Flow around nanospheres and nanocylinders

Abstract

1 Introduction

For many micro or nano flow problems, the boundary surface roughness may well be significant in comparison to the dimensions of the molecules or nanoparticles involved in the flow. In this event the usual no-slip boundary condition of fluid mechanics clearly does not apply, and must be replaced by a condition which reflects boundary slip as a consequence of the tangential shear at the boundary. The simplest linear boundary condition to accommodate this is the so-called Navier boundary condition, which has been in the literature for many years, and postulates that the boundary slip is directly proportional to the tangential shear at the boundary, and the constant of proportionality is referred to as the slip length. In this paper we consider the two model problems of low-Reynolds-number flow around nanospheres and nanocylinders, assuming the Navier boundary condition with a constant slip length. For the sphere a corresponding macroscopic problem might be the flow around a dimpled golf ball, for which it is well known that the surface indentations generate reduced drag, which not only makes the ball go further but its flight is more predictable than that of an equivalent ball with a smooth surface. Indeed the reduced drag is one consequence predicted from the analysis presented here.

The steady flow of a Newtonian fluid past a sphere and a circular cylinder is a fundamental problem, first considered by Stokes (1) in the limit of zero Reynolds number R, that is, the inertial forces are neglected with respect to viscous forces in the Navier–Stokes equations. Solutions were obtained by Stokes for the sphere, but for the circular cylinder no solution was possible, leading to the well-known Stokes' paradox. Later, Whitehead (2) attempted to improve on Stokes' original solution for the sphere by obtaining higher approximations for the flow when the Reynolds number is not negligibly small, by using an iterative procedure on the full Navier–Stokes equations, beginning the process with the solution of Stokes. Whitehead found that no second approximation could be found, leading to Whitehead's paradox.

Both paradoxes were eventually overcome by Oseen, who took the inertial terms of the Navier–Stokes equation partially into account by assuming that the components of velocity could be represented as the sum of a constant (the far-field velocity) and a perturbation term (see Schlichting (3)). Higher-order terms of the Oseen solution may be found, but their value is minimal due to the underlying assumptions required to transform the Navier–Stokes equations into the relevant Oseen equations. Since the flow around a sphere and circular cylinder for small Reynolds number may be considered a singular perturbation problem, it can be addressed using the method of matched asymptotic expansions, which begin with the full Navier–Stokes equations, and assuming appropriate inner expansions valid near the sphere or circular cylinder and outer expansions valid far from the sphere or circular cylinder. Following the terminology of Proudman and Pearson (4), Kaplun (5) and Van Dyke (6), the inner and outer expansions are termed the Stokes and Oseen expansions, respectively.

The above-mentioned analysis employs the no-slip boundary condition at the surface of the sphere and circular cylinder, that is, zero relative velocity is assumed between the fluid and solid, which is a fundamental notion in classical fluid mechanics (see for example Batchelor (7)). The microscopic origin of the no-slip boundary condition has remained elusive, and cannot be derived from first principles. As pointed out by Vinogradova (8), in most cases the no-slip boundary condition yields the correct results for simple flows, and experiments at macroscopic scales are consistent with the no-slip boundary condition. However, there exist both experimental and theoretical reasons for relaxing the no-slip hypothesis. In particular, experiments which probe molecular scales for which surface roughness may be of comparable magnitude indicate that the boundary condition at the surface of a solid may be different. An improved understanding of the effect of the fluid–solid interaction on flows at this scale is essential and has received a lot of attention (see for example Thompson and Robbins (9)).

Molecular dynamics studies by Qian et al. (10) have shown relative slipping between the fluid and the solid, in violation of the no-slip boundary condition. These findings cast doubt on the general applicability of the continuum hydrodynamics model of micro- and nanofluidics, where the boundary conditions may have macroscopic implications. Detailed studies of the fluid–solid interaction by Thompson and Troian (11) showed that the degree of slip depends on a number of interfacial parameters including unequal wall and fluid densities and the strength of the weak wall–fluid interaction. Together these effects prevent the locking of fluid layer(s) to the solid wall, thus allowing slip to occur (Qian and Wang (12)). Recent studies by Sokhan et al. (13, 14) of the fluid flow in carbon nanopores and nanotubes has shown it to be characterized by a large slip length.

Although the amount of slip depends on molecular details, Hocking (15) and Lamb (16) have suggested that the overall effect on the flow on a macroscopic scale can be obtained by postulating a dynamic boundary condition in which the relative velocity at the solid boundary is proportional to the tangential viscous stress there, that is, the net tangential force per unit area exerted on the solid due to the hydrodynamic motion of the surrounding fluid (10). The constant of proportionality which expresses the ratio of the relative velocity to the tangential viscous stress is termed the slip length, denoted by (note that Lamb (16) refers to the slip length as the coefficient of sliding friction). As demonstrated by Qian and Wang (12), for a Newtonian fluid, the tangential viscous stress is proportional to the shear rate; consequently, the amount of slip is proportional to the shear rate. This linear slip–shear relation is the simplest known boundary condition used to improve the no-slip condition, with different expressions for the slip length due to different physical backgrounds (see for example Shikhmurzaev (17)). A linear slip condition frequently makes the problem tractable (Huh and Scriven (18)), and was first introduced by Navier in 1823 (19) and later independently by Maxwell in 1879 (20) (see Dussan (21) for a historical review). The so-called Navier boundary condition is applicable whenever the molecular mean free path approaches the dimensions of the flow (18). That is, for flows at the macro or nano scale, or when mathematical results predict an unrealistic stress divergence (see Hocking (15, 22) for an account of this usage), and when there exists surface roughness which may significantly affect the flow. Usually, such a change in the no-slip boundary would in most circumstances be undetectable, since the additional term would only change the macroscopic flow by an amount proportional to the ratio of the molecular to the macroscopic length scales.

