Accelerated slip flow past a cylinder

This study deals with boundary layer flow along the entire length of a stationary semi-infinite cylinder under a steady, accelerated free-stream. Considering flow at reduced dimensions, the no-slip boundary condition is replaced with a Navier boundary condition. Asymptotic series solutions are obtained for the shear stress coefficient in terms of the Bingham number that corresponds to prescribed values of both the slip coefficient and the index of acceleration. By investigating motion at small and large axial distances, the series solutions are presented. For flow in the intermediate distances, exact and interpolated numerical solutions are obtained. Using these results, the shear stress along the entire cylinder wall is evaluated in terms of the parameters of acceleration and slip.     

Uniform Slip Flow on a Cylinder

In this paper, we determine the shear stress along the entire length of a stationary cylinder having radius a, under uniform axisymmetric flow, with velocity U0, in the slip regime. Investigating motion at small and large axial distances, we employ asymptotic series techniques to obtain the shear stress coefficient in non-dimensional form. For intermediate regions, we obtain a solution using the Rayleigh approximation method. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micro/nano sliding plate problem with Navier boundary condition

For Newtonian flow through micro or nano sized channels, the no-slip boundary condition does not apply and must be replaced by a condition which more properly reflects surface roughness. Here we adopt the so-called Navier boundary condition for the sliding plate problem, which is one of the fundamental problems of fluid mechanics. When the no-slip boundary condition is used in the study of the motion of a viscous Newtonian fluid near the intersection of fixed and moving rigid plane boundaries, singular pressure and stress profiles are obtained, leading to a non-integrable force on each boundary. Here we examine the effects of replacing the no-slip boundary condition by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. The Navier boundary condition, possesses a single parameter to account for the slip, the slip length ℓ, and two solutions are obtained; one integral transform solution and a similarity solution which is valid away from the corner. For the former the tangential stress on each boundary is obtained as a solution of a set of coupled integral equations. The particular case solved is right-angled corner flow and equal slip lengths on each boundary. It is found that when the slip length is non-zero the force on each boundary is finite. It is also found that for a suffciently large distance from the corner the tangential stress on each boundary is equal to that of the classical solution. The similarity solution involves two restrictions, either a right-angled corner flow or a dependence on the two slip lengths for each boundary. When the tangential stress on each boundary is calculated from the similarity solution, it is found that the similarity solution makes no additional contribution to the tangential stress of that of the classical solution, thus in agreement with the findings of the integral transform solution. Values of the radial component of velocity along the line θ = π /4 for increasing distance from the corner for the similarity and integral transform solutions are compared, confirming their agreement for sufficiently large distances from the corner. (Received: November 9, 2005)     

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.     

幂律速度运动表面上磁流体在驻点附近的滑移流动

从理论上研究了具有非线性延伸表面的磁流体在滑移流区的动量传输问题.通过Lie群变换把控制方程组转化为常微分方程组,利用同伦分析方法求得了问题的近似解析解.获得的级数解与文献中的数值解吻合得较好.另外,利用级数解分析滑移参数、磁场强度、速度比率参数、吸入喷注参数和幂律指数对流动的影响.结果显示这些参数对壁剪切力和边界层内流场有较大的影响.     

A moving-wall boundary layer flow of a slightly rarefied gas free stream over a moving flat plate

In the current work, the boundary layer flow of a slightly rarefied gas free stream over a moving flat plate is presented and solved numerically. The first-order slip boundary condition is adopted in the derivation. The dimensionless velocity and shear stress profiles are plotted and discussed. A theoretical derivation of the estimated solution domain is developed, which will give a very close estimation to the exact solution domain obtained numerically. The influences of velocity slip at the wall on the velocity and shear stress are also addressed.     

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.     

Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress

The velocity field and the adequate shear stress corresponding to the flow of a Maxwell fluid with fractional derivative model, between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is due to the inner cylinder that applies a longitudinal time dependent shear to the fluid. The solutions that have been obtained, presented under integral and series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. They can be easy particularizes to give the similar solutions for ordinary Maxwell and Newtonian fluids. Finally, the influence of the relaxation time and the fractional parameter, as well as a comparison between models, is shown by graphical illustrations.     

Nano boundary layers over stretching surfaces

In this paper, we present similarity solutions for the nano boundary layer flows with Navier boundary condition. We consider viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface. The resulting nonlinear ordinary differential equations are solved analytically by the Homotopy Analysis Method. Numerical solutions are obtained by using a boundary value problem solver, and are shown to agree well with the analytical solutions. The effects of the slip parameter K and the suction parameter s on the fluid velocity and on the tangential stress are investigated and discussed. As expected, we find that for such fluid flows at nano scales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter K.     

A note on the unsteady flow of a generalized second-grade fluid through a circular cylinder subject to a time dependent shear stress

The unsteady flow of a generalized second-grade fluid through an infinite straight circular cylinder is considered. The flow of the fluid is due to the longitudinal time dependent shear stress that is prescribed on the boundary of the cylinder. The fractional calculus approach in the governing equation corresponding to a second-grade fluid is introduced. The velocity field and the resulting shear stress are obtained by means of the finite Hankel and Laplace transforms. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method is used. The corresponding solutions for ordinary second-grade and Newtonian fluids, performing the same motion, are obtained as limiting cases of our general solutions. Finally, the influence of the material constants and of the fractional parameter on the velocity and shear stress variations is underlined by graphical illustrations.     

