Squeezing a viscoelastic fluid from a cone

The paper reports an exact kinematics for the squeezing flow from a cone of a general viscoelastic fluid. To obtain numerical values for the stresses, a network model that allows stress overshoot and shear-thinning in the start-up of a shear flow is adopted. Both these features are important in this flow. For the special case of an Oldroyd-B fluid it is shown that there is a limiting Weissenberg number above which at least one component of the stresses increases unboundedly with time.     

Elastic wedge impact onto a liquid surface: Wagner's solution and approximate models

The problem of elastic wedge impact onto the free surface of an ideal incompressible liquid of infinite depth is considered. The liquid flow is two-dimensional, symmetric and potential. The side walls of the wedge are modelled as Euler beams, which are either simply supported or connected to the main structure by linear springs. The liquid flow, the deflection of wedge walls and the size of wetted region are determined simultaneously within the Wagner theory of water impact. We are concerned with the impact conditions of strong coupling between the hydrodynamic loads and the structural response. The coupling is well pronounced for elastic wedges with small deadrise angles. This is the case when the fully nonlinear models fail and approximate models based on the Wagner approach are used. In contrast to the existing approximate models, we do not use any further simplifications within the Wagner theory. Calculations of the velocity potential are reduced to analytical evaluation of the added-mass matrix. Hydrodynamic pressures are not evaluated in the present analysis. In order to estimate the maximum bending stresses, both stages when the wedge surface is partially and totally wetted are considered.Three approximate models of water impact, which are frequently used in practical computations, are examined and their predictions are tested against the present numerical solution obtained by the normal mode method within the Wagner theory. It is shown that the decoupled model of elastic wedge impact, which does not account for the beam inertia, provides a useful formula for estimating the maximum bending stress in thick wedge platings.     

Squeezing of an Oldroyd-B fluid from a tube: Limiting Weissenberg number

It is shown that the squeezing flow of an Oldroyd-B fluid from a tube with a prescribed time-dependent radius has an exact separable solution. In the special case where the tube radius varies exponentially with time a similarity solution exists. However, in this case there is a critical Weissenberg number above which a component of the stress tensor increases without bound in time.     

Coaxial-disk flow of an Oldroyd-B fluid: exact solution and stability

This paper reports an exact solution for the coaxial disk flow of an Oldroyd-B fluid. The flow is approximately generated by the parallel-plate viscometer. Asymptotic and numerical solutions are reported showing that there is a critical Weissenberg number based on the angular velocity and the Maxwellian relaxation time, above which the flow is unstable. A linearized stability analysis for the basic inertialess flow confirms this numerical instability and yields the critical Weissenberg number.     

MHD mixed convection for viscoelastic fluid past a porous wedge

A magnetic hydrodynamic (MHD) mixed convective heat transfer problem of a second-grade viscoelastic fluid past a wedge with porous suction or injection has been studied. Governing equations include continuity equation, momentum equation and energy equation of the fluid. It has been analyzed by a combination of a series expansion method, the similarity transformation and a second-order accurate finite-difference method. Solutions of wedge flow on the wedge surface have been obtained by a generalized Falkner-Skan flow derivation. Some important parameters have been discussed by this study, which include the Prandtl number (Pr), the elastic number (E), the free convection parameter (G) and the magnetic parameter (M), the porous suction and injection parameter (C) and the wedge shape factor (β). Results indicated that elastic effect (E) in the flow could increase the local heat transfer coefficient and enhance the heat transfer of a wedge. In addition, similar to the results from Newtonian fluid flow and conduction analysis of a wedge, better heat transfer is obtained with a larger G and Pr.     

Rivulet flow of a viscoelastic liquid

A theoretical investigation is made of the rivulet flow of a viscoelastic liquid down an inclined plane. It is shown that elasticity causes a significant change in the shape of the rivulet, with height rise at the center. There is also a change in the relationship between the flow rate and the geometry of the rivulet. Elasticity is found to cause a flow in the cross-sectional plane in the form of counter-rotating vortices. In the Newtonian case the flow is purely axial.     

Break-up of a viscoelastic liquid jet

Applying Green's continuum theory of a slender body, the process of liquid jet break-up is analysed for a viscoelastic upper-convected Jeffreys fluid. In contrast to a Newtonian liquid an enforced growth of the perturbation is received from a linear analysis. A non-linear numerical analysis shows the viscosity-dependent filament formation between growing droplets of the viscoelastic liquid. The radius of these filaments decreases in an uniaxial extensional flow.     

Thermal convection in a viscoelastic liquid

The onset of convection in a layer of viscoelastic liquid heated from below is investigated. It is shown that the nature of the convective solution depends strongly on the particular constitutive relation used to characterize the viscoelasticity. For certain models and certain parameter ranges the convection is supercritical and stable, while for other models and parameter ranges it can be subcritical and unstable. It is suggested that observations of convective behavior can provide a test for constitutive relations proposed for a particular liquid. A Fourier representation of the solution to the nonlinear problem is developed which is shown to admit aperiodic, or chaotic, solutions in a specific truncation that generalizes the classical Lorenz system for the Newtonian Bénard problem.     

