Transient electro-osmotic and pressure driven flows of two-layer fluids through a slit microchannel

By method of the Laplace transform, this article presents semi-analytical solutions for transient electroosmotic and pressure-driven flows (EOF/PDF) of two-layer fluids between microparallel plates. The linearized Poisson-Boltzmann equation and the Cauchy momentum equation have been solved in this article. At the interface, the Maxwell stress is included as the boundary condition. By numerical computations of the inverse Laplace transform, the effects of dielectric constant ratio ε , density ratio ρ , pressure ratio p, viscosity ratio μ of layer II to layer I, interface zeta potential difference △ψ, interface charge density jump Q, the ratios of maximum electro-osmotic velocity to pressure velocity α , and the normalized pressure gradient B on transient velocity amplitude are presented.We find the velocity amplitude becomes large with the interface zeta potential difference and becomes small with the increase of the viscosity. The velocity will be large with the increases of dielectric constant ratio; the density ratio almost does not influence the EOF velocity. Larger interface charge density jump leads to a strong jump of velocity at the interface. Additionally, the effects of the thickness of fluid layers (h1 and h2 ) and pressure gradient on the velocity are also investigated.     

GPU accelerated simulations of three-dimensional flow of power-law fluids in a driven cube

Newtonian fluid flow in two- and three-dimensional cavities with a moving wall has been studied extensively in a number of previous works. However, relatively a fewer number of studies have considered the motion of non-Newtonian fluids such as shear thinning and shear thickening power law fluids. In this paper, we have simulated the three-dimensional, non-Newtonian flow of a power law fluid in a cubic cavity driven by shear from the top wall. We have used an in-house developed fractional step code, implemented on a Graphics Processor Unit. Three Reynolds numbers have been studied with power law index set to 0.5, 1.0 and 1.5. The flow patterns, viscosity distributions and velocity profiles are presented for Reynolds numbers of 100, 400 and 1000. All three Reynolds numbers are found to yield steady state flows. Tabulated values of velocity are given for the nine cases studied, including the Newtonian cases.     

Non-isothermal developing flow of a power-law fluid in a converging slit

A numerical solution of the developing non-isothermal flow of a generalised power-law fluid in a slightly converging slit is presented, a problem which is relevance to some polymer processing operations.The results are presented in graphical form. They indicate that inertia forces and kinetic energy increases along the slit and these factors affect the development of the velocity and stress distributions. The effect of other parameters on the process is also brought out.     

Mass transfer during catalytic reaction in electroosmotically driven flow in a channel microreactor

Analytical solution for concentration profile in a microreactor is obtained during heterogeneous catalytic reaction. Reaction occurs in rectangular microchannel with catalyst-coated walls. Flow is induced electroosmotically in the microchannel. A general solution is obtained for first order reaction using a power series solution. Profiles of conversion, cup-mixing concentration of reactant, etc. and variation of Sherwood number is analyzed as function of operating variables. Analytical solution is compared with numerical results.     

Electroosmotic flow in slit microchannel containing salt-free solution

A theoretical study on the electroosmotic flow through a uniformly charged planar slit microchannel containing a salt-free solution is presented. Based on the exact analytical solutions for the electric potential distribution and the transient electroosmotic flow velocity, a systematic parametric study on the characteristics of the transient electroosmotic flow is then investigated. The results show that the general behavior of electroosmotic flow in a planar slit is similar to that in a cylindrical capillary; however, the characteristic time to reach the steady-state flow, the electroosmotic mobility and the correction factor to the Smoluchowski equation in a slit are larger than those in a cylinder with its diameter equal to the slit width. Furthermore, the osmotic pressure of counterion between two equally charged planar surfaces immersed in a salt-free solution is found to be always repulsive. In addition, the implications of these results on electroosmosis in a microchannel are discussed.     

Laminar flow of power-law fluids past a rotating cylinder

In this work, the continuity and momentum equations have been solved numerically to investigate the flow of power-law fluids over a rotating cylinder. In particular, consideration has been given to the prediction of drag and lift coefficients as functions of the pertinent governing dimensionless parameters, namely, power-law index (1 ≥ n ≥ 0.2), dimensionless rotational velocity (0 ≤ α ≤ 6) and the Reynolds number (0.1 ≤ Re ≤ 40). Over the range of Reynolds number, the flow is known to be steady. Detailed streamline and vorticity contours adjacent to the rotating cylinder and surface pressure profiles provide further insights into the nature of flow. Finally, the paper is concluded by comparing the present numerical results with the scant experimental data on velocity profiles in the vicinity of a rotating cylinder available in the literature. The correspondence is seen to be excellent for Newtonian and inelastic fluids.     

