A non-uniformly travelling shock-wave in an LMS-gas

In this paper an LMS-gas is considered. This is an ideal gas with constant specific heats but with a special entropy distribution, so that generalized Riemann invariants r* and s* exist. A flow with r* and s* constant has the property that the gas moves as a rigid body with a constant acceleration or deceleration. It is shown that a flow composed of two domains, in each of which r* and s* are constant, separated by a plane normal shock-wave is admitted by the Rankine-Hugoniot equations. The path of the shock-wave is calculated exactly and a reasonably realistic situation is suggested to generate this composite flow.     

Stability of plane Poiseuille flow of a slightly viscoelastic fluid

Consideration is given to the stability of planePoiseuille flow of a slightly viscoelastic fluid which has a constant viscosity and normal stress differences varying nearly with the shear rate. It is shown that the presence of elasticity lowers the criticalReynolds number at which instability occurs.     

On Anisotropic Elastic Materials for which Young''s Modulus E ( n ) is Independent of n or the Shear Modulus G ( n , m ) is Independent of n and m

It is shown that, among anisotropic elastic materials, only certain orthotropic and hexagonal materials can have Young modulus E(n) independent of the direction n or the shear modulus G(n,m) independent of n and m. Thus the direction surface for E(n) can be a sphere for certain orthotropic and hexagonal materials. The structure of the elastic compliance for these materials is presented, and condition for identifying if the material is orthotropic or hexagonal is given. We also study the case in which n of E(n) and n, m of G(n,m) are restricted to a plane. When E(n) is a constant on a plane so are G(n,m) and Poisson's ratio ν(n,m). The converse, however, does not necessarily hold. A plane on which E(n) is a constant can exist for all anisotropic elastic materials. In particular, existence of such a plane is assured for trigonal, hexagonal and cubic materials. In fact there are four such planes for a cubic material. For these materials, not only E(n) is a constant, two other Young's moduli, the three shear moduli and the six Poisson's ratio on the plane are also constant.     

Evaluation and comparison of routes to obtain pressure coefficients from high-pressure capillary rheometry data

A capillary rheometer equipped with a pressure chamber is used to measure the pressure-dependent viscosity of polymethylmethacrylate (PMMA), poly-α-methylstyrene-co-acrylonitrile (PαMSAN), and low-density polyethylene (LDPE). Data analysis schemes are discussed to obtain pressure coefficients at constant shear rate and at constant shear stress. It is shown that the constant shear stress pressure coefficients have the advantage of being shear stress independent for the three polymers. The constant shear rate pressure coefficients, on the other hand, turn out to depend on shear rate, which makes them less suitable for use, e.g., in process simulations. In addition to the commonly used superposition method, a direct calculation method for the pressure coefficients is tested. Values obtained from both methods are equivalent. However, the latter requires less experimental and calculational efforts. From the obtained pressure coefficients, it is clear that PMMA and PαMSAN have a very similar pressure dependence, while LDPE is less sensitive to pressure.     

Solution of two-dimensional problems of unsteady heavy-fluid seepage into unsaturated porous soil on the basis of an instantaneous saturation model

Two-dimensional problems of the unsteady seepage of a heavy fluid into a uniform unsaturated porous soil from a single channel and from an infinite periodic system of identical channels are solved. The fluid level in the channel is a known function of timet, which simulates the propagation of the fluid along a furrow normal to the plane of the problem. Seepage into the soil, which occupies the space outside (mainly below) the channel, takes place under the influence of the force of gravity and begins at the momentt=0 after it is instantaneously filled with fluid. On the interval 0 <t < t₁ the height of the fluid in the channel is constant, and whent=t ₁ it falls instantaneously to some other level, also constant. This may be the zero level, which coincides with the bottom of the channel. The problem in which the fluid level in the channel rises is also considered.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–94, January–February, 1992.In conclusion the authors wish to thank V. A. Vostretsov for his assistance.     

Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness

For a spherical shell of arbitrary thickness which is subjected to an external hydrostatic pressure, symmetrical buckling takes place at a value of μ₁ which depends on and the mode number, where A₁ and A₂ are the undeformed inner and outer radii, and μ₁ is the ratio of the deformed inner radius to the undeformed inner radius. In the large mode number limit, we find that the dependence of μ₁ on has a boundary layer structure: it is a constant over almost the entire region of and decreases sharply from this constant value to unity as tends to unity (the thin-shell limit). Simple asymptotic expressions for the bifurcation condition are obtained. The classical result for thin shells is recovered directly from the equations of finite elasticity, and an asymptotic critical neutral curve (which envelops the neutral curves corresponding to different mode numbers) is obtained.     

