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Squeeze flow of a dry foam
J. Non-Newtonian Fluid Mech. 112 (2003) 115–128
Squeeze flow of a dry foam J.D. Sherwood∗ Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, UK Received 13 December 2002; received in revised form 5 March 2003
Abstract Squeeze flow of a dry compressible foam between two circular disks is studied by means of a lubrication analysis. Two simple cases are considered. If the foam behaves as a compressible Newtonian fluid, the approach of the plates can be accommodated by compression of the foam as well as by extrusion. Pressure is governed by a non-linear diffusion-like equation, and at high Peclet number (Pe) the effect of an outer boundary at which pressure remains at ambient does not diffuse far into the test sample. At low Pe pressures remain low and results differ little from those for an incompressible fluid, until the gap between the plates becomes very small. The second analysis concerns a perfectly plastic material, with a yield stress which is inversely proportional to bubble size and which therefore increases at high pressure. An analytic solution is found for this case. Close to the axis of the disks, where pressures are high, the yield stress is large and the foam is compressed rather than extruded. Close to the edge of the plates the foam is extruded. As the gap between the discs is reduced, the central region of stationary foam becomes larger and eventually the test measures elastic compression of the portion of the test sample remaining between the plates. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Foam; Squeeze flow; Yield stress; Compressible fluid
1. Introduction Foams are often considered to have a yield stress [1,2] and rheological characterization of such fluids is impeded by slip at rheometer walls [3,4]. A test which has proved useful for fluids with a yield stress is squeeze flow, in which the fluid sample is squeezed between two circular plates. Slip is not necessarily eliminated in such a test [5–11], but intimate contact between the sample and the plates is ensured by the motion of the plates towards each other. Squeeze flow tests are typically performed either by measuring the velocity of the plates towards each other when a constant force is applied [10–12] or by measuring the force required to push the plates together at a constant velocity [6–8,13]. We shall consider the (more usual) controlled velocity test. Interpretation of the measured force is not straightforward, since shear rates vary throughout the sample. The total force is related only to an average pressure between the plates, ∗
Tel.: +44-1223-325363; fax: +44-1223-315486. E-mail address: [email protected] (J.D. Sherwood). 0377-0257/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-0257(03)00065-X
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and gives no indication of the distribution of pressure over the surface of the plates. Interpretation schemes usually assume the test material to be incompressible. Here we consider the opposite limit in which the test material is a highly compressible foam. We shall first (in Section 2.1) consider how the density of the foam varies with pressure, and then (in Section 2.2) discuss the equation of continuity. Further analysis is based upon lubrication theory and depends upon the constitutive relation that describes the foam rheology. In Section 3, we consider a simple Newtonian rheology in order to highlight the effect of compressibility. In Section 4, we consider a more realistic perfectly plastic material with a negligibly small viscosity after yielding has occurred. 2. Governing equations 2.1. Gas and liquid volume fractions We suppose that at pressure p0 the foam contains a volume fraction φ0 ≈ 1 of perfect gas, of negligible density, and a volume 1 − φ0 1 of incompressible liquid at density ρl . Gas with volume φ0 at pressure p0 occupies a volume φ0 p0 /p at pressure p, at which pressure the volume fraction of gas in the foam is φ=
φ0 p0 φ0 p0 /p = 1 − φ0 + φ0 p0 /p p(1 − φ0 ) + φ0 p0
(1)
and the density of the foam is ρf =
(1 − φ0 )ρl . 1 − φ0 + φ0 p0 /p
(2)
If the gas bubbles within the foam have typical size a, the radius of curvature rc of the Plateau borders scales as rc ∼ a(1 − φ)1/2 when φ → 1 [14]. The liquid pressure within the Plateau borders will be smaller than the gas pressure by an amount of order γ/rc , where γ is the interfacial tension between liquid and gas. However, the liquid phase occupies only a fraction 1 − φ of the total volume and its contribution to the total pressure is smaller than that of the gas by a factor (1 − φ)1/2 [14,15]. We therefore neglect the effect of interfacial tension, except in so far as this gives the foam a yield stress in shear, as discussed in Section 4. We shall assume that squeezing is isothermal, rather than adiabatic. Squeeze tests are usually slow, and as the gap between the plates becomes smaller, so the timescale required for diffusion of heat across the gap is reduced. At sufficiently high pressures the volume fraction of gas becomes small, as does its compressibility, and it is no longer appropriate to neglect the compressibility of the liquid. However, we shall restrict our attention to pressures p p0 φ0 /(1 − φ0 ) such that the liquid volume fraction remains small. We may then approximate the density (2) by ρf ≈ ρ0
p , p0
p φ0 , p0 1 − φ0
(3)
where ρ0 = (1 − φ0 )ρl is the density of the foam at pressure p0 . This convenient approximation fails when the gap width becomes sufficiently small and pressures become high. The above results are valid only if there is no motion of gas relative to the liquid. Such relative motion between the various components of a multiphase fluid can sometimes occur in a squeeze test [16–19].
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Fig. 1. Squeeze flow of foam between two disks of radius R with separation h and relative velocity dh/dt = −U.
