Effect of Wall Roughness on Laminar Flow of Bingham Plastic Fluids through Microtubes

Abstract

Contributed by the Fluids Engineering Division for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering Division June 6, 2003; revised manuscript received March 4, 2004. Associate Editor: P. Signier. Miniaturization of traditional devices is in high demand for micro-electro-mechanical systems (MEMS). Examples include but are not limited to optics, communication and information systems, fluidics, biotechnology, medicine, automotive, and aerospace. An area of interest in several engineering fields is the control of fluid flow within microchannels and microtubes. The applications include, but are not limited to, fields such as vibration control of structures and systems using small devices, reactors for modification and separation of biological cells, energy systems as a mobile power supply, heat exchangers for micro and macro devices, and propulsion engines 123. In recent years, there has been growing attention given to the liquid flow in microchannels in parallel with the development of miniaturized devices and systems. The understanding of flow characteristics, such as velocity distribution and pressure loss is necessary in design and process control of microfluidic devices. Peng, Peterson, and Wang 4 experimentally studied the flow characteristics of water flowing through rectangular microchannels having hydraulic diameters of 0.133 to 0.367 mm and height to width ratios of 0.333 to 1. Their results indicated that the laminar flow transition occurred for the range of the Re number between 200 and 700. They also claimed that friction behavior for both laminar and turbulent flow depart from classical thermofluid correlations, and the friction factor is proportional to Re−1.98 rather than Re−1. Mala and Li 5 investigated the water flow through the microtubes with diameters ranging from 50 to 254 μm. In terms of flow friction, they observed significant departure from the conventional laminar flow theory. They also proposed a roughness-viscosity model to interpret the data. Weilin, Mala, and Li 6 conducted experiments to investigate the flow characteristics of water through silicon microchannels with hydraulic diameters ranging from 51 to 169 μm. They also reported considerable deviations from conventional theory. They concluded that the measured higher-pressure gradient and flow friction could be due to the effect of surface roughness of the microchannels. Judy, Maynes, and Webb 7 carried out experiments for pressure-driven liquid flow through round and square microchannels fabricated from fused silica and stainless steel. They argued that there was no distinguishable deviation from the conventional laminar flow theory for microchannels they tested. Papautsky 8 experimentally investigated the flow of water through microchannels with widths ranging from 150 to 600 μm, heights from 22.7 to 26.35 μm, and with a surface roughness of about 3.3×10−4 μm. Their results show a friction factor increase of about 20% above macroscale theoretical values within 0.001<Re<10. A recent literature review of microchannel flows has been done by Garimella and Sobhan 9. A class of materials exhibits little or no deformation up to a certain level of stress, called the yield stress. These materials are often known as Bingham plastics. Paints, slurries, pastes, and some food substances such as margarine, mayonnaise, and ketchup are good examples of Bingham plastics. Although there are numerous studies on Bingham plastic flow in macrosized channels, there is no comprehensive study dealing with the behavior of fluids that exhibit Bingham plastic properties. The present work is a preliminary study of the flow behavior of non-Newtonian fluid through microtubes using Bingham plastic constitutive model. This study focuses on the effects of the wall roughness and yield stress on the flow behavior. The analytical solution of the laminar Bingham plastic fluid is introduced, and a roughness-viscosity model proposed by Mala 10 is adapted to account for variation of dynamic viscosity of Bingham fluid across a microtube. The governing differential equation describing the flow in the microtubes is solved numerically using a finite difference method (FDM). The effect of tube wall roughness on the laminar flow in macroscale circular tubes has been ignored and for a Newtonian fluid the friction factor is assumed to be a function of only the Reynolds number. However, the presence of surface roughness affects the laminar velocity profile when the fluid is flowing through microscale tubes or channels. This phenomenon has been illustrated by a number of experiments and a comprehensive review can be found in the literature 101112. In order to consider the effects of surface roughness on laminar flow in microtubes, Mala 10 proposed a roughness-viscosity function based on the Merkle-Kubota-Ko 11 modified viscosity model as follows: (1)μRμ0=A Reε rε 1−exp−ReεRerε2,where μR is the roughness viscosity, ε is the wall roughness, and the roughness Reynolds number is given by Reε=ε2/vdu/drr=R and the coefficient A is defined as (2)A=0.1306Rε0.3693 expRe6×10−5 Rε−0.0029.The analytical model of Merkle, Kubota, and Ko 11 describes the manner in which distributed surface roughness affects transition from laminar to turbulent regime, and pictures an additional momentum transport near the wall that augments the transfer of momentum by molecular viscosity. This additional momentum transfer was taken into account by means of an effective viscosity that was large near the wall and diminishes to the molecular viscosity far from the wall. By adding the roughness viscosity in the momentum equation in a manner similar to the eddy viscosity in turbulent flow, the momentum equation becomes (3)−τy+μ0+μR dudr=−12 r dpdz.By dividing both sides of Eq. (3) by μ0 and combining with Eq. (1) one can obtain (4)1+A Reε rε 1−exp−ReεRerε2 dudr−12μ0dpdzr−τyμ0=0.Equation (4) can be expressed in terms of nondimensional parameters. By multiplying both sides of Eq. (4) by D2ρ/μ0, and after some manipulations one obtains (5)81+A Reεr¯ du¯dr¯r¯=1 r¯ε¯ 1−exp−εr¯ du¯dr¯r¯=12 du¯dr¯−Re dp¯dz¯ r−¯4 HeRe=0,where the Reynolds number is Re=ρUmDμ0,the Hedsrom number is He=D2ρτyμ02,and u¯=uUm,r¯=rD/2,ε¯=εD,p¯=p12ρUm2.As can be seen from the selected dimensionless parameters, Re represents the Newtonian behavior of the fluid and tube geometry, and He represents the effect of yield stress. Equation (5) is a modified momentum equation that includes the effect of tube wall roughness in a laminar flow of