The validity of the Navier boundary condition has been shown by Qian and Wang (12) for the driven cavity problem through molecular dynamic and continuum hydrodynamic simulation for a Newtonian fluid, where the no-slip boundary condition causes unphysical stress divergence. The slip length is determined from molecular dynamics simulations, usually ranging from one to a few nanometers, and when used as an input for the continuum hydrodynamic simulation it has been shown by Qian et al. (23, 24) that molecular dynamics results can be successfully reproduced. This and other work has shown that for flow at the micro or nano scale the no-slip boundary condition is not a valid postulate. We propose that a different postulate, namely the Navier boundary condition, may be a more accurate reflection of physical reality at a fluid/solid interface at micro or nano scales. The no-slip boundary condition is in fact an approximation to the Navier boundary condition, corresponding to = 0. (The limit of infinite slip length corresponds to the conventional condition for a liquid/gas interface, that is perfect slip, as demonstrated in the analysis of film drainage by Vinogradova (8).) As the amount of slip is extremely small for macroscopic flows, the Navier boundary condition is practically indistinguishable from the no-slip condition in most cases on the macroscopic scale. In particular, Richardson (25) has shown that for a general Newtonian fluid flow in the immediate neighbourhood of a rough solid wall the no-slip boundary condition is the relevant macroscopic boundary condition, in the sense that deviations between the results for no-slip and infinite slip are of the order of the slip length. However, for micro or nano scale problems, the difference between the solutions for no-slip and infinite slip are of an order comparable to the dimensions of the flow, and that is the dimension of interest in this study.

The Navier boundary condition assumes that the degree of slip is independent of the shear rate, that is is constant, although recent molecular dynamic simulations by Thompson and Troian (11) have indicated that is a nonlinear function of the shear rate over a large range of shear rates. At low shear rates, the Navier boundary condition is valid, that is is constant, but at high shear rates the Navier boundary condition breaks down as increases rapidly with the shear rate. Thus, the Navier boundary condition is a low-shear-rate limit of a more generalized universal relationship which is significantly nonlinear and divergent at a (high) critical shear rate. Modelling flows based on a functional dependence of the slip length on the tangential viscous stress would be inherently difficult, and possibly intractable analytically except for specific problems.

In this paper the linear Navier boundary condition with a constant slip length is applied to the steady flow of a Newtonian fluid past a nanosphere and a nanocylinder (or nanotube), a scale at which the slip length is relevant, using the method of matched asymptotic expansions. Since the phenomena of the motion of small particles depend critically on first-order effects arising from the inertia of the fluid as pointed out by Proudman and Pearson (4), the analysis is carried through to first order. The aim of the analysis is to determine the effect of a non-zero slip length on this most basic of flows, and to determine whether or not the Navier boundary condition may be employed to study more complicated flows at the nanoscale. In the following section, the equations describing the Navier boundary condition are presented. In sections 3 and 4 the analysis for the flow past a nanosphere and a nanocylinder are presented, and are subsequently solved using the method of matched asymptotic expansions with the Reynolds number assumed a small parameter. In section 5 some general features of the solutions obtained are discussed, as well as concluding remarks.

2 The Navier boundary condition

The standard no-slip boundary condition at the surface of the sphere and the circular cylinder is replaced by the Navier boundary condition, where the slip velocity is assumed to be proportional to the tangential viscous stress and the degree of slip is measured by a constant slip length. For an incompressible Newtonian fluid the viscous portion of the stress tensor or the extra stress is given by S = 2μD, where μ is the viscosity and the rate of deformation tensor is

\[\mathbf{\mathrm{D}}{=}\frac{1}{2}\left[\mathrm{{\nabla}}\mathbf{\mathrm{v}}{+}(\mathrm{{\nabla}}\mathbf{\mathrm{v}})^{T}\right],\]

that is, S ∝ D. For both the sphere and the circular cylinder, the Navier boundary condition at the corresponding surface implies v_θ ∝ D_rθ, where from Slattery (26)

\[D_{r{\theta}}{=}\frac{1}{2}\left[\frac{{\partial}v_{{\theta}}}{{\partial}r}{-}\frac{1}{r}\left(v_{{\theta}}{-}\frac{{\partial}v_{r}}{{\partial}{\theta}}\right)\right],\]

in both spherical and cylindrical coordinate systems. At the surface the normal velocity is v_r = 0 and therefore ∂v_r/∂θ = 0, so that the Navier boundary condition at the surface of the sphere and circular cylinder is given by

where is the slip length.

