UNSTEADY MHD BOUNDARY LAYER FLOW WITH DIFFUSION AND FIRST-ORDER CHEMICAL REACTION OVER A PERMEABLE STRETCHING SHEET WITH SUCTION OR BLOWING

In this study, unsteady MHD boundary layer flow with diffusion of chemically reactive species undergoing first-order chemical reaction over a permeable stretching sheet with suction or blowing and also with power-law variation in wall concentration is investigated. Using similarity transformation, the governing partial differential equations are converted into nonlinear self-similar ordinary differential equations. The transformed equations are then solved by the finite difference method using the quasi-linearization technique. Due to the increase in the unsteadiness parameter, the velocity initially decreases, but after a certain point it increases. A similar effect is also observed in case of concentration distribution. The increase in magnetic parameter causes a decrease in velocity and an increase in concentration. For increasing strength of applied suction both momentum and concentration boundary layer thicknesses decrease. On the other hand, applied blowing has reverse effects. Moreover, the mass transfer from the sheet is enhanced with increasing values of Schmidt number, reaction rate parameter, and also power-law exponent (related to wall concentration distribution). For high negative values of the power-law exponent, mass absorption at the sheet occurs. Moreover, due to increase of unsteadiness, this mass absorption is prevented.     

Boundary-layer analysis of power-law fluids

An exact numerical solution to the 2-D(Two-Dimensional) laminar boundary-layer equations of power-law non-newtonian fluids is obtained using a finite difference technique. No limitation has been imposed on the flow behavior index(n) or generalized Prandli number As a test case, velocity and temperature fields around a circular cylinder in crossflow were calculated The result clearly indicated that heat transfer of power-law materials is governed by shear dependent viscosity.     

AN ADHESIVE PROBLEM FOR A POWER-LAW FLUID

The flow caused by the pulling of a circular disk away from a plane rigid surface is investigated when the disk and the surface are separated by a thin layer of power-law fluid of thickness h(t). It is observed that the force exerted by the fluid on the disk is a suction force F, which tends to make the disk adhere to the rigid surface. The variation of F/k (where k is the power-law consistency index) with time t is computed for h(t) = t + 5.0 for both pseudoplastic (with power-law exponent n satisfying 0 < n < 1) and dilatant fluids (with n > 1). It is found that at a given instant, |F|/k for any pseudoplastic fluid is larger than that for a dilatant fluid. Further, the values of F for several realistic pseudoplastic fluids are computed at a given instant for both linear and quadratic variation of h(t) with t. The analysis further reveals that for a realistic pseudoplastic fluid, |F| at a given instant corresponding to a linear variation of h(t) is larger for a layer of given thickness than |F| at that instant for a layer of larger thickness.     

Laminar boundary layers on moving continuous surfaces

A parametric study of steady, compressible boundary-layer flows induced by a moving continuous flat plate or an axisymmetric body is conducted employing an implicit finite-difference numerical scheme. Exact solutions are sought in the framework of boundary-layer theory. First order compressible boundary-layer equations are solved for the flat plate case while for the axisymmetric body, the first order effect of transverse curvature is included. The similarity solutions are first obtained and a step by step integration technique marching along the body is used to find the non-similar solutions in the whole flow field. A polyester plane sheet or fiber with velocity and temperature gradients in the direction of motion is selected as a typical example. Boundary layer velocity and temperature profiles as well as frictional drag and heat transfer on the body are determined along with other relevant boundary-layer parameters. Comparisons to previous investigations are made for coefficient of friction for both planar and axisymmetric bodies, and for heat transfer on plane bodies.     

Unsteady flow and heat transfer on an accelerating surface with blowing or suction in the absence and presence of a heat source or sink

The problem of unsteady flow and heat transfer in the laminar boundary layer on a linearly accelerating surface with suction or blowing in the absence and presence of a heat source or sink is considered. The governing partial differential equations for this investigation are transformed into the non-dimensional equations by using pseudo-similarity time and pseudo-similarity coordinate. The resulting two points boundary-value problem is solved numerically by the central finite difference method associated with Newton's iteration from the initial stage (ξ=0) to a steady state (ξ=1) completely. A parametric study is performed to illustrate the effects of Prandtl number, power-law surface temperature (PLST) or power-law heat flux (PLHF), heat sink or heat source, and suction or blowing parameter on the dynamic velocity and temperature fields as well as the transient development of the skin-friction coefficients and the Nusselt number. These results are depicted graphically to display special aspects of unsteady flow and heat transfer characteristics in all time.     

