Thin film limit and film rupture of the visco--capillary gravity--driven channel flow

Starting from an exact solution of a visco-capillary gravity--driven film flow of a Newtonian fluid in an inclined channel, we discuss the special case of thin films. The shape of the free surface, the velocity field and the flow rate are obtained from a pure analytical treatment. Making use of an adequate rescaling with a generalized capillary length we pay special attention to vanishingly thin films where capillary effects become dominant. Our investigations deliver a necessary condition for the flow rate in order to avoid a film rupture.     

Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow

We present a rigorous numerical proof based on interval arithmetic computations categorizing the linearized and nonlinear stability of periodic viscous roll waves of the KdV–KS equation modeling weakly unstable flow of a thin fluid film on an incline in the small-amplitude KdV limit. The argument proceeds by verification of a stability condition derived by Bar–Nepomnyashchy and Johnson–Noble–Rodrigues–Zumbrun involving inner products of various elliptic functions arising through the KdV equation. One key point in the analysis is a bootstrap argument balancing the extremely poor sup norm bounds for these functions against the extremely good convergence properties for analytic interpolation in order to obtain a feasible computation time. Another is the way of handling analytic interpolation in several variables by a two-step process carving up the parameter space into manageable pieces for rigorous evaluation. These and other general aspects of the analysis should serve as blueprints for more general analyses of spectral stability.     

Thin film flow over a non-linear stretching sheet in presence of uniform transverse magnetic field

A thin viscous liquid film flow is developed over a stretching sheet under different non-linear stretching velocities in presence of uniform transverse magnetic field. Evolution equation for the film thickness is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. It is observed that all types of stretching produces film thinning, but non-monotonic stretching produces faster thinning at small distance from the origin. Effect of the transverse magnetic field is to slow down the film thinning process. Observed flow behavior is explained physically.     

Thin film flow over a non-linear stretching sheet in presence of uniform transverse magnetic field

A thin viscous liquid film flow is developed over a stretching sheet under different non-linear stretching velocities in presence of uniform transverse magnetic field. Evolution equation for the film thickness is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. It is observed that all types of stretching produces film thinning, but non-monotonic stretching produces faster thinning at small distance from the origin. Effect of the transverse magnetic field is to slow down the film thinning process. Observed flow behavior is explained physically.     

Thin film tribology of pharmaceutical elastomeric seals

The primary purpose of valve seals in inhalation and other drug dispensing devices is to inhibit leakage of highly volatile formulation from pressurised canisters. This requirement often conflicts with smooth operation of valves because of poor lubrication of seals. The repercussions of this can be variability in dispensed dose as well as loss of prime and gradual wear of seals. Although a good volume of literature is available for general purpose o-ring seals, the characteristic behaviour of those used in pharmaceutical devices deviate from this significantly. The paper studies tribology of such seals, subjected to global fitment and canister pressure deformation and localised conjunctional elastohydrodynamic pressures. It is shown that ideally smooth seals would operate under iso-viscous elastic (soft EHL) regime of lubrication. However, the predicted ultra-thin films are insufficient to ensure fluid film lubrication because of rough micro-scale nature of elastomeric seal surface and poor lubricity of the usual bio-compatible formulations. The paper also shows that siliconisation of elastomeric contacting surface only marginally improves its tribological performance.     

Attractor, chain recurrent set and limit set of flow

The properties of attractor, chain recurrent set and limit set of the flow on a compact metric space are studied.     

Micromorphic description of turbulent channel flow

The simple microfluid theory of Eringen is employed to determine the velocity profile and microgyrations in a steady flow of viscous fluids between two parallel walls. The shearing stress, microrotations, microinertia and Reynold stresses are determined. The results are compared with the experimental work of Laufer.     

Global renormalization group flow and thermodynamic limit for the hierarchical fermionic model

The renormalization group transformation for the hierarchical fermionic model is presented as a rational map in the plane of the coupling constants. We investigate the global dynamics of this transformation. The existence of the thermodynamic limit in the domain under investigation is proved. The automodel limit of the constructed fields is described.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 201–212, May, 1996.Translated by L. O. Chekhov.     

Viscosity vanishing limit of the nonlinear pipe magnetohydrodynamic flow with diffusion

We establish viscosity vanishing limit of the nonlinear pipe magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed diffusion. The convergence is verified under various Sobolev norms, including the L^∞(H¹) norm.     

Rarefied gas channel flow using the BGK model

Zusammenfassung Das Problem der Strömungsbildung in Couetteströmung suspendierter Partikel in Flüssigkeiten wird mit Hilfe der Theorie der Flüssigkeiten mit Mikrostruktur entwickelt. Die Reaktion der Mikrostruktur auf das entwickelte Geschwindigkeitsfeld, die Grenzschicht und die resultierende Schubspannung werden untersucht.     

