Start-up Flow of a Bingham Fluid in a Pipe

We present an analytical solution of axisymmetric motion for a Bingham fluid initially at rest subjected to a constant pressure gradient applied suddenly. Using the Laplace transform, we obtain expressions which allow the calculation of the instantaneous velocity, plug radius and rate of flow as a function of time. We also give a relation for the shear stress in the plug and in the region where the behaviour of the fluid is Newtonian.     

Large deviations and concentration properties for ∇ϕ interface models

We consider the massless field with zero boundary conditions outside D _N ≡D∩ (ℤ ^d /N) (N∈ℤ⁺), D a suitable subset of ℝ ^d , i.e. the continuous spin Gibbs measure ℙ _N on ℝ ^ℤd/N with Hamiltonian given by H(ϕ) = ∑ _{x,y:|x−y|=1} V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D _N ^C . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D _N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ _N = ϕ/N. We study various concentration and relaxation properties of the family of random surfaces {ξ _N } and of the induced family of gradient fields ∇ ^N ξ _N as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξ _N } and show that the corresponding rate function is given by ∫ _D σ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙ _N under the volume constraint, i.e. the constraint that (1/N ^d ) ∑ _x∈DN ξ _N (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξ _N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ _N of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ ^N ξ _N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ? _p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sj?strand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields. Received: 3 March 1999 / Revised version: 9 August 1999 / Published online: 30 March 2000     

Unsteady simple shear flow in a viscoplastic fluid: comparison between analytical and numerical solutions

In this paper, an unsteady flow of a viscoplastic fluid for simple shear flow geometry is solved numerically using two regularizing functions to overcome the discontinuity for zero shear rate of the Bingham constitutive law. The adopted models are the well-known Papanastasiou relation and one based on the error function. The numerical results are compared with the analytical solution of the same problem obtained by Sekimoto (J Non-Newton Fluid Mech 39:107–113, 1991). The analysis of the results emphasizes that the errors are much smaller in the yielded than in the unyielded region. The models approximate closer the ideal Bingham model as the regularization parameters increase. The differences between the models tend to vanish as the regularization parameters are at least greater than 10⁵.     

Fractional Moment Bounds and Disorder Relevance for Pinning Models

We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n ^−α-1 L(n), with α ≥ 0 and L(·) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For α < 1/2 disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [3,28]. The same has been proven also for α = 1/2, but under the assumption that L(·) diverges sufficiently fast at infinity, a hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [12,17], where L(·) is asymptotically constant. Here we prove that, if 1/2 < α < 1 or α > 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case α = 1/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered in [12,17] is out of our analysis and remains open. The results are achieved by setting the parameters of the model so that the annealed system is localized, but close to criticality, and by first considering a quenched system of size that does not exceed the correlation length of the annealed model. In such a regime we can show that the expectation of the partition function raised to a suitably chosen power is small. We then exploit such an information to prove that the expectation of the same fractional power of the partition function goes to zero with the size of the system, a fact that immediately entails that the quenched system is delocalized.     

Log-periodic Critical Amplitudes: A Perturbative Approach

Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.     

A Geometrical Proof of Shiota's Theorem on a Conjecture of S. P. Novikov

We give a new proof of Shiota's theorem on Novikov's consecture, which states that the K.P. equation characters Jacobians among all indecomposable principally polarized abelian varieties.