Equations of motion for a rotor blade,including gravity,pitch action and rotor speed variations

This paper extends Hodges–Dowell's partial differential equations of blade motion, by including the effects from gravity, pitch action and varying rotor speed. New equations describing the pitch action and rotor speeds are also derived. The physical interpretation of the individual terms in the equations is discussed. The partial differential equations of motion are approximated by ordinary differential equations of motion using an assumed mode method. The ordinary differential equations are used to simulate a sudden pitch change of a rotating blade. This work is a part of a project on pitch blade interaction, and the model will be extended to include an aerodynamic model and be used for analysis of basic properties of pitch blade interaction. Copyright © 2007 John Wiley & Sons, Ltd.     

Thermocapillarity and magnetic field effects in a thin liquid film on an unsteady stretching surface

In this paper, the effects of thermocapillarity and a magnetic field on the flow and heat transfer in a liquid film over an unsteady elastic stretching surface is analyzed. Similarity transformations are used to transform the governing equations to a set of coupled ordinary differential equations. The differential equations are solved analytically by the homotopy analysis method (HAM). The effects of various parameters in this study are discussed and presented graphically.     

Symmetry analysis of MHD Casson fluid flow for heat and mass transfer near a stagnation point over a linearly stretching sheet with variable viscosity and thermal conductivity

In this article, the impacts of variable viscosity and thermal conductivity on magnetohydrodynamic, heat transfer, and mass transfer flow of a Casson fluid are analyzed on a linearly stretching sheet inserted in a permeable medium along with heat source/sink and viscous dissipation. To reduce the ascendant partial differential equations into ordinary differential equations, Lie group transformation is utilized. Further, the fourth-order Runge–Kutta strategy is utilized to solve the ordinary differential equations numerically. The numerical results obtained for various parameters by employing coding in MATLAB programming are investigated and considered through graphical representation and tables. We anatomize the impacts of distinctive parameters on velocity, temperature, and concentration distributions.     

Influence of heat transfer on the nonorthogonal stagnation point flow of a third‐order fluid towards a stretching surface

The present article looks at the theoretical analysis of a steady stagnation‐point flow with heat transfer of a third‐order fluid towards a stretching surface. The formulation of the problem has been carried out for a third order fluid and constructed partial differential equations are rehabilitated into ordinary differential equations. The consequential ordinary differential equations are solved analytically using the homotopy analysis method (HAM). Graphical illustrations are shown for various parameters involved in the flow equations. Numerical values of skin friction coefficients and heat flux are computed and presented through tables. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21042     

Non‐Darcian Unsteady Flow of a Micropolar Fluid over a Porous Stretching Sheet with Thermal Radiation and Chemical Reaction

An analysis is presented to investigate the effects of a chemical reaction on an unsteady flow of a micropolar fluid over a stretching sheet embedded in a non‐Darcian porous medium. The governing partial differential equations are transformed into a system of ordinary differential equations by using similarity transformation. The resulting nonlinear coupled differential equations are solved numerically by using a fourth‐order Runge–Kutta scheme together with shooting method. The influence of pertinent parameters on velocity, angular velocity (microrotation), temperature, concentration, skin friction coefficient, Nusselt number, and Sherwood number has been studied and numerical results are presented graphically and in tabular form. Comparisons with previously published work are performed and the results are found to be in excellent agreement. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21090     

Effect of heat generation or absorption on thermophoretic free convection boundary layer from a vertical flat plate embedded in a porous medium

This paper is focused on the study of coupled heat and mass transfer by boundary-layer free convection over a vertical flat plate embedded in a fluid-saturated porous medium in the presence of thermophoretic particle deposition and heat generation or absorption effects. The governing partial differential equations are transformed into ordinary differential equations by using special transformations. The resulting similarity equations are solved numerically by an efficient implicit tri-diagonal finite-difference method. Comparisons with previously published work are performed and the results are found to be in excellent agreement. Many results are obtained and a representative set is displayed graphically to illustrate the influence of the heat generation or absorption coefficient, buoyancy ratio and the Lewis number on the temperature and concentration profiles and the wall thermophoretic deposition velocity.     

Micropolar fluid flow towards a stretching/shrinking sheet in a porous medium with suction

This paper presents a numerical analysis of a micropolar fluid flow towards a permeable stretching/shrinking sheet in a porous medium. The governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, before being solved numerically by a finite-difference scheme known as the Keller-box method. The effects of the governing parameters on the fluid flow and heat transfer characteristics are illustrated graphically. It is found that dual solutions exist for the shrinking case, whereas for the stretching case, the solution is unique.     

The influence of thermal radiation on MHD flow of a second grade fluid

The present analysis deals with the steady magnetohydrodynamic (MHD) flow of a second grade fluid in the presence of radiation. By means of similarity transformation, the arising non-linear partial differential equations are reduced to a system of four coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. The effects of various involved interesting parameters on the velocity and temperature fields are shown and discussed.     

