Transfer of a microfluid to a stochastic fibre network

The transfer of a microscopic fluid droplet from a flat surface to a deformable stochastic fibre network is investigated. Fibre networks are generated with different levels of surface roughness, and a two-dimensional, two-phase fluid-structure model is used to simulate the fluid transfer. In simulations, the Navier-Stokes equations and the Cahn-Hilliard phase-field equations are coupled to explicitly include contact line dynamics and free surface dynamics. The compressing fibre network is modelled as moving immersed boundaries. The simulations show that the amount of transferred fluid is approximately proportional to the contact area between the fluid and the fibre network. However, areas where the fluid bridges and never actually makes contact with the substrate must be subtracted.     

Diffusion of a line vortex in a second-order fluid

In this paper, the diffusion of a line vortex in a second-order fluid is considered. The Hankel transform is used to solve this problem and an exact solution for the velocity distribution is found in terms of a definite integral. The integrand is an oscillatory function and the integration is performed by a numerical technique. It is found that there are pronounced effects of viscoelastic properties on the velocity distribution with respect to that of the Newtonian fluid.     

Unsteady non-linear convection in a second-order fluid

Linear and weakly non-linear analyses of convection in a second-order fluid is investigated. The Rivlin-Ericksen constitutive equation is considered to give viscoelastic correction to the momentum equation. The linear and non-linear analyses are, respectively, based on the normal mode technique and truncated representation of Fourier series. The linear theory reveals that the critical eigenvalue is independent of viscoelastic effects and the principle of exchange of stabilities holds. An autonomous system of differential equations representing cellular convection arising in the non-linear study is solved numerically. The non-linear analysis reveals that finite amplitudes have random behaviour. The effect of viscoelasticity on the non-linear solutions is analysed by considering different projections in the phase-space. Also, the transient behaviour concerning the variations of the Nusselt number with time has been investigated. The onset of chaotic motion is also discussed in this paper.     

Rotating flow of a second-order fluid on a porous plate

A study is made of the unsteady flow engendered in a second-order incompressible, rotating fluid by an infinite porous plate exhibiting non-torsional oscillation of a given frequency. The porous character of the plate and the non-Newtonian effect of the fluid increase the order of the partial differential equation (it increases up to third order). The solution of the initial value problem is obtained by the method of Laplace transform. The effect of material parameters on the flow is given explicitly and several limiting cases are deduced. It is found that a non-Newtonian effect is present in the velocity field for both the unsteady and steady-state cases. Once again for a second-order fluid, it is also found that except for the resonant case the asymptotic steady solution exists for blowing. Furthermore, the structure of the associated boundary layers is determined.     

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

Fluctuating flow of a second-order fluid near a stagnation point

Summary The boundary layer equations for axisymmetric flow of an incompressible second-order fluid have been deduced. The flow of such a fluid near a stagnation point when the main stream outside the boundary layer fluctuates in magnitude but not in direction has been discussed. The velocity distribution is found for various values of the steady mean in two limiting cases of small and large values of the frequency of the oscillation of the main stream. The frequency for which two approximate solutions overlap has been calculated in each case.     

Feedback stabilization of a class of nonlinear second-order systems

The goal of this paper is to study stabilization techniques for a system described by nonlinear second-order differential equations. The problem is to determine the feedback control as a function of the state variables. It is shown that the following controllers can asymptotically stabilize the system: linear position feedback, linear velocity feedback and a group of nonlinear feedbacks. The stability of the corresponding closed-loop system is proved by imposing a suitable Lyapunov function and then using LaSalle’s invariance principle. The results of numerical computations are included to verify theoretical analysis and mathematical formulation. Some application examples from robotics, mechanics and electronics are presented.     

Plane flow of a second-order fluid past submerged boundaries

We consider the plane flow of a second-order fluid past submerged obstacles such as circular and elliptical cylinders in situations where inertia effects cannot be neglected. The effect of the short memory of the fluid upon the flow features is analyzed in detail. In particular, it is found that the viscoelasticity of the fluid reduces the drag coefficient for very low Reynolds numbers, while the opposite is true for large Reynolds numbers.     

Unsteady flow of a second-order fluid between concentric cylinders

The unsteady motion of an incompressible second-order fluid contained between two finite coaxial cylinders is examined when the outer cylinder is held fixed while the inner cylinder is constrained to execute an arbitrary angular velocity. A solution is obtained in closed form with the aid of transforms and an expression is obtained for the couple experienced. The particular case of a periodic angular velocity is then examined and some numerical work done. There is a marked difference between the results obtained and their classical counterparts.     

