For normalized starlike functions f : 𝔻 → ℂ, we consider the analytic functions g : 𝔻 → ℂ defined by g(z) = (1 + z(f″(z))/f′(z))/(zf′(z)/f(z)) and g(z) = (1 − α)(zf′(z))/f(z) + α(1 + (zf″(z))/f′(z)), 0 ≤ α ≤ 1. We determine the largest radius ρ with 0 < ρ ≤ 1 such that g(ρ z) is subordinate to various functions with positive real part.

We use the theory of differential subordination to explore various inequalities that are satisfied by an analytic function p defined on the unit disc so that the function p is subordinate to the function ez. These results are applied to find sufficient conditions for the normalised analytic functions f defined on the unit disc to satisfy the subordination zf'(z)/f(z) ≺ ez.

The objective of this study is to analyze the mixed convective bi-component hybrid nanofluid flow over a sphere. The equations governing the flow are transformed via dimensionless coordinates using non-similar transformations. The obtained partial differential equations, that are highly nonlinear, are linearized using the quasilinearization technique, and an efficient implicit finite difference method is adopted to solve for the unknown parameters. The consequences of various values of nanoparticle volume fractions, mixed convection parameter and viscous dissipation parameter on essential parameters such as skin friction and heat transfer coefficients are graphically depicted and the results were discussed. The temperature and heat transfer coefficient values are observed to be higher for the hybrid nanofluid than the base fluid and nanofluids silica/water and alumina/water. Silica–alumina/water hybrid nanofluid depicts the highest heat transfer rate for a higher nanoparticle volume fraction of alumina than that of silica.

We find the radius of Ma-Minda starlikeness of normalised analytic functions of the form 1(z) = z(f?(z))?, ? > 0 where f is in the classCV[A, B] of Janowski convex functions and 1(z) = z(zf?(z)/ f (z))?, ? > 0 where f is in the class CV? defined. As particular cases, we obtain criteria for these functions to belong to certain Ma-Minda classes.

The investigation of how sinusoidal mass transfer, variable viscosity, and Prandtl number affect unsteady magnetohydrodynamic flow over a yawed cylinder is the core intention of this work. The coupled non-linear partial differential equations of the flow field are linearized via quasilinearization, and an implicit finite difference scheme is employed to yield non-similar solutions. In both steady and unsteady cases, an enhancement in the magnetohydrodynamic parameter or a reduction in yaw angle boosts the skin friction coefficient in the axial direction and the heat transfer coefficient. The aforesaid effects, however, are reversed for skin friction coefficient in the spanwise direction. The skin friction and heat transfer coefficients grow as the time parameter rises, regardless of the magnetic field or yaw angle. When a sinusoidal slot suction is used in the unsteady case, zero skin friction in the axial direction is averted.

This paper investigates the unsteady magnetohydrodynamic (MHD) mixed convective fluid flow over a rotating sphere. An implicit finite difference scheme, together with quasi-linearization, is used to find non-similar solutions for the governing equations. The impact of variable physical properties and viscous dissipation are included. It is observed that the skin friction coefficient in the axial direction and the heat transfer coefficient are increasing with an increase in MHD, mixed convection and rotation parameters and with time, whereas the effect is just the opposite for the skin friction coefficient in the rotational direction. The non-uniform slot suction(injection) and the slot movement influence the point of vanishing skin friction to move in the axial direction downstream (upstream).

This study presents non-similar solutions for the magnetohydrodynamic hybrid nanofluid copper-alumina/water flow over an infinite yawed cylinder, featuring an emphasis on entropy generation owing to heat transfer, fluid friction, and joule heating. Non-similar transformations are used to convert non-linear governing equations and boundary conditions into a non-dimensional form, which is subsequently linearized using the quasi-linearization approach. Implicit finite differentiation is used to solve the equations that arise. The influence of viscous dissi-pation is considered and entropy generation analysis is done for various values of yaw angle, magnetohydrody-namic parameter and viscous dissipation parameter. The results show that when the magnetic field is increased, the ordinary separation is delayed. The thermal boundary layer of the hybrid nanofluid copper-alumina/water is found to be thicker than the thermal boundary layer of the nanofluids copper/water and alumina/water as well as the working fluid water. As the viscous dissipation and magnetic field increase, the overall entropy generation increases. To lower overall entropy generation, the cylinder's yaw angle must be increased.

