Journal of Computational Applied Mechanics

Langlois’ Recursive Approach to Non-Creeping Inertial Viscoelastic Corner Flow in Thin Films

Document Type : Research Paper

Authors

1 Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan

2 Center for Modeling & Computer Simulation, Research Institute, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

3 Center for Communications and IT Research, Research Institute, King Fahd University of Petroleum & Minerals, Dhahran-31261, Saudi Arabia

4 Interdisciplinary Research Center for Smart Mobility and Logistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

Abstract

Dip coating is a key technique in thin film fabrication, widely applied in protective coatings, and material surface engineering. The coating quality depends strongly on the fluid dynamics near substrate edges, where viscoelastic effects and inertial forces can lead to stress concentration and flow instabilities. A viscoelastic fluid model is formulated based on conservation of mass and momentum, with nonlinear governing equations solved using the Langlois recursive approach and the inverse method. Analytical solutions of the stream function provide insight into velocity fields, pressure distribution, and stress behavior near the substrate surface. Results show that stresses and pressure diverge near sharp substrate corners, which can compromise coating durability. Variations in the interface angle significantly alter stress distributions on both the substrate and free surface. Furthermore, inertial forces amplify fluid velocities in the corner region, directly influencing film thickness uniformity and mechanical performance of coated layers.

Keywords

Main Subjects

[1]          D. R. Ceratti, B. Louis, X. Paquez, M. Faustini, D. Grosso, A New Dip Coating Method to Obtain Large-Surface Coatings with a Minimum of Solution, Advanced Materials, Vol. 27, No. 34, pp. 4958-4962, 2015.
[2]          M. Dey, F. Doumenc, B. Guerrier, Numerical simulation of dip-coating in the evaporative regime, The European Physical Journal E, Vol. 39, No. 2, pp. 19, 2016/02/25, 2016.
[3]          L. Landau, B. Levich, Dragging of a Liquid by a Moving Plate,  in: P. Pelcé, Dynamics of Curved Fronts, Eds., pp. 141-153, San Diego: Academic Press, 1988.
[4]          R. B. Bird, R. C. Armstrong, O. Hassager, Dynamics of polymeric liquids. Vol. 1: Fluid mechanics, 1986.
[5]          J. N. Goodier, LXVI. Additional note on an analogy between the slow motions of a viscous fluid in two dimensions and systems of plane stress, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 17, No. 114, pp. 800-803, 1934/04/01, 1934.
[6]          K. Moffatt, Viscous and Resistive Eddies Near a Sharp Corner, Journal of Fluid Mechanics Digital Archive, Vol. 18, pp. 1-18, 01/01, 1964.
[7]          S. Taneda, Visualization of Separating Stokes Flows, Journal of the Physical Society of Japan, Vol. 46, No. 6, pp. 1935-1942, 1979/06/15, 1979.
[8]          K. Moffatt, B. Duffy, Local Similarity Solutions and their Limitations, Journal of Fluid Mechanics, Vol. 96, pp. 299-313, 01/29, 1980.
[9]          S. Betelu, J. Diez, L. Thomas, R. Gratton, B. Mariño, Instantaneous viscous flow in a corner bounded by free surfaces, Physics of Fluids, Vol. 8, pp. 2269, 09/01, 1996.
[10]        C. D. Han, L. H. Drexler, Studies of converging flows of viscoelastic polymeric melts. III. Stress and velocity distributions in the entrance region of a tapered slit die, Journal of Applied Polymer Science, Vol. 17, No. 8, pp. 2369-2393, 1973.
[11]        M. Renardy, Stress integration for the constitutive law of the upper convected Maxwell fluid near the corners in a driven cavity, Journal of non-newtonian fluid mechanics, Vol. 112, No. 1, pp. 77-84, 2003.
