On MHD and slip flow over a rotating porous disk with variable properties

The steady laminar magnetohydrodynamics (MHD) flow of a viscous Newtonian and electrically conducting fluid over a rotating disk with slip boundary condition is investigated taken into account the variable fluid properties (density, (ρ), viscosity, (μ) and thermal conductivity, (κ)). These fluid properties are taken to be dependent on temperature. The governing equations, which are partial and coupled, are transformed to ordinary ones by utilizing the similarity variables introduced by von Karman and the resulting equation system is solved numerically by using a shooting method. The resulting velocity and temperature distributions are shown graphically for different value of parameters entering into the problem. The numerical values of the radial and tangential skin-friction coefficients and the rate of heat transfer coefficient are shown in tabular form.     

Effect of catch cup geometry on 3D boundary layer flow over the wafer surface in a spin coating

Recently, development of high technology has been required for the formation of thin uniform film in manufacturing processes of semiconductor as the semiconductor instruments become more sophisticated. Spin coating is usually used for spreading photoresist on a wafer surface. However, since rotating speed of the disk is very high in spin coating, the dropped photoresist scarers outward and reattaches on the film surface. A catch cup is set up outside the wafer in spin coating, and scattered photoresist mist is removed from the wafer edge by the exhaust flow generated at the gap between the wafer edge and the catch cup. In the dry process of a spin coating, it is a serious concern that the film thickness increases near the wafer edge in the case of low rotating speed. The purpose of this study is to make clear the effect of the catch cup geometry on the 3D boundary layer flow over the wafer surface and the drying rate of liquid film.     

Prediction of turbine blade heat transfer and aerodynamics using a new unsteady boundary layer transition model

The effects of periodic unsteady flow on heat transfer and aerodynamic characteristics, particularly on the boundary layer transition along the suction and the pressure surfaces of a typical gas turbine blade, are experimentally and theoretically investigated. Comprehensive aerodynamic and heat transfer experimental data are collected for different unsteady passing frequencies that are typical of gas turbines. To predict the effect of the impinging periodic unsteady flow on the heat transfer and the aerodynamics of turbine blades, a new unsteady boundary layer transition model is developed. The model is based on a universal unsteady intermittency function and utilizes an inductive approach that implements the results of comprehensive experimental and theoretical studies of unsteady wake development and the boundary layer flow. Three distinct quantities are identified as primarily responsible for the transition of an unsteady boundary layer: (1) the universal relative intermittency function, (2) maximum intermittency, and (3) minimum intermittency. The analysis of the experimental results and the comparison with the model prediction confirm the validity of the model and its capability to accurately predict the unsteady boundary layer transition.     

Steady flow and heat transfer of the power-law fluid over a rotating disk

This paper deals with the steady flow and heat transfer of a viscous incompressible power-law fluid over a rotating infinite disk. Assumed the thermal conductivity follows the same function as the viscosity, the governing equations in the boundary layer are transformed into a set of ordinary differential equations by generalized Karman similarity transformation. The corresponding nonlinear two-point boundary value problem was solved by multi-shooting method. Numerical results indicated that the parameters of power-law index and Prandtl number have significant effects on velocity and temperature fields. The thickness of the boundary layer decays with power-law index. The peak of the radial velocity changes slightly with power- law index. The values near the boundary are affected dramatically by the thickness of the boundary layer. With the increasing of the Prandtl number the heat conducts more strongly.     

An experimental investigation of the dissipation mechanisms in the suction side boundary layer of a turbine blade

The present work is part of an extensive experimental activity carried out by the authors in recent years aimed at investigating the boundary layer transition phenomenon in turbine blades. The large scale of the cascade and the use of advanced LDV instrumentation and precision probe traversing mechanism resulted in high degree of spatial resolution and high accuracy of measurements. The main dissipation mechanism determining the profile losses in turbomachinery blades is the work of deformation of the mean motion within the boundary layer operated by both viscous and turbulent shear stresses. In the present paper, the local viscous and turbulent deformation works have been directly evaluated from the detailed measurements of boundary layer mean velocity and Reynolds shear stress. The results show the distributions and the relative importance of the viscous and turbulent contributions to the loss production, in relation with the boundary layer states occurring along the turbine profile.     