Electroviscous effects in steady fully developed flow of a power-law liquid through a cylindrical microchannel

Electroviscous effects in steady, fully developed, pressure-driven flow of power-law liquids through a uniform cylindrical microchannel have been investigated numerically by solving the Poisson–Boltzmann and the momentum equations using a finite difference method. The pipe wall is considered to have uniform surface charge density and the liquid is assumed to be a symmetric 1:1 electrolyte solution. Electroviscous resistance reduces the velocity adjacent to the wall, relative to the velocity on the axis. The effect is shown to be greater when the liquid is shear-thinning, and less when it is shear-thickening, than it is for Newtonian flow. For overlapping electrical double layers and elevated surface charge density, the electroviscous reduction in the near-wall velocity can form an almost stationary (zero shear) layer there when the liquid is shear-thinning. In that case, the liquid behaves approximately as if it is flowing through a channel of reduced diameter. The induced axial electrical field shows only a weak dependence on the power-law index with the dependence being greatest for shear-thinning liquids. This field exhibits a local maximum as surface charge density increases from zero, even though the corresponding electrokinetic resistance increases monotonically. The magnitude of the electroviscous effect on the apparent viscosity, as measured by the ratio of the apparent and physical consistency indices, decreases monotonically as the power-law index increases. Thus, overall, the electroviscous effect is stronger in shear-thinning, and weaker in shear-thickening liquids, than it is when the liquid is Newtonian.     

Transient spherical flow of non-newtonian power-law fluids in porous media

The transient spherical flow behavior of a slightly compressible non-Newtonian, power-law fluids in porous media is studied. A nonlinear partial differential equation of parabolic type is derived. The diffusivity equation for spherical flow is a special case of the new equation. We obtain analytical, asymptotic and approximate solutions by using the methods of Laplace transform and weighted mass conservation. The structures of asymptotic and approximate solutions are similar, which enriches the theory of one-dimensional flow of non-Newtonian fluids through porous media.     

Combined convection in power-law fluids along a nonisothermal vertical plate in a porous medium

The problem of combined convection from vertical surfaces in a porous medium saturated with a power-law type non-Newtonian fluid is investigated. The transformed conservation laws are solved numerically for the case of variable surface heat flux conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented. The viscosity index ranged from 0.5 to 2.0.     

Electroosmotic flow of Eyring fluid in slit microchannel with slip boundary condition

In consideration of the electroosmotic flow in a slit microchannel, the con-stitutive relationship of the Eyring fluid model is utilized. Navier＇s slip condition is used as the boundary condition. The governing equations are solved analytically, yielding the velocity distribution. The approximate expressions of the velocity distribution are also given and discussed. Furthermore, the effects of the dimensionless parameters, the electrokinetic parameter, and the slip length on the flow are studied numerically, and appropriate conclusions are drawn.     

Boundary layer flow at a three-dimensional stagnation point in power-law non-Newtonian fluids

Similarity solutions for the three-dimensional flow and heat transfer of a power-law fluid near a stagnation point of an isothermal surface are presented. The results of the numerical integrations are given in tables and shown on graphs for some different values of the power-law index n, geometric parameter c, and the Prandtl number Pr. Whenever possible, these results are compared with available analytical solutions and found to be highly accurate.     

Electromagnetohydrodynamic (EMHD) flow of fractional viscoelastic fluids in a microchannel

This study investigates the electromagnetohydrodynamic (EMHD) flow of fractional viscoelastic fluids through a microchannel under the Navier slip boundary condition. The flow is driven by the pressure gradient and electromagnetic force where the electric field is applied horizontally, and the magnetic field is vertically (upward or downward). When the electric field direction is consistent with the pressure gradient direction, the changes of the steady flow rate and velocity with the Hartmann number Ha are irrelevant to the direction of the magnetic field (upward or downward). The steady flow rate decreases monotonically to zero with the increase in Ha. In contrast, when the direction of the electric field differs from the pressure gradient direction, the flow behavior depends on the direction of the magnetic field, i.e., symmetry breaking occurs. Specifically, when the magnetic field is vertically upward, the steady flow rate increases first and then decreases with Ha. When the magnetic field is reversed, the steady flow rate first reduces to zero as Ha increases from zero. As Ha continues to increase, the steady flow rate (velocity) increases in the opposite direction and then decreases, and finally drops to zero for larger Ha. The increase in the fractional calculus parameter α or Deborah number De makes it take longer for the flow rate (velocity) to reach the steady state. In addition, the increase in the strength of the magnetic field or electric field, or in the pressure gradient tends to accelerate the slip velocity at the walls. On the other hand, the increase in the thickness of the electric double-layer tends to reduce it.

