Fourth-order Nonlinear Partial Differential Equation Research Articles

The effect of the Coriolis force on axisymmetric rotating thin film flows

The axisymmetric flow of a thin Newtonian fluid layer subject to centrifugal and Coriolis forces, surface tension and gravity is considered. Employing lubrication theory the mathematical problem is reduced to the solution of a fourth-order nonlinear partial differential equation for the film height, which requires solving numerically. At the moving contact line a precursor film model is adopted. Once the film height is known other quantities, such as fluid velocities and pressure may be easily determined. Of particular interest, is the fact that, within the restrictions of lubrication theory the Coriolis term in the radial velocity equation is of the same order as the inertia terms and is therefore negligible. This means the velocity equations are not fully coupled and, when the flow is axisymmetric, the Coriolis force has no effect on the height of the fluid film.

Transformations and Equation Reductions in Finite Elasticity IV: Illustration of the General Integral for a Plane Strain Similarity Deformation

In three previous parts of this work, for plane strain, plane stress, and axially symmetric deformations, a number of new first integrals are deduced for the so-called perfectly elastic Varga materials. These results constitute a considerable advance in the theory of finite elastic deformations, there being no results similar to these in existing theory. The new integrals, together with the constraint of incompressibility, mean that certain highly nonlinear fourth-order partial differential equations admit second-order systems, every solution of which is a solution of the corresponding fourth order problem. Moreover, many of the second-order partial differential equations admit linearization to either the harmonic equation or the Helmholtz equation, thus giving rise to the possibility of generating quite general solutions. However, it is not immediately clear how such solutions relate to solutions of the full system. This part of the work (IV) attempts to discover the extent to which the solutions of these new first integrals span the solutions of the full space. In this paper, the general integral for plane strain deformations, given in Part III, is illustrated with reference to a specific similarity deformation, which maps one wedge-shaped region into another such region and for which the general solution of the full system can be obtained in closed form.

Exact solutions of non-Newtonian fluid flows with prescribed vorticity

The equations of motion of a non-Newtonian second-grade fluid flow are highly nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions. Due to the complexity of the equations, inverse methods described by Nemenyi [1] have become attractive in the study of non-Newtonian fluids. In these methods, certain physical or geometrical properties of the flow field are assumed a priori. Lin and Tobak [2] studied steady plane viscous incompressible flows for a chosen vorticity function by decomposing the nonlinear fourth-order partial differential equation in the streamfunction. This excellent approach yielded two second-order linear partial differential equations in the streamfunction. Hui [3] used this approach to study unsteady plane viscous incompressible flows. During the past decade, there has been substantial interest in flows of viscoelastic liquids due to the occurrence of these flows in industrial processes. In this paper, we study the steady and unsteady incompressible viscous non-Newtonian second-grade fluid flows in which the vorticity is proportional to the streamfunction perturbed by a uniform stream. The solutions obtained are exact solutions and represent various non-parallel flows of second-grade fluids. The plan of this paper is as follows: In Sect. 2, the equations of motion of an unsteady plane incompressible second-grade fluid are given, and the vorticity function is assumed to be∇ 2 ψ=A(ψ−Ux−BUy 2). In Sect. 3, solutions for the steady flow are obtained. In Sect. 4, solutions for unsteady flow are obtained.

An Integral Equation Approach to the Incompressible Navier--Stokes Equations in Two Dimensions

We present a collection of methods for solving the incompressible Navier--Stokes equations in the plane that are based on a pure stream function formulation. The advantages of this approach are twofold: first, the velocity is automatically divergence free, and second, complicated (nonlocal) boundary conditions for the vorticity are avoided. The disadvantage is that the solution of a nonlinear fourth-order partial differential equation is required. By recasting this partial differential equation as an integral equation, we avoid the ill-conditioning which hampers finite difference and finite element methods in this environment. By using fast algorithms for the evaluation of volume integrals, we are able to solve the equations using O(M) or O(M log M) operations, where M is the number of points in the discretization of the domain.

Internal variable modelling of wave propagation in laminates with nonlinearly elastic constituents

The phenomena of wave propagation normal to the layers of a periodically layered elastic media comprised of nonlinear elastic constituents is modeled by employing an internal variable theory. Using the model a fourth-order nonlinear partial differential equation is derived for the evolution of the mean displacement in the laminate; properties of the solutions of the relevant evolution equation are then studied both in a neighborhood of the precursor wave as well as in a neighborhood of a characteristic traveling with the main or composite wave speed.

The Computation of One-Parameter Families of Bifurcating Elastic Surfaces

The problem of constructing the middle surface of a deformed elastic shell from its first and second fundamental forms $\hat{a} _{\alpha \beta } $ and $\hat{b} _{\alpha \beta } $ is considered. The undeformed shell is a spherical cap of radius R and thickness h with an angular width $2\theta _0 $, where$0 < \theta _0 < \pi/2$. The cap is subjected to a constant uniform load $\lambda $ and is simply supported at its edge. The authors seek to compute the one-parameter families of buckled states that branch from the unbuckled state of the shell. This is accomplished in two steps..First, a finite element method is used to solve the governing shell equations, a pair of fourth-order nonlinear partial differential equations (PDEs). A solution of this system is a curvature potential w, a stress potential f, and the load $\lambda $. Using Liapunov– Schmidt reduction, it can be shown that solutions possessing a variety of symmetries bifurcate from the unbuckled state of the shell. In the work presented here, these local branches will be numerically continued. Solution branches are parameterized in terms of a pseudo-arc-length parameter $\rho $ (i.e., $( \lambda , f, w) = ( \lambda ( \rho ), f_\rho , w_\rho ) $), enabling them to be tracked around turning points. The second step in the solution process is to solve numerically for the parameterization $\hat{\bf X} _\rho $ corresponding to the middle surface of the buckled shell $\hat{\mathcal{S}} _\rho $. This is done by integrating the PDEs of $\hat{\mathcal{S}} _\rho $. The coefficients in these differential equations involve the first and second fundamental forms of the deformed shell SP, which can be computed from $( \lambda ( \rho ), f_\rho ,w_\rho )$. A number of bifurcation diagrams corresponding to the first three branch points of a spherical cap of size $\theta _0 = 12.85^\circ $ are presented. For this example, a secondary bifurcation point was found connecting two distinct nonaxisymmetric solution branches. Computer graphics are used to display images of various buckled surfaces that branch from the unbuckled state of the shell.

