Isothermal flow past a blowing sphere

Steady, axisymmetric, isothermal, incompressible flow past a sphere with uniform blowing out of the surface is investigated for Reynolds numbers in the range 1 to 100 and surface velocities up to 10 times the free stream value. A stream-function-velocity formulation of the flow equations in spherical polar co-ordinates is used and the equations are solved by a Galerkin finite-element method. Reductions in the drag coefficients arising from blowing are computed and the effects on the viscous and pressure contributions to the drag considered. Changes in the surface pressure, surface vorticity and flow patterns for two values of the Reynolds number (1 and 40) are examined in greater detail. Particular attention is paid to the perturbation to the flow field far from the sphere.     

Breakaway self-similar flows in a laminar magnetohydrodynamic boundary layer with blowing and suction

A study was made of self-similar flows in a magnetohydrodynamic boundary layer in the presence of a pressure gradient and with the blowing and suction of a conducting liquid. The region of the existence of self-similar solutions for breakaway flow conditions, characterized by a friction at the wall equal to zero, was determined. The regions of a change in the determining parameters, with which nonbreakaway flows are established at impermeable and permeable surfaces, are indicated. It is noted that under diffusion flow conditions the self-similar equation of a magnetohydrodynamic boundary layer with fixed boundary conditions at the wall and at infinity permits an innumerable set of solutions. The article proposes a method for selecting a solution and indicates a calculating method for determining it. It demonstrates the possibility of a considerable broadening of the region of flows without breakaway with the imposition of electric and magnetic fields.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 38–46, May–June, 1972.     

Flow around and heat and mass transfer of a sphere with blowing at medium reynolds numbers

The Navier-Stokes and heat- and mass-transfer equations are solved numerically for a sphere with uniform blowing over the surface in the Reynolds number range up to 20. A method of refining the boundary conditions far from the sphere is proposed in both problems. A difference scheme from other authors is used to solve the hydrodynamic problem, and an explicit difference scheme with a second order of approximation is used for the heat problem. It is shown that blowing diminishes the aerodynamic drag of the sphere and the temperature or concentration gradient at its surface, i.e., the heat- and mass-transfer intensity.Translated from Zhurnal Priladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 148–156, July–August, 1975.     

The influence of rotation in slow viscous flows

We investigate the secondary streaming motion due to the slow rotation of an axisymmetric body in a rotating viscous fluid. We find in general that for all such bodies the velocities tend to zero at a distance $\text{0(E}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ from the body, where E is the (large) Ekman number. Four specific geometries are considered: sphere, spheroid, spherical cap and the double sphere; in all except the first case a small Rossby number has been assumed. The resultant translational force on the spherical cap, where there is no fore-aft symmetry, has been calculated. Further, separated flow in the last two cases can be displayed.     

Steady slow rotation of a sphere in a visco-elastic liquid

Summary In this paper, we obtain the flow due to slow steady rotation of a sphere in a visco-elastic liquid characterized by the constitutive relation given by Rivlin. The non-Newtonian effects are strongly dependent on a non-dimensional parameter K independent of the angular velocity of the sphere. If 1<K3, we notice four vortices symmetrically placed around the sphere. When K lies outside this range, the direction of the flow pattern is the same as that in the Newtonian case but displaced towards the sphere as K decreases. Also the expression for the couple on the sphere has been obtained which depends on K.     

Görtler vortices in Falkner–Skan flows with suction and blowing

In this paper, we use nonlinear calculations to study curved boundary‐layer flows with pressure gradients and self‐similar suction or blowing. For an accelerated outer flow, stabilization occurs in the linear region while the saturation amplitude of vortices is larger than for flows with a decelerating outer flow. The combined effects of boundary‐layer suction and a favourable pressure gradient can give a significant stabilization of the flow. Streamwise vortices can be amplified on both concave and convex walls for decelerated Falkner–Skan flow with an overshoot in the velocity profile. The disturbance amplitude is generally lower far downstream compared with profiles without overshoot. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution behavior of the falling sphere problem in viscoelastic flows

Thehp-finite element method and pseudo arc-length method are combined to track solution behavior of the steady flow of a sphere falling in a tube filled with viscoelastic fluids. The computations is proved to be convergent and stable by a posteriori error analyses; the solutions smooth and with extremely steep stress gradient are obtained by using appropriate high-order interpolation distributions. The commonly used drag coefficient is proved not a reliable indicator for the approximation error. The solution curves of the upper convected Maxwell (UCM) and the Oldroyd-B fluids clearly indicate that there exist limiting points, parameterized by the Deborah number, where the magnitude of the solutions and the axial normal stress gradients behind the rear stagnation rise up rapidly, maybe go to infinity. The limiting points can be simply removed by employing the modified Chilcott and Rallison (MCR) fluid, which replaces the infinitely extendable, linear Gaussian chain with the finitely extendable nonlinear elastic spring (the FENE chain). Therefore the limiting behavior is caused by the physical model; it is not a numerical artifact. The project supported by the National Natural Science Foundation of China (20274041) and the Cooperative Research Center for Polymers, Australia     

On local solutions to non-Newtonian slow viscous flows

The eigenfunction expansion method is used to obtain local solutions to some non-Newtonian slow viscous flows. The forms of viscosity variation amenable to such analysis are restricted but do include power-law fluids. Power-law flow near a sharp corner between plane boundaries is analysed and results are obtained for the critical corner angle for eddy formation. Flows near a 90° corner with either a moving boundary or a finite flow rate at the corner are also considered. The “stick-slip” behaviour of a power-law fluid at a plane solid boundary is shown to obey a simple law.     