However, if the foam remains sufficiently dry, the foam permeability to movement of liquid will be low, and relative motion can be neglected. 2.2. Equation of continuity The foam is squeezed between two plates of radius R (see Fig. 1), separated by a gap h which decreases with time t as dh = −U. (4) dt We proceed by means of a standard lubrication analysis. Let Q(r, t) be the volumetric flow rate (per unit length in the circumferential direction) at radius r. Then by continuity ∂ r 2πr ρf (r , t)h dr = −2πrρf (r, t)Q(r, t). (5) ∂t 0 The volumetric flow rate Q(r, t) depends upon the local pressure gradient ∂p/∂r, in a way which depends upon the rheology of the foam. We now restrict our attention to two simple rheologies, namely (i) a Newtonian fluid and (ii) a Bingham fluid with a very low viscosity after yielding has taken place, i.e. a perfectly plastic material. Lubrication analyses of Bingham fluids in squeeze flow predict that there is a region of plug flow around the center plane mid-way between the two plates, but that this plug deforms in extension. This inconsistency was pointed out by Lipscomb and Denn [20], and has been further studied (and resolved) both analytically [21] and in numerical work [22]. In Section 4, when considering a perfectly plastic material, we shall allow slip to occur at the walls [8] and the pressure is then given by a simple force balance similar to that used in [19]. 3. A Newtonian fluid We suppose that the foam behaves as a Newtonian fluid with a viscosity µ independent of the pressure (and hence of the density). This is unlikely to be true for a foam, but enables us to concentrate our attention on the effects of compressibility. The volumetric flow rate Q in a channel of width h between two parallel plates depends upon the pressure gradient ∂p/∂r. We assume a no-slip boundary condition at the plates, so that h3 ∂p Q=− . (6) 12µ ∂r
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Hence, the equation of continuity becomes rρf h3 ∂p ∂ r r ρf h dr = . ∂t 0 12µ ∂r Taking a partial derivative with respect to r gives ∂p h3 ∂ ∂ ∂ rρf = (rρf h) = hr ρf − Urρf . 12µ ∂r ∂r ∂t ∂t
(7)
(8)
If the foam is incompressible, the first term on the right-hand side of (8) vanishes and we obtain the standard result for an incompressible Newtonian fluid: p = p0 + p1 (r, t),
(9)
where p0 is the ambient pressure at the outer periphery r = R of the plates and p1 =
3µU 2 (R − r 2 ). h3
(10)
3.1. Small changes in pressure If changes in pressure are sufficiently small, the foam density ρf = ρ0 p/p0 (3) varies little. If we take the Newtonian pressure (9) as a first approximation to the pressure field, the foam density is predicted to be ρ0 3µU 2 2 ρf = p0 + (R − r ) + · · · . (11) p0 h3 We look for a correction p2 to the Newtonian pressure (9) such that p = p0 + p1 + p2 + · · · . The equation of continuity (8) gives an equation for p2 : h3 ∂ ∂p2 ∂(rp1 h) ∂p1 rp0 , + rp1 = 12µ ∂r ∂t ∂r ∂r
(12)
(13)
so that −2 p0 h3 ∂ ∂p2 h3 3µU ∂ 2 2 ∂(h ) 2 2 ∂p1 r = 3µUr(R − r ) − r(R − r ) 12µ ∂r ∂r ∂t 12µ h3 ∂r ∂r 2 2 6µU 3µU ∂ = (rR2 − r 3 ) 3 + r2 R2 − r 4 . h 2h3 ∂r We integrate this once to obtain 2 2 r R r4 6µU 2 3µU 2 2 2 p0 h3 r ∂p2 r R − r4 . = + − 3 3 12µ ∂r 2 4 h 2h
(14)
(15)
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A second integration gives p0 h3 p2 3µU 2 =− (2R2 − r 2 )(R2 − r 2 ), 3 12µ 4h
(16)
where we have chosen the constants of integration to ensure that p2 is well-behaved at r = 0, with p2 = 0 at the outer periphery r = R. The pressure correction p2 is negative in 0 < r < R. The compressibility of the test material has reduced the radial velocities. We see that p2 p1 µUR2 =O , (17) =O 3 p1 h p0 p0 so that the expansion (12) breaks down when h becomes sufficiently small. The ambient pressure p0 acts on the external surface of the plates, so that the additional force required to push the plates together is R 3πµUR4 15πµ2 U 2 R6 F = 2π r(p1 + p2 + · · · ) dr = − + ··· . (18) 2h3 2p0 h6 0 3.2. Higher pressures, p0 < p p0 φ0 /(1 − φ0 ) At higher pressures we expect the effect of compressibility to become more important. The reduction in volume between the plates can be accommodated by compression of the foam, rather than by radial extrusion. Pressures need only increase as h−1 , rather than as h−3 for an incompressible fluid. There will thus be solutions in which pressure gradients are small away from the edge of the plates where the pressure must drop rapidly to p0 . The evolution of the pressure within the foam is related to diffusion of pressure in a compressible fluid between two plates, with the added complication that h is changing. Only if the test is sufficiently slow does the fixed pressure p = p0 at the boundary r = R have time to affect the pressure in the interior of the foam. Further progress can be made numerically. We scale pressures by p0 , the gap width by h0 , lengths in the radial direction by R, and time by h0 /U. Using the approximate density (3), the non-dimensional form of the governing equation (8) is ˆ ˆh3 ∂ rˆ pˆ ∂pˆ = Pe ∂(ˆr pˆ h) , (19) ∂ˆr ∂ˆr ∂ˆt where Pe =
12µR2 U p0 h30
(20)
and where non-dimensional quantities have been denoted by a caret. In the absence of motion of the plates, the evolution of small changes in pressure between the plates is controlled by a diffusion equation with diffusivity D = ph2 /12µ. Convective radial velocities are of order UR/ h, so that the Peclet number Pe (20) represents a ratio of convective fluxes of pressure to diffusive flux. The governing equation (19) was solved numerically for hˆ pˆ by the method of lines. In the absence of any diffusion of pressure, we expect hˆ pˆ = 1 everywhere, apart from at the edge rˆ = 1 of the plate where pˆ = 1 so that pˆ hˆ = hˆ = 1 − ˆt .
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Figs. 2–4 show scaled pressures hˆ pˆ as a function of radial position. Fig. 2 corresponds to Pe = 104 , and the effect of the ambient pressure at the edge rˆ = 1 has diffused only a short way into the foam. When Pe = 100 (Fig. 3), pressure diffuses further into the foam, and by Pe = 1 (Fig. 4) the effect of the boundary is felt all the way to the axis rˆ = 0.
Fig. 2. hˆ pˆ as a function of non-dimensional radial position rˆ , for Pe = 104 : (a) hˆ = 0.8, (b) hˆ = 0.6, (c) hˆ = 0.4, (d) hˆ = 0.2, (e) hˆ = 0.02.
Fig. 3. hˆ pˆ as a function of non-dimensional radial position rˆ , for Pe = 100: (a) hˆ = 0.8, (b) hˆ = 0.6, (c) hˆ = 0.4, (d) hˆ = 0.2, (e) hˆ = 0.02.
Fig. 4. hˆ pˆ as a function of non-dimensional radial position rˆ , for Pe = 1: (a) hˆ = 0.8, (b) hˆ = 0.6, (c) hˆ = 0.4, (d) hˆ = 0.2, (e) hˆ = 0.02.
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The excess force required to push the plates together is 1 F = 2πR2 p0 (pˆ − 1)ˆr dˆr.
121
(21)
0
In the limit of very large Peclet numbers we expect pˆ = hˆ −1 and 1 F 1 Fˆ = = 2 (pˆ − 1)ˆr dˆr ∼ − 1, Pe 1, 2 πR p0 hˆ 0 so that hˆ Fˆ + hˆ = 1,
Pe 1.
