Analytical solutions of laminar swirl decay in a straight pipe

In this work, the laminar swirl flow in a straight pipe is revisited and solved analytically by using prescribed axial flow velocity profiles. Based on two axial velocity profiles, namely a slug flow and a developed parabolic velocity profiles, the swirl velocity equation is solved by the separation of variable technique for a rather general inlet swirl velocity distribution, which includes a forced vortex in the core and a free vortex near the wall. The solutions are expressed by the Bessel function for the slug flow and by the generalized Laguerre function for the developed parabolic velocity. Numerical examples are calculated and plotted for different combinations of influential parameters. The effects of the Reynolds number, the pipe axial distance, and the inlet swirl profiles on the swirl velocity distribution and the swirl decay are analyzed. The current results offer analytical equations to estimate the decay rate and the outlet swirl intensity and velocity distribution for the design of swirl flow devices.     

Convergent expansions of the Bessel functions in terms of elementary functions

We consider the Bessel functions J _ν (z) and Y _ν (z) for R ν > ?1/2 and R z ≥ 0. We derive a convergent expansion of J _ν (z) in terms of the derivatives of \((\sin z)/z\), and a convergent expansion of Y _ν (z) in terms of derivatives of \((1-\cos z)/z\), derivatives of (1 ? e ^?z )/z and Γ(2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments.     

Heat transfer by magnetohydrodynamic laminar flow in circular pipe

This paper presents the influence of magnetic field on heat due to viscous and electrical dissipations for an incompressible, viscous, electrically conducting fluid through a circular pipe in the presence of an applied (transverse) uniform magnetic field. The walls of the pipe are assumed to be non-conducting and kept at uniform temperature gradient in one case and at a constant temperature gradient in another case. The heat equation governing the present problem is solved exactly in hypergeometric series. The temperature at the centre of the pipe T_e, unweighted mean temperature T_m and weight mean temperature T_M are calculated. The temperature profiles are shown graphically for different values of Hartmann number M, Brinkman number B_r and a non-dimensional number S. Numerical calculations are made for the Nusselt number and are entered in the table.     

New analytical and CWENO numerical solutions for axisymmetric horizontal flows with heat sources

Some new nonlinear analytical solutions are found for axisymmetric horizontal flows dominated by strong heat sources. These flows are common in multiscale atmospheric and oceanic flows such as hurricane embryos and ocean gyres. The analytical solutions are illustrated with several examples. The proposed exact solutions provide analytical support for previous numerical observations and can be also used as benchmark problems for validating numerical models. A central weighted essentially non-oscillatory (CWENO) reconstruction is also employed for numerical simulation of the corresponding integro-differential equations. Due to the use of the same polynomial reconstruction for all derivatives and integral terms, the balance between those terms is well preserved, and the method can precisely reproduce the exact solutions, which are hard to capture by traditional upwind schemes. The developed analytical solutions were employed to evaluate the performance of the numerical method, which showed an excellent performance of the numerical model in terms of numerical diffusion and oscillation.     

The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid in a circular pipe

Summary An exact solution of pulsating laminar flow superposed on the steady motion in a circular pipe is presented under the assumption of parallel flow to the axis of pipe. Total mass of flow on time average is found to be identified with that given byHagen-Poiseuille's low calculated on the steady component of pressure gradient. The phase lag of velocity variation from that of pressure gradient increases from zero in the steady motion to 90° in the pulsation of infinite frequency. Integration of work for changing kinetic energy of fluid through one period is vanished, while that of dissipation of energy by internal friction remains finite and excess amount caused by the components of periodic motion is added to the components of steady flow.It is found that the given rate of mass flow is attained in pulsating motion by giving the same amount of average gradient of pressure as in steady flow, but that excess works to the steady case are necessary for maintenance of this motion.

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Zusammenfassung Eine exakte Lösung der pulsierenden laminaren Strömung in einem Kreisrohr wird angegeben mit der Annahme, dass die Richtung dem Geschwindigkeitsvektor der Rohrachse parallel ist. Die Durchflussmenge stimmt überein mit der aus der stationären Druckgefällekomponente gerechneten Menge. Für die Erhaltung der Bewegung dagegen ist die der Dissipation entsprechende Extraarbeit notwendig. Die Quantität dieser Arbeit hängt ab von den Frequenzen der Stromschwingungen.

     

Finite-zone solutions in terms of automorphic functions the Kadomtsev-petviashvili equation

One obtains an expression for the finite-zone solutions of the Kadomtsev-Petviashvili equation in terms of the Poincaré theta-series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 5–16, 1983.The author expresses his sincere gratitude to A. R. Its for useful discussions and for his constant interest in the paper.     

Temperature distribution in a laminar circular jet

The effect of frictional heat on the temperature distribution in a laminar circular jet has been studied. It is found from the analysis and the graphs that as the Prandtl number decreases from unity the overall temperature difference near the axis of the jet increases but as we move away from the axis it goes on decreasing. The reverse phenomenon happens in the case of increasing Prandtl number.     

Existence of solutions of laminar boundary layer equations with decelerating external flows

New results on the existence and nonexistence of solutions of laminar boundary layer equations with decelerating external flows are obtained. Some previous results treated these equations with accelerating or constant external flows. Our approach is to establish a system of two integral equations with singularities which will be proved to be equivalent to the laminar boundary layer equations and prove that the system has solutions by using the Leray–Schauder fixed point theorem and the Helly selection principle.     

