The Mathematics Department is delighted to welcome Prof Saleh Tanveer (The Ohio State University) as a Visiting Nelder Fellow in the Applied Mathematics Section from 27 May until 30 July 2022.  

 Saleh Tanveer has been a Professor at the Ohio State University since 1989. He works in asymptotics, complex variables and PDEs and their applications, primarily in fluid mechanics. His interests range from theoretical to applied aspects of a problem. 

During his visit Prof Saleh Tanveer will held lecture series on:

"Asymptotics and analysis of a few nonlinear problems arising in applications"

 

Talk 1: 9 June 2022

Time: 14:00 - 15:30, Room 140

Singularities of P-1 and proof of Dubrovin Conjecture

Abstract: Painl/'eve equations arise often in applications, particularly from reductions of integrable PDEs. In connection to a universal scenario for blow up in the Nonlinear Schroedinger Equation proposed by Dubrovin, Grava and Klein, it is necessary that the Tri-Tronquee solution of P-1 does not have any singularity in a sector of width 8 \pi/5. This result of Dubrovin, Grava and Klein, which was supported by numerical evidence, usually referred to as Dubrovin Conjecture, was proved in 2014.  It was also extended in 2016 to determine prove previous results on location of nearest singularity of P-1 from the origin. 

The method relies on precise exponential asymptotics, and linearization about an empirical solution, a method introduced earlier by Costin et al in 2012.

Collaborators: O. Costin, M. Huang and A. Adali

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Talk 2: 10 June 2022

Time: 14:00 - 15:00, Room 139

Results for a nonlocal thin-film model in two fluid shear flow.

Abstract:  Kaligirou et al (2016) recently derived a nonlocal thin-film equation model for two fluid shear flows in a channel. We discuss results on whether or not the 2+1 model equations allow smooth solutions for all time, or if there is singularity in finite time. Furthermore, we discuss existence of different branches of steady traveling wave solutions in one dimension that initially bifurcate from a flat state and present their stability and bifurcation properties, including Hopf-bifurcation to time periodic states. We also present techniques for justifying these results mathematically. We also present results on how other effects such as slip can be accommodated in this mathematical framework. 

Joint work with D. Papageorgiou

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Talk 3: 16 June 2022 

Time: 14:00 - 15:30, Room 140

 

On existence of solution to general Nonlinear Two point boundary value problems, and some applications.

Abstract: Two point boundary value problems arise in many applications, such as determining homoclinic and heteroclinic orbits in dynamical systems,  or even determining periodic solutions to PDEs. We introduce a method for analysis for a general system of nonlinear ODEs, and apply them in particular to a Cholera epidemic model. We also show how method extends to infinite domains, for instance the Blasius similarity solution. 

Collaborators: A. Chowdhury, X. Wang, O. Costin, T.E. Kim

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Talk 4: 23 June 2022

Time: 14:00 - 15:30, Room 140
 

Saleh Tanveer (The Ohio State University): Traveling waves in pipe flow.

Abstract: It is well-known that Poiseuille flow in a cylindrical pipe is linearly stable for any Reynolds number R, yet one observes transition to turbulence in experiments due to nonlinear instability threshold decreasing with R. In the Navier-Stokes dynamical system, special role is played in the transition process by traveling wave states that in the large Reynolds number limit approach the Pouseille flow in some sense, and yet disconnected from Poisueille flow in the parameter space.

In the talk, we present numerical calculations of two new traveling wave states, characterized by a shrinking core towards the pipe center with increasing R. We also present arguments to identify all possible scaling solutions as R → ∞. Within the solution class where the axial wave length is independent of R, we identify two families. One is a nonlinear viscous core (NVC) solution characterized by a fully nonlinear interaction between rolls, streaks and waves in a shrinking core of radius δ = R^(-1/4) where all axial velocity components scale as R^(-1/2) , while perpendicular plane components of velocity scale at R^(−3/4) . In this case, the wave speed satisfies  (1 − c) = O(R^(−1/2) ). The streak for two-fold azimuthally symmetric states, with or without additional shift and rotate symmetry, does not decay as one exits the core until wall effects become important. The asymptotics also suggests possibility of a second class of solutions that we call vortex wave interaction (VWI) solutions: For δ = 1 or δ = R^(-1/6) , a linear wave of size δ^(-4/3) R^(−5/6) traveling with wave speed c where 1 − c = O(δ^2 ) concentrated in a critical layer of thickness δ(δ 4R)^(−1/3) around the critical curve drives rolls of size R^(−1) δ^(−1/3) localized in a core of radius δ, which in turn drives a streak of size δ^2 . We identify previous calculations by other researchers as finite R realization of VWI states for δ=1. We also present linear stability results that suggests that these states have a low dimensional unstable manifold. In some cases, the unstable eigenvalues scale with inverse powers of R explaining why these states can form meta-stable coherent structures around which intermediate R turbulent flow appear to organize themselves. 

Joint work with O.Ozcakir, P. Hall, E. Overman

Host: Prof Demetrios Papageorgiou