3 Flow past a nanosphere

The relevant boundary conditions are given by

\[v_{z}{=}U{\Rightarrow}v_{r}{=}U{\,}\mathrm{cos}{\theta},{\ }v_{{\theta}}{=}{-}U{\,}\mathrm{sin}{\theta},{\ }r{\rightarrow}{\infty},\]

where U is the uniform velocity of the flow far from the sphere. It is assumed that

\[v_{r}{=}v_{r}(r,{\theta}),{\ }v_{{\theta}}{=}v_{{\theta}}(r,{\theta}),{\ }v_{{\phi}}{=}0,\]

so that a stream function ψ (r, θ) defined by

\[v_{r}{=}\frac{1}{r^{2}{\,}\mathrm{sin}{\,}{\theta}}\frac{{\partial}{\psi}}{{\partial}{\theta}},{\ }v_{{\theta}}{=}{-}\frac{1}{r{\,}\mathrm{sin}{\,}{\theta}}\frac{{\partial}{\psi}}{{\partial}r},\]

may be introduced such that the differential mass conservation equation is automatically satisfied. In terms of this stream function the boundary conditions may be written

\[{\psi}{=}\frac{1}{2}Ur^{2}{\,}\mathrm{sin}^{2}{\theta},{\ }r{\rightarrow}{\infty}.\]

The Navier–Stokes equation in spherical coordinates with a stream function defined as above is given in Slattery (26)

\[{\nu}E_{r}^{4}{\psi}{=}{-}\frac{1}{r^{2}{\,}\mathrm{sin}{\theta}}\left[\frac{{\partial}({\psi},E_{r}^{2}{\psi})}{{\partial}(r,{\theta})}{-}2E_{r}^{2}{\psi}\left(\mathrm{cot}{\theta}\frac{{\partial}{\psi}}{{\partial}r}{-}\frac{1}{r}\frac{{\partial}{\psi}}{{\partial}{\theta}}\right)\right],\]

where ν = μ/ρ is the kinematic viscosity and ρ is the density, and the operator

\(E_{r}^{2}\)

is defined by

\[E_{r}^{2}{=}\frac{{\partial}^{2}}{{\partial}r^{2}}{+}\frac{1}{r^{2}}\left(\frac{{\partial}^{2}}{{\partial}{\theta}^{2}}{-}\mathrm{cot}{\theta}\frac{{\partial}}{{\partial}{\theta}}\right).\]

By introducing the dimensionless variables r^* = r/a, v^* = v/U, ψ^* = ψ/Ua², ^* = /a the Navier–Stokes equation and boundary conditions become

\[E_{r^{{\ast}}}^{4}{\psi}^{{\ast}}{=}{-}\frac{R}{r^{{\ast}2}{\,}\mathrm{sin}{\theta}}\left[\frac{{\partial}({\psi}^{{\ast}},{\,}E_{r^{{\ast}}}^{2}{\psi}^{{\ast}})}{{\partial}(r^{{\ast}},{\theta})}{-}2E_{r^{{\ast}}}^{2}{\psi}^{{\ast}}\left(\mathrm{cot}{\theta}\frac{{\partial}{\psi}^{{\ast}}}{{\partial}r^{{\ast}}}{-}\frac{1}{r^{{\ast}}}\frac{{\partial}{\psi}^{{\ast}}}{{\partial}{\theta}}\right)\right],\]

\[{\psi}^{{\ast}}{=}\frac{1}{2}r^{{\ast}2}{\,}\mathrm{sin}^{2}{\theta},{\ }r^{{\ast}}{\rightarrow}{\infty},\]

where R is the dimensionless Reynolds number defined by R = Ua/ν.

Hence the Stokes expansion satisfies

\[E_{r^{{\ast}}}^{4}{\psi}^{{\ast}}{=}{-}\frac{R}{r^{{\ast}2}{\,}\mathrm{sin}{\theta}}\left[\frac{{\partial}({\psi}^{{\ast}},{\,}E_{r^{{\ast}}}^{2}{\psi}^{{\ast}})}{{\partial}(r^{{\ast}},{\theta})}{-}2E_{r^{{\ast}}}^{2}{\psi}^{{\ast}}\left(\mathrm{cot}{\theta}\frac{{\partial}{\psi}^{{\ast}}}{{\partial}r^{{\ast}}}{-}\frac{1}{r^{{\ast}}}\frac{{\partial}{\psi}^{{\ast}}}{{\partial}{\theta}}\right)\right],\]

while the Oseen expansion satisfies

\begin{eqnarray*}&&E_{{\rho}^{{\ast}}}^{4}\mathrm{{\Psi}}^{{\ast}}{=}{-}\frac{1}{{\rho}^{{\ast}2}{\,}\mathrm{sin}{\theta}}\left[\frac{{\partial}(\mathrm{{\Psi}}^{{\ast}},E_{{\rho}^{{\ast}}}^{2}\mathrm{{\Psi}}^{{\ast}})}{{\partial}({\rho}^{{\ast}},{\theta})}{-}2E_{{\rho}^{{\ast}}}^{2}\mathrm{{\Psi}}^{{\ast}}\left(\mathrm{cot}{\theta}\frac{{\partial}\mathrm{{\Psi}}^{{\ast}}}{{\partial}{\rho}^{{\ast}}}{-}\frac{1}{{\rho}^{{\ast}}}\frac{{\partial}\mathrm{{\Psi}}^{{\ast}}}{{\partial}{\theta}}\right)\right],\\&&\mathrm{{\Psi}}^{{\ast}}{=}\frac{1}{2}{\rho}^{{\ast}2}{\,}\mathrm{sin}^{2}{\theta},{\ }r^{{\ast}}{\rightarrow}{\infty}.\end{eqnarray*}

