear boundary value problem. As the permeability of the porous body diminishes the important functionals, e.g. the force and torque resisting the rolling motion of the sphere, are divergent. The nature of this divergence is found and the dependence on the permeability of the porous solid is explicitly evaluated.

Canal Seepage with Boundary at Finite Depth

Closed-form solution for the problem of seepage from channels in homogeneous and isotropic material extending up to finite depth with drainages at a finite distance from the canal and at finite depth has been obtained by the use of conformal mapping. Equations for seepage discharge and the shape of the phreatic line are given. The analysis indicates that the phreatic surface rises with decrease in the depth of impervious layer, and increase in the width of the channel and the drainage distance. For larger values of T/n, the phreatic surface is seen to curve continuously and is S-shaped. With a decrease in the value of T/n, the curvature of the phreatic surface is reduced. It is also seen that the seepage discharge increases with an increase in the depth of the pervious stratum and approaches the value for the case of a canal with depth of pervious stratum extending to infinity.

Fundamental frequency of vibration of clamped plates of arbitrary shape subjected to a hydrostatic state of in-plane stress

The present paper makes use of conformal mapping and Galerkin's method to determine the fundamental frequency of vibration of clamped plates of arbitrary shape subjected to uniform in-plane stress. Numerical results are presented for several plates of regular polygonal shape. Frequency curves obtained by means of Lurie's expression are also given for simply supported plates.

Stability of Structures with Intermediate Filters

Until now no solution was available to determine the effect of plane drainage located anywhere between the two cutoffs. The problem is solved exactly by the use of conformal mapping. The equations for seepage discharge to the intermediate filter and uplift pressures below the floor are given. These equations have been solved numerically to obtain pressures at key points. The results are given in the form of easy-to-use design curves. The uplift pressures reduce considerably along the entire profile of the structure with the provision of filter of even very small length. The uplift pressures decrease on the downstream side and increase on the upstream as the filter is moved from upstream to the downstream side. The filter should be located such that total uplift on the floor downstream of the gateline is minimum. With increase in the depth of downstream cutoff, the uplift pressures increase only along the floor downstream of the filter.

Accelerated-convergence technique and analyticity

The use of conformal mappings in elementary particle physics and the resulting accelerated convergence of expansions are discussed. If there is a discontinuity across a cut to be mapped, the mapping technique will be unable to completely describe the situation. The implications for the analytic properties of the amplitude are studied in this case.

Analysis of Canal Seepage to Interceptor Drain

No solution for the problem of seepage from trapezoidal channels in homogenous and isotropic material to an interceptor drain and the main drainage at finite distance is available so far. This general problem is solved exactly by the use of conformal mapping. Equations for seepage discharge to the interceptor drain and the main drainage and shape of free surface are given. These equations are solved numerically. The results are given in the form of dimensionless curves. The analysis indicates that the effect of lowering of phreatic surface by provision of interceptor drain extends to a considerable distance on either side of the drain. The location of the drain farther away from the canal results in greater lowering of free surface beyond the drain although this results in a slight rise in the elevation of free surface between the canal and the drain.

Phase shift analysis of K +p scattering

We have analyzed data on K +p elastic scattering at 21 momenta covering the range 0.5–2.5 GeV/ c. We present results of three phase shift analyses. One of these is an energy independent accelerated convergence expansion (ACE) analysis in which high partial waves are included through use of conformal mapping as suggested by analytic approximation theory. The other two are an energy independent and an energy smoothed conventional analysis. The ACE method gave, with the same number of parameters, a 20% average improvement in x 2 as well as more unique phase shift solutions. The high partial waves gave significant contributions to total and inelastic cross sections. The ACE method is also sensitive to the value of g ΛΣ 2 = g K ΛN 2 + g K ΣN 2; we have obtained the value g ΛΣ 2 = 15.2 ± 2.3.

Buckling of Rhombic Plates

The present note deals with the determination of buckling loads for clamped rhombic plates. The solution is accomplished by use of conformal mapping and a variational approach to solve the transformed partial differential equation.