Influences of the Exhaust Flow on the Boundary Layer Flow on the Wafer Surface in Spin Coating System

Recently, development of high technology has been required for the formation of thin uniform film in manufacturing processes of semiconductor as the semiconductor become more sophisticated. Spin coating is usually used for spreading photoresist on a wafer surface. However, since rotating speed of the disk is very high in spin coating, the dropped resist scatters outward and reattaches to the film surface. So, the scattered resist is removed by the exhaust flow generated at the gap between the wafer edge and the catch cup. It is seriously concerned that the stripes called Ekman spiral vortices appears on the disk in the case of high rotating speed and the film thickness increases near the wafer edge in the case of low rotating speed, because it prevent the formation of uniform film. The purpose of this study is to make clear the generation mechanism of Ekman spiral vortices and the influence of exhaust flow on it Moreover the influence of the catch cup geometry on the wafer surface boundary layer flow is investigated.     

Mechanical and thermal stresses in a functionally graded rotating disk with variable thickness due to radially symmetry loads

Rotating disks have many applications in the aerospace industry such as gas turbines and gears. These disks normally work under thermo mechanical loads. Minimizing the weight of such components can help reduce the overall payload in aerospace industry. For this purpose, a rotating functionally graded (FG) disk with variable thickness under a steady temperature field is considered in this paper. Thermo elastic solutions and the weight of the disk are related to the material grading index and the geometry of the disk. It is found that a disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared to a uniform thickness disk. Maximum radial stress due to centrifugal load in the solid disk with parabolic thickness profile may not be at the center unlike uniform thickness disk. Functionally graded disk with variable thickness has smaller stresses due to thermal load compared to those with uniform thickness. It is seen that for a given value of grading index, the FG disk having concave thickness profile is the lightest in weight whereas the FG disk with uniform thickness profile is the heaviest. Also for any given thickness profile, the weight of the FG disk lies in between the weights of the all-metal and the all-ceramic disks.     

Effect of convection on boundary layer structures in finite thermoelasticity

Abstract

Till now, all of the research on boundary layer structures in thermoelasticity has focused on conduction as the primary mode of heat transfer. In this article, we investigate the additional effect of convection on the deformation field boundary layer structures formed within a thin infinite slab made of a neo-Hookean material. We find that additionally introducing convection in finite thermos-elasticity shifts the boundary layer vertically, while retaining its shape.     

Drying of flowing liquid film on rotating disk

Recently,development of high technology has been required for the formation of uniform thin film
in manufacturing processes of semiconductor as the precision instruments become more
sophisticated.A method called spin coating is often used for spreading photoresist on a wafer surface
and drying photoresist film.In spin coating process,photoresist is uniformly spread on the wafer surface
by centrifugal force caused by rotating wafer.However,it is a serious concern that streaky lines,which
are caused by spiral vortices,appear on the wafer surface and prevent the formation of uniform film in
the case of high rotating speed.On the other hand,in the case of low rotating speed,a small hump of
the film is formed near the wafer edge.The main purpose of this study is to make clear the drying
characteristics of the flowing liquid film on the rotating wafer.Temperature distribution of the flowing
liquid film is captured by an infrared thermal video camera and radial gradient of the film temperature is
introduced in order to evaluate the drying characteristic of the flowing film under steady state
condition.Effects of the flow rate of the liquid film on the film temperature are investigated.The film
temperature gradually decreases in the radial direction in all cases.At low rotating speed,the radial
gradient of the film temperature is almost constant widely.On the other hand,at high rotating
speed,the radial gradient of the film temperature takes a certain maximum value.It is found that the
location of the gradient peak corresponds with the transition region of the air boundary layer,in which
spiral vortices swirl,and shifts to the inner side of the disk with an increase of the liquid flow rate.     

