Slip at the surface of a sphere translating perpendicular to a plane wall in micropolar fluid

The Stokes axisymmetrical flow caused by a sphere translating in a micropolar fluid perpendicular to a plane wall at an arbitrary position from the wall is presented using a combined analytical-numerical method. A linear slip, Basset type, boundary condition on the surface of the sphere has been used. To solve the Stokes equations for the fluid velocity field and the microrotation vector, a general solution is constructed from fundamental solutions in both cylindrical, and spherical coordinate systems. Boundary conditions are satisfied first at the plane wall by the Fourier transforms and then on the sphere surface by the collocation method. The drag acting on the sphere is evaluated with good convergence. Numerical results for the hydrodynamic drag force and wall effect with respect to the micropolarity, slip parameters and the separation distance parameter between the sphere and the wall are presented both in tabular and graphical forms. Comparisons are made between the classical fluid and micropolar fluid.      

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Viscoelastic flow in a curved channel: A similarity solution for the Oldroyd-B fluid

A similarity solution is used to analyse the flow of the Oldroyd fluid B, which includes the Newtonian and Maxwell fluids, in a curved channel modelled by the narrow annular region between two circular concentric cylinders of large radius. The solution is exact, including inertial forces. It is found that the non-Netonian kinematics are very similar to the Newtonian ones, although some stress components can become very large. At high Reynolds number a boundary layer is developed at the inner cylinder. The structure of this boundary layer is asymptotically analysed for the Newtonian fluid. Non-Newtonian stress boundary layers are also developed at the inner cylinder at large Reynolds numbers.     

Flow past a porous sphere at small Reynolds number

low of an incompressible viscous fluid past a porous sphere has been discussed. The flow has been divided in three regions. The Region-I is the region inside the porous sphere in which the flow is governed by Brinkman equation with the effective viscosity different from that of the clear fluid. In Regions II and III clear fluid flows and Stokes and Oseen solutions are respectively valid. In all the three regions Stokes stream function is expressed in powers of Reynolds number. Stream function of Region II is matched with that of Region I at the surface of the sphere by the conditions suggested by Ochao-Tapia and Whitaker and it is matched with that of Oseen’s solutions far away from the sphere. It is found that the drag on the sphere reduces significantly when it is porous and it decreases with the increase of permeability of the medium.Received: February 7, 2002; revised: April 8, 2003 / June 9, 2004     

Crocco variable formulation for uniform shear flow over a stretching surface with transpiration: Multiple solutions and stability

The simultaneous effects of transpiration through and tangential movement of a semi-infinite flat plate on the self-similar boundary layer flow driven by uniform shear in the far field is considered. Difficulties with standard shooting techniques are overcome using Crocco variables which also serve to better elucidate the solution structure. The stabilities of dual, triple and even quadruple steady flow solutions encountered in different ranges of plate stretching and wall stress are determined using a linear temporal stability analysis for the self-similar flow.      

The asymptotics of neutral curve crossing in Taylor–Dean flow

The fluid flow between a pair of coaxial circular cylinders generated by the uniform rotation of the inner cylinder and an azimuthal pressure gradient is susceptible to both Taylor and Dean type instabilities. The flow can be characterised by two parameters: a measure of the relative magnitude of the rotation and pressure effects and a non-dimensional Taylor number. This work considers the small gap, large wavenumber limit for linear perturbations when the onset of the Taylor and Dean instabilities is concurrent. A consistent, matched asymptotic solution is found across the whole annular domain and identifies five regions of interest: two boundary adjustment regions and three internal critical points. Necessary conditions for the Taylor number and wavenumber are found and interpreted with reference to the suggestion of neutral curve kinks occurring at moderate wavelengths. Received: October 21, 2003; revised: November 11, 2004     

A three-layer asymptotic analysis of turbulent channel flow

In the classical theory for large-Reynolds number fully developed channel flow, the solutions obtained by asymptotic-expansion techniques for the outer Karman defect layer and the inner Prandtl wall layer are demonstrated to match through the introduction of an intermediate layer, based on a general intermediate limit. From an examination of the results for this general intermediate layer, the distinguished intermediate limit and the corresponding intermediate layer for which the turbulent and laminar contributions to the difference of the Reynolds stress from the wall stress are of the same order of magnitude are identified. The thickness of this distinguished intermediate layer is found to be of the order of the geometric mean of the thicknesses of the outer and inner layers     

The flow due to a rough rotating disk

Von Kármáns problem of a rotating disk in an infinite viscous fluid is extended to the case where the disk surface admits partial slip. The nonlinear similarity equations are integrated accurately for the full range of slip coefficients. The effects of slip are discussed. An existence proof is also given.     

Axisymmetric flow over a backward-facing step in an annular pipe

The incompressible flow of a Newtonian fluid over a backward-facing step is investigated numerically. The geometry is an annular pipe in which the radius of the inner cylinder decreases suddenly. Keeping the radial expansion ratio fixed axisymmetric flows are computed for outlet radius ratios from 0.1 to 1 (ratio of the inner to the outer outlet radius). The Reynolds number at which the flow separates from the outer cylinder decreases as the outlet radius ratio decreases for constant inlet geometry. The growth with Reynolds number of the recirculation zone on the inner outlet cylinder just behind the step is strongly reduced when the recirculation zone on the outer cylinder is established. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wall effects on the slow steady motion of a particle in a viscous incompressible fluid

In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section. In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.     