Transient film profile of thin liquid film flow on a stretching surface

In this paper we have studied a non-planar thin liquid film flow on a planar stretching surface. The stretching surface is assumed to stretch impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically. It is observed that faster stretching causes quicker thinning of the film on the stretching surface. Velocity distribution in the liquid film and the transient film profile as functions of time are obtained. (Received: May 4, 2004; revised: February 2/August 24, 2005)     

Transient film profile of thin liquid film flow on a stretching surface

In this paper we have studied a non-planar thin liquid film flow on a planar stretching surface. The stretching surface is assumed to stretch impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically. It is observed that faster stretching causes quicker thinning of the film on the stretching surface. Velocity distribution in the liquid film and the transient film profile as functions of time are obtained.     

Minimax principle for magnetohydrodynamic channel flow

A variational principle, applicable to a wide range of linear non-self-adjoint problems, is used to derive maximum and minimum principles for a magnetohydrodynamic channel flow. Truncated Fourier series for the velocity and magnetic field are found which give reasonable approximations at small Hartmann numbers.     

Minimax principle for magnetohydrodynamic channel flow

Summary A variational principle, applicable to a wide range of linear non-self-adjoint problems, is used to derive maximum and minimum principles for a magnetohydrodynamic channel flow. Truncated Fourier series for the velocity and magnetic field are found which give reasonable approximations at small Hartmann numbers.

---

Zusammenfassung Ein auf ein breites Spektrum linearer nicht selbstadjungierter Probleme anwendbares Variationsprinzip wird benutzt, um maximale und minimale Prinzipien für einen magnetohydrodynamischen Kanalfluss abzuleiten. Verkürzte Fourierreihen für die Geschwindigkeit und für das magnetische Feld, die vernünftige Annäherungen bei kleinen Hartmannschen Zahlen geben, werden gefunden.

     

Diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow with fixed viscosity

We establish magnetic diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed viscosity. The convergence is verified under various Sobolev norms, including the L~∞(L~2)and L~∞(H~1) norm.     

Effect of contraction on turbulent channel flow

Summary Measurements in an effectively two-dimensional channel indicate that flow acceleration at constant Reynolds Number can have appreciable effects on the turbulence structure. As in wakes, the structure approaches equilibrium exponentially after the acceleration. The effect of acceleration appears to be (at least qualitatively) of the same kind as is observed when turbulence is subjected to rapid distortion.

---

Zusammenfassung Messungen in einem effektiv zwei-dimensionalen Kanal zeigen, daß eine Strömungsbeschleunigung bei konstanter Reynolds-Zahl einen wesentlichen Einfluß auf die Turbulenz-Struktur haben kann. Nach der Beschleunigung findet man eine exponentielle Annäherung an das Gleichgewicht, wie bei Dellen. Der Beschleunigungs-Effekt auf die Turbulenz erscheint, wenigstens qualitativ, von der gleichen Art zu sein wie der Einfluß von raschen Verformungen.

---

Symbols c _f skin-friction coefficient = - D half-width of the channel - L length of contraction=12 inches - dp/dx pressure gradient - V mean velocity in thex-direction,U=U ₀ aty=D - U mean velocity - average of the mean velocity - U ^* friction velocity = (₀/)^± - root mean square value of the turbulent velocity fluctuations inx, y andz directions - X coordinate in the direction of the flow;X=0 at the end of contraction - Y coordinate perpendicular to the surface of the wall on which measurements are madey=0 at the wall andy=D at the centre of the channel - ₀ Wall shear stress - density of the fluid - viscosity of the fluid - kinematic viscosity of the fluid=/     

Statistical modeling of compressible turbulent channel flow

Several questions that are relevant to turbulence modeling are addressed on the basis of recently obtained direct numerical simulation results of turbulent supersonic channel flow. In particular, this concerns the turbulence frequency production mechanism, wall damping effects on turbulence model parameters, and the relevance of compressibility effects. Limited support is found for usually applied models for the turbulence frequency production and wall damping effects. In contrast to that it is shown that turbulence frequency production mechanisms and wall damping effects may be explained very well on the basis of a frequency scaling that characterizes mean flow changes. The influence of compressibility is found to be relevant. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compressible gas flow inside a cylindrical channel with a source on the channel axis

The article solves the problem of compressible nonviscous gas flow inside a cylindrical channel in the presence of a source on the channel axis. The cases considered include a supersonic spherically symmetrical source and a hypersonic source modeling jet flow. The numerical solution is obtained by two methods: one method treats the flow singularities (shocks, etc.) in explicit form, while the other uses the “through calculation” procedure. The solution is applied to analyze the physical flow pattern and the distribution of gasdynamic parameters in the shock layer, in particular on the channel walls. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 169–180.     

Numerical appoximation of a channel flow problem

Our object in this article is to describe some numerical schemes for the approximation of a Channel Flow problem, and to study the stability of the schemes. Discretization is performed by the combination of spectral methods and finite difference methods.