Suction, viscous dissipation and thermal radiation effects on the flow and heat transfer of a power-law fluid past an infinite porous plate

This paper presents a study of the flow and heat transfer of an incompressible Ostwald de-Waele power-law fluid past an infinite porous plate, subject to suction at the plate. The power-law index n satisfies 0 < n < 1 (shear-thinning fluid only) provided that there is suction at the plate. Three cases are studied, namely, (i) the plate with constant surface temperature (CST case), (ii) the plate with prescribed surface temperature (PST case), and (iii) the plate with prescribed heat flux (PHF case). The effects of viscous dissipation and thermal radiation are also considered in the energy equation and the variations of dimensionless surface temperature and dimensionless surface temperature gradient with various parameters are graphed and tabulated.     

MHD FREE CONVECTION FLOW OF A NANOFLUID PAST A VERTICAL PLATE IN THE PRESENCE OF HEAT GENERATION OR ABSORPTION EFFECTS

This work is focused on the numerical solution of steady natural convection boundary-layer flow of a nanofluid consisting of a pure fluid with nanoparticles along a permeable vertical plate in the presence of magnetic field, heat generation or absorption, and suction or injection effects. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing boundary-layer equations of the problem are formulated and transformed into a non-similar form. The obtained equations are then solved numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. Comparisons with previously published work are performed and are found to be in excellent agreement. Representative results for the longitudinal velocity, temperature, and nanoparticle volume fraction profiles as well as the local heat transfer rates for various values of the physical parameters are displayed in both graphical and tabular forms.     

Calendering poly(vinyl chloride): Theory and experiments

An isothermal model for the calendering of power-law fluids which exhibit a slip boundary condition is presented. This model is based on lubrication approximation and Gaskell's theorxy. It Predicts sheet thickness, pressure distribution, velocity profiles, torque and power from rheological data for the melt and machine parameters. The predictions (sheet thickness, pressure distribution and torque) are compared with a set of experimental data for rigid poly(vinyl chloride)(PVC) provided to the authors by Solvay & Cie S.A.     

BLASIUS AND SAKIADIS FLOW WITH UNIFORM BLOWING OR SUCTION IN NON-NEWTONIAN POWER-LAW FLUIDS

The Blasius and Sakiadis flows of a non-Newtonian power-law fluid are considered. The plate is porous and fluid can be either injected or sucked through it. The boundary layer equations are transformed into a nondimensional form and are solved with a finite difference method. For the case of uniform suction, new results have been found, although this problem has been investigated in the past. Among them are analytical solutions for dilatant fluids of the Blasius flow and analytical solutions of the Sakiadis flow for all values of the power-law index. For the case of uniform injection, the characteristics of the flow until a separation state are investigated and discussed.     

Three-dimensional modelling of Non-Newtonian fluid flow in a coat-hanger die

The three-dimensional model of isothermal flow of power-law fluid in a coat-hanger die has been developed using finite element method. The shape of coat-hanger die used in the present model was determined according to the previous analytical design equation which is based on one-dimensional flow model in the manifold and the slot. Because uniform flow rate across the die outlet is most important to achieve uniform thickness of extruded polymer sheet or film, flow rate distribution is mainly examined to determine the valid process condition for the design equation as the design parameters are changed. The effects of fluid property in terms of power-law index and process parameters not considered in one-dimensional design equation such as die inlet size and the presence of land were analyzed. Results show that the manifold angle is the most influencing design parameter on flow rate distribution. When the material of different power-law index from design value is processed, the change of power-law index affects the uniformity of flow rate appreciably.     

FURTHER RESULTS ON HYDROMAGNETIC BOUNDARY-LAYER FLOW OF A NON-NEWTONIAN POWER-LAW FLUID OVER A CONTINUOUSLY MOVING SURFACE WITH SUCTION

The flow of a non-Newtonian, power-law conducting fluid under the effect of a constant transverse magnetic field is considered. The flow is produced by a plate moving with constant velocity in a calm fluid. The plate is porous and fluid can either be sucked or injected through it. The boundary layer equations are transformed into a nondimensional form and are solved with a finite difference method. Part of this problem has been investigated in the past but only for suction and pseudo-plastic fluids. However, all flows of the present work reach an asymptotic state and exact analytical solutions exist for Newtonian fluids. In the present work we extend the investigation to both pseudo-plastic Newtonian and dilatant fluids in both suction and injection cases.     