An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces

The fourth-order nonlinear partial differential equation for surface diffusion is approximated by a new integrable nonlinear evolution equation. Exact solutions are obtained for thermal grooving, subject to boundary conditions representing a section of a grain boundary. When the slope m of the groove centre is large, the linear model grossly overestimates the groove depth. In the linear model dimensionless groove depth increases linearly with m, but in the nonlinear model it approaches an upper limit. A nontrivial similarity solution is found for the limiting case of a thermal groove whose central slope is vertical

A Saint-Venant principle for a theory of nonlinear plane elasticity

A formulation of Saint-Venant’s principle within the context of a restricted theory of nonlinear plane elasticity is described. The theory assumes small displacement gradients while the stress-strain relations are nonlinear. We consider plane deformations of such an elastic material occupying a rectangular region. The lateral sides are traction-free while the far end is subjected to a uniformly distributed tensile traction τ ≥ 0 \tau \ge 0 . The near end is subjected to a prescribed normal and shear traction. If this end loading were also that of uniform tension, then one possible corresponding stress state throughout the rectangle is that of uniform tension. When the near end loading is not uniform, the resulting stress field is expected to approach a uniform tensile state with increasing distance from the near end. This result is established here using differential inequality techniques for quadratic functionals. It is shown that, under certain constitutive assumptions, an energy-like quadratic functional, defined on the difference between the deformation field and the uniform tensile state, decays exponentially with distance from the near end. The estimated decay rate (which is a lower bound on the actual rate of decay) is characterized in terms of the load level τ \tau , the domain geometry, and material properties. The results predict a progressively slower decay of end effects with increasing load level τ \tau . The mathematical issues of concern involve spatial decay of solutions of a fourth-order nonlinear elliptic partial differential equation.

Bending-wave propagation by microtubules and flagella

The movement of an elastic filament in a viscous medium can be computed from the fourth-order nonlinear partial differential equation obtained by balancing bending moments at all points along the length of the filament. These bending moments result from active forces, elastic resistance to bending, and viscous resistance to movement through the medium. I have studied numerical solutions obtained for two situations of biological interest: For the movement of individual microtubules, the active force is generated by interaction between the microtubule and the substratum over which it is moving, and is directed along the axis of the microtubule. The computations can reproduce the gliding movement of unrestrained microtubules, and also the periodic bending and bend propagation seen when the leading end of the microtubule is restrained. No modulation of active force is required to generate bending waves. For the movement of flagella, the active forces are generated internally as sliding forces between adjacent members of a cylinder of nine microtubular doublets. Without some additional control assumptions, the forces will be balanced and no bending moments will be generated. The problem faced by investigators of flagellar motility is to determine the control mechanisms that operate to make the system asymmetric, so that active bending moments are generated. Computations with models in which the curvature of the flagellum modulates the active-force generators have indicated that this control specification is sufficient to generate oscillation and bend propagation, but is insufficient to completely determine the movement.

Converging nonlinear resonance cones

The theory of nonlinear plasma resonance cones due to a circular ring exciter is presented. The ring source excites a converging resonance cone which produces a maximum electric field at its focus where the strongest nonlinear effects occur. The nonlinearity is assumed to be due to the ponderomotive force. From a numerical integration of the fourth-order nonlinear partial differential equation, it is shown that the resonance cone trajectory is modified by the nonlinearities in a direction opposite to that caused by thermal effects, especially near and after the focus. Steepening of the electric field occurs near the cones and focal region, and the peak electric field on the cones and at the focus is enhanced due to the nonlinearity. The location of the focal point is shifted toward the exciter due to nonlinearities in contrast to the shift due to the thermal effects which is away from the exciter.

Notes on elastic-plastic analysis

Abstract A fourth-order nonlinear partial differential equation is obtained for plane stress conditions in terms of a stress function. Suggested methods of solution include direct substitution and perturbation based on the elastic solution. The perfectly plastic problem is considered as a limiting case for the Ramberg-Osgood stress-strain relation. It is pointed out that the compatability equation can be used to find the secant modulus if the perfectly plastic part is contained by a perfectly elastic region.

A computational method for viscous flow problems

A digital computer method for solving certain problems involving two-dimensional incompressible viscous flow is described. The time-dependent case is treated; the mathematical problem is thus that of solving a non-linear fourth-order partial differential equation in three variables. The choice of difference equations, of relaxation procedure, the kind of approximation to boundary conditions, and the resulting computational stability, speed, and accuracy are considered. Most experience so far has been for a rectangular region for which boundary velocities are prescribed as certain functions of time; an example of one such problem showing vortex formation and break-up is given.