3.1 Leading-order expansions

The leading-order solution for the Stokes expansion satisfies the equation

\(E_{r^{{\ast}}}^{4}{\psi}_{0}{=}0,\)

which has solution

\[{\psi}_{0}(r^{{\ast}},{\theta}){=}{[}C_{1}r^{{\ast}{-}1}{+}C_{2}r^{{\ast}}{+}C_{3}r^{{\ast}2}{+}C_{4}r^{{\ast}4}]\mathrm{sin}^{2}{\theta},\]

where throughout C₁, C₂, C₃ and C₄ denote arbitrary integration constants. Invoking the ‘principle of minimum singularity’ (see (6)) gives C₄ = 0, and the boundary conditions

imply

Hence

and

The leading-order solution for the Oseen expansion which satisfies the following equation and boundary condition

\begin{eqnarray*}&&E_{{\rho}^{{\ast}}}^{4}\mathrm{{\Psi}}_{0}{=}{-}\frac{1}{{\rho}^{{\ast}2}{\,}\mathrm{sin}{\theta}}\left[\frac{{\partial}(\mathrm{{\Psi}}_{0},{\,}E_{{\rho}^{{\ast}}}^{2}\mathrm{{\Psi}}_{0})}{{\partial}({\rho}^{{\ast}},{\theta})}{-}2E_{{\rho}^{{\ast}}}^{2}\mathrm{{\Psi}}_{0}\left(\mathrm{cot}{\theta}\frac{{\partial}\mathrm{{\Psi}}_{0}}{{\partial}{\rho}^{{\ast}}}{-}\frac{1}{{\rho}^{{\ast}}}\frac{{\partial}\mathrm{{\Psi}}_{0}}{{\partial}{\theta}}\right)\right],\\&&\mathrm{{\Psi}}_{0}{=}\frac{1}{2}{\rho}^{{\ast}2}\mathrm{sin}^{2}{\theta},{\,}{\rho}^{{\ast}}{\rightarrow}{\infty},\end{eqnarray*}

is found to be

\[\mathrm{{\Psi}}_{0}({\rho}^{{\ast}},{\theta}){=}\frac{1}{2}{\rho}^{{\ast}2}\mathrm{sin}^{2}{\theta}.\]

Expanding the leading-order Stokes expansion in terms of the Oseen variable ρ^* = Rr^* yields

The asymptotic matching principle (see (6)) requires

hence the leading-order Stokes expansion is given by

which reduces to the solution of Proudman and Pearson (4) when ^* = 0 and agrees with the solution which appears in Basset (27) and Lamb (16).

3.2 First-order expansions

The first-order solution to the Stokes expansion satisfies the following equation and boundary conditions:

\[\frac{f_{1}(R)}{R}E_{r^{{\ast}}}^{4}{\psi}_{1}{=}{-}\frac{1}{r^{{\ast}2}{\,}\mathrm{sin}{\theta}}\left[\frac{{\partial}({\psi}_{0},{\,}E_{r^{{\ast}}}^{2}{\psi}_{0})}{{\partial}(r^{{\ast}},{\theta})}{-}2E_{r^{{\ast}}}^{2}{\psi}_{0}\left(\mathrm{cot}{\theta}\frac{{\partial}{\psi}_{0}}{{\partial}r^{{\ast}}}{-}\frac{1}{r^{{\ast}}}\frac{{\partial}{\psi}_{0}}{{\partial}{\theta}}\right)\right],\]

Setting

\(f_{1}(R){=}R\)

and substituting the leading-order Stokes solution yields

A particular solution ψ_1P is assumed to be of the form

\({\psi}_{1P}(r^{{\ast}},{\theta}){=}f(r^{{\ast}}){\,}\mathrm{sin}^{2}{\theta}{\,}\mathrm{cos}{\theta},\)

which implies

which has solution

Invoking the ‘principle of minimum singularity’ (see (6)) gives C₃ = 0 = C₄, and the boundary conditions

imply

Hence

The complementary solution ψ_1C which satisfies the equation

\(E_{r^{{\ast}}}^{4}{\psi}_{1C}{=}0\)

is of the form

so that ψ₁ = ψ_1C + ψ_1P.

Expanding the leading-order Stokes expansion in terms of the Oseen variable ρ^* = Rr^* yields

and the full Oseen expansion for small ρ^* is given by

\[\mathrm{{\Psi}}^{{\ast}}{=}\mathrm{{\Psi}}_{0}{+}F_{1}(R)\mathrm{{\Psi}}_{1}{\sim}\frac{1}{2}{\rho}^{{\ast}2}{\,}\mathrm{sin}^{2}{\theta}{+}F_{1}(R)(\mathrm{{\Psi}}_{1C}{-}C_{2}{\rho}^{{\ast}}{\,}\mathrm{sin}^{2}{\theta}).\]

The asymptotic matching principle (see (6)) requires

and thus the full Oseen expansion is given by

which reduces to the solution of Proudman and Pearson (4) when ^* = 0.

Expanding the full Stokes expansion in terms of the Oseen variable ρ^* = Rr^* yields

and the full Oseen expansion for small ρ^* is given by

Hence the asymptotic matching principle (see (6)) requires

and the full Stokes expansion is given by

(3.1)

which reduces to the solution of Proudman and Pearson (4) when ^* = 0.

3.3 Frictional drag

There are two methods for calculating the frictional drag on a solid body immersed in a fluid—either the viscous stresses exerted by the fluid on the surface of the body are summed, or the theorem of momentum (or energy dissipation) is applied to an infinite mass of fluid surrounding the body, as mentioned by Tomotika and Aoi (28). Here the first method is used so that only the Stokes expansions and the relevant Navier–Stokes equation with assumptions consistent with the analysis are required to calculate the stresses, instead of a complementary solution of the Stokes and Oseen expansions and much more complicated form of the Navier–Stokes equation.