Dissipation in a dilute suspension of spheres in a second-order fluid

We examine the effective medium properties of a dilute suspension of spheres in a second-order fluid under linear shear. Since the second-order fluid is the first step toward the general viscoelastic fluid, the results obtained may provide a qualitative feel for the problem in which the suspending fluid obeys a more complicated (and realistic) constitutive relation.The dissipation in the medium is calculated by determining the rate of working by surface forces; this is compared to the dissipation in a homogeneous fluid to give the effective properties. The results show that the term linear in volume fraction increases the corresponding rheological coefficient, just as in the Newtonian case. It is to be noted that the second-order dissipation is zero for simple shear and other weak flows, whereas for strong flows the small correction may increase or decrease the overall dissipation.     

On unsteady motions of a second-order fluid over a plane wall

In this paper, three types of unsteady flows of second-order fluids are considered, namely, flow caused by impulsive motion of a flat plate, flow induced by a constantly accelerating plane and flow imposed by a flat plate that applies a constant tangential stress to the fluid. The previous attempts made regarding these problems, by using the Laplace transform, have failed. In this paper, the sine and the cosine transforms are used to solve these problems and exact solutions for the velocity distributions are found in terms of definite integrals. It is shown that these exact solutions satisfy the initial and the boundary conditions and the governing equation.     

Stable second-order accurate iterative solutions for second-order elliptic problems

The paper describes a numerical scheme for solving a convection–diffusion elliptic system with very small diffusion coefficients. This iterative numerical procedure is unconditionally stable and converges very rapidly. Although only linear equations are considered here, this technique can be easily extended to non-linear equations, while keeping its main features as for the linear case. The numerical experiments presented are quite general and confirm most of these features. These examples also show a good way of implementing this scheme.     

Free surface instability of a rotating liquid film of a second-order fluid

Summary The free surface instability of a liquid film of an incompressible second-order fluid attached to a rotating circular cylinder is studied with respect to rotationally symmetric infinitesimal disturbances. The surface mode is examined in detail; the shear wave mode is qualitatively discussed. The results show that for any Reynolds numberR and any surface tensionT, there is always a range of wave numberm for which the flow is unstable. The dependence of the growth rate as well as the critical wave numberm, on the normal stress coefficients of the fluid is studied quantitatively for small as well as for finiteR. It is found that while the first normal stress coefficient has no effect on the stability of the flow, the presence of the second normal stress coefficient always tend to make the flow less stable (i.e., larger growth rate). The critical wave length (which corresponds to the mode of maximum instability and thereby determines the wave length of the breaking apart of the rings formed on the cylinder) is longer for a second-order fluid than for a Newtonian one. These effects (larger growth rate and longer critical wave length) are more pronounced asR increases and/orb (the thickness parameter) increases for a fixed surface tension parameterS. Surface tension is always stabilizing. Decreasing surface tension increases the growth rate, more so for larger magnitude of the second normal stress coefficient.

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Zusammenfassung Die Instabilität der freien Oberfläche eines einem rotierenden Kreiszylinder anhaftenden Flüssigkeitsfilms einer inkompressiblen Flüssigkeit zweiter Ordnung wird gegenüber rotationssymmetrischen infinitesimalen Störungen untersucht. Dabei werden Oberflächenwellen eingehend, Scherwellen dagegen nur qualitativ behandelt. Für jede ReynoldszahlR und für jede OberflächenspannungT gibt es einen Bereich von Wellenzahlenm, für welche die Strömung instabil ist. Für kleine und für endlicheR wird die Abhängigkeit der Zuwachsrate und der kritischen Wellenzahlm _cr vom Normalspannungskoeffizienten der Flüssigkeit quantitativ untersucht. Während der erste Normalspannungskoeffizient keinen Einfluß auf die Stabilität der Strömung zeigt, hat der zweite Normalspannungskoeffizient die Tendenz, die Strömung weniger stabil zu machen (z. B. größere Zuwachsraten zu liefern). Die kritische Wellenlänge (die der maximalen Instabilität entspricht und infolgedessen die Trennung der sich auf dem Zylinder bildenden Ringe verursacht) ist für die Flüssigkeit zweiter Ordnung größer als für die newtonsche Flüssigkeit. Diese Erscheinungen (größere Zuwachsrate und größere kritische Wellenlänge) sind für konstanten OberflächenspannungsparameterS mit zunehmendemR und/oder zunehmendemb (Dickenparameter des Flüssigkeitsfilms) stärker ausgeprägt. Die Oberflächenspannung wirkt immer stabilisierend. Eine Abnahme in der Oberflächenspannung erhöht die Zuwachsrate, dieser Effekt ist für große Werte des zweiten Normalspannungskoeffizienten stärker ausgeprägt.

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With 12 figures     

Solvability of a class of second-order quasilinear boundary value problems

The second-order quasilinear boundary value problems are considered when the nonlinear term is singular and the limit growth function at the infinite exists. With the introduction of the height function of the nonlinear term on a bounded set and the consideration of the integration of the height function, the existence of the solution is proven. The existence theorem shows that the problem has a solution if the integration of the limit growth function has an appropriate value.