Numerical modeling of viscoelastic flows is of great importance for complex engineering applications involving blood, paints, adhesives or foodstuff. When considering viscoelastic flows, the velocity, pressure and stress must satisfy the mass and momentum equation, supplemented with a constitutive equation involving the velocity and stress.

Viscoelastic fluid can undergo **longer stretching** before breaking

Stretching of a newtonian fluid

Stretching of a viscoelastic fluid

One of the simplest model is the so-called **Oldroyd-B** constitutive relation which can be derived from the kinetic theory of polymer dilute solutions.

The transient flow of a jet injected into a 2D rectangular cavity is reproduced here (from left to right: newtonian fluid, Deborah 1, Deborah 10, Deborah 100).

The transient flow of a jet injected into a parallelepiped cavity is reproduced here (left: newtonian fluid, right: 1,Deborah 100).

A fluid at rest contained between two parallel coaxial circular disks is considered. At the initial time, the top end-plate is moved vertically (left:newtonian fluid, right: viscoelastic fluid).

When the aspect ratio (height / diameter) is small, instabilities appear.

- BONITO, A. AND BURMAN, E.; A continuous interior penalty method for viscoelastic flows, SIAM: J. Sci. Comput., 30(3), 1156--1177, 2008.
- BONITO, A. AND CLÉMENT, PH. AND PICASSO, M.; Viscoelastic flows with complex free surfaces: Numerical analysis and simulations, Handbook of Numerical Analysis, eds P.G. Ciarlet, vol. XVI: Numerical Methods for Non-Newtonian Fluids, 305--370, 2011.
- BONITO, A. AND BURMAN, E.; A continuous interior penalty method for viscoelastic flows, SIAM: J. Sci. Comput., 30(3), 1156--1177, 2008.
- BONITO, A. AND CLÉMENT, PH. AND PICASSO, M.; Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow, Numer. Math., 107(2), 213--255, 2007.
- BONITO, A. AND CLÉMENT, PH. AND PICASSO, M.; Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows, Math. Model. Numer. Anal., 40(4), 785--814, 2006.
- BONITO, A. AND PICASSO, M. AND LASO, M.; Numerical simulation of 3D viscoelastic flows with free surfaces, J. Comput. Phys., 215(2), 691--716, 2006.

In all cases reported below, the bouncing jets appear after a **thin air layer** is formed between the jet and the rest of the fluid. This air layer is sustained in the newtonian case by the bath horizontal velocity and in the shampoo case by its ability to shear-thinning.

A newtoninan fluid is poored on a bath of the same fluid flowing with a horizontal velocity (S. Lee PhD 2014). The left computation takes into account the surface tension effect.

When a stream of shampoo is fed onto a pool in ones hand a jet can leap sideways or rebound from the liquid surface in an intriguing phenomenon known as the Kaye effect (S. Lee PhD 2014).

- BONITO, A. AND GUERMOND, J.-L. AND LEE, S.; Numerical Simulations of Bouncing Jets, Internat. J. Numer. Methods Fluids, 80(1), 53--75, 2015.
- LEE, S. AND LI, E.Q. AND MARSTON, J.O. AND BONITO, A. AND THORODDSEN, S.T; Leaping shampoo glides on a lubricating layer, Phys. Rev. E. (Rapid Communication), 87(6), 4 pages, 2013.