Heat transfer from a slowly rotating sphere

The problem of steady state forced convection heat transfer in a viscous incompressible fluid occupying the annular region between two concentric spheres is considered. The inner sphere is maintained at a constant temperatureT ₀ and rotates slowly around an axis through the centre. The outer sphere is at rest and the temperature of its surface is prescribed as a function of the spherical coordinates and. It is shown that, when viscous dissipation is small, the overall rate of heat transfer from the rotating sphere into the fluid is unaffected by convection from the sphere surface, in case of a slow rotation, where the Stokes solution holds.     

Derivation of the equations of slow nonisothermal gas flows

In recent years, some new phenomena have been predicted theoretically on the basis of the Burnett approximation. These include thermal-stress and concentration-stress convection [1–3], and also effects due to the influence of a magnetic field in a multiatomic gas (viscomagnetic heat flux, etc., [4]). It has been shown theoretically (see [5]) that under certain conditions various terms of the Burnett approximation must be taken into account in the expression for barodiffusion. The conclusions relating to a viscomagnetic heat flux have recently been confirmed experimentally [4]. The predicted phenomena follow rigorously from the Burnett equations. However, many hydrodynamicists adopt a sceptical attitude to these equations, which is due partly perhaps to attachment to the classical Navier-Stokes equations, which have served theoreticians without fail for a century and a half. In this connection, we discuss the evolution of ideas relating to the validity of the Burnett approximation. We discuss the minimal assumptions which must be made in order to derive the equations of slow [Reynolds number R = 0(1)], essentially nonisothermal [ ln T = 0(1)] flows of a gas as a continuous medium (Knudsen number K O) in the case when the derivatives of the thermal Burnett stresses in the momentum equation have the same order of magnitude as the Euler and Navier-Stokes terms of this equation [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 77–84, November–December, 1979.We thank G. I. Petrov and L. I. Sedov for discussions that stimulated the above analysis.     

The very slow flow of a Powell-Eyring fluid around a sphere

Summary The very slow flow of a Powell-Eyring type non-Newtonian fluid around a sphere is investigated by a variational technique. The result, a correction factor that is applied to the Stokes' equation, is given as a plot and as an equation which is empirically fit to the plot. Also, a comparison of the very slow flows of a simplified viscoelastic Oldroyd fluid and the Powell-Eyring fluid is made which indicates that in a certain restricted region of the very slow flows, both models give essentially the same results. The Oldroyd and Powell-Eyring model parameters are interrelated by forcing both models to fit the same tube flow viscosity data.Nomenclature B dimensionless quantity, v /R - C dimensionless second invariant - c ₁ constant determined by variational method - D dimensionless variational integral - D _2j , D _j+k position-independent variables used in specification of trial functions - E _2j , E _j+k position-independent variables used in specification of trial functions - f friction factor - f _corr friction factor correction - F _drag drag force on sphere - g, g ₀, g ₁ general trial function; first and second terms in the general trial function - G, H terms in the expression for C - j index - J variational integral - k index - K term in the expression for C - p, q integers - r integer, radial coordinate - R radius of sphere - Re Reynolds number - Re ₀ Reynolds number at point of zero shear rate - Re Reynolds number at infinite distance from sphere - Re _NN Reynolds number based on variable part of viscosity - u, v dimensionless position coordinates - V volume considered - v _i ith velocity component - v _r , v , v _z velocity components in the r, , and z-directions - v approach velocity of the fluid - x/ parameter in Powell-Eyring model - x _i i-position coordinate - parameter in Powell-Eyring model - rate of deformation - , _c , _N , ₀ coefficient of viscosity; cross viscosity; parameter in Powell-Eyring model; viscosity in limit of zero shear rate - spherical coordinate - , _ij rate of deformation tensor; ij-component of rate of deformation tensor - ₁, ₂ parameters in Oldroyd model - Newtonian viscosity - ₁, ₂ parameters in Oldroyd model - dimensionless radial coordinate, r/R - second invariant - fluid density - spherical coordinate - stream function     

Effect of blowing on the resistance of a sphere in laminar flow of viscous fluid

Using the method of matching asymptotic expansions [1–3], a stationary field of velocities is obtained in the vicinity of a sphere for Reynolds numbers R and R computed from the blowing velocity and the fluid flow, respectively; they satisfy the relations R² 1 and R 1. It is also shown that for intensive blowing (R 1), the resistive force is considerably smaller than that found by using the Stokes formula. For weak blowing the results are in good agreement with the solution of Oseen.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskai Fiziki, No. 3, pp. 110–114, May–June, 1972.     

Supersonic dusty-gas flows with Knudsen effect in interphase momentum exchange

On the basis of the two-continuum model of dilute gas-solid suspensions, the dynamic behavior of inertial particles in supersonic dusty-gas flows past a blunt body is studied for moderate Reynolds numbers, when the Knudsen effect in the interphase momentum exchange is significant. The limits of the inertial particle deposition regime in the space of governing parameters are found numerically under the assumption of the slip and free-molecule flow regimes around particles. As a model problem, the flow structure is obtained for a supersonic dusty-gas point-source flow colliding with a hypersonic flow of pure gas. The calculations performed using the full Lagrangian approach for the near-symmetry-axis region and the free-molecular flow regime around the particles reveal a multi-layer structure of the dispersed-phase density with a sharp accumulation of the particles in some thin regions between the bow and termination shock waves. The project supported by the National Natural Science Foundation of China (90205024), and the Russian Foundation for Basic Research (RFBR grant No. 02-01-00770 and joint RFBR-NSFC grant No. 03-01-39004)     

Flow of rarefied gas past a sphere in the presence of blowing from its surface

Flow of a rarefied gas past a sphere with various conditions of blowing on the surface is considered. The investigation is based on numerical solution of model kinetic equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 182–185, September–October, 1982.