SLIP EFFECT ON DIFFUSION OF CHEMICALLY REACTIVE SPECIES IN BOUNDARY LAYER FLOW OVER A VERTICAL STRETCHING SHEET WITH SUCTION OR BLOWING

In the present investigation, we study the effects of slip boundary condition on the diffusion of chemically reactive species in steady boundary layer flow of viscous incompressible fluid over a vertical stretching sheet with suction or blowing. The first-order chemical reaction is considered and wall concentration varies linearly along the sheet. The self-similar equations are obtained using similarity transformations and are solved numerically using shooting method. Our study reveals that due to the increase of diffusion parameter and blowing, the velocity increases, and it decreases with suction, Schmidt number, and reaction rate parameter. Importantly, for increase of slip parameter, the boundary layer thickness increases. In contrast, the concentration at a point increases only for increasing slip and blowing, while it decreases for increase of all other parameters.     

SLIP FLOW AND HEAT TRANSFER OF A SECOND-GRADE FLUID IN A POROUS MEDIUM OVER A STRETCHING SHEET WITH POWER-LAW SURFACE TEMPERATURE OR HEAT FLUX

This article deals with the study of boundary layer flow of a second-grade fluid in a porous medium past a stretching sheet and heat transfer characteristics with power-law surface temperature or heat flux. The flow in the boundary layer is considered to be generated solely by the linear stretching of the boundary sheet adjacent to a porous medium, and boundary wall slip condition is assumed. In the energy equation effects of viscous dissipation, work done due to deformation and internal heat generation/absorption is taken into account. Closed form solutions are obtained for this problem.     

Calculation of the phase separation rate in two-phase flow of non-Newtonian power-law fluid and gas bubbles flowing inside the T- and Y-junctions using random vortex method (RVM)

In this paper, the phase separation rate in a two-phase flow of non-Newtonian power-law fluid and gas bubbles flowing inside T- and Y-junction channels with branch angles of 45° and 135° is numerically investigated. The numerical simulation is carried out using the random vortex method (RVM) for the Reynolds number of liquid phase at the inlet of the channel Re_m = 250 for various power-law indexes (n = 0.2–1.4). The result of this research represents valuable information about the effect of branch angle and also the power-law index on the phase separation. The result shows that when increasing the branch angle, the phase separation is increased, while when increasing the power-law index, the phase separation is decreased. The acceptable conformation between this study and the experimental results shows the capability of the evaluated method.     

GROUP THEORY AND DIFFERENTIAL TRANSFORM ANALYSIS OF MIXED CONVECTIVE HEAT AND MASS TRANSFER FROM A HORIZONTAL SURFACE WITH CHEMICAL REACTION EFFECTS

Viscous, laminar mixed convection boundary-layer flow over a horizontal plate, with chemical reaction, is considered. The governing equations are expressed in nondimensional form. Group theory is employed to determine the invariant solutions of these equations under a particular continuous one-parameter group. Series solutions of the transformed coupled system of equations are then generated for velocity, temperature, and concentration functions using the Differential Transform Method (DTM) with Padé approximants. The influence of thermal buoyancy parameter, species buoyancy parameter, chemical reaction parameter, order of chemical reaction, Prandtl number, and Schmidt number on the flow characteristics is evaluated in detail The obtained solutions are verified by comparison with the numerical shooting quadrature results. Applications of the study arise in sheet materials processing, bio-reactors, and catalytic systems in chemical engineering.     

A simplified method for analyzing mold filling dynamics Part I: Theory

A graphical method based on dimensional analysis is presented for estimating the injection pressure and clamp force required for injection molding amorphous polymers to form disk-shaped parts with a constant wall thickness. A procedure is suggested for estimating clamp force when the projected area of the mold cavity is smaller than the surface area of one side of the molded part. The results reported here are based on a numerical simulation of a power-law fluid filling a cold mold at a constant injection rate. The dimensionless bulk temperature and the ratios of the nonisothermal injection pressure (clamp force) to the isothermal injection pressure (clamp force) are given as functions of the dimensionless cooling time τ, the Brinkman number Br which characterizes viscous heating, the power-law exponent n, and a dimensionless temperature β which includes the inlet melt and mold wall temperatures and the temperature coefficient for viscosity.     