Magnetohydrodynamic Slip Flow and Heat Transfer in a Porous Medium over a Stretching Cylinder: Homotopy Analysis Method

This study considers magnetohydrodynamic flow and heat transfer outside a hollow stretching cylinder immersed in a fluid saturated porous medium of sparse distribution of particles with high permeability. Partial slip boundary conditions for the velocity and temperature fields are assumed at the stretching surface of the cylinder. Using similarity transformations, the nonlinear partial differential equations governing the flow and heat transfer are converted into nonlinear ordinary differential equations which are then solved by the homotopy analysis method. The effects of the pertinent parameters on the velocity and temperature profiles are investigated and discussed graphically.     

Heat transfer in a second grade fluid through a porous medium from a permeable stretching sheet with non-uniform heat source/sink

In the present article an analysis is carried out to study the boundary layer flow and heat transfer characteristics of a second grade, non-Newtonian fluid through a porous medium. The stretching sheet is assumed to be permeable so that suction effects come into play. The effects of viscous dissipation, non-uniform heat source/sink on heat transfer are addressed. The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. Analytical solutions are obtained for the resulting boundary value problems. The effects of viscous dissipation and non-uniform heat source/sink, Prandtl number, Eckert number and suction/injection on heat transfer are shown in several plots for two different heating processes (CST and PST cases). Dimensionless surface temperature gradient is tabulated for various values of the governing the parameters.     

Nonlinear radiation on Maxwell fluid in a convective heat transfer with viscous dissipation and activation energy

The present analysis addresses linear and nonlinear radiation effects in hydrodynamic viscous Maxwell fluid flow on a unidirectional stretching surface through viscous dissipation. The relaxation effect is considered in the mathematical model, which elucidates mass transport mechanisms under binary chemical reaction and activation energy. Mathematical modeling contains nonlinear partial differential equations using boundary conditions. Appropriate transformations convert the partial differential equations into ordinary differential equations. Numerical solutions for regular differential equations are brought by Runge–Kutta–Fehlberg numerical quadrature and a shooting method with a tolerance level of 10⁻⁹. The influence of physical variables, such as Deborah relaxation number, rotation parameter, Biot number, activation energy parameter, reaction rate parameter, Eckert number, and Prandtl number are investigated. Increasing the Biot number improves the temperature region in the boundary layer. With high rotation, the increasing Deborah number enhances the fluid temperature substantially throughout the boundary layer.     

Convection heat and mass transfer in a hydromagnetic flow of a second grade fluid in the presence of thermal radiation and thermal diffusion

The convection heat and mass transfer in a hydromagnetic flow of a second grade fluid past a semi-infinite stretching sheet in the presence of thermal radiation and thermal diffusion are considered. The governing coupled non-linear partial differential equations describing the flow problem are transformed into non-linear ordinary differential equations by method of similarity transformation. The resulting similarity equations are solved numerically using Runge-Kutta shooting method. The results are presented as velocity, temperature and concentration fields for different values of parameters entering into the problem. The skin friction, rate of heat transfer and mass transfer are presented numerically in tabular form. In addition, the results obtained showed that these parameters have significant influence on the flow, heat and mass transfer.     

Numerical investigation of dual solutions for double diffusive convection from a permeable horizontal flat plate

Numerical simulations are carried out to study the simultaneous effects of thermal and concentration diffusions on a mixed convection boundary layer flow over a permeable horizontal flat plate with suction/injection in a viscous incompressible fluid. The non-linear coupled partial differential equations governing the flow, thermal and concentration fields are first transformed into a set of non-linear coupled ordinary differential equations by a set of suitable similarity transformations. The resulting system of coupled non-linear differential equations is solved using shooting method by converting into initial value problem. In this method, system of equations is converted into a set of first order system which is solved by fourth order Runge–Kutta method. Flows with both assisting and opposing buoyancy forces are considered in the present investigation. The study reveals that dual solutions of velocity, temperature and concentration profiles exist for certain values of suction/injection and buoyancy parameters. Suction/injection parameter, Prandtl and Schmidt numbers strongly affect thermal and concentration boundary layers, respectively.     

Inherent irreversibility in a reactive and radiating magnetic nanoliquid film along a slippery inclined heated surface with convective cooling

In this study, the electrically conducting nanoliquid film on a slippery inclined heated surface with convective cooling under the effect of the reactive and radiating magnetic field is numerically investigated. The nonlinear constructing governing differential equations are then converted into a set of dimensionless ordinary differential equations by similarity variables, and the systems are numerically solved by shooting method together with the Runge‐Kutta method. Some of the parameters involved in the flow system are then examined graphically and discussed with respect to the velocity, temperature, entropy generation rate, skin friction, Nusselt number, and the Bejan number profile.     