Converging flow of a viscoelastic liquid

Two-dimensional creeping sink flow of a Maxwell fluid is an accurate approximation to a bounded converging flow of contraction ratio at least 5 : 1 and covergence half-angles of up to 45°. In this range, a perturbation expansion in Weissenberg number can be used over most of the flow field in the range where stable processing can be carried out.     

Squeezing flows of Newtonian liquid films an analysis including fluid inertia

A theoretical study is made of the flow behavior of thin Newtonian liquid films being squeezed between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations.Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.Nomenclature a half of the distance, or gap, between the two plates - a ₀ the value of a at time t=0 - adot da/dt - ä d² a/dt ² - d³ a/dt ³ - a ⁱ components of a contravariant acceleration vector - f unknown function of z ₀ and t defined in (6) - f ⁱ function defined in (9) f ¹=r ₀ g(z ₀, t) f ²= ₀ f ³=f(z ₀, t) - F force applied to the plates - g unknown function of z ₀ and t defined in (6) - g g/z ₀ - h grid dimension in the z ₀ direction (see Fig. 5) - Christoffel symbol - i, j, k, l indices - k grid dimension in the t direction (see Fig. 5) - r radial coordinate direction defined in Fig. 1 - r ₀ radial convected coordinate - R radius of the circular plates - t time - v _r fluid velocity in the r direction - v _z fluid velocity in the z direction - v fluid velocity in the direction - x ⁱ cylindrical coordinate x ¹=r x²= x³=z - z vertical coordinate direction defined in Fig. 1 - z ₀ vertical convected coordinate - tangential coordinate direction - ₀ tangential convected coordinate - viscosity - kinematic viscosity, / - ⁱ convected coordinate ¹=r₀ ²=₀ ³=z₀ - density     

An asymptotic solution of an initial value problem for a non-linear viscoelastic rod

Stress wave propagation in a one-dimensional materai, described by the non-linear constitutive law

$\text{?σ}\text{?t}\text{? E}\text{?ε}\text{?t}\text{+ k[σ ? E'g3 + βε}{}^2\text{] = 0}$

, is analyzed by a two-time variable perturbation method. Secular terms are “cast out” by employing special orthogonality conditions. The resulting expansion reveals a “mode coupling”, due to the non-linearity, and displays the significance of the three parameters present in the constitutive law. Numerical values for the parameters are assumed and the resulting displacements computed.     

Extension of a cylinder of viscoelastic liquid

The basic rheological relations for the motion of viscoelastic media and a thixotropic viscoelastic medium in a constant longitudinal velocity gradient field were developed in [1] for the case of plane flow with a stagnation point. The results of that study showed that, in contrast with simple shear deformation, in the case of uniaxial extension steady-state flow of the liquid is not possible after reaching a deformation rate exceeding some critical value, and the liquid will undergo quasibrittle failure. In this case the approximation of the rheological relations by the Maxwell equation gives good qualitative agreement with the behavior of media of a more complex rheological structure. Below we investigate the kinematic and dynamic characteristics of liquid motion in a longitudinal velocity gradient field which is not constant in time, and we study some particular cases using as an example a Maxwellian liquid. The results of the study may be used to analyze the technological processes of forming and drawing fibers, and also for determining the rheological parameters of polymers by the extension method proposed by Kargin and Sogolova [2].     

Exact solution for standing monochromatic waves within a wedge

Exact solutions for standing internal waves are obtained in a region bounded by horizontal and inclined planes in the linear, ideal stratified fluid approximation at a constant Brent-V?is?l? frequency. The solutions are expressed in terms of the Macdonald functions and their derivatives. Nonuiform and uniform asymptotics are also constructed in the case of a small slope of the lower plane. The latter asymptotics are expressed in terms of either the Airy function or the Macdonald function. The exact and asymptotic solutions are numerically compared.     

求解动荷载作用下多层粘弹性半空间轴对称问题的精确刚度矩阵法

利用Laplace—Hankel联合积分变换，推导出了单层粘弹性半空间轴对称问题在动荷载作用下，层间完全接触情况的刚度矩阵，然后按传统的有限元方法组成总体刚度矩阵。通过求解由总体刚度矩阵所构成的代数方程就可解出动荷载作用下多层粘弹性半空间轴对称问题的矩阵。由于刚度矩阵的元素中只含有负指数项，计算时不会出现溢出的现象。本文还成功地应用了Durbin的Laplace逆变换的数值方法，求解出了多层粘弹性体的时域解。最后，文中还给出了路面动弯沉的计算结果与实测结果的对比来证明推导结果的准确性。