Viscous liquid sheets and operability bounds in extrusion coating

The steady-state extrusion coating of an extremely viscous power-law liquid from a slot and onto a moving substrate across a narrow gap is examined. A key feature of this configuration is a two-dimensional liquid sheet that bridges the gap and might be subjected to an ambient pressure drop across its wide faces. The local thickness of this sheet may be viewed as gradually thinning from the slot to the substrate, and simplified equations governing its shape are derived. Analytical solutions for the shape of the liquid sheet, and the flow within, are obtained in the limit where viscous and pressure forces dominate all others. It is found that the centerline shape of the liquid sheet is circular when a pressure drop is applied, despite the fact that surface tension forces, which typically give rise to circular shapes in narrow gaps, are neglected. Nevertheless, it is analytically shown that the stretching of the viscous-dominated liquid sheet gives rise to an effective tension that plays an analogous role to surface tension. The range of possible applied pressures is thus deduced via geometrical considerations of a circular shape constrained in the narrow gap, as in the analogous surface-tension-dominated analysis (Ruschak, 1976. Chemical Engineering Science 31, 1057); extrusion coating fails when a liquid sheet cannot be constructed successfully.     

Output Flow Determination of Reverse Flow Diverters in Reverse Flow Mode Based on Pressure Data

The key fluidic component of a pneumatic pulse jet pump is a reverse flow diverter (RFD), which consists of a driving nozzle and diffuser, with a suction gap separating the two. Because the suction gap is open to the storage tank, the mass flow through a RFD is not constant, and it is difficult to determine its pumping capacity in the reverse flow mode. For symmetrical RFDs, we experimentally investigated the effects of the driving nozzle exit diameter d₀, suction gap length d_c, liquid viscosity μ, and pipeline load impedance on the RFDs’ performance in the reverse flow mode. A dimensionless performance curve insensitive to d₀, d_c, μ, and the pipeline load impedance was found based on these experimental data. The output flows of RFDs with suction factors higher than 0.9 could be determined using this dimensionless performance curve.     

Stability of fiber spinning of power-law fluids

A stability analysis of fiber spinning of isothermal power-law fluids has been carried out. The analysis for purely viscous fluids indicates that the critical extension ratio increases with an increase in power-law constant q above 1. For q greater than approximately 1.5, very high values of critical extension ratio are obtained. A stability analysis in the presence of viscous and inertial forces indicates that for q > 1 critical extension ratio can be correlated to a quantity Rq = q ? 1 + 3Re, wherein Re is the Reynolds number. For the values of Rq greather than approximately 0.5, very high values of critical extension ratios are obtained.     

Fouling Control in a Submerged Flat Sheet Membrane System: Part I – Bubbling and Hydrodynamic Effects

Abstract

Submerged flat sheet membranes are mostly used in membrane bioreactors for wastewater treatment. The major problems for these modules are concentration polarization and subsequent fouling. By using gas‐liquid two‐phase flow, these problems can be ameliorated. This paper describes a study of the use of gas‐liquid two‐phase flow as a fouling control mechanism for submerged flat sheet membrane bioreactors. The effect of various hydrodynamic factors such as airflow rate, nozzle size, intermittent filtration, channel gap width, feed concentration, imposed flux, and the use of membrane baffles were investigated. Experiments conducted on model feeds showed that fouling reduction increased with air flow rate up to a given value and beyond this flowrate no further enhancement was achieved. The effect of bubbling was also found to increase with nozzle size at constant airflow. Using intermittent filtration as an operating strategy was found to be more effective than continuous filtration and it also reduced energy requirements. The study showed the importance of the size of the gap between the submerged flat sheet membranes. As the gap was increased from 7 mm to 14 mm, the fouling became worse and the degree of fouling reduction by two‐phase flow decreased by at least 40% based on suction pressure rise (dTMP/dt). This is the first study which has reported the effects of baffles in improving air distribution across a flat sheet submerged membrane. It was found that baffles could decrease the rate of fouling by at least a factor of 2.0 based on the dTMP/dt data, and significantly increase critical flux.