Micropolar fluid flows relative to a swarm of spherical porous shells Dr. Curtis Boodoo 1 1 Utilities
and Sustainable Engineering, The University of Trinidad and Tobago, Trinidad
and Tobago
1. INTRODUCTION Micropolar fluid flow has various applications in different fields. It is used in heat exchangers, cooling nuclear reactors, designing energy systems, and the casting and injection processes of fluids Kocic et al. (2023). Micropolar fluids are also relevant in the flow of human or animal blood, the extraction of crude oil, and the flow of polymer fluids or liquid crystals Kocic et al. (2023). They are utilized in biomedical applications, such as biological, physical, and chemical processes, as well as lubrication systems and hydrodynamic-fluid problems Kethireddy et al. (2022). Micropolar fluids are also important in convective heat and mass transfer, polymer production, and the cooling of particles for metallic sheets Awan et al. (2022). Additionally, micropolar fluids containing small conductive particles have potential applications in drug delivery and localized heating Narla et al. (2022). This paper focuses on the application of micropolar fluid and porous spheres in drug delivery. The analytical investigation of micropolar fluid flow past a porous sphere has been carried out in several papers. El-Sapa analyzes the axisymmetric creeping flow of micropolar fluid past a porous surface saturated with micropolar fluid using an analytical method El-Sapa (2022). Maurya and Deo study the flow of micropolar fluid through a porous cylinder enclosing a solid cylindrical core under an externally applied magnetic field Khanukaeva (2022). Aparna et al. consider the uniform flow of a viscous fluid past a spherical ball filled with porous medium saturated by micropolar fluid and observe the effects of porosity and micropolarity parameters on the flow and drag Maurya & Deo (2022). Khanukaeva models the flow of micropolar fluid through a spherical cell consisting of a solid core, porous layer, and liquid envelope using the cell model technique Slattery & Bird (1961). Prasad investigates the low Reynolds number flow of an incompressible micropolar fluid past and through a porous sphere and studies the variation of drag force with the permeability parameter Podilay et al. (2022). Khanukaeva analyzed the micropolar liquid filtration through spherical cell membranes with porous layers using micropolar and Brinkman-type equations, investigating hydrodynamic permeability based on media characteristics Khanukaeva (2022). For the application of oral drug delivery, a collection or swarm of porous particles is involved. The study that follows is aimed at modeling and exploring the hydrodynamics of this swarm of porous particles. Complications arise when modeling complex geometry comprising an assemblage of particles. The two primary methods for handling boundary value problems involving a collection of particles are: the method of reflections and the unit cell technique. The concept behind the unit cell technique is that a collection of particles can be divided into a number of identical cells, with one particle occupying each cell Happel & Brenner (1983). The boundary value problem is now reduced to a single particle bounded by a fictitious envelope. The cell model is most applicable for a concentration of periodic particles where the effect of the container wall can be neglected. Various cell shapes can be implemented, but it is more convenient to use a spherical envelope as the fictitious surface of the cell. The boundary conditions imposed on the surface of the envelope represent the interactions with the other porous particles of the assembly. The thickness of the surrounding fluid layer is adjusted so the ratio of the solid volume to the volume of the liquid envelope represents exactly the solid volume fraction of the porous medium. An early sphere-in-cell model was proposed by Cunningham (1910). He considered particle sedimentation and postulated that the movement of each spherical particle was allowed only within a concentric mass of fluid boundaries. Mehta and Morse Mehta & Morse (1975) adopted Cunningham's approach by assuming the tangential velocity as a component of the fluid velocity, signifying the homogeneity of the flow on the cell boundary. The importance of the Mehta and Morse boundary conditions is that the average flow variables over a cell volume can be scaled to obtain large scale behavior. Happel (1958) and Kuwabara (1959) independently presented sphere-in-cell models. The major differences between these two models are in the boundary conditions imposed on the outer envelope surface. The Happel model assumes a uniform velocity condition and no tangential stress at the cell envelope surface. This formulation results in an analytically closed solution that is axisymmetric and can be easily used for heat and mass transfer calculations. The Kuwabara cell model uses zero vorticity condition on the cell surface. The Kuwabara formulation requires a small exchange of mechanical energy with the environment. The mechanical power given by the sphere to the fluid is not all consumed by viscous dissipation in the fluid layer Vasin et al. (2008). Neale & Nader (1974) considered the spherical cell embedded in an unbounded, continuous, homogeneous, and isotropic permeable medium of the same porosity and permeability as the porous particles that make up the swarm. They used Brinkmans's equation to model the porous regions. Kvashnin (1980) implemented a tangential component of velocity that reaches a minimum with respect to radial distance on the cell surface. This was done to represent the symmetry of the cell. This study investigates the creeping axisymmetric flow of an incompressible micropolar