Exact self-similar solutions of the magnetohydrodynamic boundary layer system for power-law fluids

In a particular self-similar case, the magnetohydrodynamic boundary layer system for an electrically conducting power-law fluid together with certain boundary conditions can be transformed into a boundary value problem for a third-order nonlinear ordinary differential equation, only whose (generalized) normal solutions possess the physical meaning of the original problem. Uniqueness, existence and nonexistence results are established for the problem. Representations are also given for all (generalized) normal solutions. The project was supported by the Natural Science Foundation of Fujian Province of China (No. Z0511005) and NNSF of china(No. 10501037).     

Frequencies of Free Acoustic Oscillations in a Fluid Separating Rigid and Elastic Tube Walls of Finite Length

A solution of the problem of determining the frequencies and mode shapes of free nonsymmetric oscillations in an annular volume filled with an ideal compressible fluid is constructed. The inner tube and the end plane walls are ideally rigid. A thin elastic shell with edges clamped to the end walls is located on the outer tube boundary. A phenomenon of a decrease in the fundamental frequency as the thickness of a fluid layer adjacent to the elastic wall decreases is confirmed. Bibliography: 8 titles.     

Linear stability of radially-heated circular Couette flow with simulated radial gravity

The stability of circular Couette flow between vertical concentric cylinders in the presence of a radial temperature gradient is considered with an effective “radial gravity.” In addition to terrestrial buoyancy − ρg e _z we include the term − ρg _m f(r)e _r where g _m f(r) is the effective gravitational acceleration directed radially inward across the gap. Physically, this body force arises in experiments using ferrofluid in the annular gap of a Taylor–Couette cell whose inner cylinder surrounds a vertical stack of equally spaced disk magnets. The radial dependence f(r) of this force is proportional to the modified Bessel function K ₁(κr), where 2π/κ is the spatial period of the magnetic stack and r is the radial coordinate. Linear stability calculations made to compare with conditions reported by Ali and Weidman (J. Fluid Mech., 220, 1990) show strong destabilization effects, measured by the onset Rayleigh number R, when the inner wall is warmer, and strong stabilization effects when the outer wall is warmer, with increasing values of the dimensionless radial gravity γ = g _m /g. Further calculations presented for the geometry and fluid properties of a terrestrial laboratory experiment reveal a hitherto unappreciated structure of the stability problem for differentially-heated cylinders: multiple wavenumber minima exist in the marginal stability curves. Transitions in global minima among these curves give rise to a competition between differing instabilities of the same spiral mode number, but widely separated axial wavenumbers.     

Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions

In this paper, we investigate the heat transfer of a viscous fluid flow over a stretching/shrinking sheet with a convective boundary condition. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameter on the temperature profiles and wall heat flux are presented and discussed. The solution is given as an incomplete Gamma function. It is found the convective boundary conditions results in temperature slip at the wall and this temperature slip is greatly affected by the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameters. The temperature profiles in the fluid are also quite different from the prescribed wall temperature cases.     

Magnetohydrodynamics flow of a Burgers’ fluid in an orthogonal rheometer

The magnetohydrodynamics flow of an electrically conducting, incompressible Burgers’ fluid in an orthogonal rheometer is investigated. An exact solution is obtained. The effects of various dimensionless parameters existing in the model on the velocity field, vorticity and traction are studied graphically. It is noted that boundary layers form for a variety of reasons. It form as the Reynolds number increases. Also, as the Weissenberg number increases a distinct boundary layer formation is observed. It can develop at low Reynolds number provided the Weissenberg number is sufficiently high, however, it is not possible in the case of a Newtonian fluid. It is shown that no torque is exerted by the fluid on one of the disks. Results are compared with Oldroyd-B fluid.     

Unsteady flow of a Maxwell fluid with fractional derivatives in a circular cylinder moving with a nonlinear velocity

Abstract

In this paper we determine the velocity field and the shear stress corresponding to the unsteady flow of a Maxwell fluid with fractional derivatives driven by an infinite circular cylinder that slides along its axes with a velocity At^a. The general solutions, obtained by means of integral transforms, satisfy all imposed initial and boundary conditions. They can be easily particularized to give the similar solutions for ordinary Maxwell and Newtonian fluids. Finally, the influence of the parameters α and β on the fluid motion as well as a comparison between models is underlined by graphical illustrations.     

Finite element calculations of flow past a spherical bubble rising on the axis of a cylindrical tube

The velocity and pressure fields of a Newtonian fluid with homogeneous and constant physical properties flowing around a sphere on the axis of a cylindrical tube with no slip, free slip and partial slip at the sphere surface and no slip at the cylinder wall have been calculated by solving the Navier-Stokes equations and the continuity equation using the finite element technique with the penalty function method. Terminal rise velocities of spherical air bubbles in water have been calculated as function of the bubble radius and some conclusions have been drawn about the nature of the interface. Finally, the influence of the presence of a cylindrical wall on the drag force has been determined and a new empirical equation is derived for the wall correction factor for a sphere rising with free slip at its surface at low Reynolds number.     

Axial Couette flow of a generalized Oldroyd-B fluid due to a longitudinal time-dependent shear stress

Abstract

The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid in an infinite circular cylinder are determined by means of Hankel and Laplace transforms. The solutions that have been obtained, written in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The similar solutions for generalized Maxwell fluids as well as those for ordinary fluids are obtained as limiting cases of our general solutions.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

Local-in-time well-posedness for compressible MHD boundary layer

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work [27], we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.