(22)
(23)
Results for hˆ Fˆ + hˆ are shown in Figs. 5 and 6. We see that (23) breaks down for Pe < 100. In the limit of very small Peclet numbers, we expect the excess force to be given by (18), i.e. Fˆ =
Pe 5Pe2 − + ··· , 8hˆ 3 96hˆ 6
ˆ (a) Pe = 104 , (b) Pe = 103 . Fig. 5. hˆ Fˆ + hˆ against non-dimensional gap width h:
ˆ (a) Pe = 100, (b) Pe = 10, (c) Pe = 1, (d) Pe = 0.1. Fig. 6. hˆ Fˆ + hˆ against non-dimensional gap width h:
(24)
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ˆ (a) Pe = 10−4 , (b) Pe = 10−3 , (c) Pe = 10−2 , (d) Pe = 10−1 , (e) Pe = 1. Fig. 7. 8Fˆ hˆ 3 /Pe against non-dimensional gap width h: Broken lines show the asymptotic prediction (25) for Pe 1: (f) Pe = 10−4 , (g) Pe = 10−3 , (h) Pe = 10−2 .
so that it is useful to plot 8Fˆ hˆ 3 5Pe =1− + ··· . Pe 12hˆ 3
(25)
In this limit, the pressure profile is given by (10),(12),(16) and is pˆ = 1 + Pe
(1 − rˆ 2 ) (2 − 3ˆr 2 + rˆ 4 ) − Pe2 + ··· . 4hˆ 3 4hˆ 6
(26)
Numerical solutions for 8Fˆ hˆ 3 /Pe are shown in Fig. 7, together with the perturbation expansion (25). When Pe > 0.01 diffusion is slow and the incompressible Newtonian pressure profile is not established before the effects of compressibility are felt. Expansions (24) and (26) both break down when hˆ ∼ Pe1/3 . When h is sufficiently small the left-hand side of (19) becomes small (away from regions near the periphery of the disc where pressures re-adjust over a small radial distance to match the boundary condition). Hence, towards the end of the test the ˆ derivative ∂(pˆ h)/∂t on the right-hand side of (19) is small, and we have compression of the foam, rather than extrusion, even when Pe 1. Fig. 8 replots the numerical results of Fig. 7, together with results for higher Peclet numbers. When h is small, the differences between the various curves in Fig. 8 relate to the different amounts of foam remaining between the plates. When the test is stopped and U = 0, in the absence of any yield stress the foam will expand until the pressure p = p0 everywhere. If no foam emerges after the test, then either effects of compressibility were small, so that the test may be interpreted by an incompressible analysis, or yield stresses dominate any viscous stresses and prevent post-test expansion. In the next section we consider a foam with a yield stress which is sufficiently high that viscous stresses due to flow after yield may be neglected. Note that once we have adopted the linear approximation (3) between pressure and density, the analysis becomes identical to that for an isothermal perfect gas. Perfect gases are usually considered to be Newtonian, with viscosity independent of pressure [23]. The results presented above may therefore have been obtained previously, though it is unlikely that anyone would use squeeze-flow rheometry to characterize a low-viscosity perfect gas.
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ˆ (a) Pe = 10−4 , (b) Pe = 10−3 , (c) Pe = 10−2 , (d) Pe = 10−1 , (e) Pe = 1, Fig. 8. 8Fˆ hˆ 3 /Pe against non-dimensional gap width h: (f) Pe = 10, (g) Pe = 100, (h) Pe = 1000.
4. A yield stress fluid There is evidence that foams have a yield stress [1,2]. If the typical bubble size is a and interfacial tension between liquid and gas is γ, then on dimensional grounds we expect the yield stress τy to be of order γ τy ∼ . (27) a The mass of gas in each bubble remains constant, so that the volume of each bubble is proportional to p−1 and hence a typical bubble size a ∼ p−1/3 . We assume that the yield stress is 1/3 p , (28) τy = τ0 p0 where τ0 is the yield stress at ambient pressure p0 . As the pressure increases, so the gas/liquid interfacial area (per unit volume of foam) increases, and deformation of the foam becomes more difficult. This scaling (28) will go wrong once the volume fraction of the bubbles is sufficiently small that the bubbles no longer touch. If we consider bubbles on a regular cubic array, the bubbles touch when the volume fraction is φ = π/6 and this occurs when, by (1), p φ0 6 = −1 , (29) p0 1 − φ0 π which is not a restriction since we have already assumed that pressures are sufficiently small that (3) holds. However, experiments by Princen [2] suggest that even when φ > π/6 the yield stress is a strong function of liquid content. We must make the more restrictive assumption that the foam remains dry (i.e. that the liquid volume fraction 1 − φ 1), and even in this limit, the scaling (28) must be treated with caution. Note that bubbles will not remain isotropic during the extensional deformation of the foam. We shall find, as in Section 4, that p ∼ h−1 when h becomes sufficiently small. The lateral dimension of an
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isotropic bubble would decrease only as h−1/3 , making it impossible for the bubble to remain isotropic. We shall ignore any anisotropy that develops. Coarsening will occur in a real foam: bubbles grow and shrink due to diffusion of gas through liquid films, and bubbles may coalesce when films break. We ignore such structural changes. If we assume that the foam behaves as a Bingham fluid, we must also specify the viscosity of the fluid after yielding. Here we shall simply assume that the squeezing is sufficiently slow that viscous stresses are small compared to yield stresses and may be neglected, as in a perfectly plastic material. This may well be true during most of the squeeze test, but is likely to fail when the gap h becomes very narrow and shear rates become large. We assume that when sliding occurs at the wall the wall shear stress τw is equal to the yield stress τy of the foam. Sherwood and Durban [8] considered wall shear stresses τw = mτy , with constant m in the range 0 ≤ m ≤ 1. We conclude from their analysis that if h/R m < 1 the results presented below are valid if we replace τy by τw . A simple stress balance gives the pressure gradient h
∂p = −2τy (r). ∂r
(30)
If the yield stress τy were constant the pressure would be p = p0 +
2τy (R − r) h
(31)
and the total excess force required to push the plates together would be 2πτy R3 F = 2π (p − p0 )r dr = . 3h However, if the yield stress varies with pressure (28), the stress balance becomes 1/3 ∂p p h = −2τy (r) = −2τ0 , ∂r p0
(32)
(33)
so that 2/3
p2/3 − p0 i.e.
p = p0
=
4τ0 1/3
3hp0
(R − r),
(34)
3/2
4τ0 (R − r) + 1 3hp0
.
(35)
The pressure (31) is linear in the distance R − r from the periphery of the plates when the yield stress is assumed constant. However, in a foam the higher pressure in the central region r < R reduces the typical bubble size and thereby increases the effect of interfacial tension. The yield stress increases, and in (35) the pressure p increases more rapidly than linearly. When the plates are pushed together, if the rise in pressure given by (35) is more than is required for simple compression of the material, there will be no flow. The pressure will increase merely because
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the foam is compressed. The rate of change of the pressure (35) predicted by the simple lubrication theory is 1/2 ∂p 4τ0 2τ0 U(R − r) = (R − r) + 1 , (36) ∂t 3hp0 h2 whereas the rate of increase assuming straightforward compaction comes from considering that ρf h would then be constant, so that 3/2 ∂p pU Up0 4τ0 = = (R − r) + 1 . (37) ∂t h h 3hp0 These expressions (36) and (37) are equal when 2τ0 (R − r) = p0 , 3h which occurs at 3hp0 r = r1 = R − 2τ0
(38)
(39)
when h = h1 (r1 ) =
2τ0 (R − r1 ) , 3p0
(40)
at which gap width the pressure at r1 is p1 = 33/2 p0 . The no-flow region first occurs on the axis r = 0 when h = 2τ0 R/3p0 ; prior to this all the foam is in motion, and r1 , as defined by (39), is negative. We set 3hp0 r2 = , (41) 2τ0 so that r1 + r2 = R.
