• Volume 88, Issue 2

May 1979,   pages  93-189

• On α-close-to-convex functions

LetP(α) denote the class of functionsf analytic in the unit discE, withf(0)=0,f(z)≠0 (0&lt;|z|&lt;1) andf′(z)≠0 inE, satisfying the condition$$\int\limits_{\theta _1 }^{\theta _2 } {\operatorname{Re} } \left\{ {a\left( {1 + \frac{{zf''\left( z \right)}}{{f'\left( z \right)}}} \right) + \left( {1 - a} \right)\frac{{zf'\left( z \right)}}{{f\left( z \right)}}} \right\}d\theta &gt; - \pi$$ whenever 0≤θ1≤θ2≤θ1+2π,z=rer&lt;1 and α is any positive real number. The functions inP(α) unify the classes of close-to-starlike (α=0) and close-to-convex (α=1) functions. We callfP(α) and α-close-to-convex function. In this paper we investigate certain properties of the classP(α).

• Existence and uniqueness of solutions of boundary value problems for third order nonlinear differential equations

Existence and uniqueness results for three-point boundary value problems for the third order nonlinear differential equations are obtained. The results are better than those obtained recently by Barr and Sherman, using extra conditions.

• Limitation theorems for absolute Nörlund summability

A limitation theorem concerning absolute Nörlund summability was proved recently. In the present paper limitation theorems concerning |N, p, q| summability are proved and corresponding theorems for |E, δ|, |C, α|, |C, α, β|,$$\left| {\bar N,p_n } \right|$$, |N, pnα| methods are deduced as special cases.

• A class of algebraically special null Einstein-Maxwell fields in Newman-Penrose formalism

An algebraically special null Einstein-Maxwell space-time has been obtained. It contains shear-free, geodetic and hypersurface orthogonal congruences and more over the space-time is twist-free.

• On propagation and attenuation of Love waves

The period equation for Love waves is derived for a layered medium, which is composed of a compressible, viscous liquid layer sandwiched between homogeneous, isotropic, elastic solid layer and homogeneous, isotropic half space. In general, the period equation will admit complex roots and hence Love waves will be dispersive and attenuated for this type of model. The period equation is discussed in the limiting case when thicknessH2 and coefficient of viscosity, η2, of the liquid layer tend to zero so as to maintain the ratioP=H22 constant. Numerical values for phase velocity, group velocity, quality factor (Q) and displacement in the elastic layer and half space have been computed as a function of the frequency for first and second modes for various values of the parameterP. It is shown that Love waves are not attenuated whenP=0 and ∞. The computed values ofQ for first and second modes indicate that whenP≠0 or ∞ the value ofQ attains minimum value as a function of dimensionless angular frequency.

• On Stoke's problem in magnetohydrodynamics

Stoke's classic problem involving the impulsive motion of an infinite flat plate in an unbounded viscous incompressible fluid is investigated under the additional specification that the fluid is electrically conducting and the motion is developed in the presence of uniform transverse magnetic field. For the fluids with arbitrary magnetic Prandtl number, the compact expression for the skin friction coefficient at the plate is given in terms of exponential and error functions of complex arguments. For the fluids with unit magnetic Prandtl number, expressions for the induced magnetic field, velocity, current density and induced electric field in the viscous boundary layer region set up near the plate are obtained. The effect of the magnetic field on the skin friction is to make it approach the steady state faster than in nonmagnetic case.

• Steady flow of a second-order thermo-viscous fluid over an infinite plate

The steady flow of a second-order thermo-viscous fluids over a moving infinite flat plate is examined. It is observed that in the absence of any body force in the flow region, a constant temperature gradient is generated, which depends upon the plate velocity and material constants. The mean velocity, mean bulk temperature and Nusselt number are calculated.

• Effect of porous lining on the flow between two concentric rotating cylinders

The effect of non-erodible porous lining on the flow between two concentric rotating cylinders is investigated using Beavers and Joseph slip boundary condition. It is shown that the shearing stress at the walls increases with the porous lining thickness parameter ε.

• Dusty viscous flow through a cylinder of triangular cross-section

Unsteady laminar flow of a dusty viscous, incompressible fluid through a cylindrical tube of triangular cross-section is considered in two cases: (i) when the pressure gradient varies harmonically with time and (ii) when it varies exponentially. The velocity fields for the fluid and dust particles have been determined. Flux and skin-friction drag on the walls of the cylinder have been calculated and particular cases discussed.

• Temperature distribution in a laminar plane wall jet due to variably heated wall

The heat transfer in a laminar incompressible plane wall jet due to a variably heated wall has been studied. It is assumed that the difference of temperatures between the wall and the issuing jet is inversely proportional to an arbitrary exponent of the distance from the slit. A similar solution of the energy equation is possible. The solutions, for arbitrary values of the Prandtl number and of the exponent are obtained. It is found that in some cases the heat transfer at the wall may become zero or negative.

• # Proceedings – Mathematical Sciences

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