The frictional drag is calculated from the force that the fluid exerts on the nanosphere beyond the force attributable to the ambient and hydrostatic pressure. This force is calculated from the tangential and normal components of stress T = – pI + 2μD integrated over the surface of the nanosphere, where p is the mean pressure and I is the identity tensor. The z component of the stress that the fluid exerts on the nanosphere is T_rr cos θ – T_rθ sin θ, so that the z component of the force is

\[F_{z}{=}{{\int}_{0}^{2{\pi}}}{{\int}_{0}^{{\pi}}}(T_{rr}{\,}\mathrm{cos}{\theta}{-}T_{r{\theta}}{\,}\mathrm{sin}{\theta}){\vert}_{r{=}a}a^{2}{\,}\mathrm{sin}{\theta}d{\theta}d{\phi},\]

where from Slattery (26)

\begin{eqnarray*}&&T_{rr}{=}{-}p{+}2{\mu}D_{rr}{=}{-}p{+}2{\mu}\frac{{\partial}v_{r}}{{\partial}r},\\&&T_{r{\theta}}{=}2{\mu}D_{r{\theta}}{=}{\mu}\left[r\frac{{\partial}}{{\partial}r}\left(\frac{v_{{\theta}}}{r}\right){+}\frac{1}{r}\frac{{\partial}v_{r}}{{\partial}{\theta}}\right].\end{eqnarray*}

The mean pressure near the nanosphere is calculated from the Navier–Stokes equation consistent with the approximation employed for the Stokes expansion of the stream function. Recall that

\({\psi}^{{\ast}}{=}{\psi}_{0}{+}R{\psi}_{1}{=}{\psi}_{0}{+}R({\psi}_{1C}{+}{\psi}_{1P}),\)

where ψ₀ and ψ_1C satisfy

\(E_{r^{{\ast}}}^{4}{\psi}_{0,1C}{=}0,\)

while ψ_1P satisfies a more complicated expression. However, because of symmetry ψ_1P contributes nothing to the frictional drag, so that the mean pressure may be calculated from ψ^* = ψ₀ + Rψ_1C only. The Navier–Stokes equations consistent with the approximations for ψ₀ and ψ_1C, that is, neglecting the convective inertial term, is given in Slattery (26)r component:

\[\frac{{\partial}p}{{\partial}r}{=}\frac{{\mu}}{r^{2}}\left[\frac{{\partial}}{{\partial}r}\left(r^{2}\frac{{\partial}v_{r}}{{\partial}r}\right){+}\frac{{\partial}^{2}v_{r}}{{\partial}{\theta}^{2}}{+}\mathrm{cot}{\theta}\left(\frac{{\partial}v_{r}}{{\partial}{\theta}}{-}2v_{{\theta}}\right){-}2\left(v_{r}{+}\frac{{\partial}v_{{\theta}}}{{\partial}{\theta}}\right)\right],\]

θ component:

\[\frac{{\partial}p}{{\partial}{\theta}}{=}\frac{{\mu}}{r}\left[\frac{{\partial}}{{\partial}r}\left(r^{2}\frac{{\partial}v_{{\theta}}}{{\partial}r}\right){+}\frac{{\partial}^{2}v_{{\theta}}}{{\partial}{\theta}^{2}}{+}\mathrm{cot}{\theta}\frac{{\partial}v_{{\theta}}}{{\partial}{\theta}}{+}2\frac{{\partial}v_{r}}{{\partial}{\theta}}{-}\frac{v_{{\theta}}}{\mathrm{sin}^{2}{\theta}}\right],\]

while the ϕ component yields ∂p/∂ϕ = 0. By calculating the r and θ components of velocity from the definition of the stream function, returning to dimensional variables (except for ^*, which we keep dimensionless), substituting into the Navier–Stokes equations and integrating, yields the following expression for p evaluated at r = a:

Substituting this expression and the expressions for the r and θ components of velocity into the expression for F_z and performing the required integrations yield

The drag coefficient c_D is defined as the force referenced by the dynamic head ρU²a², that is,

(3.2)

which reduces to the solution of Proudman and Pearson (4) when ^* = 0, and agrees with the solution which appears in Basset (27) and Lamb (16) to leading order.

4 Flow past a nanocylinder

Consider uniform streaming flow past a nanocylinder of radius a. A cylindrical coordinate system r, θ, z will be used such that

\[v_{r}{=}v_{x}{\,}\mathrm{cos}{\theta}{+}v_{y}{\,}\mathrm{sin}{\theta},{\ }v_{{\theta}}{=}{-}v_{x}{\,}\mathrm{sin}{\theta}{+}v_{y}{\,}\mathrm{cos}{\theta}.\]

The relevant boundary conditions are given by

\[v_{x}{=}U{\Rightarrow}v_{r}{=}U{\,}\mathrm{cos}{\theta},{\,}v_{{\theta}}{=}{-}U{\,}\mathrm{sin}{\theta},{\,}r{\rightarrow}{\infty}.\]

It is assumed that

\[v_{r}{=}v_{r}(r,{\theta}),{\ }v_{{\theta}}{=}v_{{\theta}}(r,{\theta}),{\ }v_{z}{=}0,\]

so that a stream function ψ (r, θ), defined by

\[v_{r}{=}\frac{1}{r}\frac{{\partial}{\psi}}{{\partial}{\theta}},{\,}v_{{\theta}}{=}{-}\frac{{\partial}{\psi}}{{\partial}r},\]

may be introduced such that the differential mass conservation equation is automatically satisfied. In terms of this stream function the boundary conditions may be written

\[{\psi}{=}Ur{\,}\mathrm{sin}{\theta},{\ }r{\rightarrow}{\infty}.\]

