Some exact solutions for rotating flows of a generalized Burgers’ fluid in cylindrical domains

The velocity field and the adequate shear stress corresponding to the flow of a generalized Burgers’ fluid model, between two infinite co-axial cylinders, are determined by means of Laplace and finite Hankel transforms. The motion is due to the inner cylinder that applies a time dependent torsional shear to the fluid. The solutions that have been obtained, presented in series form in terms of usual Bessel functions J₁( ? ), J₂( ? ), Y₁( ? ) and Y₂( ? ), satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for Burgers’, Oldroyd-B, Maxwell, second grade, Newtonian fluids and large-time transient solutions for generalized Burgers’ fluid are also obtained as special cases of the present general solutions. The effect of various parameters on large-time and transient solutions of generalized Burgers’ fluid is also discussed. Furthermore, for small values of the material parameters, λ₂ and λ₄ or λ₁, λ₂, λ₃ and λ₄, the general solutions corresponding to generalized Burgers’ fluids are going to those for Oldroyd-B and Newtonian fluids, respectively. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.     

Exact solution of electroosmotic flow in generalized Burgers fluid

An exact solution is developed for the time periodic electroosmotic flow of a non-Newtonian fluid between the micro-parallel plates. The constitutive equations of a generalized Burgers fluid are used in the mathematical formulation. The resulting problem is solved by a Fourier transform technique. Graphs are plotted and discussed for various emerging parameters of interest.     

Properties of chaotic fluid oscillations in cylindrical basins

Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 28, No. 6, pp. 52–61, June, 1992.     

Exact solutions of MHD second Stokes flow of generalized Burgers fluid

This work concerns with the exact solutions of magnetohydrodynamic (MHD) flow of generalized Burgers fluid describing the second Stokes problem. The modified Darcy law is taken into account. The related velocity distribution and shear stress are expressed as a combination of steady-state and transient solutions computed by means of integral transformations. The effects of various parameters on the flow field are investigated. The MHD flow results in reduction of velocity distribution and associated thickness of the boundary layer.     

Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains

The exact solutions for the motion of a Maxwell fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of the Laplace transform. These solutions are presented as sum of the steady-state and transient solutions and describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the motion is described by the steady-state solution which is periodic in time and independent of the initial conditions. Finally, by means of graphical illustrations, the required times to reach the steady-state are determined for sine, cosine and combined oscillations of the boundary.     

Forced nonlinear oscillations of cylindrical shells interacting with fluid flow

The dynamic interaction of thin cylindrical shells with the fluid flow inside them under external periodic loads is studied. A technique is proposed to calculate the parameters of forced nonlinear oscillations of shells with a fluid moving with nearly critical velocities. The amplitude-frequency characteristics of the fluid-shell system under steady-state oscillation are plotted __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 91–99, April 2006.     

Influence of surface tension on damping of the oscillations of a viscous incompressible fluid in a cylindrical channel

A numerical calculation is made of small oscillations of a viscous incompressible fluid that fills half of a horizontal cylindrical channel. The calculation is made with and without allowance for surface tension. The results of the calculation show that allowance for surface tension increases the damping of the oscillations. The general properties of problems of the normal oscillations of a heavy and capillary viscous incompressible fluid were studied in [1–3], in which the possibility of applying the Bubnov-Galerkin method to these problems was pointed out. A method for calculating the oscillations of a viscous incompressible fluid that partly fills an arbitrary vessel at large Reynolds numbers was developed in [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–132, May–June, 1980.     

Calculation of the decay rate of oscillations of a low-viscosity fluid rotating in a cylindrical vessel

A study is made of the problem of small oscillations of a low-viscosity fluid rotating under conditions of weightlessness in a cylindrical vessel. For fixed volume of the fluid and different values of the angular velocity and the wetting angle the decay rate and frequency of the characteristic oscillations are calculated. For an ideal fluid, the shapes of the characteristic oscillations of the free surface of the fluid are also calculated.     

Analysis of nonlinear wave motions of a fluid in a cylindrical vessel performing given angular oscillations

Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 6, pp. 95–101, June, 1990.     

Free oscillations of a fluid in a cylindrical container with arbitrary axisymmetric bottom and elastic elements on the free surface of the fluid

We consider the problem of free oscillations of an ideal fluid in a container that has the form of a right circular cylinder with arbitrary axisymmetric bottom in the case where the unperturbed free surface of the fluid is covered by an elastic membrane or plate. Using the expansion in eigenfunctions of an auxiliary spectral problem with a parameter in boundary conditions and the method of decomposition of the domain of meridional cross-section of a container, we obtain an analytic solution of the problem. Individual examples of mechanical systems are considered, for which we construct solutions by using the proposed algorithm, analyze these solutions, and compute the frequencies and forms of oscillations.     

Transversal oscillations of a fluid in a long cylindrical container with membrane or elastic plate on the free surface

We propose approximate solutions of two-dimensional hydroelastic problems that describe free oscillations of an ideal fluid in a horizontal long cylindrical container with arbitrary symmetric cross section. The free surface of the fluid is covered by a plane membrane or an elastic plate. Using specific examples, we analyze the obtained solutions and the results of computation of frequencies and forms of oscillations of the mechanical system under consideration.     

On the unsteady rotational flow of fractional Oldroyd-B fluid in cylindrical domains

This paper concerned with the unsteady rotational flow of fractional Oldroyd-B fluid, between two infinite coaxial circular cylinders. To solve the problem we used the finite Hankel and Laplace transforms. The motion is produced by the inner cylinder that, at time t=0⁺, is subject to a time-dependent rotational shear. The solutions that have been obtained, presented under series form in terms of the generalized G functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, fractional and ordinary Maxwell, fractional and ordinary second grade, and Newtonian fluids, performing the same motion, are obtained as limiting cases of general solutions.     

Small oscillations of a liquid in a partitioned cylindrical vessel

The equations of motion of a rigid body whose cavity is partially filled with an ideal fluid have been obtained in works of Moiseev [1, 2, 3], Okhotsimskii [4], Narimanov [5], and Rabinovich [6]. All the equation coefficients have been calculated for a cavity in the form of a circular cylinder or two concentric cylinders.The problem of fluid motion in a partitioned cylindrical cavity was considered by Rabinovich [7]. It was also considered by Bauer [8], who analyzed the particular case of vessel motion in the plane of one of the partitions.In the following we consider the two-dimensional motion of a cylinder with radial and annular baffles, and a definition is given of the velocity potential in the case of arbitrary positioning of the radial baffles with respect to the motion plane. Formulas are obtained for determining the parameters of a mechanical analog of the wave oscillations, which consists of two mathematical pendulum subsystems.