[12]        A. Siddiqui, A. Ansari, A. Ahmad, N. late, On Taylor's scraping problem and flow of a Sisko fluid, Mathematical Modelling and Analysis, Vol. 14, pp. 515-529, 12/31, 2009.
[13]        J. E. Sprittles, Y. D. Shikhmurzaev, Viscous flow in domains with corners: Numerical artifacts, their origin and removal, Computer Methods in Applied Mechanics and Engineering, Vol. 200, No. 9, pp. 1087-1099, 2011/02/01/, 2011.
[14]        S. T. Chaffin, J. M. Rees, Carreau fluid in a wall driven corner flow, Journal of Non-Newtonian Fluid Mechanics, Vol. 253, pp. 16-26, 2018/03/01/, 2018.
[15]        A. Mahmood, Effect of Inertia on Viscous Fluid in a wall-driven Corner Flow with Leakage, Appl. Math, Vol. 13, No. 3, pp. 353-360, 2019.
[16]        R. Ellahi, Recent trends in coatings and thin film: Modeling and application, 8, MDPI, 2020, pp. 777.
[17]        N. Okechi, S. Asghar, Viscoelastic flow through a wavy curved channel, Chinese Journal of Physics, Vol. 74, 09/01, 2021.
[18]        M. Turkyilmazoglu, Three dimensional MHD flow and heat transfer over a stretching/shrinking surface in a viscoelastic fluid with various physical effects, International Journal of Heat and Mass Transfer, Vol. 78, pp. 150-155, 2014/11/01/, 2014.
[19]        M. Bhatti, F. Ishtiaq, R. Ellahi, S. Sait, Novel Aspects of Cilia-Driven Flow of Viscoelastic Fluid through a Non-Darcy Medium under the Influence of an Induced Magnetic Field and Heat Transfer, Mathematics, Vol. 11, pp. 2284, 05/14, 2023.
[20]        S. Shaheen, K. Maqbool, R. Ellahi, S. Sait, Metachronal propulsion of non-Newtonian viscoelastic mucus in an axisymmetric tube with ciliated walls, Communications in Theoretical Physics, Vol. 73, pp. 035006, 03/01, 2021.
[21]        F. J. Awan, K. Maqbool, S. M. Sait, R. Ellahi, Buoyancy effect on the unsteady diffusive convective flow of a Carreau fluid passed over a coated disk with energy loss, Coatings, Vol. 12, No. 10, pp. 1510, 2022.
[22]        A. Majeed, A. Zeeshan, S. Z. Alamri, R. Ellahi, Heat transfer analysis in ferromagnetic viscoelastic fluid flow over a stretching sheet with suction, Neural Comput. Appl., Vol. 30, No. 6, pp. 1947–1955, 2018.
[23]        H. Alfadil, A. E. Abouelregal, Ö. Civalek, H. F. Öztop, Effect of the photothermal Moore–Gibson–Thomson model on a rotating viscoelastic continuum body with a cylindrical hole due to the fractional Kelvin-Voigt model, Indian Journal of Physics, Vol. 97, No. 3, pp. 829-843, 2023.
[24]        A. Zeeshan, H. Javed, N. Shehzad, S. Sait, R. Ellahi, An integrated numerical and analytical investigation on cilia-generated MHD flow of Jeffrey fluid through a porous medium, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 34, 08/30, 2024.
[25]        M. M. Bhatti, R. Ellahi, S. Sait, R. Ullah, Exact solitary wave solutions of time fractional nonlinear evolution models: a hybrid analytic approach, Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science, pp. 83-98, 09/03, 2024.
[26]        W. Langlois, A RECURSIVE APPROACH TO THE THEORY OF SLOW, STEADY-STATE VISCOELESTIC FLOW,  pp. 1962.
[27]        W. Langlois, The recursive theory of slow viscoelastic flow applied to three basic problems of hydrodynamics, Journal of Rheology, 1964.
[28]        A. Mahmood, A. Siddiqui, Two dimensional inertial flow of a viscous fluid in a corner, Applied Mathematical Sciences, Vol. 11, pp. 407-424, 01/01, 2017.
[29]        G. Taylor, On scraping viscous fluid from a plane surface, The Scientific Papers of Sir Geoffrey Ingram Taylor, No. 1971, pp. 410-413, 1962.
Journal of Computational Applied Mechanics
Volume 57, Issue 2
April 2026
Pages 326-347
  • Receive Date: 01 February 2026
  • Accept Date: 01 February 2026