Entropy and radiation on a pipe MHD flow with variable viscosity

The present work, the entropy generation due to radiation and variable viscosity magnetohydrodynamic effects with a porous medium in a circular pipe, has been obtained and studied numerically. The governing continuity, momentum, and energy equations in cylindrical coordination are converted into a system of nonlinear ordinary differential equations by means of similarity transformation. The resulting system of coupled nonlinear ordinary differential equations is solved numerically by a Runge-Kutta method and shooting technique. Numerical results are presented for velocity, temperature profiles, pressure profile, entropy generation rates, and Bejan number for different physical parameters of the problem. Also, the effects of the pertinent parameters on the skin friction and the rate of heat transfer are obtained and discussed numerically and illustrated graphically.     

Three-dimensional viscous flow and heat transfer over a permeable shrinking sheet

This paper presents a numerical analysis of a steady three-dimensional fluid flow and heat transfer towards a permeable shrinking sheet. The governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, which are then solved numerically by a shooting method. The effects of the governing parameters on the skin friction and heat transfer from the surface of the shrinking surface are illustrated graphically. It is found that dual solutions exist for the shrinking case. A comparison with known results from the open literature has been done and it is shown to be in excellent agreement.     

Heat transfer in a liquid film over an unsteady stretching sheet

The effects of viscous dissipation, non-uniform heat source/sink, magnetic field, and thermal radiation on heat transfer characteristics of a thin liquid film flow over an unsteady stretching sheet are analyzed. A similarity transformation is used to reduce the governing time dependent momentum and energy equations into non-linear ordinary differential equations. The resulting differential equations with the appropriate boundary conditions are solved by an efficient shooting algorithm with fourth order Runge–Kutta technique. The effects of the physical parameters on the flow and heat transfer characteristics are presented through graphs and analyzed. The numerical results for the wall temperature gradient (Nusselt number) are calculated and presented through tables. Also, the effects of the physical parameters on the heat transfer characteristics are brought out: suggestions are made for efficient cooling. Furthermore, the limiting cases are obtained and are found to be in good agreement with the previously published results.     

Effect of thermal radiation on magnetohydrodynamic three‐dimensional motion of a nanofluid past a shrinking surface under the influence of a heat source

An analytical technique known as the homotopy analysis method is used to acquire solutions for magnetohydrodynamic 3‐D motion of a viscous nanofluid over a saturated porous medium with a heat source and thermal radiation. The governing nonlinear partial differential equations are changed to ordinary differential equations employing appropriate transformations. Validation of the present result is done with the help of error analysis for flow and temperature. The influences of pertinent parameters on momentum, energy, and Nusselt number are studied and discussed. The major findings are: the velocity of the nanofluid is affected by the nanoparticle volume fraction and the thickness of the thermal boundary layer becomes thinner and thinner subject to sink, whereas the effect is revered in case of the source.     

Analytical and numerical solutions of heat and mass transfer of boundary layer flow in the presence of a transverse magnetic field

The present investigation aims to study the effect of a transverse magnetic field with the presence of an adverse pressure gradient on the two‐dimensional laminar incompressible boundary layer flow over a flat plate. Using appropriated similarity transformations, the partial differential equations governing the studied problems are transformed into the ordinary nonlinear differential equations. Thereafter, these equations are solved numerically and analytically using the fourth‐order Runge‐Kutta method featuring shooting technique and the Adomian decomposition method, respectively. Obtained results reveal an excellent agreement between analytical and numerical data for temperature and concentration profiles.     

Bioconvection in a squeezing flow of a micropolar fluid in a horizontal channel

The bioconvection flow of an incompressible micropolar fluid containing microorganisms between two infinite stretchable parallel plates is considered. A mathematical model, with a fully coupled nonlinear system of equations describing the total mass, momentum, thermal energy, mass diffusion, and microorganisms is presented. The governing equations are reduced to a set of nonlinear ordinary differential equations with the help of suitable transformations. The resulting nonlinear ordinary differential equations are linearized using successive linearization method, and the resulting system of linear equations is solved using the Chebyshev collocation method. The detailed analysis illustrating the influences of various physical parameters, such as the micropolar coupling number, squeezing parameter, the bioconvection Schmidt number, Prandtl numbers, Lewis number, and bioconvection Peclet number on the velocity, microrotation, temperature, concentration and motile microorganism distributions, skin friction coefficient, Nusselt number, Sherwood number, and density number of motile microorganism, is examined. The influence of the squeezing parameter is to increase the dimensionless velocities and temperature and to decrease the local Nusselt number and local Sherwood number. The density number of motile microorganism is decreasing with squeezing parameter, bioconvection Lewis number, bioconvection Peclet number, and bioconvection Schmidt number.