fluid past a porous shell. The porous region is modeled using Darcy equation sandwiched between two transition Brinkman regions. A stream function formulation is used to solve the system. The accompanying boundary conditions used are continuity of velocity, normal and tangential stresses, and non-homogeneous microrotations across the fluid porous region interfaces. At the Brinkman- Darcy interfaces continuity of velocity, normal stresses, microrotations and the Beavers and Joseph condition is implemented. An analytical expression for the dimensionless drag, for the bounded fluid cases as outlined by Happel, Kuwabara, Kvashnin and Mehta, and Morse is derived, and plots of the dimensionless drag as it varies with hydraulic resistivity, porous layer thickness, and porosity are presented. The study's insights into the hydrodynamics of swarms of porous particles have direct implications for designing more efficient oral drug delivery systems. By understanding the interactions between drug-carrying particles and the surrounding fluid, pharmaceutical engineers can optimize the design of these systems for targeted delivery El-Sapa (2022). The research on micropolar fluid flows through porous shells can be applied to the optimization of heat exchangers, cooling systems for nuclear reactors, and other energy systems. By manipulating parameters such as hydraulic resistivity, porous layer thickness, and porosity, engineers can control fluid resistance and enhance the efficiency of these systems Maurya & Deo (2022). 2. MODEL FORMULATION The unit cell technique will be used to model a swarm of porous shells in an incompressible micropolar fluid. The porous region is modeled as a transition Brinkman region overlying a Darcy region. The problem geometry is presented in Figure 1. Figure 1
The boundary conditions implemented in the model for the
single porous shell in a micropolar fluid will be used, with the exception of
the specific boundary conditions at The following unit cell techniques will be implemented at 1) Happel:
2) Kuwabara:
3) Kvashnin:
4) Mehta
and Morse In the analysis for the single porous shell the radius volume fraction of the porous medium In dimensionless form this is: All unit cell techniques utilize the boundary condition 1) Happel: the dimensionless tangential stress is given by: Therefore: 2) Kuwabara: 3) Kvashnin: 4) Mehta
and Morse: 2.1. Dimensionless Drag An expression for the dimensionless drag on a porous shell in a micropolar fluid was derived and given by Happel & Brenner (1983) as: Where for a micropolar fluid [19] which gives in dimensionless form: where: The analysis, plots and discussion were all for an unbounded micropolar fluid. In using the various unit cell techniques, boundary conditions are specified on the outer envelope. This represents a bounding of the micropolar fluid. The stream functions for the various regions were solved and given below. The outer micropolar fluid stream function for the unit cell technique is given from as: The inner micropolar region stream function is given from as: and the Brinkman regions as: where The Darcy region from is: where The constants are solved using the boundary conditions given in the analysis for the single porous shell, which is replaced by boundary condition for Happel, Kuwabarra and Kvashnin unit cell techniques respectively. The dimensionless drag, 3. RESULTS AND DISCUSSIONS Plots of the
dimensionless drag Figure 2
Figure 3
Figure 4
The values of There is a
decrease in As the
thickness of the porous layer 4. CONCLUSIONS and RECOMMENDATIONS The analysis of the dimensionless drag Figure 2 further elaborates on
the effects of porous layer thickness, The dependency of drag on porosity, ϕ, is visually captured
in Figure 3. As porosity approaches
unity, we observe a steep decline in drag, indicating that higher porosity
levels are conducive to reduced fluid resistance. This trend aligns with the
theoretical expectations for unbounded fluids, where higher porosity equates to
lesser obstruction to fluid flow. The consistent inverse relationship between hydraulic resistivity and
drag, across all unit cell models, as shown in Figure 1, is particularly
relevant for optimizing heat exchanger designs and improving the efficiency of
cooling nuclear reactors. By strategically manipulating hydraulic resistivity,
it may be possible to control fluid flow resistance to achieve desired temperature
regulation more effectively. Figure 2’s illustration of the
impact of porous layer thickness on drag, especially the unique trend exhibited
by the Mehta and Morse model, has direct implications for the design of energy
systems and the casting and injection processes of fluids. The understanding of
how drag changes with layer thickness can aid in refining these processes to
minimize energy loss and enhance the precision of fluid manipulation. Moreover, the significant reduction in drag with increasing porosity, as
depicted in Figure 3, underpins the
applications in biological systems, such as blood flow, where the reduction in
drag can potentially lead to less damage to blood cells in artificial devices.
This insight also extends to the extraction of crude oil, where increased
porosity could be beneficial for reducing the energy required for extraction. In the specific context of drug delivery, where the study focuses, the ability to predict and control the hydrodynamics around porous particles becomes critically important. The cell models and corresponding drag behaviours analysed here can inform the design of drug delivery systems that rely on the micro-scale interactions between the drug-carrying particles and the surrounding fluid. This can lead to more efficient and targeted delivery, potentially improving therapeutic outcomes.