Seepage from Trapezoidal Channels

No solution for the problem of seepage from trapezoidal channels in homogeneous and isotropic material with drainages at a finite distance from the canal and at a finite depth is available so far. This general problem is solved exactly by the use of conformal mapping. Equations for seepage discharge and the shape of the free surface are given. These equations are solved numerically. The results are given in the form of easy-to-use curves. The analysis indicates that the phreatic surface has a continuous curve from the canal to the drain. This surface rises with increase in bed width, water depth and drainage distance, though the effect of side slopes is very little. The seepage discharge increases with an increase in bed width, water depth, side slopes, and with a decrease in the drainage distance.

A theory for free outflow beneath radial gates

An analysis is made of the free outflow of fluid from those underflow gates known as radial or Tainter gates. Attention is focused on a correct treatment of the effects of gate curvature and of gravity on the flow. The rapidly convergent, iterative solution is based on the combined use of conformal mapping and the Riemann-Hilbert solution to a mixed boundary-value problem. A limited comparison with some experimental results shows agreement to be good.

Jets from two-dimensional symmetric nozzles of arbitrary shape

A unified approach to the problem of jet efflux from symmetrical channels of finite width and possessing a general curvilinear nozzle shape is presented. The nozzle may be composed of polygonal and/or curved-arc segments. Precise nozzle shapes cannot be initially prescribed, however. The solution is based on the combined use of conformal mapping and the Riemann-Hilbert solution to a mixed boundary-value problem. The selection of an appropriate curvature function is described; examples show possible applications.

Application of Complex-Variable Theory to the Determination of the Fundamental Frequency of Vibrating Plates

This study deals with the determination of the fundamental frequency of vibration of plates of complicated shape. The method makes use of conformal mapping and variational techniques. In the case of a square plate, the results obtained are in good agreement with the exact values. Numerical results are also presented for a circular plate with two flat sides and several plates of regular polygonal shape.

Seepage from an Array of Triangular Channels

The problem of seepage losses from a system of parallel triangular channels is solved exactly by use of conformal mapping. Formulas for the discharge and the shape of the free surface are given. In general, the system is underlain by a drainage layer at a finite depth, and the case of an infinitely deep aquifer is solved separately. Numerical examples are given with the results presented in the form of graphs and tables.

Use of Conformal Mapping to Improve Field Uniformity of Electromagnet

Conformal mapping is used to determine the cross section of a shim in order to improve the uniformity of the field in the air gap of a magnet. Experimental measurements on a laboratory electromagnet demonstrate the improvement in field uniformity.

Approximate Method for the Study of Heat Conduction in Bars of Arbitrary Cross Section

The method to be described consists in use of conformal mapping for solving certain unsteady state temperature distribution problems.

A linearized theory for rotational supercavitating flow

In this paper methods are given for establishing qualitative and quantitative measures of the effects of rotation in supercavitating flows past slender bodies. A linearized theory is developed for steady, two-dimensional flow under the assumption that the flow has a constant rotation throughout. The stream function of the rotational flow satisfies Poisson's equation. By using a particular solution of this equation, the rotational problem is reduced to a problem involving Laplace's equation and harmonic perturbation velocities. The resulting boundary-value problem is solved by use of conformal mapping and singularities from thinairfoil theory. The theory is then applied to asymmetric shear flow past wedges and hydrofoils and to symmetric shear flow past wedges. The presence of rotation is shown to create significant changes in the forces acting on the slender bodies and in the shape and size of the trailing cavities.

Use of Conformal Mapping for the Analysis of Wave Propagation in Inhomogeneous Media

Solutions to the two-dimensional wave equation for a variable velocity medium may be found by a coordinate transformation which renders the transformed medium homogeneous. Any conformal transformation w = w(z) from the complex w to the complex z plane will furnish such solutions with |dw/dz| = n, where n is the (variable) index of refraction of the medium. An exponential velocity variation and a medium in which the velocity is a function of two variables will be treated by this method. The transformation has an interesting physical significance in connection with Fermat's principle.

On the use of Conformal Mapping in Shaping Wing Profiles

(1940). On the use of Conformal Mapping in Shaping Wing Profiles. The American Mathematical Monthly: Vol. 47, No. 6, pp. 362-373.