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Wall effects with slip
ZAMM - Journal of Applied Mathematics and Mechanics
Abstract
The creeping motion of a spheroidal particle at the instant it passes the centre of a spherical container has been examined using a general slip condition at the surface of the particle. The shape of the spheroid deviates slightly from that of a sphere at its surface. A first-order expression has been obtained for the stream function characterizing the flow field and this expression has been utilized to evaluate the drag on the spheroid. Special cases are deduced and wall effects then examined.
... In all the above mentioned problems, the authors utilized no-slip condition on the surface of the inner sphere. [18] studied the motion of solid spheroidal particle in a spherical container using a slip condition at the surface of the inner particle and evaluated the expression for drag on it and examined the wall effects. They concluded that the wall effects increase as the spheroidal particle becomes more spheroidal. ...
... The unknowns appearing in (11) and (13) can be deduced by substituting these equations into the boundary conditions (15)- (18), which lead to the following system of algebraic equations: ...
An analysis is carried out to study the flow characteristics of creeping motion of an inner non-Newtonian Reiner-Rivlin liquid spheroid r = 1+ ∑_{k=2}^∞α_kG_k(cos θ), here α_k is very small shape factor and G_k is Gegenbauer function of first kind of order k, at the instant it passes the centre of a rigid spherical container filled with a Newtonian fluid. The shape of the liquid spheroid is assumed to depart a bit at its surface from the shape a sphere. The analytical expression for stream function solution for the flow in spherical container is obtained by using Stokes equation. While for the flow inside the Reiner-Rivlin liquid spheroid, the expression for stream function is obtained by expressing it in a power series of S, characterizing the cross-viscosity of Reiner-Rivlin fluid. Both the flow fields are then determined explicitly by matching the boundary conditions at the interface of Newtonian fluid and non-Newtonian fluid and also the condition of impenetrability and no-slip on the outer surface to the first order in the small parameter ε, characterizing the deformation of the liquid sphere. As an application, we consider an oblate liquid spheroid r = 1+2εG_2(cos θ) and the drag and wall effects on the body are evaluated. Their variations with regard to separation parameter, viscosity ratio λ, cross-viscosity, i.e., S and deformation parameter are studied and demonstrated graphically. Several well-noted cases of interest are derived from the present analysis. Attempts are made to compare between Newtonian and Reiner-Rivlin fluids which yield that the cross-viscosity μ_c is to decrease the wall effects K and to increase the drag D_N when deformation is comparatively small. It is observed that drag not only varies with λ, but as η increases, the rate of change in behavior of drag force increases also.
... The essence of the problem is to assume a single particle in a cell and solve it. Ramkissoon and Rahaman [2] investigated the slow motion of a spheroidal particle in a concentric spherical container using slip at the surface of the particle. The different features of the hydrodynamic cell models are discussed by Zholkovskiy et al. [3]. ...
The problem of the steady, axisymmetric flow past a rigid sphere in a hypothetical spherical cavity under the effect of the magnetic field based on Happel’s and Kuwabara’s cell model has been investigated analytically. The inner sphere is assumed to be rigid, and the outer one is fictitious. Slip boundary condition applied on the surface of the inner sphere. Happel’s and Kuwabara’s boundary conditions are applied at the outer sphere surface. The flow field inside the cavity is governed by the Stokes equation. The expression for drag acting on the sphere is obtained. Variation of drag coefficient against various parameters like slip parameter and Hartmann number is studied graphically. Some special cases are also discussed.
The creeping motion of a non-Newtonian (Reiner-Rivlin) liquid sphere at the instant it passes the center of an approximate spherical container is discussed. The flow in the spheroidal container is governed by the Stokes equation, while for the flow inside the Reiner-Rivlin liquid sphere, the expression for the stream function is obtained by expressing it in the power series of a parameter S, characterizing the cross-viscosity. Both the flow fields are then determined explicitly by matching the boundary conditions at the interface of Newtonian fluid and non-Newtonian fluid, and also the condition of imperviousness and no-slip on the outer surface. As an application, we have considered an oblate spheroidal container. The drag and wall effects on the liquid spherical body are evaluated. Their variations with regard to the separation parameter ℓ, viscosity ratio λ, cross-viscosity S, and deformation parameter ε are studied and demonstrated graphically. Several renowned cases are derived from the present analysis. It is observed that the drag not only varies with ε, but as ℓ increases, the rate of change in behavior of drag force also increases. The influences of these parameters on the wall effects has also been studied and presented in a table.
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This paper investigates first the Stokes’ axisymmetrical translational motion of a spheroid particle, whose shape differs slightly from that of a sphere, in an unbounded micropolar fluid. A linear slip, Basset-type, boundary condition has been used. The drag acting on the spheroid is evaluated and discussed for the various parameters of the problem. Also, the terminal velocity is evaluated and tabulated for the slip, deformity, and micropolarity parameters. Secondly, the motion of a spheroidal particle at the instant it passes the centre of a spherical envelope filled with a micropolar fluid is investigated using the slip condition at the surface of the particle. The analytical expressions for the stream function and microrotation component are obtained to first order in the small parameter characterizing the deformation. As an application, we consider an oblate spheroidal particle and the drag acting on the body is evaluated. Its variation with respect to the diameter ratio, deformity, micropolarity, and slip parameters is tabulated and displayed graphically. Well-known cases are deduced, the wall effect is then examined and comparisons are attempted between the classical fluid and micropolar fluid.PACS Nos.: 47.45.Gx, 47.15.Gf, 47.50.–d
Wall effects for rigid and fluid spheres in slow motion with a moving liquid
- W L Haberman
- R M Sayre
W. L. Haberman and R. M. Sayre, Wall effects for rigid and fluid spheres in slow motion with a moving liquid. David Taylor model, Basin Report 1143, Washington D.C. (1958).