Crack problems of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in piezoelectrics or on the interfaces of piezoelectric bimaterials. A class of boundary problems involving many cracks is also solved. For homogeneous materials it is found that the normal electric displacementD ₂ induced by the crack is constant along the crack faces which depends only on the applied remote stress field. Within the crack slit, the electric fields induced by the crack are also constant and not affected by the applied electric field. For the bimaterials with realH, the normal electric displacementD ₂ is constant along the crack faces and electric fieldE ₂ has the singularity ahead of the crack tip and a jump across the interface. The project is supported by the National Natural Science Foundation of China(No. 19704100) and the Natural Science Foundation of Chinese Academy of Sciences(No. KJ951-1-201).     

Unique Global Solvability for¶Initial-Boundary Value Problems¶in One-Dimensional Nonlinear Thermoviscoelasticity

The balance laws of mass, momentum and energy are considered for a broad class of one-dimensional nonlinear thermoviscoelastic materials. For the initial-boundary value problem corresponding to pinned endpoints held at constant temperature, we establish existence and uniqueness of temporally global classical solutions for initial data of unrestricted size. Our approach also applies to all boundary conditions involving pinned or stress-free endpoints which are either held at constant temperature or insulated. An additional and novel feature of the theory is that solid-like and gaseous materials are treated in a unified way. Accepted: June 24, 1999     

Experimental free coating flows at high capillary and Reynolds number

 Experimental studies are carried out to enhance the fundamental understanding of coating processes over a broad parametric range. Experiments herein identify the phenomena leading to the formation of an asymptotic meniscus profile, which eventually develops a cusp at the interface. The non-dimensional parameters that describe these phenomena are identified. In addition, flow visualization is carried out to reveal the entire flow structure using a visible laser. Two phenomena of free coating are identified depending on a parameter called the property number(Po). When Po is larger than about 0.5, the non-dimensional final film thickness (T ₀) becomes constant beyond the capillary number(Ca) of about unity. When Po is less than about 0.1, T ₀ depends on Ca and the Reynolds number(Re) but it becomes constant beyond the Weber number(=Ca Re) of about 0.2. In both cases T ₀ becomes constant as the effect of surface tension on the meniscus becomes relatively unimportant. The cusp formation is due to the effect of inertia(Re). The effect of applicator dimensions on T ₀ is also investigated for large Re flows. Received: 12 May 1998/Accepted: 19 January 1999     

Wave resistance of a viscous liquid

In this paper we examine the resistance encountered by a system of normal stresses during its rectilinear motion along the surface of a viscous liquid of infinite depth. The problem is solved in the linear formulation, i.e., it is assumed that amplitudes of the waves which arise are small and the waves are shallow. The solution for the two-and three-dimensional problems is obtained in the general case in closed form. In the two-dimensional case a detailed study is made of the case when a constant pressure p₀, moving with the constant velocity U, is given on a segment of length 2l. In the three-dimen-sional problem the case is studied when the normal stress is concentrated on a segment of a straight line of length 2l, which can replace a ship moving along a straight course with the constant velocity U. The integrals obtained in both cases are studied using the stationary phase method, the application of which for the three-dimensional integrals with respect to a volume with boundaries is justified in §1 of the paper. As a result we obtain equations for the wave resistance in the two- (§2) and three-dimensional (§3) cases.     

A flux-based modified method of characteristics

A flux-based modified method of characteristics (MMOC) methodology in 1D is described which has the following properties: unconditional stability (though explicit), exact answers for integer CFL (Courant) numbers, completely conservative (locally and globally) and able to utilize various flux limiters and various characteristic- (trajectory-) tracking algorithms. The use of characteristics based on cell-wise constant characteristic velocities results in considerable code simplification, and Van Leer's MUSCL is an accurate and cost-effective flux limiter. For CFL≥1 the flux limiter is applied only to the non-integer part of CFL, whereas the integer part is exact for constant velocities; therefore accuracy improves with larger CFL. It is not a cheap algorithm, although explicit, because the operation count per time step increases with the integer part of CFL, but it is much more accurate than the commonly used implicit upstream differencing. This flux-based MMOC method is well suited for groundwater flow calculations in which large local Courant numbers arise owing to grid clustering.