The Navier–Stokes equation in cylindrical coordinates with a stream function defined as above is given by Slattery (26)

\[{\nu}\mathrm{{\nabla}}_{r}^{4}{\psi}{=}{-}\frac{1}{r}\frac{{\partial}({\psi},\mathrm{{\nabla}}_{r}^{2}{\psi})}{{\partial}(r,{\theta})},\]

where the Laplacian

\(\mathrm{{\nabla}}_{r}^{2}\)

is given by

\[\mathrm{{\nabla}}_{r}^{2}{=}\frac{{\partial}^{2}}{{\partial}r^{2}}{+}\frac{1}{r}\frac{{\partial}}{{\partial}r}{+}\frac{1}{r^{2}}\frac{{\partial}^{2}}{{\partial}{\theta}^{2}}.\]

By introducing the dimensionless variables r^* = r/a, v^* = v/U, ψ^* = ψ/Ua, ^* = /a the Navier–Stokes equation and boundary conditions become

\[\mathrm{{\nabla}}_{r^{{\ast}}}^{4}{\psi}^{{\ast}}{=}{-}\frac{R}{r^{{\ast}}}\frac{{\partial}({\psi}^{{\ast}},\mathrm{{\nabla}}_{r^{{\ast}}}^{2}{\psi}^{{\ast}})}{{\partial}(r^{{\ast}},{\theta})},\]

\[{\psi}^{{\ast}}{=}r^{{\ast}}{\,}\mathrm{sin}{\theta},{\ }r^{{\ast}}{\rightarrow}{\infty}.\]

As before, the above set of equations are solved using the method of matched asymptotic expansions, where R is assumed to be a small parameter. Near the sphere we have a Stokes expansion defined by

\[{\psi}^{{\ast}}(r^{{\ast}},{\theta}){=}f_{0}(R){\psi}_{0}(r^{{\ast}},{\theta}){+}f_{1}(R){\psi}_{1}(r^{{\ast}},{\theta}){+}O[f_{2}(R)],\]

where

\(f_{0}(R){\gg}R\)

and

\(f_{n{+}1}(R)/f_{n}(R){\rightarrow}0\)

as

\(R{\rightarrow}{\infty},\)

and far from the sphere we have an Oseen expansion defined by

\[\mathrm{{\Psi}}^{{\ast}}({\rho}^{{\ast}},{\theta}){=}\mathrm{{\Psi}}_{0}({\rho}^{{\ast}},{\theta}){+}F_{1}(R){\Psi}_{1}({\rho}^{{\ast}},{\theta}){+}O[F_{2}(R)],\]

where

\(F_{n{+}1}(R)/F_{n}(R){\rightarrow}0\)

as

\(R{\rightarrow}{\infty},\)

and Ψ^* and ρ^* are suitably defined Oseen variables such that the right-hand side of the Navier–Stokes equation is the same order as the left-hand side. The Oseen variables are found to be ρ^* = Rr^*, Ψ^* = Rψ^*.

Hence the Stokes expansion satisfies

\[\mathrm{{\nabla}}_{r^{{\ast}}}^{4}{\psi}^{{\ast}}{=}{-}\frac{R}{r^{{\ast}}}\frac{{\partial}({\psi}^{{\ast}},\mathrm{{\nabla}}_{r^{{\ast}}}^{2}{\psi}^{{\ast}})}{{\partial}(r^{{\ast}},{\theta})},\]

while the Oseen expansion satisfies

\begin{eqnarray*}&&\mathrm{{\nabla}}_{{\rho}^{{\ast}}}^{4}\mathrm{{\Psi}}^{{\ast}}{=}{-}\frac{1}{{\rho}^{{\ast}}}\frac{{\partial}(\mathrm{{\Psi}}^{{\ast}},\mathrm{{\nabla}}_{{\rho}^{{\ast}}}^{2}\mathrm{{\Psi}}^{{\ast}})}{{\partial}({\rho}^{{\ast}},{\theta})},\\&&\mathrm{{\Psi}}^{{\ast}}{=}{\rho}^{{\ast}}{\,}\mathrm{sin}{\theta},{\ }r^{{\ast}}{\rightarrow}{\infty}.\end{eqnarray*}

4.1 Leading-order expansions

The leading-order solution for the Stokes expansion satisfies the biharmonic equation

\(\mathrm{{\nabla}}_{r^{{\ast}}}^{4}{\psi}_{0}{=}0,\)

which has solution

\[{\psi}_{0}(r^{{\ast}},{\theta}){=}[C_{1}r^{{\ast}{-}1}{+}C_{2}r^{{\ast}}{+}C_{3}r^{{\ast}3}{+}C_{4}r^{{\ast}}{\,}\mathrm{ln}{\,}r^{{\ast}}]{\,}\mathrm{sin}{\theta}.\]

