Analysis of viscous flow due to a stretching sheet with surface slip and suction

The viscous flow due to a stretching sheet with slip and suction is studied. The Navier–Stokes equations admit exact similarity solutions. For two-dimensional stretching a closed-form solution is found and uniqueness is proved. For axisymmetric stretching both existence and uniqueness are shown. The boundary value problem is then integrated numerically.     

A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for steady compressible viscous flow

In this paper, the authors consider the Navier–Stokes equations for steady compressible viscous flow in three-dimensional cylindrical domain. A differential inequality for appropriate energy associated with the solutions of the Navier–Stokes isentropic flow in semi-infinite pipe is derived, from which the authors show a Phragmén–Lindelöf alternative result, i.e. the solutions for steady compressible viscous N–S flow problem either grow or decay exponentially as the distance from the entry section tends to infinity. In the decay case, the authors indicate how to bound explicitly the total energy in terms of data.     

Semi‐invariant solutions of the Navier‐Stokes equations

We present new exact solutions and reduced differential systems of the Navier‐Stokes equations of incompressible viscous fluid flow. We apply the method of semi‐invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier‐Stokes equations are, in fact, semi‐invariant and that the reduced differential systems we derive using semi‐invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi‐invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier‐Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier‐Stokes equations or find their approximations, and physical interpretation.     

Decay of the 2D Periodic Stationary Viscous Surface Waves

We consider the evolution of viscous fluids in a 2D horizontally periodic slab bounded above by a free top surface and below by a fixed flat bottom. This is a free boundary problem. The dynamics of the fluid are governed by the incompressible stationary Navier–Stokes equations under the influence of gravity and the effect of surface tension. We develop the global theory of solutions in low regularity Sobolev spaces for small data by nonlinear energy estimates.     

Arbitrary Motions of a Viscous Incompressible Liquid in a Finite Region

By using an entirely constructive analysis it is shown that there exist solutions of the viscous flow equations in which two of the Cartesian components of velocity are arbitrary and the third component can be constructed uniquely by the application of integrability conditions. The solutions are in general defined in a finite region of the fluid and can be matched onto a well-defined deterministic solution of the Navier–Stokes equations such that all of the relevant physical quantities are continuous at the interface.     

Elliptical flows perturbed by shear waves

We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin–Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier–Stokes fluid in an orthogonal rheometer).     

Singularities on Free Surfaces of Fluid Flows

Isolated singularities on free surfaces of two-dimensional and axially symmetric three-dimensional steady potential flows with gravity are considered. A systematic study is presented, where known solutions are recovered and new ones found. In two dimensions, the singularities found include those described by the Stokes solution with a 120° angle, Craya's flow with a cusp on the free surface, Gurevich's flow with a free surface meeting a rigid plane at 120° angle, and Dagan and Tulin's flow with a horizontal free surface meeting a rigid wall at an angle less than 120°. In three dimensions, the singularities found include those in Garabedian's axially symmetric flow about a conical surface with an approximately 130° angle, flows with axially symmetric cusps, and flows with a horizontal free surface and conical stream surfaces. The Stokes, Gurevich, and Garabedian flows are exact solutions. These are used to generate local solutions, including perturbations of the Stokes solution by Grant and Longuet-Higgins and Fox, perturbations of Gurevich's flow by Vanden-Broeck and Tuck, asymmetric perturbations of Stokes flow and nonaxisymmetric perturbations of Garabedian's flow. A generalization of the Stokes solution to three fluids meeting at a point is also found.     

A three-component generalization of Camassa–Holm equation with N-peakon solutions

A three-component generalization of Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3×3 matrix spectral problem with three potentials. With the aid of the zero-curvature equation, we derive a hierarchy of new nonlinear evolution equations and establish their Hamiltonian structures. The three-component generalization of Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N-peakons and an infinite sequence of conserved quantities.     

A Super Camassa–Holm Equation with N‐Peakon Solutions

A super Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials. With the aid of the zero‐curvature equation, we derive a hierarchy of super Harry Dym type equations and establish their Hamiltonian structures. It is shown that the super Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N peakons. As an example, exact 1‐peakon solutions of the super Camassa–Holm equation are given. Infinitely many conserved quantities of the super Camassa–Holm equation and the super Harry Dym type equation are, respectively, obtained.     

Doubly periodic and multiple pole solutions of the sinh-Poisson equation: Application of reciprocal transformations in subsonic gas dynamics

Vortex patterns associated with the sinh-Poisson equation arise in a remarkable manner as relaxation states of the Navier–Stokes equations. Here, doubly periodic and multiple-pole solutions of the sinh-Poisson equation are generated via the Hirota bilinear operator formalism and exploitation of the phenomenon of coalescence of wave numbers. It is then shown how the multi-parameter reciprocal transformations of gas dynamics may be applied to a seed doubly periodic solution of the sinh-Poisson equation to generate associated periodic vortex structures valid in the subsonic flow of a generalized Kármán–Tsien gas.     