(42)
Once slip at the wall has ceased, we assume that subsequent elastic deformation of the foam is uniaxial compression. The pressure within the region of no-flow is √ h1 (r) 2 3τ0 (R − r) p(r) = p1 = (43) h h and the pressure gradient in this region is √ 2 3τ0 ∂p =− , r < r1 . (44) ∂r h The local wall shear stress therefore has magnitude √ |τ| = 3τ0 . (45) The local yield stress in r < r1 is √ 1/3 1/3 √ 2 3τ0 (R − r) p τy = τ0 = τ0 ≥ 3τ0 = |τ|, p0 p0 h
(46)
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with |τ| = τy at r = r1 . The magnitude of the wall shear stress |τ| in r < r1 is less than the local yield stress τy , so there can be no flow in this inner region, as postulated. The total excess force required to push the plates together is R F = 2π (p(r) − p0 )r dr. (47) 0
If h > 2τ0 R/3p0 the foam flows even at r = 0 and the pressure is given by (35) everywhere. Hence, 3/2 R 4τ0 F = 2πp0 (R − r) + 1 r dr − πR2 p0 . (48) 3hp0 0 Now r(a1 − a2 r)3/2 dr =
2 2a1 (a − a2 r)7/2 − 2 (a1 − a2 r)5/2 2 1 7a2 5a2
and hence, setting a2 = 4τ0 /3hp0 = 2/r2 and a1 = 2(R/r2 ) + 1 we find
7/2 7R 2R 2πp0 r22 +1 − − 1 − πR2 p0 . F= 35 r2 r2 When h is very large 2πτ0 R3 F∼ , 3h
h
τ0 R , p0
(49)
(50)
(51)
which agrees with the result (32) for a fluid with a constant yield stress τ0 . When hp0 τ0 R pressures remain close to p0 and the yield stress (28) varies little. As in Section 3, we non-dimensionalize forces by πR2 p0 and lengths in the radial direction by R, so that (51) becomes 1 3h0 p0 Fˆ ∼ , rˆ2 = 1. (52) rˆ2 2τ0 R When 2τ0 R/3hp0 > 1 the foam flows in the region r > r1 and is static in r < r1 . The pressure is 3/2 4τ0 (53a) p = p0 (R − r) + 1 , r > r1 3hp0 √ p = 2τ0 (R − r) 3h−1 , r < r1 (53b) and hence the total force is √ 3/2 R r1 4 3πτ0 4τ0 F = r(R − r) dr + 2πp0 (R − r) + 1 r dr − πR2 p0 h 3hp0 0 r1 √
6 3πp0 r12 R r1 37/2 1 2R 2 2 1 5/2 = + 1 [1 − 3 ] . − − πR p0 + πp0 r2 − − r2 2 3 7 7 5 r2
(54)
Note that (50) and (54) agree when r1 = 0, as expected. In the limit h → 0 we see from (39) that r1 → R and the no-flow region extends over the whole plate. The force tends asymptotically to 2πτ0 R3 F∼ √ , 3h
(55)
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Fig. 9. Foam with a yield stress. Non-dimensional excess force Fˆ = F/πR2 p0 , plotted as Fˆ rˆ2 = 3Fh/2πR3 τ0 against the non-dimensional parameter rˆ2 = 3hp0 /2τ0 R, as predicted by (50) and (54).
i.e.
√ 3 Fˆ ∼ , rˆ2
rˆ2 1,
(56)
√ which is larger than the incompressible result (52) by a factor 3. Fig. 9 shows Fˆ rˆ2 = Fr2 /p0 πR3 , as given by (50) and (54) as a function of rˆ2 = r2 /R. We see that Fˆ rˆ2 varies smoothly between the limits (52) and (56). Given the approximation (3), the mass M of foam remaining between the plates is equal to R 2πhρ0 R hρ0 M = 2πh rρf dr = rp dr = (F + πR2 p0 ) (57) p p 0 0 0 0 M∼
2πR3 ρ0 τ0 √ 3p0
as h → 0.
(58)
Thus the final stages of the squeeze flow test investigate the compressive elastic properties of the mass M of foam which remains in the device. However, since M depends upon the yield stress of the foam, the test nevertheless leads to an estimate for the foam’s yield stress. If no material were to leave the gap between the plates, and if the original plate separation had been h0 , the pressure within the plates would be p0 h0 / h and the force F = πR2 p0 h0 / h would vary, like (51) and (55), as h−1 . It is therefore important to verify that the material behaves plastically, rather than elastically, by observing that material is indeed squeezed out from the gap between the plates. When the foam has a yield stress, stresses will not drop to zero when the plate velocity U = 0 at the end of the test, and foam will not be subsequently extruded. It should be possible to measure the mass of foam remaining at the end of the test, though an accurate determination of this mass may be difficult when h is small.