Invoking the ‘principle of minimum singularity’ (see (6)) gives C₃ = 0, and the boundary conditions

imply

Hence

and

The leading-order solution for the Oseen expansion which satisfies the following equation and boundary condition:

\begin{eqnarray*}&&\mathrm{{\nabla}}_{{\rho}^{{\ast}}}^{4}\mathrm{{\Psi}}_{0}{=}{-}\frac{1}{{\rho}^{{\ast}}}\frac{{\partial}(\mathrm{{\Psi}}_{0},\mathrm{{\nabla}}_{{\rho}^{{\ast}}}^{2}\mathrm{{\Psi}}_{0})}{{\partial}({\rho}^{{\ast}},{\theta})},\\&&\mathrm{{\Psi}}_{0}{=}{\rho}^{{\ast}}{\,}\mathrm{sin}{\theta},{\ }{\rho}^{{\ast}}{\rightarrow}{\infty},\end{eqnarray*}

is found to be

\[\mathrm{{\Psi}}_{0}({\rho}^{{\ast}},{\theta}){=}{\rho}^{{\ast}}{\,}\mathrm{sin}{\theta}.\]

Expanding the leading-order Stokes expansion in terms of the Oseen variable ρ^* = Rr^* yields

The asymptotic matching principle (see (6)) requires

or following the method of Kaplun (5), in general

\[f_{0}(R){=}[\mathrm{ln}(1/R){+}k]^{{-}1},\]

where

\(k{\ll}\mathrm{ln}(1/R)\)

is any constant. Hence the leading-order Stokes expansion is given by

which reduces to the solution of Proudman and Pearson (4) when ^* = 0.

4.2 First-order expansions

The first-order solution to the Stokes expansion satisfies the biharmonic equation

Expanding the leading-order Stokes expansion in terms of the Oseen variable ρ^* = Rr^* yields

\[f_{0}(R){\psi}_{0}{\sim}R^{{-}1}[1{+}f_{0}(R){\,}\mathrm{ln}{\rho}^{{\ast}}]{\rho}^{{\ast}}{\,}\mathrm{sin}{\theta},\]

and the full Oseen expansion for small ρ^* is given by

\[\mathrm{{\Psi}}^{{\ast}}{=}\mathrm{{\Psi}}_{0}{+}F_{1}(R)\mathrm{{\Psi}}_{1}{\sim}{\rho}^{{\ast}}{\,}\mathrm{sin}{\theta}{+}F_{1}(R)(\mathrm{{\Psi}}_{1C}{+}C_{1}{\rho}^{{\ast}}{\,}\mathrm{ln}{\rho}^{{\ast}}{\,}\mathrm{sin}{\theta}).\]

The asymptotic matching principle (see Van Dyke (6)) requires

\[F_{1}(R){=}f_{0}(R),{\,}C_{1}{=}1,{\,}\mathrm{{\Psi}}_{1C}{=}0,\]

thus the full Oseen expansion is given by

\begin{eqnarray*}&&\mathrm{{\Psi}}^{{\ast}}({\rho}^{{\ast}},{\theta}){=}{\rho}^{{\ast}}{\,}\mathrm{sin}{\theta}{-}f_{0}(R){{\sum}_{n{=}1}^{{\infty}}}\left\{2K_{1}\left(\frac{1}{2}{\rho}^{{\ast}}\right)I_{n}\left(\frac{1}{2}{\rho}^{{\ast}}\right)\right.\ \\&&\left.\ {+}K_{0}\left(\frac{1}{2}{\rho}^{{\ast}}\right)\left[I_{n{-}1}\left(\frac{1}{2}{\rho}^{{\ast}}\right){+}I_{n{+}1}\left(\frac{1}{2}{\rho}^{{\ast}}\right)\right]\right\}{\rho}^{{\ast}}n^{{-}1}{\,}\mathrm{sin}n{\theta},\end{eqnarray*}

which reduces to the solution of Proudman and Pearson (4) and Kaplun (5) when ^* = 0.

Expanding the leading-order Stokes expansion further in terms of the Oseen variable ρ^* = Rr^* yields

and the full Oseen expansion for small ρ^* is given by

\[\mathrm{{\Psi}}^{{\ast}}{\sim}\left\{1{+}f_{0}(R)\left[\mathrm{ln}\left(\frac{1}{4}{\rho}^{{\ast}}\right){+}{\gamma}{-}1\right]\right\}{\rho}^{{\ast}}{\,}\mathrm{sin}{\theta},\]

where γ is the Euler–Mascheroni constant. For these solutions to match perfectly, the asymptotic matching principle (see (6)) requires

so that

(4.2)

which reduces to the solution of Kaplun (5) when ^* = 0, and the full Stokes expansion is given by

(4.3)

4.3 Frictional drag

The frictional drag is calculated from the force that the fluid exerts on the nanocylinder per unit length beyond the force attributable to the ambient and hydrostatic pressure. This force is calculated from the tangential and normal components of stress T = – pI + 2μD integrated around the surface of the nanocylinder. The x component of the stress that the fluid exerts on the nanocylinder is T_rr cos θ – T_rθ sin θ, so that the x component of the force is

\[F_{x}{=}{{\int}_{0}^{2{\pi}}}(T_{rr}{\,}\mathrm{cos}{\theta}{-}T_{r{\theta}}{\,}\mathrm{sin}{\theta}){\vert}_{r{=}a}ad{\theta},\]

where from (26)

\begin{eqnarray*}&&T_{rr}{=}{-}p{+}2{\mu}D_{rr}{=}{-}p{+}2{\mu}\frac{{\partial}v_{r}}{{\partial}r},\\&&T_{r{\theta}}{=}2{\mu}D_{r{\theta}}{=}{\mu}\left[r\frac{{\partial}}{{\partial}r}\left(\frac{v_{{\theta}}}{r}\right){+}\frac{1}{r}\frac{{\partial}v_{r}}{{\partial}{\theta}}\right].\end{eqnarray*}

