Fluid slip at walls with mesoscopic surface roughness

Physica B 165&166 (1990) 555-556 North-Holland

FLUID

SLIP

AT WALLS

P. Panzer+, t Institut fiir Theoretische 1: Walther-MeiBner-Institut

WITH MESOSCOPIC

D. Einzel”

SURFACE

ROUGHNESS

and M. Liu+

Physik, Univ. Hannover, D-3000 Hannover, FRG fiir Tieftemperaturforschung, D-8046 Garching, FRG

The traditional picture of fluid slip at boundaries as a manifestation of a mean free path effect has to be revised whenever the boundary acquires a finite curvature on mesoscopic length scales. Our results explain discrepancies between experimentally ob-~ served fluid flow and conventional slip theory.

1. INTRODUCTION In this Note we consider the problem of hydrodynamic flow of fluids past solid boundaries. Its purpose is to go beyond the traditional description of fluid slip at the container walls as a manifestation of mean free path effects. This is motivated by the results of two torsional oscillator experiments performed in normal fluid 3He at low temperatures. The first of these 111 concludes from Poiseuille flux data, that the experimentally determined slip length C (defined as the fictive distance at which the macroscopic fluid velocity extrapolates to zero behind the wall) is (at low pressure) smaller than the for diffuse scattering of theoretical result quasiparticles. The second finds from surface impedance data [Zl, that in the presence of a ‘He coverage on the cell walls (i. e. enhanced specularity) the ratio of imaginary to real part of the surface impedance Y/X z 0.6 in the hydrodynamic limit, where one expects 1. An assumption, common to almost all attempts to calculate slip lengths from a Maxwellor Landau-Boltzmann equation, is, that in the case of diffuse scattering the fluid particles are reemitted with an azimuthally isotropic momensurface. However, tum distribution from a flat the surfaces of measuring cells are rough on mesoscopic length scales 121 (typically A lpm). A finite boundary curvature gives rise to additional quasiparticle backscattering events, which enhance the transfer of transverse momentum to the surface with respect to the case of purely diffuse scattering off a flat surface. As a con sequence, the experimentally determined effective slip length c is reduced below its value i0 for diffusely scattering flat surfaces. In what follows, we discuss this new effect on the level of phenomenological hydrodynamic theory using a simple model for mesoscopic surface roughness. 2. MODEL FOR THE SURFACE ROUGHNESS We consider now 2-D fluid flow between parallel plates of distance L. The surface roughness is mimiced by the assumption of a symmetric periodic variation of the surface amplitude h(x) in

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the x-direction periodicity length h(x)

number

with wave A = 2n/q:

-- E hb cosi@qx P,’

- in/21

;

i

U,i

q

and

ili

Here h,‘, denote the Fourier components of the amplitude variation. ‘The fluid is then considered to be entrained between the upper wall at y”(x) = L t h(x) and a lower wall at y‘(x) = -h(x). of 2-D flow betwcerl A convenient description these boundaries is possible in terms of a ntrt-a ming function Y!(x,y), which is reinted to tile components v,(x,y) and vy(x,y) of the macros-topic velocity field in the ustinl way :is iX T: away, vti = -awax. The streaming function is obtained hy solving the linearized Navier-Stokes equations for v?, and vy, amended, by a generalized set of bouu dary conditions [31. They arc: c:hc~sensuch that (i) there is no velocity component. normal to the surface, or equivalently: q x,y”‘,Xi:

; %s,ytix))

= const

(2j

characterized by a and (ii) there is slip, the of independent) length co, (roughness tangential component of the velocity field:

Here t,, n, , i = x, y are the components of the surface tangent and normal vectors, respects tively, and the plus (minus) sign refers to the lower (upper) surface. The slip length i0 has been determined from microscopic theory [41 to be related to the transport mean free palh h with a0 a slip through co = [ae(l+s)/(l-s)lx coefficient for the case of diffuse quasiparticle scattering (SO). The specularity enhancement factor in the brackets accounts for the presence of a fraction s (0 < s r; 1) of specularly reflected fluid particles. In the case of Poiseuille flow, the streaming function Y is related to the total flux Q be-tween the two plates:

@ 1990 - Elsevier Science Publishers B.V. (North-Holland)

P. Panzer, D. Einzel, M. Liu

556

Q = P[WX,Y"(X)) s - g

(E]

- Y(X,YL(X))I [l

+ 6

F,

(4)

where p denotes the fluid density, 7) the shear viscosity of the fluid, aP/ax the external linear pressure gradient parallel to the plates. Eq. (4) defines an effective macroscopic slip length ( which has to be determined as a function of cO. Clearly, if h(x) = 0, then we recover the result of conventional slip theory, C ? ? Co. It can be shown [4,51 that the same form of the macroscopic slip length C (to be specified below) also goes into the transverse surface impedance Zl which characterizes the dynamics of the plates, oscillating at a frequency w in the limit where the viscous penetration depth 6 = [2~/p~l’/~ d: L. The hydrodynamic result for Zl is found to be of the well known form:

K 2

1

0

-1 however, with the macroscopic slip length C replacing the microscopic one Co. We determine the slip length < by solving both the stationary and time dependent NavierStokes equation with the boundary conditions (2) and (3) and a wall profile of the form (1) by Fourier analysis. The details of this calculation will be published elsewhere [51.

3. DISCUSSION

OF RESULTS

!<.,, < is a mor10tOI:0lls l”~~r~ctl0rl SIIKW! hly L~r~nr~tx:ts !hr+ limiting values (‘$ < < I <;. ‘This behavior is illustrated c’ase of a wsint: profile in Fig. i fill, the special the exact M’e have plotted h(s) = h,, cos(qxi. numerical result for tho dimensionless quantity K q(’ in this case ~.~sc;<.,,. The parameter VdUeS range from weak to strong variation of ? ? qh,, the periodic: roughness. We believe that the almost temperature indem pendent ratios Y/X = 0.6, T/X, = 0.7 and Y/Y,, r 0.45 (where the index p refers to pure 3He) of real and iniagirlcu-y parts of the surface impe dance observed in the presence of a 4He cover At

3rbitr;xy

<.I I- <(,, ;t irir:h

Fig.

1: Effective slip length C vs. cosine surface profile

to for a

age by Ritchie, Saunders and Brewer [21 are a consequence of the finiteness of the slip length in the specular limit. The 4He film serves as to introduce specularity (s + 1). The measured slip length in this specular limit does not diverge like (l-s)-‘, but is dominated by CE, which, assuming q-’ = 1.8 pm and h, = lprn is of the order of 6pm. Inserting this into the expression (5) for the impedance, we obtain at 40 mK Y/X = 0.56, X/X, ? ? 0.86 and Y/Y, = 0.5 in fair agreement with the experimental observation. In summary, we have demonstrated how a finite boundary curvature alters the flow properties (Poiseuille flow and shear impedance) and in particular the slip length of a fluid. For increasing values of the microscopic slip length co the macroscopic slip length C varies from negative values in the stick limit to a finite asymptotic value in the specular limit, which is characterized by the details of the wall shape function. We conjecture that this effect. is the cause for discrepancies between surface impe-dance experiments performed with specular walls and conventional slip theory. REFERENCES [ll D. Einzel and J. M. Parpia, Phys. Rev. Lett. 58, 1937 (1987) [Zl D. A. Ritchie, J. Saunders and D. Brewer, Phys. Rev. Lett. 59, 465 (1987) E31 M. Grabinski and M. Liu Phys. Rev. Lett 58, 800 (1987) [41 H. Hojgaard Jensen, H. Smith, P. Wijlfle, K. Nagai and T. Maak Bisgaard, J. Low Temp. Phys. 41, 473 (1980) [51 P. Panzer, M. Liu and D. Einzel, to be published