Stokes flow solutions in infinite and semi-infinite porous channels

We derive a class of exact solutions for Stokes flow in infinite and semi-infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no-slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi-infinite channel domain, the ground-state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane-based water purification, heat and mass transfer, and fuel cells.     

A numerical procedure for viscous free surface flows

We present a numerical procedure for two-dimensional unsteady viscous free surface flow problems with surface tension. The procedure is based on a finite difference approach to a primitive variable formulation; a coordinate transformation is used to transform the irregularly shaped flow domain onto a fixed rectangular domain. The procedure is tested on a standing wave problem and a cavity flow problem with a free surface. Satisfactory numerical solutions are obtained for both problems for Reynolds numbers up to 200.     

Analysis of unsteady stagnation‐point flow over a shrinking sheet and solving the equation with rational Chebyshev functions

This paper investigates the nonlinear boundary value problem, resulting from the exact reduction of the Navier–Stokes equations for unsteady laminar boundary layer flow caused by a stretching surface in a quiescent viscous incompressible fluid. We prove existence of solutions for all values of the relevant parameters and provide unique results in the case of a monotonic solution. The results are obtained using a topological shooting argument, which varies a parameter related to the axial shear stress. To solve this equation, a numerical method is proposed based on a rational Chebyshev functions spectral method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. Copyright © 2016 John Wiley & Sons, Ltd.     

Stokes flow past a porous sphere using Brinkman's model

A general non-axisymmetric Stokes flow past a porous sphere in a viscous, incompressible fluid is considered. The flow inside the sphere is governed by Brinkman's equations. A representation for velocity and pressure for the Brinkman's equations is suggested and a method of finding the flow quantities is given. Faxén's laws for drag and torque for the flow past a porous sphere are also given.     

A Class of Solutions for the Equations Governing the Steady Two-Dimensional Motion of a Viscous Incompressible Liquid

A general viscosity dependent solution for the stream function is found satisfying the simplest nondegenerate form of the steady flow Navier-Stokes equations for a viscous incompressible liquid. The solution is two-dimensional and is expressed in terms of arbitrary analytic functions in the fluid domain. This class of flows is generated by complex stream functions, and the region of definition is restricted by an inequality containing these analytic functions. A general potential flow, and degenerate Stokes or creeping flows are recovered as particular solutions in limiting cases.     

The quasistationary approximation in the problem of the motion of an isolated volume of a viscous incompressible capillary liquid

The equations of the quasistationary approximation in the problem of the motion of an isolated volume of a viscous incompressible capillary liquid are derived from the exact equations using an expansion in a small quasistationary parameter, which is equal to the ratio of the Stokes time to the capillary time. The problem contains yet another dimensionless parameter, which is proportional to the modulus of the conserved angular momentum of the liquid volume, which is also assumed to be small. Depending on the relation between these parameters, three versions of the limiting problem are obtained: the traditional version and two new versions. Asymptotic solutions of the problems which arise when the quasistationary parameter tends to zero are constructed.     

Exact and approximate expressions for resistance coefficients of a colloidal sphere approaching a solid surface at intermediate Reynolds numbers

A mathematical model has been developed to describe the force of liquid flow acting on a colloidal spherical particle as it approaches a solid surface at intermediate-Reynolds-number-flow regime. The model has incorporated bispherical coordinates to determine a stream function for the flow disturbed by the sphere. The stream function was then used to derive the flow force on the particle as a function of the inter-surface separation distance. The force equation was related to the modified Stokes equation to obtain an exact analytical expression for the correction factor to the Stokes law. Finally, a rational approximation is presented, which is in good agreement with the exact numerical result, and can be readily applied to more general particle–surface interactions involving short-range hydrodynamics associated with colloidal particles in the near vicinity of a large solid collector surface at intermediate Reynolds number of the supporting flow.     

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol.142, No. 2, pp. 197–217, February, 2005.     

Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers

This paper presents a slender body theory for the dynamics of a curved inertial viscous Newtonian fiber. Neglecting surface tension and temperature dependence, the fiber flow is modeled as a three‐dimensional free boundary value problem in terms of instationary incompressible Navier–Stokes equations. From regular asymptotic expansions in powers of the slenderness parameter, leading‐order balance laws for mass (cross‐section) and momentum are derived that combine the unrestricted motion of the fiber centerline with the inner viscous transport. The physically reasonable form of the one‐dimensional fiber model results thereby from the introduction of the intrinsic velocity that characterizes the convective terms. For the numerical investigation of the viscous, gravitational and rotational effects on the fiber dynamics, a finite volume approach on a staggered grid with implicit upwind flux discretization is applied. Copyright © 2007 John Wiley & Sons, Ltd.