Exact solutions of two dimensional flows of second order incompressible fluids

Summary Some two dimensional exact solutions have been obtained describing non-steady flows of second order incompressible fluids. The results are expressed in terms of a non-dimensional parameter which depends on the non-Newtonian coefficient and the frequency of excitation of the external disturbance. It is noticed that for a critical value of K (= K _c), the flow properties are identical with those in the Newtonian case.Before communicating this note, it has been noticed from a recent paper: Non-steady helical flows of second-order flows by H. Markovitz and B. D. Coleman Cf: Physics of Fluids 7 (1964) 833 that Truesdell considered some problems involving standing waves in second order fluids (unpublished). The results of Truesdell are, however, not known to the present author.     

Unsteady flows of inhomogeneous incompressible fluids

In this paper, we study the unsteady motion of inhomogeneous incompressible viscous fluids. We present the results corresponding to Stokes' second problem and for the flow between two parallel plates where one is oscillating.     

Global solutions of two-dimensional Navier-Stokes and euler equations

Long-time solutions to the Navier-Stokes (NS) and Euler (E) equations of incompressible flow in the whole plane are constructed, under the assumption that the initial vorticity is in L ¹(²) for (NS) and in L ¹(²) L ^r(²) for some r>2 for (E). It is shown that the solution to (NS) is unique, smooth and depends continuously on the initial data, and that the (velocity) solution to (E) is Hölder continuous in the space and time coordinates. It is shown that as the viscosity vanishes, there is a subsequence of solutions to (NS) converging to a solution of (E).     

Smallest scale estimates for the Navier-Stokes equations for incompressible fluids

We consider solutions of the Navier-Stokes equations for incompressible fluids in two and three space dimensions. We obtain improved estimates, in the limit of vanishing viscosity, for the Fourier coefficients. The coefficients decay exponentially fast for wave numbers larger than the square root of the maximum of the velocity gradients divided by the square root of the viscosity. This defines the minimum scale, the size of the smallest feature in the flow.The work of Kreiss was supported in part by National Science Foundation under Grant DMS-8312264 and Office of Naval Research under Contract N-00014-83-K-0422.     

Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids

Summary We perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.     

Nonsimilar incompressible laminar boundary-layer flows in micropolar fluids

Summary The solution of the steady laminar incompressible nonsimilar boundary-layer problem for micropolar fluids over two-dimensional and axisymmetric bodies has been presented. The partial differential equations governing the flow have been transformed into new co-ordinates having finite range. The resulting equations have been solved numerically using implicit finite-difference scheme. The computations have been carried out for a cylinder and a sphere. The results indicate that the separation in micropolar fluids occurs at earlier streamwise locations as compared to Newtonian fluids. The skin friction and velocity profiles depend on the shape of the body and are almost insensitive to microrotation or coupling parameter, provided the coupling parameter is small. On the other hand, the microrotation profiles and microrotation gradient depend on the microrotation parameter and they are insensitive to the coupling parameter.

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Zusammenfassung Es wird die Lösung des stationären Grenzschichtproblems inkompressibler mikropolarer Flüssigkeiten für den Fall der Nichtähnlichkeit bei zweidimensionalen und achsensymmetrischen Körpern vorgelegt. Die dem Problem zugrunde liegenden partiellen Differentialgleichungen werden durch Einführung neuer Koordinaten auf ein endliches Gebiet transformiert. Die so erhaltenen Gleichungen werden mit Hilfe eines impliziten Differenzenverfahrens numerisch gelöst. Die Rechnung wird für den Zylinder und die Kugel durchgeführt. Die Ergebnisse zeigen, daß die Grenzschichtablösung früher erfolgt als bei vergleichbaren newtonschen Flüssigkeiten. Wandreibung und Geschwindigkeitsprofile hängen von der Gestalt des Körpers ab und sind nahezu unempfindlich gegen Mikrorotation und Kopplungsparameter, vorausgesetzt, daß der letztere klein ist. Dagegen hängen das Profil und der Gradient der Mikrorotation vom Parameter der Mikrorotation ab und sind ebenfalls unempfindlich gegen die Kopplungsparameter.

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With 6 figures     

Finite element solution of the streamfunction-vorticity equations for incompressible two-dimensional flows

The streamfunction-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method. The vorticity at no-slip boundaries is evaluated in the framework of the streamfunction equation. The resulting scheme achieves convergence, even for very high values of the Reynolds number, without the traditional need for upwinding. The stability and accuracy of the approach are demonstrated by the solution of two well-known benchmark problems: flow in a lid-driven cavity at Re ? 10,000 and flow over a backward-facing step at Re = 800.     