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References [1] A.M. Kraynik, Foam flows, Ann. Rev. Fluid Mech. 20 (1988) 325–357. [2] H.M. Princen, Rheology of foams and highly concentrated emulsions, J. Colloid Interface Sci. 105 (1985) 150–171. [3] D.M. Kalyon, P. Yaras, B. Aral, U. Yilmazer, Rheological behavior of a concentrated suspension: a solid rocket fuel simulant, J. Rheol. 37 (1993) 35–53. [4] H.A. Barnes, A review of slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure, J. Non-Newtonian Fluid Mech. 56 (1995) 221–251. [5] S. Chatraei, C.W. Macosko, H.H. Winter, Lubricating squeezing flow: a new biaxial extensional rheometer, J. Rheol. 25 (1981) 433–443. [6] J.D. Sherwood, G.H. Meeten, C.A. Farrow, N.J. Alderman, Squeeze-film rheometry of non-uniform mudcakes, J. Non-Newtonian Fluid Mech. 39 (1991) 311–334. [7] M.J. Adams, B. Edmondson, D.G. Caughey, R. Yahya, An experimental and theoretical study of the squeeze-film deformation and flow of elastoplastic fluids, J. Non-Newtonian Fluid Mech. 51 (1994) 61–78. [8] J.D. Sherwood, D. Durban, Squeeze flow of a power-law viscoplastic solid, J. Non-Newtonian Fluid Mech. 62 (1996) 35–54. [9] H.M. Laun, M. Rady, O. Hassager, Analytical solutions for squeeze flow with partial wall slip, J. Non-Newtonian Fluid Mech. 81 (1999) 1–15. [10] G.H. Meeten, Constant-force squeeze flow of soft solids, Rheol. Acta 41 (2002) 557–566. [11] G.H. Meeten, Squeeze flow of soft solids between rough surfaces, Rheol. Acta (accepted). [12] G.H. Meeten, Yield stress of structured fluids measured by squeeze flow, Rheol. Acta 39 (2000) 399–408. [13] G.H. Covey, B.R. Stanmore, Use of the parallel-plate plastometer for the characterization of viscous fluids with a yield stress, J. Non-Newtonian Fluid Mech. 8 (1981) 249–260. [14] H.M. Princen, Osmotic pressure of foams and highly concentrated emulsions. 1. Theoretical considerations, Langmuir 2 (1986) 519–524. [15] J.D. Sherwood, A model for static filtration of emulsions and foams, Chem. Eng. Sci. 48 (1993) 3355–3361. [16] N. Delhaye, A. Poitou, M. Chaouche, Squeeze flow of highly concentrated suspensions of spheres, J. Non-Newtonian Fluid Mech. 94 (2000) 67–74. [17] Y. Mélinge, C. Lanos, F.X. Mortreuil, M. Laquerbe, Heterogeneity and electrical properties of sheared mineral paste, in: D.M. Binding, et al. (Eds.), Proceedings of the 13th International Congress Rheology, vol. 3, Cambridge, 20–25 August 2000, British Society of Rheology, 2000, pp. 95–97. [18] A. Poitou, G. Racineux, A squeezing experiment showing binder migration in concentrated suspensions, J. Rheol. 45 (2001) 609–626. [19] J.D. Sherwood, Liquid–solid relative motion during squeeze flow of pastes, J. Non-Newtonian Fluid Mech. 104 (2002) 1–32. [20] G.G. Lipscomb, M.M. Denn, Flow of Bingham fluids in complex geometries, J. Non-Newtonian Fluid Mech. 14 (1984) 337–346. [21] C.J. Lawrence, G.M. Corfield, Non-viscometric flow of viscoplastic materials: squeeze flow, in: M.J. Adams, R.A. Mashelkar, J.R.A. Pearson, A.R. Rennie (Eds.), Dynamics of Complex fluids, Imperial College Press, The Royal Society, London, 1998. [22] D.N. Smyrnaios, J.A. Tsamopoulos, Squeeze flow of Bingham plastics, J. Non-Newtonian Fluid Mech. 100 (2001) 165–189. [23] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed. Pergamon Press, Oxford, 1987.
Squeeze flow of a dry foam J.D. Sherwood∗ Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, UK Received 13 December 2002; received in revised form 5 March 2003
Abstract Squeeze flow of a dry compressible foam between two circular disks is studied by means of a lubrication analysis. Two simple cases are considered. If the foam behaves as a compressible Newtonian fluid, the approach of the plates can be accommodated by compression of the foam as well as by extrusion. Pressure is governed by a non-linear diffusion-like equation, and at high Peclet number (Pe) the effect of an outer boundary at which pressure remains at ambient does not diffuse far into the test sample. At low Pe pressures remain low and results differ little from those for an incompressible fluid, until the gap between the plates becomes very small. The second analysis concerns a perfectly plastic material, with a yield stress which is inversely proportional to bubble size and which therefore increases at high pressure. An analytic solution is found for this case. Close to the axis of the disks, where pressures are high, the yield stress is large and the foam is compressed rather than extruded. Close to the edge of the plates the foam is extruded. As the gap between the discs is reduced, the central region of stationary foam becomes larger and eventually the test measures elastic compression of the portion of the test sample remaining between the plates. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Foam; Squeeze flow; Yield stress; Compressible fluid
1. Introduction Foams are often considered to have a yield stress [1,2] and rheological characterization of such fluids is impeded by slip at rheometer walls [3,4]. A test which has proved useful for fluids with a yield stress is squeeze flow, in which the fluid sample is squeezed between two circular plates. Slip is not necessarily eliminated in such a test [5–11], but intimate contact between the sample and the plates is ensured by the motion of the plates towards each other. Squeeze flow tests are typically performed either by measuring the velocity of the plates towards each other when a constant force is applied [10–12] or by measuring the force required to push the plates together at a constant velocity [6–8,13]. We shall consider the (more usual) controlled velocity test. Interpretation of the measured force is not straightforward, since shear rates vary throughout the sample. The total force is related only to an average pressure between the plates, ∗
Tel.: +44-1223-325363; fax: +44-1223-315486. E-mail address: [email protected] (J.D. Sherwood). 0377-0257/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0377-0257(03)00065-X
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and gives no indication of the distribution of pressure over the surface of the plates. Interpretation schemes usually assume the test material to be incompressible. Here we consider the opposite limit in which the test material is a highly compressible foam. We shall first (in Section 2.1) consider how the density of the foam varies with pressure, and then (in Section 2.2) discuss the equation of continuity. Further analysis is based upon lubrication theory and depends upon the constitutive relation that describes the foam rheology. In Section 3, we consider a simple Newtonian rheology in order to highlight the effect of compressibility. In Section 4, we consider a more realistic perfectly plastic material with a negligibly small viscosity after yielding has occurred. 2. Governing equations 2.1. Gas and liquid volume fractions We suppose that at pressure p0 the foam contains a volume fraction φ0 ≈ 1 of perfect gas, of negligible density, and a volume 1 − φ0 1 of incompressible liquid at density ρl . Gas with volume φ0 at pressure p0 occupies a volume φ0 p0 /p at pressure p, at which pressure the volume fraction of gas in the foam is φ=
φ0 p0 φ0 p0 /p = 1 − φ0 + φ0 p0 /p p(1 − φ0 ) + φ0 p0
(1)
and the density of the foam is ρf =
(1 − φ0 )ρl . 1 − φ0 + φ0 p0 /p
(2)
If the gas bubbles within the foam have typical size a, the radius of curvature rc of the Plateau borders scales as rc ∼ a(1 − φ)1/2 when φ → 1 [14]. The liquid pressure within the Plateau borders will be smaller than the gas pressure by an amount of order γ/rc , where γ is the interfacial tension between liquid and gas. However, the liquid phase occupies only a fraction 1 − φ of the total volume and its contribution to the total pressure is smaller than that of the gas by a factor (1 − φ)1/2 [14,15]. We therefore neglect the effect of interfacial tension, except in so far as this gives the foam a yield stress in shear, as discussed in Section 4. We shall assume that squeezing is isothermal, rather than adiabatic. Squeeze tests are usually slow, and as the gap between the plates becomes smaller, so the timescale required for diffusion of heat across the gap is reduced. At sufficiently high pressures the volume fraction of gas becomes small, as does its compressibility, and it is no longer appropriate to neglect the compressibility of the liquid. However, we shall restrict our attention to pressures p p0 φ0 /(1 − φ0 ) such that the liquid volume fraction remains small. We may then approximate the density (2) by ρf ≈ ρ0
p , p0
p φ0 , p0 1 − φ0
(3)
where ρ0 = (1 − φ0 )ρl is the density of the foam at pressure p0 . This convenient approximation fails when the gap width becomes sufficiently small and pressures become high. The above results are valid only if there is no motion of gas relative to the liquid. Such relative motion between the various components of a multiphase fluid can sometimes occur in a squeeze test [16–19].
