Navier Stroke Equation (NSE) in the Transport Phenomena (Momentum, Heat, and Mass Transfer Balances) for the Movement of Water in A Horizontal Pipe from a Pumping Station to A Holding Facility Before Distribution

References

[1] Apostol, M. (2024). The Navier-Stokes Equation and a fully developed Turbulence. Journal of Applied Math, 2 (5), 1594. https://doi.org/10.59400/jam159

[2] Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2007). Transport Phenomena (2nd ed.). Wiley.

[3] C.P Kothanadaraman, S. Subramanyan. (1970) Heat and Mass Transfer Data Book (8th edition). New Age international publishers

[4] Cengel, Y. A., & Cimbala, J. M. (2006). Fluid Mechanics: Fundamentals and Applications. McGraw-Hill Education.

[5] Chen, K. T., Yarn, K. F., Chen, H. Y., Tsai, C. C., Luo, W. J., & Chen, C. N. (2017). Aspect ratio effect on laminar flow bifurcations in a curved rectangular tube driven by pressure gradients. Journal of Mechanics, 33 (6), 831-840. https://doi.org/10.1017/jmech.2017.93

[6] Davidzon, M. Y. (2017). About Navier-Stokes Equation in the Theory of Convective Heat Transfer. Journal of Physics: Conference. Series, 891, 012041.

[7] Dumitrescu, H., Cardos, V., & Bogateanu, R. (2023). The Physical vs. Mathematical Problem of Navier-Stokes Equations (NSE). BULLETIN, 15(1), 21-34.

[8] Fefferman, C. L. (2000). Existence and Smoothness of the Navier-Stokes Equations.

[9] Fox, R. W., McDonald, A. T., & Pritchard, P. J. (2011). Introduction to Fluid Mechanics (8th ed.). John Wiley & Sons.

[10] Hu, K. S., & Shih, T. I. P. (2024). Large-Eddy vs. Reynolds-Averaged Navier–Stokes Simulations of Flow and Heat Transfer in a U-Duct with Unsteady Flow Separation. Energies, 17, 2414. https://doi.org/10.3390/en17102414

[11] Jha, B. K., & Gambo, D. (2020). Effect of an oscillating time-dependent pressure gradient on Dean flow: transient solution. Beni-Suef University Journal of Basic and Applied Sciences, 9 (1), 39. https://doi.org/10.1186/s43088-020-00066-8

[12] Kays, W. M., & Crawford, M. E. (1993). Convective heat and mass transfer (2nd ed.). McGraw-Hill.

[13] Lighthill, M. J. (1986). An informal introduction to theoretical fluid mechanics. Oxford: Clarendon Press.

[14] Linot, A. J., Schmid, P. J., & Taira, K. (2024). On the laminar solutions and stability of accelerating and decelerating channel flows. Journal of Fluid Mechanics, 999 (1), A43. https://doi.org/10.1017/jfm.2024.709

[15] Munson, B. R., Rothmayer, A. P., & Okiishi, T. H. (2012). Fundamentals of Fluid Mechanics (7th ed.). Wiley.

[16] NPTEL Course on Transport Phenomena in Materials. Discusses Navier-Stokes equations for pipe flow in cylindrical coordinates, with practical engineering applications. Retrieved from [NPTEL Transport Phenomena](https://archive.nptel.ac.in).

[17] Panton, R. L. (2013). Incompressible fluid flow (3rd ed.). Wiley.

[18] Peter, & Bala. (2022). Comparison Review of Methods of Water Distribution System For Efficient Water Supply. International Journal of Mechanical Engineering, 7 (5), 0974-5823.

[19] Physics LibreTexts. Transport Phenomena. Explains the theoretical framework of the Navier-Stokes equations in modeling fluid flow, particularly in pipe geometries. Retrieved from [Physics LibreTexts](https://phys.libretexts.org).

[20] Prandtl, L., & Tietjens, O. G. (2013). Applied hydrodynamics (3rd ed.). Dover Publications.

[21] Reynolds, O. (1883). An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall be Direct or Sinuous, and the Law of Resistance in Parallel Channels. Philosophical Transactions of the Royal Society of London, 174, 935-982.

[22] Romero-Gomez, P., Choi, C. Y., Lansey, K. E., Preis, A., & Ostfeld, A. (2008). Transport Phenomena in Drinking Water Systems, Sensor Network design with improved water quality models at cross junctions. 10th Annual Water Distribution Systems Analysis Symposium, Kruger National Park, South Africa

[23] Schlichting, H., & Gersten, K. (2016). Boundary-layer Theory (9th ed.). Springer.

[24] Sharma, V., Tyagi, S., Rastogi, S., Kumar, K., Siddiqui, M. A., & Kumar, M. (1968). Boundary Layer Theory. McGraw-Hill Book Company.

[25] Shankar Subramanian R., (2020). Notes on Transport Phenomena. Department Of Chemical and Biomolecular Engineering, Clarkson University Potsdam, New York 13699-570

[26] Tennekes, H., & Lumley, J. L. (1972). A first course in turbulence. The MIT Press.

[27] Uruba, V. (2009). Turbulence. CTU Praha.

[28] Uruba, V. (2018). On Reynolds number physical interpretation. AIP Conference Proceedings. https://doi.org/10.1063/1.50499

[29] Veldman, A. E. P. (1976). Boundary Layer flow past a finite flat plate. Dissertation, Rijksuniversiteit, Groningen, Holland.

[30] White, F. M. (2011). Fluid mechanics (7th ed.). McGraw-Hill.

[31] White, F. M. (2016). Viscous fluid flow (3rd ed.). McGraw-Hill.

@article{1706866,
      author = {Dr. Adeola Grace Olugbenga, Henry Akintola Akpa, Amali Paul Elaine, Oyelade Oluwabukoye Enoch, Salihu Fatima Aliyu},
      title = {Navier Stroke Equation (NSE) in the Transport Phenomena (Momentum, Heat, and Mass Transfer Balances) for the Movement of Water in A Horizontal Pipe from a Pumping Station to A Holding Facility Before Distribution},
      journal = {Iconic Research And Engineering Journals},
      year = {2025},
      volume = {8},
      number = {7},
      pages = {314-324},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1706866.pdf},
      abstract = {This study investigate the application of  Navier Stokes equation in transport phenomenon, specifically examining water flow in horizontal pipe. The Navier-Stokes equations, which describe the mechanics of fluid motion, primarily control the flow behavior when water flows through horizontal pipes.  These equations offer a mathematical model for explaining the motion of incompressible, viscous fluids and are derived from the basic ideas of mass, momentum, and energy conservation. The equations show the forces operating on the fluid, such as viscous forces, pressure gradients, and external body forces (if any, like gravity). Typically, in a horizontal pipe, the considerations are the cylindrical coordinates (r,?,z), where r is the radial distance, ? is the angular position, and z is the axial direction along the pipe. Velocity profile equation for laminar flow was used with  assumed range of radial distance of  r  values from 0 (Centre) to R (pipe wall) to compute the velocity for each r in which the values obtained are tabulated and a plot of the velocity against radial distance demonstrating the inverse proportionality between them. The pipe radius R was used to calculate the volumetric flow rate Q and the data obtained was used to plot the graph of flowrate against pipe  radius, The curve is steep and nonlinear, showcasing the R4 dependence. For a fixed pressure gradient and fluid viscosity, the flow rate increases exponentially as the pipe radius increase. The results highlight the significance of Navier Stokes equation in understanding and predicting fluid flow behavior in transport phenomenon.},
      month = {January},
  }