A finite element for incompressible plane flows of fluids with memory

Flows of fluids with single-integral memory functionals are considered. Evaluation of the stress at a material point involves the deformation history of that point, and a dominant computational cost in finite element approximation is the construction of streamlines. It is shown that the simple crossed-triangle macro-element is in many ways an ideal finite element for the difficult non-linear, non-self-adjoint problem. The question as to whether this element produces convergent velocity and pressure solutions is addressed in the light of its failure to satisfy the discrete LBB condition. The effect of the element's ill-disposed (‘spurious’) pressure modes is discussed, and a pressure smoothing scheme is given which gives good results in Newtonian and non-Newtonian flows at various Reynolds and Deborah numbers. As an example of the element's success in modelling such flows, the problem of pressure differences in flows over transverse slots is studied numerically. The results are compared with experimental observations of such flows. The effect of fluid memory on the relation between first normal-stress differences and pressure differences is investigated.     

On numerical solutions of the time-dependent Euler equations for incompressible flow

This paper presents finite element methods for the non-stationary Euler equations of a two dimensional inviscid and incompressible flow. For the time discretization, we compare numerical results obtained by the use of a leap-frog scheme and a semi-implicit scheme of order two.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅱ)

The solvability of the Euler equations about incompressible inviscid fluid based on the stratification theory is discussed. And the conditions for the existence of formal solutions and the methods are presented for calculating all kinds of ill-posed initial value problems. Two examples are given as the evidences that the initial problems at the hyper surface does not exist any unique solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅰ)

The ill posed initial value problem of the Euler equations and the formal solvability of ill posed problem based on stratification theory are discussed. For some ill posed initial value problems, the existence conditions of formal solutions and the methods of how to construct a formal solution are given. Finally, an example is given to discuss the ill posedness of the initial value problem on hyper plane {t=0} in R⁴, and explain that the problem has more than one solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

A class of exact solutions for N-dimensional incompressible magnetohydrodynamic equations

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic(MHD)equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations are directly constructed.     

A two-step godunov-type scheme for the euler equations

A second-order Godunov-type scheme for the Euler equations in conservation form is derived. The method is based on the ENO formulation proposed by Harten et al. The fundamental difference lies in the use of a two-step scheme to compute the time evolution. The scheme is TVD in the linear scalar case, and gives oscillation-free solutions when dealing with nonlinear hyperbolic systems. The admissible time step is twice that of classical Godunovtype schemes. This feature makes it computationally cheaper than one-step schemes, while requiring the same computer storage.

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Sommario Viene data una nuova estensione al secondo ordine del metodo di Godunov per la soluzione delle equazioni di Eulero in forma conservativa. Il metodo é basato sulla formulazione ENO proposta da Harten et al. La differenza fondamentale consiste nel calcolo dell'evoluzione temporale, ottenuta mediante uno schema a due passi. Questo consente l'uso di un passo di integrazione nel tempo doppio rispetto agli altri schemi alla Godunov ad un solo passo. Il metodo proposto risulta quindi piú efficiente e puó inoltre essere implementato senza alcun aumento dell'occupazione di memoria. Viene dimostrato che lo schema é TVD nel caso lineare, e che fornisce soluzioni prive di oscillazioni spurie nel caso di sistemi non-lineari.

     

Filtering non-solenoidal modes in numerical solutions of incompressible flows

Solving the incompressible Navier–Stokes equations requires special care if the velocity field is not discretely divergence-free. Approximate projection methods and many pressure Poisson equation methods fall into this category. The approximate projection operator does not dampen high frequency modes that represent a local decoupling of the velocity field. For robust behavior, filtering is necessary. This is especially true in two instances that were studied: long-term integrations and large density jumps. Projection-based filters and velocity-based filters are derived and discussed. A cell-centered velocity filter, in conjunction with a vertex-projection filter, was found to be the most effective in the widest range of cases. © 1998 John Wiley & Sons, Ltd.     

Asymptotics of solutions of the three-dimensional elasticity equations for compressible and incompressible bodies

We analyze the leading terms in the general asymptotic expansions of solutions of the first boundary value problem of three-dimensional elasticity in displacements. The cases of compressible and incompressible bodies, which have substantially different statements, are considered separately. The minimum-to-maximum ratio of characteristic dimensions of the elastic body is a natural small asymptotic parameter. The third dimension can be of any “intermediate” order, including the endpoints. For example, such a geometry is typical of bodies that simultaneously have characteristic macro-, micro-, and nano-dimensions in three coordinate axes.     

Time-dependent solutions of viscous incompressible flows in moving co-ordinates

A time-accurate solution method for the incompressible Navier-Stokes equations in generalized moving coordinates is presented. A finite volume discretization method that satisfies the geometric conservation laws for time-varying computational cells is used. The discrete equations are solved by a fractional step solution procedure. The solution is second-order-accurate in space and first-order-accurate in time. The pressure and the volume fluxes are chosen as the unknowns to facilitate the formulation of a consistent Poisson equation and thus to obtain a robust Poisson solver with favourable convergence properties. The method is validated by comparing the solutions with other numerical and experimental results. Good agreement is obtained in all cases.     

Exact solutions of the boundary layer equations for power-law fluids

Exact analytic solutions of the equations of the hydrodynamic boundary layer are obtained for pseudoplastic fluids with exponents n=1/5, 1/4, 1/2, 3/5, 5/7 flowing longitudinally over a flat plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 39–42, September–October, 1989.