CONFLICT OF INTERESTS None. ACKNOWLEDGMENTS None. REFERENCES Awan, A. U., Ahammad, N. A., Ali, B. M., Tag-Eldin, E., Guedriand, K., Gamaoun, F. (2022, Jun.). Significance of Thermal Phenomena and Mechanisms of Heat Transfer through the Dynamics of Second-Grade Micropolar Nanofluids, 14(15). https://doi.org/10.3390/su14159361 Cunningham, E. (1910, Mar.). On the Velocity of Steady Fall of Spherical Particles through Fluid Medium, Proc. R. Soc. Lond. Ser. A, 83(563), 357-365. https://doi.org/10.1098/rspa.1910.0024 El-Sapa, S. (2022, Jun.). Cell Models for Micropolar Fluid Past a Porous Micropolar Fluid Sphere with Stress Jump Condition, 34(8). https://doi.org/10.1063/5.0104279 Happel, J. (1958, Jun.). Viscous Flow in Multiparticle Systems: Slow Motion of Fluids Relative to Beds of Spherical Particles, AIChE J., 4(2), 197-201. https://doi.org/10.1002/aic.690040214 Happel, J., & Brenner, H. (1983). Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media. Springer. https://doi.org/10.1007/978-94-009-8352-6 Kethireddy, B., Reddyand, G. J., Basha, H. (2022, Jun.). Bejan's Thermal and Mass Flow Visualization in Micropolar Fluid. https://doi.org/10.1080/17455030.2022.2088887 Khanukaeva, D. (2022, Jun.). Filtration of Micropolar Liquid Through a Membrane Composed of Spherical Cells with Porous Layer, 34(3). https://doi.org/10.1007/s00162-020-00527-x Kocic, M. M., Stamenkovic, Z., Petrovićand, J., Bogdanović-Jovanović, J. (2023, Jun.). MHD Micropolar Fluid Flow in Porous Media, 15(6). https://doi.org/10.1177/16878132231178436 Kocic, M. M., Stamenkovic, Z., Petrovićand, J., Bogdanović-Jovanović, J. (2023, Mar.). Control of MHD Flow and Heat Transfer of a Micropolar Fluid through Porous Media in a Horizontal Channel, 8(3). https://doi.org/10.3390/fluids8030093 Kuwabara, S. (1959, Apr.). The Forces Experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Numbers, J. Phys. Soc. Jpn., 14, 527. https://doi.org/10.1143/JPSJ.14.527 Kvashnin, A. G. (1980). Cell Model of Suspension of Spherical Particles, Fluid Dyn., 14(4), 598-602. https://doi.org/10.1007/BF01051266 Maurya, P. K., & Deo, S. (2022, Jun.). MHD Effects on Micropolar Fluid Flow Through a Porous Cylinder Enclosing an Impermeable Core, 13(5). https://doi.org/10.1615/SpecialTopicsRevPorousMedia.2022042199 Mehta, G. D., & Morse, T. F. (1975, Sep.). Flow Through Charged Membranes, J. Chem. Phys., 63(5), 1878-1889. https://doi.org/10.1063/1.431575 Narla, V. K., Tripathiand, D., & Bhandari, D. S. (2022, Jun.). Thermal Analysis of Micropolar Fluid Flow Driven by Electroosmosis and Peristalsis in Microchannel, 43(1). https://doi.org/10.1080/01430750.2022.2091034 Neale, G., & Nader, W. (1974, Aug.). Practical Significance of Brinkman's Extension of Darcy's Law: Coupled Parallel Flows within a Channel and a Bounding Porous Medium, Can. J. Chem. Eng., 52(4), 475-478. https://doi.org/10.1002/cjce.5450520407 Podila, A., Podila, P., Pothannaand, N., & Ramana Murthy, J. V. (2022, Feb.). Uniform Flow of Viscous Fluid Past a Porous Sphere Saturated with Micro Polar Fluid, 13(1). https://doi.org/10.33263/BRIAC131.069 Saad, E. I. (2008). Motion of a Spheroidal Particle in a Micropolar Fluid Contained in a Spherical Envelope, Can. J. Phys., 86(9), 1039-1056. https://doi.org/10.1139/p08-045 Slattery, J. C., & Bird, R. B. (1961, Dec.). Non-Newtonian Flow Past a Sphere, 16. https://doi.org/10.1016/0009-2509(61)80034-1 Vasin, S. I., Filippov, A. N., & Starov, V. M. (2008, Jun.). Hydrodynamic Permeability of Membranes Built up by Particles Covered by Porous Shells: Cell Models. Adv. Colloid Interface Sci., 139(1-2), 83-96. https://doi.org/10.1016/j.cis.2008.01.005
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