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Fig. 1. Squeeze flow of foam between two disks of radius R with separation h and relative velocity dh/dt = −U.
However, if the foam remains sufficiently dry, the foam permeability to movement of liquid will be low, and relative motion can be neglected. 2.2. Equation of continuity The foam is squeezed between two plates of radius R (see Fig. 1), separated by a gap h which decreases with time t as dh = −U. (4) dt We proceed by means of a standard lubrication analysis. Let Q(r, t) be the volumetric flow rate (per unit length in the circumferential direction) at radius r. Then by continuity ∂ r 2πr ρf (r , t)h dr = −2πrρf (r, t)Q(r, t). (5) ∂t 0 The volumetric flow rate Q(r, t) depends upon the local pressure gradient ∂p/∂r, in a way which depends upon the rheology of the foam. We now restrict our attention to two simple rheologies, namely (i) a Newtonian fluid and (ii) a Bingham fluid with a very low viscosity after yielding has taken place, i.e. a perfectly plastic material. Lubrication analyses of Bingham fluids in squeeze flow predict that there is a region of plug flow around the center plane mid-way between the two plates, but that this plug deforms in extension. This inconsistency was pointed out by Lipscomb and Denn [20], and has been further studied (and resolved) both analytically [21] and in numerical work [22]. In Section 4, when considering a perfectly plastic material, we shall allow slip to occur at the walls [8] and the pressure is then given by a simple force balance similar to that used in [19]. 3. A Newtonian fluid We suppose that the foam behaves as a Newtonian fluid with a viscosity µ independent of the pressure (and hence of the density). This is unlikely to be true for a foam, but enables us to concentrate our attention on the effects of compressibility. The volumetric flow rate Q in a channel of width h between two parallel plates depends upon the pressure gradient ∂p/∂r. We assume a no-slip boundary condition at the plates, so that h3 ∂p Q=− . (6) 12µ ∂r
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Hence, the equation of continuity becomes rρf h3 ∂p ∂ r r ρf h dr = . ∂t 0 12µ ∂r Taking a partial derivative with respect to r gives ∂p h3 ∂ ∂ ∂ rρf = (rρf h) = hr ρf − Urρf . 12µ ∂r ∂r ∂t ∂t
(7)
(8)
If the foam is incompressible, the first term on the right-hand side of (8) vanishes and we obtain the standard result for an incompressible Newtonian fluid: p = p0 + p1 (r, t),
(9)
where p0 is the ambient pressure at the outer periphery r = R of the plates and p1 =
3µU 2 (R − r 2 ). h3
(10)
3.1. Small changes in pressure If changes in pressure are sufficiently small, the foam density ρf = ρ0 p/p0 (3) varies little. If we take the Newtonian pressure (9) as a first approximation to the pressure field, the foam density is predicted to be ρ0 3µU 2 2 ρf = p0 + (R − r ) + · · · . (11) p0 h3 We look for a correction p2 to the Newtonian pressure (9) such that p = p0 + p1 + p2 + · · · . The equation of continuity (8) gives an equation for p2 : h3 ∂ ∂p2 ∂(rp1 h) ∂p1 rp0 , + rp1 = 12µ ∂r ∂t ∂r ∂r
(12)
(13)
so that −2 p0 h3 ∂ ∂p2 h3 3µU ∂ 2 2 ∂(h ) 2 2 ∂p1 r = 3µUr(R − r ) − r(R − r ) 12µ ∂r ∂r ∂t 12µ h3 ∂r ∂r 2 2 6µU 3µU ∂ = (rR2 − r 3 ) 3 + r2 R2 − r 4 . h 2h3 ∂r We integrate this once to obtain 2 2 r R r4 6µU 2 3µU 2 2 2 p0 h3 r ∂p2 r R − r4 . = + − 3 3 12µ ∂r 2 4 h 2h
(14)
(15)
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A second integration gives p0 h3 p2 3µU 2 =− (2R2 − r 2 )(R2 − r 2 ), 3 12µ 4h
(16)
where we have chosen the constants of integration to ensure that p2 is well-behaved at r = 0, with p2 = 0 at the outer periphery r = R. The pressure correction p2 is negative in 0 < r < R. The compressibility of the test material has reduced the radial velocities. We see that p2 p1 µUR2 =O , (17) =O 3 p1 h p0 p0 so that the expansion (12) breaks down when h becomes sufficiently small. The ambient pressure p0 acts on the external surface of the plates, so that the additional force required to push the plates together is R 3πµUR4 15πµ2 U 2 R6 F = 2π r(p1 + p2 + · · · ) dr = − + ··· . (18) 2h3 2p0 h6 0 3.2. Higher pressures, p0 < p p0 φ0 /(1 − φ0 ) At higher pressures we expect the effect of compressibility to become more important. The reduction in volume between the plates can be accommodated by compression of the foam, rather than by radial extrusion. Pressures need only increase as h−1 , rather than as h−3 for an incompressible fluid. There will thus be solutions in which pressure gradients are small away from the edge of the plates where the pressure must drop rapidly to p0 . The evolution of the pressure within the foam is related to diffusion of pressure in a compressible fluid between two plates, with the added complication that h is changing. Only if the test is sufficiently slow does the fixed pressure p = p0 at the boundary r = R have time to affect the pressure in the interior of the foam. Further progress can be made numerically. We scale pressures by p0 , the gap width by h0 , lengths in the radial direction by R, and time by h0 /U. Using the approximate density (3), the non-dimensional form of the governing equation (8) is ˆ ˆh3 ∂ rˆ pˆ ∂pˆ = Pe ∂(ˆr pˆ h) , (19) ∂ˆr ∂ˆr ∂ˆt where Pe =
12µR2 U p0 h30
(20)
and where non-dimensional quantities have been denoted by a caret. In the absence of motion of the plates, the evolution of small changes in pressure between the plates is controlled by a diffusion equation with diffusivity D = ph2 /12µ. Convective radial velocities are of order UR/ h, so that the Peclet number Pe (20) represents a ratio of convective fluxes of pressure to diffusive flux. The governing equation (19) was solved numerically for hˆ pˆ by the method of lines. In the absence of any diffusion of pressure, we expect hˆ pˆ = 1 everywhere, apart from at the edge rˆ = 1 of the plate where pˆ = 1 so that pˆ hˆ = hˆ = 1 − ˆt .