By using the theory of differential subordination, we investigate differential inequalities, satisfied by an analytic function [Formula: see text] defined on the unit disc, that imply the boundedness of the function [Formula: see text]. Our results are applied to find sufficient conditions for the normalized analytic functions to be either starlike or close-to-convex.

This paper examines the steady magnetohydrodynamic (MHD) flow of water over a yawed cylinder with variable fluid properties and non-uniform mass transfer. The impact of viscous dissipation is taken into consideration. The velocity and temperature fields are governed by coupled nonlinear partial differential equations together with boundary constraints. These governing equations are converted to dimensionless form with suitable non-similar transformations and then solved using an implicit finite difference method and the quasi-linearization technique. The results indicate that the yaw angle enhancement declines the skin friction coefficient in the axial direction and the heat transfer coefficient. It is also ascertained that the separation can be delayed by enhancing the MHD effect, the suction parameter with slot movement in the downstream direction.

This paper examines the unsteady magnetohydrodynamic (MHD) mixed convection flow over a sphere combined with variable fluid properties. An implicit finite difference scheme, together with the quasi-linearization, is used to find non-similar solutions for the governing equations. The vanishing skin friction is prevented or at least delayed by enhancing the mixed convection in both the cases of steady and unsteady fluid flow. Both skin friction and heat transfer coefficients are found to be increasing with an increase in time or MHD parameter.

The present study is on mixed convection nanofluid flow with an exponentially decreasing velocity distribution embedded in a Darcy Forchheimer permeable medium. The nanofluid saturates the porous medium through Darcy Forchheimer relation. In a high flow situation the effect of inertia is necessary to be considered by including an additional velocity squared term in the momentum equation known as Forchheimer extension. The equations governing the flow are made dimensionless using suitable nonsimilarity transformation. The resulting coupled nonlinear partial differential equations are solved by quasilinearization technique in combination with the implicit finite difference method. Numerical computations are done for different parameters. The effect of Forchheimer, porosity, Lewis number, thermophoresis, and Brownian motion parameters on the velocity, temperature, and concentration gradient are graphically studied for the considered unsteady nanofluid flow and compared with the existing results and are found to be in good agreement.

Cardiovascular disease (CVD) is a leading cause of mortality and morbidity in developed countries. CVD is initiated by atherosclerotic lesions that reduce arterial lumen size through plaque formation and decreasing blood flow to the heart and frequently leading to severe complications. To cure these diseases drug delivery technologies modify drug release profile, absorption and distribution for the benefit of improving product efficacy and safety. Nano-medicine is an expeditiously growing science in which nanoscale range materials are used to serve the purpose of therapeutic agents. Among all the other interesting applications of nanomaterials, the nano-drug delivery system has emerged as an outstanding platform to deliver the remedial agents to a diseased site in a more controlled and targeted manner. By sitespecificity, lowering toxicity and target-oriented delivery, nanotechnology endeavours many benefits by treating frightful diseases. The current review examines nano-drug delivery systems, nanoparticles and describes recent computational simulations of magnetic targeted nano-pharmacodynamics.

Purpose The purpose of this paper is to consider axisymmetric mixed convection flow of water over a sphere with variable viscosity and Prandtl number and an applied magnetic field. Design/methodology/approach The non-similar solutions have been obtained from the origin of the streamwise co-ordinate to the point of zero skin friction using quasilinearization technique with an implicit finite-difference scheme. Findings The effect of M is not notable on the temperature and heat transfer coefficient when λ is large. The skin friction coefficient and velocity profile are enhance with the increase of MHD parameter M when λ is small. Viscous dissipation has no significant on the skin friction coefficient under MHD effect. For M=1, the movement of the slot or slot suction or slot injection do not cause any effect on flow separation. The slot suction and the movement of the slot in downstream direction delay the point of zero skin friction for M=0. Originality/value The present results are original and new for water boundary-layer flow over sphere in mixed convection flow with MHD effect and non-uniform mass transfer. So this study would be useful in analysing the skin friction and heat transfer coefficient on sphere of mixed convection flow of water boundary layer with MHD effect.