Some new exact analytical solutions for helical flows of second grade fluids

The helical flow of a second grade fluid, between two infinite coaxial circular cylinders, is studied using Laplace and finite Hankel transforms. The motion of the fluid is due to the inner cylinder that, at time t = 0⁺ begins to rotate around its axis, and to slide along the same axis due to hyperbolic sine or cosine shear stresses. The components of the velocity field and the resulting shear stresses are presented in series form in terms of Bessel functions J₀(•), Y₀(•), J₁(•), Y₁(•), J₂(•) and Y₂(•). The solutions that have been obtained satisfy all imposed initial and boundary conditions and are presented as a sum of large-time and transient solutions. Furthermore, the solutions for Newtonian fluids performing the same motion are also obtained as special cases of general solutions. Finally, the solutions that have been obtained are compared and the influence of pertinent parameters on the fluid motion is discussed. A comparison between second grade and Newtonian fluids is analyzed by graphical illustrations.     

Multiple solutions for the laminar flow in a porous pipe with suction at slowly expanding or contracting wall

In this paper, the asymptotic solution for the similarity equation of the laminar flow in a porous pipe with suction at expanding and contracting wall has been obtained using the singular perturbation method. However, this solution neglects exponentially small terms in the matching process. To take into account these exponentially small terms, a method involving the inclusion of exponentially small terms in a perturbation series was used to find the two solutions analytically. The series involving the exponentially small terms and expansion ratio predicts dual solutions. Furthermore, the result indicates that the expansion ratio has much important influence on the solutions. When the expansion ratio is zero, it is a special case that Terrill has discussed.     

Numerical simulation of laminar flow past a circular cylinder

The present paper focuses on the analysis of two- and three-dimensional flow past a circular cylinder in different laminar flow regimes. In this simulation, an implicit pressure-based finite volume method is used for time-accurate computation of incompressible flow using second order accurate convective flux discretisation schemes. The computation results are validated against measurement data for mean surface pressure, skin friction coefficients, the size and strength of the recirculating wake for the steady flow regime and also for the Strouhal frequency of vortex shedding and the mean and RMS amplitude of the fluctuating aerodynamic coefficients for the unsteady periodic flow regime. The complex three dimensional flow structure of the cylinder wake is also reasonably captured by the present prediction procedure.     

The stability of time-dependent laminar flow: Parallel flows

Zusammenfassung In dieser Arbeit wird das allgemeine Problem der Stabilität inkompressibler, instationärer, paralleler und laminarer Strömungen zwischen zwei unendlichen Ebenen untersucht. Dabei werden sowohl die klassische linearisierte Theorie für Störungen mit kleiner Amplitude als auch eine Variationsmethode für Störungen endlicher Grösse angewandt. Die grundlegenden Lehrsätze der linearen Theorie werden, soweit dies für das allgemeine Problem möglich ist, erweitert und ihre Konsequenzen aufgezeigt. Unter Anwendung der Variationsmethode werden für mehrere besondere instationäre Strömungen quantitative Ergebnisse gezeigt, aber die hier entwickelten Gleichungen können ebenso auf jedwede parallele Strömung angewendet werden. Das stationäre Problem ist als ein besonderer Grenzfall behandelt.     

An analytical solution for time-dependent stress field of lined circular tunnels using complex potential functions in a stepwise procedure

Time-dependent behavior of surrounding mass plays a significant role in designing underground constructions. Considering simple configuration of lined circular tunnels, a lot of solution have been proposed to this problem. However, many assume hydrostatic initial stress field, and other solutions are only applicable to simple rheological models and could not account for viscosity effect in long-term time periods. In this study, an analytical plane strain solution is proposed for lined circular tunnels under non-hydrostatic initial stress field, assuming rock mass as a viscoelastic material obeying Burgers model, while concrete lining is supposed to have linear elastic behavior. The solution which employs complex variable method combined with correspondence principle benefits from time discretization approach enabling the solution to take into account the viscosity effect in the both short-term and long-term periods of time, while predicting stress components accurately. The results obtained by the proposed solution were compared with those predicted by finite element COMSOL software which exhibited a close agreement. It was found that by increasing time both the proposed analytical solution and finite element numerical method tend to an oblique asymptote due to viscosity effect of Maxwell body in the Burgers model. Finally, a parametric analysis was performed with respect to Burgers model coefficients which showed different behavior for short and long periods of time.     

Optimal recovery of functions analytical in a half-plane

On the class of functions analytical and bounded in a half-plane with bounded derivative of order n ≥ 0, we solve problems of optimal recovery of values of a function and its derivatives of order m ≥ 0, using the restriction of the spectral function. We obtain appropriate exact inequalities for functions analytical in a half-plane.     

An elementary approach to the analysis of exact solutions for the Navier-Stokes stagnation flows with slips

We deal with the exact solutions of the Navier-Stokes equations for stagnation flows with slips. The problem becomes the solvability of certain third-order ordinary differential equations (ODEs). Reducing the order of ODEs, we exhibit another elementary proof of the existence and asymptotic behavior of solutions. Numerical investigations are also provided. Received: 14 August 2003     

New exact solutions for magnetohydrodynamic flows of an Oldroyd-B fluid

This paper presents the new exact analytical solutions for magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid. The explicit expressions for the velocity field and the associated tangential stress are established by using the Laplace transform method. Three characteristic examples: (i) flow due to impulsive motion of plate, (ii) flow due to uniformly accelerated plate, and (iii) flow due to non-uniformly accelerated plate are considered. The solutions for the hydrodynamic flows are special cases of the presented solutions. Moreover, the similar solutions corresponding to Maxwell and Newtonian fluids in the presence as well as absence of a magnetic field appear as the limiting cases of our solutions. The influences of the exerted magnetic field on the flow are also graphically presented and discussed. In particular, graphical results for the Oldroyd-B fluid are compared with those of a Newtonian fluid.     

Inequalities for derivatives of functions analytical in a strip

This work was supported by the Russian Foundation for Fundamental Research, Grant No. 93-01-00237, and by the International Scientific Foundation, Grant No. MP 1000.