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Figs. 2–4 show scaled pressures hˆ pˆ as a function of radial position. Fig. 2 corresponds to Pe = 104 , and the effect of the ambient pressure at the edge rˆ = 1 has diffused only a short way into the foam. When Pe = 100 (Fig. 3), pressure diffuses further into the foam, and by Pe = 1 (Fig. 4) the effect of the boundary is felt all the way to the axis rˆ = 0.
Fig. 2. hˆ pˆ as a function of non-dimensional radial position rˆ , for Pe = 104 : (a) hˆ = 0.8, (b) hˆ = 0.6, (c) hˆ = 0.4, (d) hˆ = 0.2, (e) hˆ = 0.02.
Fig. 3. hˆ pˆ as a function of non-dimensional radial position rˆ , for Pe = 100: (a) hˆ = 0.8, (b) hˆ = 0.6, (c) hˆ = 0.4, (d) hˆ = 0.2, (e) hˆ = 0.02.
Fig. 4. hˆ pˆ as a function of non-dimensional radial position rˆ , for Pe = 1: (a) hˆ = 0.8, (b) hˆ = 0.6, (c) hˆ = 0.4, (d) hˆ = 0.2, (e) hˆ = 0.02.
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The excess force required to push the plates together is 1 F = 2πR2 p0 (pˆ − 1)ˆr dˆr.
121
(21)
0
In the limit of very large Peclet numbers we expect pˆ = hˆ −1 and 1 F 1 Fˆ = = 2 (pˆ − 1)ˆr dˆr ∼ − 1, Pe 1, 2 πR p0 hˆ 0 so that hˆ Fˆ + hˆ = 1,
Pe 1.
(22)
(23)
Results for hˆ Fˆ + hˆ are shown in Figs. 5 and 6. We see that (23) breaks down for Pe < 100. In the limit of very small Peclet numbers, we expect the excess force to be given by (18), i.e. Fˆ =
Pe 5Pe2 − + ··· , 8hˆ 3 96hˆ 6
ˆ (a) Pe = 104 , (b) Pe = 103 . Fig. 5. hˆ Fˆ + hˆ against non-dimensional gap width h:
ˆ (a) Pe = 100, (b) Pe = 10, (c) Pe = 1, (d) Pe = 0.1. Fig. 6. hˆ Fˆ + hˆ against non-dimensional gap width h:
(24)
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ˆ (a) Pe = 10−4 , (b) Pe = 10−3 , (c) Pe = 10−2 , (d) Pe = 10−1 , (e) Pe = 1. Fig. 7. 8Fˆ hˆ 3 /Pe against non-dimensional gap width h: Broken lines show the asymptotic prediction (25) for Pe 1: (f) Pe = 10−4 , (g) Pe = 10−3 , (h) Pe = 10−2 .
so that it is useful to plot 8Fˆ hˆ 3 5Pe =1− + ··· . Pe 12hˆ 3
(25)
In this limit, the pressure profile is given by (10),(12),(16) and is pˆ = 1 + Pe
(1 − rˆ 2 ) (2 − 3ˆr 2 + rˆ 4 ) − Pe2 + ··· . 4hˆ 3 4hˆ 6
(26)
Numerical solutions for 8Fˆ hˆ 3 /Pe are shown in Fig. 7, together with the perturbation expansion (25). When Pe > 0.01 diffusion is slow and the incompressible Newtonian pressure profile is not established before the effects of compressibility are felt. Expansions (24) and (26) both break down when hˆ ∼ Pe1/3 . When h is sufficiently small the left-hand side of (19) becomes small (away from regions near the periphery of the disc where pressures re-adjust over a small radial distance to match the boundary condition). Hence, towards the end of the test the ˆ derivative ∂(pˆ h)/∂t on the right-hand side of (19) is small, and we have compression of the foam, rather than extrusion, even when Pe 1. Fig. 8 replots the numerical results of Fig. 7, together with results for higher Peclet numbers. When h is small, the differences between the various curves in Fig. 8 relate to the different amounts of foam remaining between the plates. When the test is stopped and U = 0, in the absence of any yield stress the foam will expand until the pressure p = p0 everywhere. If no foam emerges after the test, then either effects of compressibility were small, so that the test may be interpreted by an incompressible analysis, or yield stresses dominate any viscous stresses and prevent post-test expansion. In the next section we consider a foam with a yield stress which is sufficiently high that viscous stresses due to flow after yield may be neglected. Note that once we have adopted the linear approximation (3) between pressure and density, the analysis becomes identical to that for an isothermal perfect gas. Perfect gases are usually considered to be Newtonian, with viscosity independent of pressure [23]. The results presented above may therefore have been obtained previously, though it is unlikely that anyone would use squeeze-flow rheometry to characterize a low-viscosity perfect gas.
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ˆ (a) Pe = 10−4 , (b) Pe = 10−3 , (c) Pe = 10−2 , (d) Pe = 10−1 , (e) Pe = 1, Fig. 8. 8Fˆ hˆ 3 /Pe against non-dimensional gap width h: (f) Pe = 10, (g) Pe = 100, (h) Pe = 1000.
4. A yield stress fluid There is evidence that foams have a yield stress [1,2]. If the typical bubble size is a and interfacial tension between liquid and gas is γ, then on dimensional grounds we expect the yield stress τy to be of order γ τy ∼ . (27) a The mass of gas in each bubble remains constant, so that the volume of each bubble is proportional to p−1 and hence a typical bubble size a ∼ p−1/3 . We assume that the yield stress is 1/3 p , (28) τy = τ0 p0 where τ0 is the yield stress at ambient pressure p0 . As the pressure increases, so the gas/liquid interfacial area (per unit volume of foam) increases, and deformation of the foam becomes more difficult. This scaling (28) will go wrong once the volume fraction of the bubbles is sufficiently small that the bubbles no longer touch. If we consider bubbles on a regular cubic array, the bubbles touch when the volume fraction is φ = π/6 and this occurs when, by (1), p φ0 6 = −1 , (29) p0 1 − φ0 π which is not a restriction since we have already assumed that pressures are sufficiently small that (3) holds. However, experiments by Princen [2] suggest that even when φ > π/6 the yield stress is a strong function of liquid content. We must make the more restrictive assumption that the foam remains dry (i.e. that the liquid volume fraction 1 − φ 1), and even in this limit, the scaling (28) must be treated with caution. Note that bubbles will not remain isotropic during the extensional deformation of the foam. We shall find, as in Section 4, that p ∼ h−1 when h becomes sufficiently small. The lateral dimension of an
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isotropic bubble would decrease only as h−1/3 , making it impossible for the bubble to remain isotropic. We shall ignore any anisotropy that develops. Coarsening will occur in a real foam: bubbles grow and shrink due to diffusion of gas through liquid films, and bubbles may coalesce when films break. We ignore such structural changes. If we assume that the foam behaves as a Bingham fluid, we must also specify the viscosity of the fluid after yielding. Here we shall simply assume that the squeezing is sufficiently slow that viscous stresses are small compared to yield stresses and may be neglected, as in a perfectly plastic material. This may well be true during most of the squeeze test, but is likely to fail when the gap h becomes very narrow and shear rates become large. We assume that when sliding occurs at the wall the wall shear stress τw is equal to the yield stress τy of the foam. Sherwood and Durban [8] considered wall shear stresses τw = mτy , with constant m in the range 0 ≤ m ≤ 1. We conclude from their analysis that if h/R m < 1 the results presented below are valid if we replace τy by τw . A simple stress balance gives the pressure gradient h
∂p = −2τy (r). ∂r
(30)
If the yield stress τy were constant the pressure would be p = p0 +
2τy (R − r) h
(31)
and the total excess force required to push the plates together would be 2πτy R3 F = 2π (p − p0 )r dr = . 3h However, if the yield stress varies with pressure (28), the stress balance becomes 1/3 ∂p p h = −2τy (r) = −2τ0 , ∂r p0
(32)
(33)
so that 2/3
p2/3 − p0 i.e.