Heat transfer on a microcylinder with slip

The purpose of this paper is to determine the heat transfer from a semi-infinite microcylinder under uniform axial flow in the slip regime. Following the treatments of Glauert and Lighthill (Proc Royal Soc 230A:188–203, 1955), and Crane (Ingenieur-Archiv 43:203–214, 1974), an asymptotic series solution is developed to account for the heat flux in terms of the Nusselt number for large axial distances along a cylinder maintained at constant temperature.     

Shear stress analysis in axially symmetric flow using the Padé approximation technique

This paper evaluates the shear stress on the surface of a circular cylinder in axially symmetric flow within the boundary layer. An extended solution is obtained by established techniques, e.g. asymptotic series or Pohlhausen method. A Padé approximation technique is applied to the extended solution obtained. The region of validity of the solution is extended and the results are then compared and examined against known approximate Pohlhausen and series solutions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

Slip flow on a microcylinder

The purpose of this paper is to determine the heat transfer from a semi-infinite microcylinder under uniform axial flow in the slip regime. Following the treatments of Glauert and Lighthill (Proc Royal Soc 230A:188–203, 1955), and Crane (Ingenieur-Archiv 43:203–214, 1974), an asymptotic series solution is developed to account for the heat flux in terms of the Nusselt number for large axial distances along a cylinder maintained at constant temperature.     

Slip flow on a microcylinder

This paper is concerned with uniform axial flow of a gas along a semi-infinite microcylinder in the slip regime. Following Glauert and Lighthill (Proc R Soc 230A:188–203, 1955), the effect of slip on the skin friction is investigated by employing an asymptotic series approach for large axial distances along the cylinder.     

Unsteady longitudinal flow of a generalized Oldroyd-B fluid in cylindrical domains

Considering a fractional derivative model the unsteady flow of an Oldroyd-B fluid between two infinite coaxial circular cylinders is studied by using finite Hankel and Laplace transforms. The motion is produced by the inner cylinder which is subject to a time dependent longitudinal shear stress at time t = 0⁺. The solution obtained under series form in terms of generalized G and R functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, generalized and ordinary Maxwell, and Newtonian fluids are obtained as limiting cases of our general solutions. The influence of pertinent parameters on the fluid motion as well as a comparison between models is illustrated graphically.     

Exterior Stokes flows with stick-slip boundary conditions

Steady two-dimensional creeping flows induced by line singularities in the presence of an infinitely long circular cylinder with stick-slip boundary conditions are examined. The singularities considered here include a rotlet, a potential source and a stokeslet located outside a cylinder and lying in a plane containing the cylinder axis. The general exterior boundary value problem is formulated and solved in terms of a stream function by making use of the Fourier expansion method. The solutions for various singularity driven flows in the presence of a cylinder are derived from the general results. The stream function representation of the solutions involves a definite integral whose evaluation depends on a non-dimensional slip parameter l₁\lambda_1. For extremal values, l₁ = 0\lambda_1 = 0 and l₁ = 1\lambda_1 = 1, of the slip parameter our results reduce to solutions of boundary value problems with stick (no-slip) and perfect slip conditions, respectively.¶The slip parameter influences the flow patterns significantly. The plots of streamlines in each case show interesting flow patterns. In particular, in the case of a single rotlet/stokeslet (with axis along y-direction) flows, eddies are observed for various values of l₁\lambda_1. The flow fields for a pair of singularities located on either side of the cylinder are also presented. In these flows, eddies of different sizes and shapes exist for various values of l₁\lambda_1 and the singularity locations. Plots of the fluid velocity on the surface show locations of the stagnation points on the surface of the cylinder and their dependencies on l₁\lambda_1 and singularity locations.     

Peristatcally induced MHD slip flow in a porous medium due to a surface acoustic wavy wall

The influences of Hall current and slip condition on the MHD flow induced by sinusoidal peristaltic wavy wall in two dimensional viscous fluid through a porous medium for moderately large Reynolds number is considered on the basis of boundary layer theory in the case where the thickness of the boundary layer is larger than the amplitude of the wavy wall. Solutions are obtained in terms of a series expansion with respect to small amplitude by a regular perturbation method. Graphs of velocity components, both for the outer and inner flows for various values of the Reynolds number, slip parameter, Hall and magnetic parameters are drawn. The inner and outer solutions are matched by the matching process. An interesting application of the present results to mechanical engineering may be the possibility of the fluid transportation without an external pressure.     

On some axial Couette flows of non-Newtonian fluids

The velocity fields corresponding to some flows of second grade and Maxwell fluids, induced by a circular cylinder subject to a constantly accelerating translation along its symmetry axis, are presented as Fourier-Bessel series in terms of the eigenfunctions of some suitable boundary value problems. These solutions satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For α or λ → 0, they are going to those for a Newtonian fluid. Finally, for comparison, some diagrams corresponding to the solutions for the flow through a circular cylinder are presented for different values of t and of the material constants.     

Unsteady helical flows of Oldroyd-B fluids

The unsteady helical flow of an Oldroyd-B fluid, in an infinite circular cylinder, is studied by using finite Hankel transforms. The motion is produced by the cylinder that, at time t = 0⁺, is subject to torsional and longitudinal time-dependent shear stresses. The solutions that have been obtained, presented under series form, satisfy all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting cases of general solutions. Finally, the influence of the pertinent parameters on the fluid motion is underlined by graphical illustrations.