Three‐dimensional boundary layer flow over a rotating disk with power‐law stretching in a nanofluid containing gyrotactic microorganisms

The nanofluid model containing microorganisms over a rotating disk with power‐law stretching is constructed in this paper. The combined effects of nanoparticles and microorganisms in nanofluid are investigated by solving the governing equations numerically. The numerical solutions of the skin friction coefficient and local Nusselt number are in agreement with the corresponding previously published results. The quantities of physical interest are graphically presented and discussed in detail. It is found that the power‐law stretching index has produced profound influence on the flow as well as the heat and mass transfer.     

Absolute and convective instabilities in the incompressible boundary layer on a rotating disk with temperature-dependent viscosity

The absolute and convective instability of Von-Kármán rotating disk flow with a temperature dependence viscosity of the form μ′ = μ_∞/[1 + ?(T − T_∞)/(T_ω − T_∞)] is investigated. With the use of a spectral method, the linear stability equations are formulated and then solved numerically. Solutions have been obtained for various values of the parameter ? which controls the temperature dependence of viscosity. It is established the stability of the flow is particularly sensitive to changes in viscosity and even for small positive values of ? the flow is much more unstable compared to the constant viscosity case.     

Effect of slip on entropy generation in a single rotating disk in MHD flow

In the present study, the effect of slip on entropy generation in magnetohydrodynamic (MHD) flow over a rotating disk is investigated by semi-numerical analytical solution technique. The nonlinear governing equations of flow and thermal fields are reduced to ordinary differential equations by the Von Karman approach, then solved via differential transform method (DTM), a recently-developed, powerful analytical method. Related entropy generation equations are derived and nondimensionalized using geometrical and physical flow field-dependent parameters. For a rotating surface the form of slip introduced into the governing equations is rarefaction. For comparison, slip and no-slip regimes in the range 0.1 > Kn > 0 and their interaction with magnetic effects are investigated by minimum entropy generation. While minimizing entropy generation, equipartitioning is encountered between fluid friction irreversibility and Joule dissipation.     

Mixed free and forced convection in the incompressible boundary layer along a rotating vertical cylinder with fluid injection

We examine the steady incompressible laminar boundary layer flow along a vertical cylinder with isothermal walls. The mixed free and forced convection regime is studied while injection/suction of fluid can take place through the cylinder surface. The two-dimensional boundary layer equations are solved using an efficient finite difference scheme, and velocity and temperature profiles, as well as skin friction, heat transfer and pressure coefficients, are calculated. It is proved that fluid injection can considerably reduce the skin friction and heat transfer at the wall. Also, significant differences are reported when the present results are compared with published results for the zero mass transfer case.     

Significance of thermal radiation on dusty fluid flow over a stretching rotating disk with convective boundary condition

This paper explores the flow of dusty fluid over a stretching rotating disk with thermal radiation. Further, the convective boundary condition is considered in this modeling. The described governing equations are reduced to ordinary differential equations by using apt similarity transformations and then they are numerically solved using Runge–Kutta–Fehlberg-45 scheme. To gain a clear understanding of the current boundary layer flow problem, the graphical results of the velocity and thermal profiles, shear stresses at the disk, and Nusselt number are drawn. Results reveal that the increase in the value of the porosity parameter reduces the velocity of both particle and fluid phases. The increase in the value of the Biot number improves the temperature gradient of both particle and fluid phases. The rise in the value of the radiation parameter advances the heat transference of both phases. The rise in the value of the Biot number improves the rate of heat transfer. Finally, increasing the value of the radiation parameter improves the rate of heat transfer.     

On the effectiveness of viscous dissipation and Joule heating on steady MHD flow and heat transfer of a Bingham fluid over a porous rotating disk in the presence of Hall and ion-slip currents

The combined effects of viscous dissipation and Joule heating on steady magnetohydrodynamics (MHD) flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid over a porous rotating disk in the presence of Hall and ion-slip currents is studied. An external uniform magnetic field is applied in the z-direction and the fluid is subjected to uniform suction. Numerical solutions are obtained for the governing momentum and energy equations. Results for the details of the velocity as well as temperature are shown graphically and the numerical values of the skin friction and the rate of heat transfer are entered in tables.     

