{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T19:35:12Z","timestamp":1761766512643,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,15]],"date-time":"2019-04-15T00:00:00Z","timestamp":1555286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article investigates the two-dimensional creeping flow of a non-Newtonian micropolar fluid in a small width permeable channel. Fluid is absorbed through permeable walls at a variable rate. This situation arises in filtration and mass transfer phenomena in industrial and engineering processes. The exact solution of the equations of motion is obtained. Graphs of the velocity profiles and pressure drop reveal the significant impact of the non-Newtonian nature of the micropolar fluid on the flow. The obtained solutions are used to discuss the hydrodynamical aspects of the physiological phenomenon of blood filtration in an artificial kidney, the flat plate dialyzer (FPD). Expressions for finding the ultrafiltration rate and mean pressure drop in an FPD are derived. Ultrafiltration rate and the mean pressure difference in an FPD are computed using derived expressions. A comparison of these with the existing empirical and experimental results shows a good agreement. For certain values of parameters, the derived form of the flow rate reveals that the axial flow rate in an FPD decays exponentially along the membrane length. This is a well-established and admitted result used by several researchers for studying the hydrodynamics of blood flow in renal tubules of kidneys. It is concluded that the presented model can be used to study the hydrodynamical aspects of blood flow in an FPD.<\/jats:p>","DOI":"10.3390\/sym11040541","type":"journal-article","created":{"date-parts":[[2019,4,15]],"date-time":"2019-04-15T11:15:58Z","timestamp":1555326958000},"page":"541","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Hydrodynamical Study of Micropolar Fluid in a Porous-Walled Channel: Application to Flat Plate Dialyzer"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6896-172X","authenticated-orcid":false,"given":"Dianchen","family":"Lu","sequence":"first","affiliation":[{"name":"Faculty of Science, Jiangsu University, Zhenjiang 212013, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6324-3884","authenticated-orcid":false,"given":"Muhammad","family":"Kahshan","sequence":"additional","affiliation":[{"name":"Faculty of Science, Jiangsu University, Zhenjiang 212013, China"},{"name":"Department of Mathematics, COMSATS University Islamabad, Abbottabad 22060, Pakistan"}]},{"given":"A. M.","family":"Siddiqui","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Pennsylvania State University, YorkCampus, 1031 Edgecomb Avenue, York, PA 17403, USA"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,15]]},"reference":[{"key":"ref_1","unstructured":"Voutchkov, N. (2012). Desalination Engineering: Planning and Design, McGraw Hill."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF02477766","article-title":"Pressure flow patterns in a cylinder with reabsorbing walls","volume":"25","author":"Macey","year":"1963","journal-title":"Bull. Math. Biophys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/BF02498766","article-title":"Hydrodynamics in the renal tubule","volume":"27","author":"Macey","year":"1965","journal-title":"Bull. Math. Biophys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1007\/BF02463260","article-title":"Flow of a Newtonian fluid through a permeable tube: The application to the proximal renal tubule","volume":"25","author":"Marshall","year":"1974","journal-title":"Bull. Math. Biophys."},{"key":"ref_5","first-page":"119","article-title":"Flow of a Newtonian fluid between parallel flat permeable plates\u2014The application to a flat-plate hemodialyzer","volume":"27","author":"Marshall","year":"1975","journal-title":"Math. Biophys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2445","DOI":"10.1016\/j.camwa.2017.12.028","article-title":"Three-dimensional lattice Boltzmann simulations of high density ratio two-phase flows in porous media","volume":"75","author":"Sadeghi","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1017\/jfm.2019.4","article-title":"Viscous fingering phenomena in the early stage of polymer membrane formation","volume":"864","author":"Hirschler","year":"2019","journal-title":"J. Fluid Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1021\/i160035a033","article-title":"Velocity profiles in porous-walled ducts","volume":"9","author":"Kozinski","year":"1970","journal-title":"Ind. Eng. Chem. Fundam."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Papanastasiou, T.C. (2000). Viscous Fluid Flow, CRC Press LLC.","DOI":"10.1201\/9781420050288"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1063\/1.1721476","article-title":"Laminar flow in channels with porous walls","volume":"24","author":"Berman","year":"1953","journal-title":"J. Appl. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1063\/1.1722948","article-title":"Laminar flow in an annulus with porous walls","volume":"29","author":"Berman","year":"1958","journal-title":"J. Appl. Phys."