An analysis is accomplished to investigate the features of heat transfer and fluid flow of a steady laminar magnetohydrodynamics (MHD) mixed convection flow of water around a rotating sphere. The transformed governing equations of the non-similar boundary layers are solved by an implicit finite difference method along with the quasi-linearization technique. It is perceived that both the local friction coefficients in the x and y directions and the heat transfer coefficient are increasing and the point of separation is delayed with increasing magnetic parameter, suction parameter, and buoyancy force.

Non-similar solutions are found numerically to a system of coupled non-linear partial differential equations indicating, unsteady laminar water boundary layer flow over yawed cylinder using implicit finite difference scheme along with Quasi-linearization technique. The fluid properties such as viscosity and Prandtl number are considered as an inverse function of temperature. Unsteadiness is caused by upstream velocity in and directions and non-uniform mass transfer (suction/injection) which is applied through slot on the surface of the geometry. The effect of yaw angle, variable fluid properties and non-uniform mass transfer on skin friction and heat transfer coefficients is analyzed. It is found that non-uniform slot suction and downstream movement of the slot cause the point of vanishing skin friction moves downstream, but non-uniform slot injection produces the opposite result of that corresponding to the suction case. When the yaw angle increases, both the skin friction coefficient in the – direction and the heat transfer coefficient decrease but the skin friction coefficient in the – direction increases for all times. The effect of the yaw angle is very little on the point of vanishing skin friction.

PurposeThe purpose of this paper is to make an analysis to study the non‐similar solution for unsteady water boundary layer flow over sphere with the influence of temperature‐dependent viscosity, Prandtl number, non‐uniform surface mass transfer and heat transfer.Design/methodology/approachThe governing quasi‐linear partial differential equations have been solved numerically using an implicit finite difference scheme along with a quasi‐linearization technique. Non‐similar solutions have been obtained from the starting point of the stream‐wise coordinate to the point where the skin friction value vanishes.FindingsIt is observed that non‐uniform suction causes the point of vanishing skin friction to move downstream. The slot injection causes the vanishing skin friction to move upstream.Originality/valueThe effect of unsteadiness is more significant on the skin friction as compared to the heat transfer.

In the present analysis an unsteady laminar forced convection water boundary layer flow is considered. The fluid properties such as viscosity and Prandtl number are taken as variables such that those are inversely proportional to temperature. By using quasi-linearization technique the nonlinear coupled partial differential equations are linearized and the numerical solutions are obtained by using implicit finite difference scheme with the appropriate selection of step sizes. Non-similar solutions have been obtained from the starting point of the stream-wise coordinate to the point where skin friction value vanishes. The effect non-uniform mass transfer along the surface of the cylinder through slot is studied on the skin friction and heat transfer coefficients. Keywords—Boundary layer, heat transfer, non-similar solution, non-uniform mass, unsteady flow.

An analysis is performed to study the influence of nonuniform double slot suction on a steady laminar boundary layer flow over a rotating sphere when fluid properties such as viscosity and Prandtl number are inverse linear functions of temperature. Nonsimilar solutions have been obtained from the starting point of the streamwise co-ordinate to the exact point of separation. The difficulties arising at the starting point of the streamwise co-ordinate, at the edges of the slot and at the point of separation have been overcome by applying an implicit finite difference scheme in combination with the quasi-linearization technique and an appropriate selection of the finer step sizes along the stream-wise direction. The present investigation shows that the point of ordinary separation can be delayed by nonuniform double slot suction if the mass transfer rate is increased and also if the slots are positioned further downstream. In addition, the investigation reveals that double slot suction is found to be more effective compared to a single slot suction in delaying ordinary separation. As rotation parameter increase the point of separation moves upstream direction. Keywords—boundary layer, suction, mass transfer, rotating sphere.