642 research outputs found
Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations
We study the propagation of two-dimensional finite-amplitude shear waves in a
nonlinear pre-strained incompressible solid, and derive several asymptotic
amplitude equations in a simple, consistent, and rigorous manner. The scalar
Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations
of motion for all elastic generalized neo-Hookean solids (with strain energy
depending only on the first principal invariant of Cauchy-Green strain).
However, we show that the Z equation cannot be a scalar equation for the
propagation of two-dimensional shear waves in general elastic materials (with
strain energy depending on the first and second principal invariants of
strain). Then we introduce dispersive and dissipative terms to deduce the
scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and
Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid
mechanics.Comment: 15 page
Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics
Scaling Navier-Stokes Equation in Nanotubes
On one hand, classical Monte Carlo and molecular dynamics (MD) simulations
have been very useful in the study of liquids in nanotubes, enabling a wide
variety of properties to be calculated in intuitive agreement with experiments.
On the other hand, recent studies indicate that the theory of continuum breaks
down only at the nanometer level; consequently flows through nanotubes still
can be investigated with Navier-Stokes equations if we take suitable boundary
conditions into account. The aim of this paper is to study the statics and
dynamics of liquids in nanotubes by using methods of non-linear continuum
mechanics. We assume that the nanotube is filled with only a liquid phase; by
using a second gradient theory the static profile of the liquid density in the
tube is analytically obtained and compared with the profile issued from
molecular dynamics simulation. Inside the tube there are two domains: a thin
layer near the solid wall where the liquid density is non-uniform and a central
core where the liquid density is uniform. In the dynamic case a closed form
analytic solution seems to be no more possible, but by a scaling argument it is
shown that, in the tube, two distinct domains connected at their frontiers
still exist. The thin inhomogeneous layer near the solid wall can be
interpreted in relation with the Navier length when the liquid slips on the
boundary as it is expected by experiments and molecular dynamics calculations.Comment: 27 page
Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane
In this paper we consider a fluid whose viscosity depends on both the mean normal stress and the shear rate flowing down an inclined plane. Such flows have relevance to geophysical flows. In order to make the problem amenable to analysis, we consider a generalization of the lubrication approximation for the flows of such fluids based on the development of the generalization of the Reynolds equation for such flows. This allows us to obtain analytical solutions to the problem of propagation of waves in a fluid flowing down an inclined plane. We find that the dependence of the viscosity on the pressure can increase the breaking time by an order of magnitude or more than that for the classical Newtonian fluid. In the viscous regime, we find both upslope and downslope travelling wave solutions, and these solutions are quantitatively and qualitatively different from the classical Newtonian solutions
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