The influence of higher order effects on the vortex instability of thermal boundary layer flow in a wedge-shaped domain

We reconsider the onset of streamwise vortices in the thermal boundary layer flow induced by an inclined upward-facing heated semi-infinite surface placed within a Newtonian fluid. Particular emphasis is laid upon how the induced flow in the isothermal region outside the boundary layer affects the boundary layer itself at higher order, and how this, in turn, affects the stability criterion for the onset of vortices. We find that the stability criterion for thermal boundary layers in air is less susceptible to changes in external geometry than for boundary layers in water. In general, we conclude that the variation of the stability criterion with wedge angle (between the heated and the outer boundary surface) is too great for the theory to predict reliably where disturbances first begin to grow.     

Thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer of a steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating

Thermo-diffusion (Soret effect) and diffusion-thermo (Dufour effect) effects on combined heat and mass transfer of a steady hydromagnetic convective and slip flow due to a rotating disk in the presence of viscous dissipation and Ohmic heating is investigated. The partial differential equations governing the problem under consideration have been transformed by a similarity transformation into a system of ordinary differential equations which are solved numerically by applying the shooting method. For fluids of medium molecular weight (H₂, air), profiles of the dimensionless velocity, temperature and concentration distributions are shown graphically for various values of slip parameter γ, magnetic field parameter M, Eckert Ec, Schmidt Sc, Dufour Du and Soret Sr numbers. Finally, numerical values of physical quantities, such as the local skin friction coefficient, the local Nusselt number and the local Sherwood number are presented in tabular form.     

Effects of boundary layer on wedge-induced oblique detonation structures in hydrogen-air mixtures

Oblique detonation wave (ODW) structures are studied widely in recent years, but most of them are solved by the Euler equations without considering viscosity and then effects of boundary layer. In this study, the Navier-Stokes Equations are used to simulate the wedge-induced ODWs in hydrogen-air mixtures, and the two types of ODW transition structures at different incident Mach number M_i are analyzed to clarify the effects of viscosity and hence the boundary layer. Results show that the effect of boundary layer on ODW structures should be classified by the types of ODW transition patterns. As for the smooth transition pattern of ODW at high Mach numbers, the effect of boundary layer can be neglected, but for the abrupt transition pattern of ODW at low Mach numbers, the effect of boundary layer is large and it changes the ODW structure greatly. Resulting from the interaction of shock and boundary layer, a recirculation zone is formed within the viscous ODW layer at M_i = 7, which leads to the phenomenon that the straight oblique shock wave evolves into two sections, with the downstream one having a larger shock angle. Additionally, the corresponding transition position moves upstream, and the initiation length becomes only one third of that in inviscid ODW. The great importance of considering viscosity in ODW simulations and future designs of combustor of oblique detonation engine has been addressed.     

On the sliding and profile of a liquid droplet on a rotating disk

This theoretical and experimental study was conducted to investigate the critical condition at which a liquid droplet starts to move on a rotating disk. The critical rotational speed ω was theoretically calculated based on the force balance between the surface tension and the centrifugal force, where ω was experimentally measured for each combination between three kinds of test plates and test liquids. The movements of droplets were judged from the careful observation of infinitesimal motion of the three‐phase contact line. The calculated rotational speeds agreed well with measured ones for arbitrary contact angle when the droplets were set on the plate. The three‐dimensional surface profiles of droplets were calculated from the approximate Laplace equation in which the contact line was assumed as the combination of two ellipses with different ratio of measure to minor axis. The measured profiles on the rotating disk were approximated well by the method proposed in this study. © 2009 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20276     

Effect of variable viscosity on mixed convection boundary layer flow over a vertical surface embedded in a porous medium

The steady mixed convection boundary layer flow over a vertical impermeable surface embedded in a porous medium when the viscosity of the fluid varies inversely as a linear function of the temperature is studied. Both cases of assisting and opposing flows are considered. The transformed boundary layer equations are solved numerically by a finite difference method. Numerical results for the flow and heat transfer characteristics are obtained for various values of the mixed convection parameter ε and the variable viscosity parameter θ_e. It has been found that in the opposing flow case, dual solutions exist and boundary separation occurs.