The mean pressure near the nanocylinder is calculated from the Navier–Stokes equation consistent with the approximation employed for the Stokes expansion of the stream function. Recall that

\({\psi}^{{\ast}}{=}f_{0}\left(R\right){\psi}_{0},\)

where ψ₀ satisfies the biharmonic equation

\(\mathrm{{\nabla}}_{r^{{\ast}}}^{4}{\psi}_{0}{=}0.\)

The Navier–Stokes equations consistent with the approximation for ψ₀, that is, neglecting the convective inertial term, is given by Slattery (26)r component:

\[\frac{{\partial}p}{{\partial}r}{=}\frac{{\mu}}{r^{2}}\left[r\frac{{\partial}}{{\partial}r}\left(r\frac{{\partial}v_{r}}{{\partial}r}\right){-}v_{r}{+}\frac{{\partial}^{2}v_{r}}{{\partial}{\theta}^{2}}{-}2\frac{{\partial}v_{{\theta}}}{{\partial}{\theta}}\right],\]

θ component:

\[\frac{{\partial}p}{{\partial}{\theta}}{=}\frac{{\mu}}{r}\left[r\frac{{\partial}}{{\partial}r}\left(r\frac{{\partial}v_{{\theta}}}{{\partial}r}\right){-}v_{{\theta}}{+}\frac{{\partial}^{2}v_{{\theta}}}{{\partial}{\theta}^{2}}{+}2\frac{{\partial}v_{r}}{{\partial}{\theta}}\right],\]

while the z component yields ∂p/∂z = 0. By calculating the r and θ components of velocity from the definition of the stream function, returning to dimensional variables (except for ^*, which we keep dimensionless), substituting into the Navier–Stokes equations and integrating yields the following expression for p evaluated at r = a:

\[p{\vert}_{r{=}a}{=}{-}2{\mu}Ua^{{-}1}{\,}\mathrm{cos}{\theta}f_{0}(R).\]

Substituting this expression and the expressions for the r and θ components of velocity into the expression for F_x and performing the required integration yields

\[F_{x}{=}4{\pi}{\mu}Uf_{0}(R).\]

The drag coefficient c_D is defined as the force referenced by the dynamic head ρU²a, that is,

\[c_{D}{=}F_{x}/{\rho}U^{2}a{=}4{\pi}R^{{-}1}f_{0}(R),\]

(4.4)

which reduces to the solution of (5) to leading order when ^* = 0, where

\(f_{0}(R)\)

is defined by (4.2).

5 Numerical results and conclusions

The expressions derived for the flow and the frictional drag on a nanosphere and nanocylinder, (3.1), (3.2), (4.3) and (4.4), may be discussed on the basis of the two extreme limits ^* = 0 and ^*→∞. The relevant results for finite ^* will then fall between these two bounding curves.

The flow past a nanosphere and nanocylinder are shown in Figs 1 and 3 for the case ψ^* = 0·5 and R = 0·5 in the limits of ^* = 0 and ^*→∞. These results show that the effect of slip has little effect on the streamlines of the flow, which move slightly closer to the nanosphere and nanocylinder for ^* > 0, indicating a reduced drag.

Fig. 1

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Flow past a nanosphere for ψ^* = 0·5 and R = 0·5

Fig. 2

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Frictional drag versus Reynolds number for a nanosphere

Fig. 3

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Flow past a nanocylinder for ψ^* = 0·5 and R = 0·5

The frictional drag on a nanosphere and nanocylinder are shown in Figs 2 and 4 as a function of the Reynolds number R. As one would intuitively expect, the frictional drag is reduced for values of ^* > 0, since the fluid layers close to the nanosphere and nanocylinder would move across the corresponding surface with less resistance when slip is allowed. An important point to note is that, for the nanocylinder, the results for ^*→∞ for the frictional drag are in complete accord with the low-Reynolds-number experimental results of Tritton (30), and this agreement is much better than for ^* = 0 and for the well-known higher approximation for the frictional drag obtained by Kaplun (5), which is given by

\[c_{D}{=}4{\pi}R^{{-}1}[f_{0}(R){-}0{\cdot}87f_{0}^{3}(R)],\]

where

\(f_{0}(R)\)

is given by (4.2) with ^* = 0, which for comparison is also shown in Fig. 4. The excellent agreement with the experimental results of Tritton (30) in some part may be attributed to the micro diameter fibres used in the experiments, confirming the applicability of the approach adopted here at least at the micro scale.

Fig. 4

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Frictional drag versus Reynolds number for a nanocylinder with experimental points

In summary, the Newtonian flow past a nanosphere and a nanocylinder is modelled using matched asymptotic expansions. The classical no-slip boundary condition is replaced by the Navier boundary condition, which states that the relative velocity at a boundary is proportional to the tangential viscous stress at the boundary, and the constant of proportionality is the slip length . Results for the streamlines and frictional drag are presented for = 0 corresponding to classical theory, and →∞ corresponding to complete slip. Numerical calculations reveal that the Navier boundary condition with a constant slip length has only a small effect on the response of both systems, implying that it only makes a small improvement over the classical no-slip boundary conditions for well-posed problems, that is, for problems that do not predict unrealistic divergence of any system response. The real advantages of using the Navier boundary condition as a replacement for the no-slip boundary condition arise when the former overcomes inherent failures of the latter, or when the slip length has a functional dependence on the shear rate.

This work is funded by the Discovery Project scheme of the Australian Research Council, and the authors gratefully acknowledge this support.

References