},{"key":"ref_12","first-page":"719","article-title":"Laminar Pipe Flow With Injection and Suction Through a Porous Wall","volume":"78","author":"Yuan","year":"1956","journal-title":"Trans. ASME"},{"key":"ref_13","first-page":"795","article-title":"Creeping Flow of Viscous Fluid through a Proximal Tubule with Uniform Reabsorption: A Mathematical Study","volume":"10","author":"Haroon","year":"2016","journal-title":"Appl. Math. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/s10483-016-2032-6","article-title":"Hydrodynamics of viscous fluid through porous slit with linear absorption","volume":"37","author":"Siddiqui","year":"2016","journal-title":"Appl. Math. and Mech."},{"key":"ref_15","unstructured":"Byron, B.R. (1987). Dynamics of Polymeric Liquids, John Wiley & Sons."},{"key":"ref_16","first-page":"1","article-title":"Theory of micropolar fluids","volume":"1","author":"Eringen","year":"1965","journal-title":"J. Math. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1919","DOI":"10.1063\/1.1692219","article-title":"Polar fluids","volume":"35","author":"Cowin","year":"1968","journal-title":"Phys. Fluids"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1017\/S0022112067001375","article-title":"Boundary conditions at a naturally permeable wall","volume":"30","author":"Beavers","year":"1967","journal-title":"J. Fluid Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1002\/sapm197150293","article-title":"On the boundary condition at the surface of a porous medium","volume":"50","author":"Saffman","year":"1971","journal-title":"Stud. Appl. Math."},{"key":"ref_20","unstructured":"Darcy, H.P.G. (1856). D\u00e8termination des Lois D\u2019\u00e8coulement de L\u2019eau \u00e0 Travers le Sable, Victor Dalamont."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"655","DOI":"10.22364\/mhd.51.4.2","article-title":"MHD flow of newtonian fluid in a permeable tubule","volume":"51","author":"Siddiqui","year":"2015","journal-title":"Magnetohydrodynamics"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1140\/epjp\/i2018-12410-6","article-title":"A micropolar fluid model for hydrodynamics in the renal tubule","volume":"133","author":"Kahshan","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1143\/JJAP.9.345","article-title":"A theoretical study of the flow of blood in a capillary with permeable wall","volume":"9","author":"Oka","year":"1970","journal-title":"Jpn. J. Appl. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4319","DOI":"10.1166\/jctn.2015.4358","article-title":"Slip Effects on the Flow of Newtonian Fluid in Renal Tubule","volume":"12","author":"Siddiqui","year":"2015","journal-title":"J. Comput. Theor. Nanosci."},{"key":"ref_25","unstructured":"(1981\u20132014). Maple 18, Waterloo Maple Inc."},{"key":"ref_26","unstructured":"Drukker, W., Parsons, F.M., and Maher, J.F. (2012). Replacement of Renal Function by Dialysis: A Textbook of Dialysis, Springer Science & Business Media."},{"key":"ref_27","first-page":"127","article-title":"A new disposable plate-kidney","volume":"15","author":"Sausse","year":"1969","journal-title":"ASAIO J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1142\/S0219519408002541","article-title":"A study of micropolar fluid in an annular tube with application to blood flow","volume":"8","author":"Muthu","year":"2008","journal-title":"J. Mech. Med. Biol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1750013","DOI":"10.1142\/S0219519417500130","article-title":"Peristaltic flow and hydrodynamic dispersion of a reactive micropolar fluid-simulation of chemical effects in the digestive process","volume":"17","author":"Kiran","year":"2017","journal-title":"J. Mech. Med. Biol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1108\/09615531311293470","article-title":"Series solution for heat transfer of continuous stretching sheet immersed in a micropolar fluid in the existence of radiation","volume":"23","author":"Shadloo","year":"2013","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1002\/aic.690140120","article-title":"Studies of intramembrane transport: A phenomenological approach","volume":"14","year":"1968","journal-title":"AIChE J."},{"key":"ref_32","first-page":"461","article-title":"An experimental method for obtaining an ultrafiltrate of the blood","volume":"12","author":"Malinow","year":"1947","journal-title":"Transl. Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/S0022-5347(17)63278-6","article-title":"An automatic peritoneal dialysis machine: Preliminary report","volume":"96","author":"McDonald","year":"1966","journal-title":"J. Urol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1097\/00002480-196204000-00042","article-title":"Prolonged hemodialysis with bath refrigeration: The influence of dialyzer membrane thickness, temperature and other variables on performance","volume":"8","author":"Brown","year":"1962","journal-title":"Trans Am. Soc. Artif. Intern. Organs"},{"key":"ref_35","first-page":"303","article-title":"A theoretical note on exponential flow in the proximal part of the mammalian nephron","volume":"24","author":"Kelman","year":"1962","journal-title":"Bull. Math. Biol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/S0092-8240(81)90013-6","article-title":"A hydrodynamical study of the flow in renal tubules","volume":"43","author":"Radhakrishnamacharya","year":"1981","journal-title":"Bull. Math. 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