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We study a liquid film that is deposited onto a flat plate that is inclinedat a constant angle to the horizontal and is extracted from a liquid bath at aconstant speed. We additionally assume that there is a constant temperaturegradient along the plate that induces a Marangoni shear stress. We analysesteady-state solutions of a long-wave evolution equation for the filmthickness. Using centre manifold theory, we first obtain an asymptoticexpansion of solutions in the bath region. The presence of the temperaturegradient significantly changes these expansions and leads to the presence oflogarithmic terms that are absent otherwise. Next, we obtain numericalsolutions of the steady-state equation and analyse the behaviour of thesolutions as the plate velocity is changed. We observe that the bifurcationcurve exhibits snaking behaviour when the plate inclination angle is beyond acertain critical value. Otherwise, the bifurcation curve is monotonic. Thesolutions along these curves are characterised by a foot-like structure that isformed close to the meniscus and is preceded by a thin precursor film furtherup the plate. The length of the foot increases along the bifurcation curve.Finally, we explain that the snaking behaviour of the bifurcation curves iscaused by the existence of an infinite number of heteroclinic orbits close to aheteroclinic chain that connects in an appropriate three-dimensional phasespace the fixed point corresponding to the precursor film with the fixed pointcorresponding to the foot and then with the fixed point corresponding to thebath.

Collapsed heteroclinic snaking near a heteroclinic chain in dragged meniscus problems