non-Newtonian fluids exhibiting Bingham plastic behavior. It is also a first-order nonlinear differential equation, which does not have a closed-form solution. It can be solved using the FDM. The dimensionless velocity gradient in Eq. (5) is written in backward difference form for the shear flow region, and the resulting nonlinear system of equations is solved by using the Newton-Raphson method. It is generally accepted that the surface roughness has an effect on laminar flow characteristics and results in a reduction in Reynolds number 1112. Based on the Merkle-Kubota-Ko 11 modified viscosity model the roughness-affected viscosity μR increases exponentially from the tube centerline to the wall. At the centerline, the roughness viscosity is assumed to be zero, and at the wall it reaches its maximum value. The average height of the surface roughness dictates the dominance of this additional frictional effect. In Fig. 1, the effect of relative roughness on the roughness viscosity is shown for a Hedsrom number of zero, which would characterize zero yield stress. As can be seen from Fig. 1, as the relative surface roughness increases, the roughness viscosity increases significantly. The total viscosity is increased by 54% near the wall with 8% roughness as compared to viscosity in the centerline of the tube. Comparison between the experimentally measured and predicted friction factors is shown in Fig. 2. It is clearly seen that the effect of wall roughness plays an important role on the friction factor even in the laminar flow conditions 101112. The dashed line indicates theoretical friction factor for Newtonian fluid, which is known as f Re=16. From Fig. 2, it can be concluded that there is a good agreement between the predictions of the present study and the experimental friction factor data. Figure 3 shows the velocity profiles for different relative surface roughness values and a constant He number of 1000. The dashed curve represents the theoretical laminar Bingham plastic flow velocity profile. As in the case of He=0, the peak velocity (or plug velocity) decreases considerably with increasing roughness. For the cases considered in Fig. 3, peak velocity reduces by 7.2%, 15.5%, and 23.2% compared to the conventional theory for relative roughness of 2%, 4%, and 8%, respectively. Figure 4 shows nondimensional velocity profiles for different He numbers at a constant relative roughness of 4%. The results indicate that the effect of surface roughness on the velocity profile increases with increasing Hedsrom number. Particularly, for high He numbers, the shear flow is trapped into a small gap between plug flow and the tube wall. This increases the velocity gradient at the wall, du¯/dr¯r¯=1 and it results in increased effect of the roughness, as it is clear from Eq. (5). From Fig. 4, with smooth tube surfaces, it is seen that for high yield stresses (or He number), deviation from the velocity profile increases. For lower He numbers, as expected, the shear flow region increases due to reduction in the plug region and the percentage deviation of the plug velocity for the rough tube from the one for smooth tube reduces. The flow behavior of a fluid through microtubes can be interpreted in terms of flow friction. Theoretically, laminar flow friction coefficient is given by (6)f=a Reb,where a=16 and b=−1, respectively, and it does not depend on surface roughness for Newtonian fluid flow in a macrotube. For microtubes, in order to examine the effect of wall roughness, the dependency between the flow friction coefficient (f) and Reynolds number (Re) is plotted for various relative wall roughnesses as shown in Fig. 5, for He=0. For comparison, the theoretical flow friction factor is also plotted in the same figure. It is clear from Fig. 5 that all predicted friction factors using the model presented in this paper are well above the conventional laminar flow line within the range of the Re numbers considered. This can be attributed to the fact that the friction factor, i.e., pressure drop, in microchannels is considerably higher than the predictions based on the conventional laminar flow theory. This is presumably due to fact that the relative roughness height becomes comparable relative to the tube size. It is also reported in the literature 611 that the departure from the conventional theory increases with increasing Reynolds number for a Newtonian fluid. Results shown in Fig. 5 indicate that a friction factor for a tube of 4% relative roughness increases by 10.6% compared to the smooth tube for Re=100, while this increase is 17.2% for Re=1000. Considering Eq. (6), the results demonstrate an increase in the multiplier a with increasing relative roughness, while b, the slope in the log-log scale, remains nearly unchanged. This is clear from the fact that f vs Re lines are parallel to each other, indicating about the same slope. Figure 6 illustrates the variation of friction coefficient with Reynolds number as a function of He. The results show that the difference between friction factors for different He decreases with increasing Re. On the other hand, deviation of friction factor for a rough tube from a smooth one increase with increasing Reynolds number with the exception of He=1000. This is probably due to the increase in the interactions between fluid particles and irregular wall surface as discussed before. The deviation also increases with increasing roughness as expected. The flow characteristics of liquids modeled by the Bingham plastic constitutive relation through microtubes were numerically studied. The wall roughness effect in a microtube was characterized by using a roughness-viscosity model. This model is used to modify the laminar Bingham plastic flow equations in microtubes by introducing a location dependent apparent viscosity. The numerical results presented in this paper show that the flow friction in microtubes is considerably larger than the predictions of the conventional theory based on uniform viscosity. The departure from the conventional laminar flow theory is also dependent upon fluid yield stress, which is characterized by the Hedsrom number. Therefore, the combined effects of wall roughness and the yield stress appear to have a considerable impact on the flow behavior through microtubes. One of us (T.E.) has been supported by The Scientific and Research Council of Turkey (TUBITAK) to conduct this research under NATO B1 international scholarship program. TUBITAK’s support is gratefully acknowledged. Also, the authors would like to thank Gregory Hitchcock for his suggestions on the manuscript.