     

Axisymmetric stagnation point flow and mass transfer for power-law fluids

The boundary-layer equations for axisymmetric stagnation point flow of a power-law fluid are solved by a similarity transformation, and values of the wall shear rate are obtained. Theoretical expressions for local and average Sherwood numbers are derived from the convective diffusion equation for systems with high Schmidt numbers. The results can be used to predict diffusion coefficients of dilute species in fluids with specified power-law characteristics.     

Pressure effects on electroosmotic flow of power-law fluids in rectangular microchannels

In this paper, the fully developed electroosmotic flow of power-law fluids in rectangular microchannels in the presence of pressure gradient is analyzed. The electrical potential and momentum equations are numerically solved through a finite difference procedure for a non-uniform grid. A complete parametric study reveals that the pressure effects are more pronounced at higher values of the channel aspect ratio and smaller values of the flow behavior index. The Poiseuille number is found to be an increasing function of the channel aspect ratio for pressure assisted flow and a decreasing function of this parameter for pressure opposed flow. It is also observed that the Poiseuille number is increased by increasing the zeta potential. Furthermore, the results show that an increase in the flow behavior index results in a lower flow rate ratio, defined to be the ratio of the flow rate to that of a Newtonian fluid at the same conditions. Moreover, whereas the flow rate ratio in the presence of an opposed pressure gradient is smaller than that of a favorable pressure force for shear thinnings, the opposite is true for shear-thickening fluids.     

Gravity-driven laminar film flow of power-law fluids along vertical walls

A theoretical analysis is presented which brings steady laminar film flow of power-law fluids within the framework of classical boundary layer theory. The upper part of the film, which consists of a developing viscous boundary layer and an external inviscid freestream, is treated separately from the viscous dominated part of the flow, thereby taking advantage of the distinguishing features of each flow region. It is demonstrated that the film boundary layer developing along a vertical wall can be described by a generalized Falkner-Skan type equation originally developed for wedge flow. An exact similarity solution for the velocity field in the film boundary layer is thus made available.Downstream of the boundary layer flow regime the fluid flow is completely dominated by the action of viscous shear, and fairly accurate solutions are obtained by the Von Karman integral method approach. A new form of the velocity profile is assumed, which reduces to the exact analytic solution for the fully-developed film. By matching the downstream integral method solution to the upstream generalized Falkner-Skan similarity solution, accurate estimates for the hydrodynamic entrance length are obtained. It is also shown that the flow development in the upstream region predicted by the approximate integral method closely corresponds to the exact similarity solution for that flow regime. An analytical solution of the resulting integral equation for the Newtonian case is compared with previously published results.     

Stability analysis of gravity driven shear flows with free surface for power-law fluids

Summary We study the stability of thin films of fluids subject to gravity along inclined planes, obeying a power-law constitutive relation of the Ostwald-de Waele type. A first analysis, in which the inertia terms are ignored, shows such flow to be stable against small, linear perturbations; a second analysis, in which the inertia terms are included, proves that there are stable and unstable regimes that are separated by a critical Ostwald-de Waele number O. Numerical computations for selected values of O demonstrate the decay and growth rate behavior of some finite amplitude disturbances. Received 12 May 1997; accepted for publication 23 July 1997     

Axial static pressure measurements of water flow in a rectangular microchannel

An experimental investigation of water flow through an aluminum rectangular microchannel with a hydraulic diameter of 169 μm was conducted over a Reynolds number (based upon mean velocity and hydraulic diameter) range from 230 to 4,740. Pressure measurements were simultaneously acquired at eight different axial locations within the channel along with pressure measurements in the inlet and outlet ports. The 27 μm pressure taps were more densely packed near the channel entrance in order to study the developing flow region. The average Poiseuille number for laminar flows was 86.4, which is in excellent agreement with the theoretical value of 86.9. The average critical Reynolds number was found to be 2,370. The limited turbulent friction factor data were in good agreement with the Haaland equation. The inlet to the channel was not well rounded and pressure distributions near the channel entrance show a region of pressure recovery. Entrance length and some minor loss coefficient data were not in agreement with theory, but the cause of these deviations were primarily a function of the inlet geometry and pressure recovery in the microchannel rather than a microscale effect.     

Downstream pressure and elastic wall reflection of droplet flow in a T-junction microchannel

This paper discusses pressure variation on a wall during the process of liquid flow and droplet formation in a T-junction microchannel. Relevant pressure in the chan-nel, deformation of the elastic wall, and responses of the droplet generation are analyzed using a numerical method. The pressure difference between the continuous and dis-persed phases can indicate the droplet-generation period. The pressure along the channel of the droplet flow is affected by the position of droplets, droplet-generation period, and droplet escape from the outlet. The varying pressures along the channel cause a nonuniform deformation of the wall when they are elastic. The deformation is a vibration and has the same period as the droplet generation arising from the process of droplet formation.