scholarly journals Steady states and dynamics of a thin-film-type equation with non-conserved mass

2019 ◽  
Vol 31 (6) ◽  
pp. 968-1001
Author(s):  
HANGJIE JI ◽  
THOMAS P. WITELSKI
Keyword(s):  
Thin Film ◽  
Gradient Flow ◽  
Type Equation ◽  
Finite Amplitude ◽  
Steady States ◽  
Parabolic Partial Differential Equation ◽  
Fluid Films ◽  
Film Type ◽  
Order Degenerate ◽  
Steady State Solutions

We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

scholarly journals Energy Levels of Steady States for Thin-Film-Type Equations

2002 ◽  
Vol 182 (2) ◽  
pp. 377-415 ◽  
Author(s):  
R.S. Laugesen ◽  
M.C. Pugh
Keyword(s):  
Thin Film ◽  
Energy Levels ◽  
Steady States ◽  
Film Type

The Significance of Tread Element Flexibility to Thin Film Wet Traction

1975 ◽  
Vol 3 (4) ◽  
pp. 215-234 ◽  
Author(s):  
A. L. Browne ◽  
D. Whicker ◽  
S. M. Rohde
Keyword(s):  
Thin Film ◽  
Time Dependent ◽  
Lateral Expansion ◽  
Wheel Slip ◽  
Fluid Films ◽  
Thin Fluid

Abstract An analysis is presented for the action of individual tire tread elements on polished sections of pavement covered by thin fluid films. Tread element flexibility, wheel slip, and time-dependent loading are incorporated. The effect of the lateral expansion of tread elements on groove closure is also studied.

A system of conservation laws with discontinuous flux modelling flotation with sedimentation

2019 ◽  
Vol 84 (5) ◽  
pp. 930-973 ◽  
Author(s):  
Raimund Bürger ◽  
Stefan Diehl ◽  
María del Carmen Martí
Keyword(s):  
Conservation Laws ◽  
Steady States ◽  
Solid Particles ◽  
Flotation Column ◽  
Drift Flux ◽  
Hydrophobic Particles ◽  
Discontinuous Flux ◽  
Scalar Conservation ◽  
Steady State Solutions ◽  
Hydrophilic Particles

Abstract The continuous unit operation of flotation is extensively used in mineral processing, wastewater treatment and other applications for selectively separating hydrophobic particles (or droplets) from hydrophilic ones, where both are suspended in a viscous fluid. Within a flotation column, the hydrophobic particles are attached to gas bubbles that are injected and float as aggregates forming a foam or froth at the top that is skimmed. The hydrophilic particles sediment and are discharged at the bottom. The hydrodynamics of a flotation column is described in simplified form by studying three phases, namely the fluid, the aggregates and solid particles, in one space dimension. The relative movements between the phases are given by constitutive drift-flux functions. The resulting model is a system of two scalar conservation laws with a multiply discontinuous flux for the aggregates and solids volume fractions as functions of height and time. The model is of triangular nature since one equation can be solved independently of the other. Based on the theory of conservation laws with discontinuous flux, steady-state solutions that satisfy all jump and entropy conditions are constructed. For the existence of the industrially relevant steady states, conditions on feed flows and concentrations are established and mapped as ‘operating charts’. A numerical method that exploits the triangular structure is formulated on a pair of staggered grids and is employed for the simulation of the fill-up and transitions between steady states of the flotation column.

Asymptotic behaviour of the thin film equation in bounded domains

2001 ◽  
Vol 12 (2) ◽  
pp. 135-157 ◽  
Author(s):  
M. BOWEN ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Contact Angle ◽  
Early Time ◽  
Similarity Solutions ◽  
Bounded Domains ◽  
Extinction Time ◽  
Degenerate Diffusion ◽  
Thin Film Equation ◽  
Order Degenerate ◽  
Mass Increase

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.

Swirling flow states in finite-length diverging or contracting circular pipes

2017 ◽  
Vol 819 ◽  
pp. 678-712 ◽  
Author(s):  
Zvi Rusak ◽  
Yuxin Zhang ◽  
Harry Lee ◽  
Shixiao Wang
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Finite Length ◽  
Vortex Breakdown ◽  
Steady States ◽  
Inviscid Limit ◽  
Circular Pipes ◽  
Outlet Flow ◽  
Separation Zones ◽  
Steady State Solutions

The dynamics of inviscid-limit, incompressible and axisymmetric swirling flows in finite-length, diverging or contracting, long circular pipes is studied through global analysis techniques and numerical simulations. The inlet flow is described by the profiles of the circumferential and axial velocity together with a fixed azimuthal vorticity while the outlet flow is characterized by a state with zero radial velocity. A mathematical model that is based on the Squire–Long equation (SLE) is formulated to identify steady-state solutions of the problem with special conditions to describe states with separation zones. The problem is then reduced to the columnar (axially-independent) SLE, with centreline and wall conditions for the solution of the outlet flow streamfunction. The solution of the columnar SLE problem gives rise to the existence of four types of solutions. The SLE problem is then solved numerically using a special procedure to capture states with vortex-breakdown or wall-separation zones. Numerical simulations based on the unsteady vorticity circulation equations are also conducted and show correlation between time-asymptotic states and steady states according to the SLE and the columnar SLE problems. The simulations also shed light on the stability of the various steady states. The uniqueness of steady-state solutions in a certain range of swirl is proven analytically and demonstrated numerically. The computed results provide the bifurcation diagrams of steady states in terms of the incoming swirl ratio and size of pipe divergence or contraction. Critical swirls for the first appearance of the various types of states are identified. The results show that pipe divergence promotes the appearance of vortex-breakdown states at lower levels of the incoming swirl while pipe contraction delays the appearance of vortex breakdown to higher levels of swirl and promotes the formation of wall-separation states.

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