# Matched-asymptotic and Singular Perturbation Approach For Inertial Coating Flow Near Exit

The department of Mathematics and Statistics at the University of Limerick invites you to a seminar by  Prof. Roger Khayat (Western University):

Title: Matched-asymptotic and Singular Perturbation Approach For Inertial Coating Flow Near Exit

Abstract: The flow of a two-dimensional steady wall jet observed in the slot and blade coating processes is examined theoretically near and far from the channel exit. The free-surface jet emerges from a channel and deposits onto an infinite moving substrate with an adverse or favorable pressure gradient applied inside the channel to describe slot and blade coating, respectively. The Reynolds number considered is within the moderate range, and fully developed Couette-Poiseuille flow conditions are assumed to prevail far upstream inside the channel. The effects of inertia, surface tension and flow rate (upstream pressure gradient) on the velocity, the shape of the free surface, and the stress are emphasized. The problem is solved using the method of matched asymptotic expansions. The small parameter involved in the expansions is the inverse cubic power of the Reynolds number. The flow field is obtained by matching the inviscid rotational core flow separately with the free-surface and the two wall layers. The upstream influenced is examined as well as the break in the symmetry between the two wall layers. The theoretical approach is closely related to the triple-deck theory (Sobey 2000), and follows closely the formulations adopted for a symmetric jet emerging from a channel (Tillett 1968) or a tube (Philippe & Dumargue 1991). We find that the jet always contracts near the channel exit regardless of the level of inertia and direction of the applied pressure gradient. Some expansion is predicted downstream of the exit for blade coating which strengthens with increasing flow rate. An adverse pressure gradient provides a thinner film thickness than a favorable one. I will also briefly discuss non-Newtonian effects (Saffari & Khayat 2009; Khayat 2014) as well as the effects of slip (Khayat 2016) on the shape of the emerging jet. The case of a jet on a stationary wall (Khayat 2017) will also be discussed.

Khayat, R. E. 2014 Free-surface jet flow of a shear-thinning power-law fluid near the channel exit. J. Fluid Mech. 748, 580-617.

Khayat, R. E. 2017 Initial development of a free-surface wall jet at moderate Reynolds number. J. Fluid Mech. 826, 235-269.

Philippe, C., & Dumargue, P. 1991 Étude de l'établissement d'un jet liquide laminaire émergeant d'une conduite cylindrique verticale semi-infinie et soumis à l'influence de la gravité. Zeitschrift für angewandte Mathematik und Physik ZAMP, 42(2), 227-242.

Saffari, A., & Khayat, R. E. 2009 Flow of viscoelastic jet with moderate inertia near channel exit. J. Fluid Mech. 639, 65-100.

Sobey, I. J. 2005 Introduction to Interactive Boundary Layer Theory. Oxford University Press.

Tillett, J. P. K. 1968 On the laminar flow in a free jet of liquid at high Reynolds numbers. J. Fluid Mech. 32, 273-292.

This seminar will take place on Friday 4th May, at 4p.m. in Room A2-002

If you have any questions regarding this seminar, please direct them to Iain Moyles (061 233726, iain.moyles@ul.ie).

A full list of upcoming seminars can be found at http://www.ulsites.ul.ie/macsi/node/48011

Supported by Science Foundation Ireland funding, MACSI - the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie), centred at the University of Limerick, is dedicated to the mathematical modelling and solution of problems which arise in science, engineering and industry in Ireland.