p = p0
=
4τ0 1/3
3hp0
(R − r),
(34)
3/2
4τ0 (R − r) + 1 3hp0
.
(35)
The pressure (31) is linear in the distance R − r from the periphery of the plates when the yield stress is assumed constant. However, in a foam the higher pressure in the central region r < R reduces the typical bubble size and thereby increases the effect of interfacial tension. The yield stress increases, and in (35) the pressure p increases more rapidly than linearly. When the plates are pushed together, if the rise in pressure given by (35) is more than is required for simple compression of the material, there will be no flow. The pressure will increase merely because
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the foam is compressed. The rate of change of the pressure (35) predicted by the simple lubrication theory is 1/2 ∂p 4τ0 2τ0 U(R − r) = (R − r) + 1 , (36) ∂t 3hp0 h2 whereas the rate of increase assuming straightforward compaction comes from considering that ρf h would then be constant, so that 3/2 ∂p pU Up0 4τ0 = = (R − r) + 1 . (37) ∂t h h 3hp0 These expressions (36) and (37) are equal when 2τ0 (R − r) = p0 , 3h which occurs at 3hp0 r = r1 = R − 2τ0
(38)
(39)
when h = h1 (r1 ) =
2τ0 (R − r1 ) , 3p0
(40)
at which gap width the pressure at r1 is p1 = 33/2 p0 . The no-flow region first occurs on the axis r = 0 when h = 2τ0 R/3p0 ; prior to this all the foam is in motion, and r1 , as defined by (39), is negative. We set 3hp0 r2 = , (41) 2τ0 so that r1 + r2 = R.
(42)
Once slip at the wall has ceased, we assume that subsequent elastic deformation of the foam is uniaxial compression. The pressure within the region of no-flow is √ h1 (r) 2 3τ0 (R − r) p(r) = p1 = (43) h h and the pressure gradient in this region is √ 2 3τ0 ∂p =− , r < r1 . (44) ∂r h The local wall shear stress therefore has magnitude √ |τ| = 3τ0 . (45) The local yield stress in r < r1 is √ 1/3 1/3 √ 2 3τ0 (R − r) p τy = τ0 = τ0 ≥ 3τ0 = |τ|, p0 p0 h
(46)
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with |τ| = τy at r = r1 . The magnitude of the wall shear stress |τ| in r < r1 is less than the local yield stress τy , so there can be no flow in this inner region, as postulated. The total excess force required to push the plates together is R F = 2π (p(r) − p0 )r dr. (47) 0
If h > 2τ0 R/3p0 the foam flows even at r = 0 and the pressure is given by (35) everywhere. Hence, 3/2 R 4τ0 F = 2πp0 (R − r) + 1 r dr − πR2 p0 . (48) 3hp0 0 Now r(a1 − a2 r)3/2 dr =
2 2a1 (a − a2 r)7/2 − 2 (a1 − a2 r)5/2 2 1 7a2 5a2
and hence, setting a2 = 4τ0 /3hp0 = 2/r2 and a1 = 2(R/r2 ) + 1 we find
7/2 7R 2R 2πp0 r22 +1 − − 1 − πR2 p0 . F= 35 r2 r2 When h is very large 2πτ0 R3 F∼ , 3h
h
τ0 R , p0
(49)
(50)
(51)
which agrees with the result (32) for a fluid with a constant yield stress τ0 . When hp0 τ0 R pressures remain close to p0 and the yield stress (28) varies little. As in Section 3, we non-dimensionalize forces by πR2 p0 and lengths in the radial direction by R, so that (51) becomes 1 3h0 p0 Fˆ ∼ , rˆ2 = 1. (52) rˆ2 2τ0 R When 2τ0 R/3hp0 > 1 the foam flows in the region r > r1 and is static in r < r1 . The pressure is 3/2 4τ0 (53a) p = p0 (R − r) + 1 , r > r1 3hp0 √ p = 2τ0 (R − r) 3h−1 , r < r1 (53b) and hence the total force is √ 3/2 R r1 4 3πτ0 4τ0 F = r(R − r) dr + 2πp0 (R − r) + 1 r dr − πR2 p0 h 3hp0 0 r1 √
6 3πp0 r12 R r1 37/2 1 2R 2 2 1 5/2 = + 1 [1 − 3 ] . − − πR p0 + πp0 r2 − − r2 2 3 7 7 5 r2
(54)
Note that (50) and (54) agree when r1 = 0, as expected. In the limit h → 0 we see from (39) that r1 → R and the no-flow region extends over the whole plate. The force tends asymptotically to 2πτ0 R3 F∼ √ , 3h
(55)
J.D. Sherwood / J. Non-Newtonian Fluid Mech. 112 (2003) 115–128
127
Fig. 9. Foam with a yield stress. Non-dimensional excess force Fˆ = F/πR2 p0 , plotted as Fˆ rˆ2 = 3Fh/2πR3 τ0 against the non-dimensional parameter rˆ2 = 3hp0 /2τ0 R, as predicted by (50) and (54).
i.e.
√ 3 Fˆ ∼ , rˆ2
rˆ2 1,
(56)
√ which is larger than the incompressible result (52) by a factor 3. Fig. 9 shows Fˆ rˆ2 = Fr2 /p0 πR3 , as given by (50) and (54) as a function of rˆ2 = r2 /R. We see that Fˆ rˆ2 varies smoothly between the limits (52) and (56). Given the approximation (3), the mass M of foam remaining between the plates is equal to R 2πhρ0 R hρ0 M = 2πh rρf dr = rp dr = (F + πR2 p0 ) (57) p p 0 0 0 0 M∼
2πR3 ρ0 τ0 √ 3p0
as h → 0.
(58)
Thus the final stages of the squeeze flow test investigate the compressive elastic properties of the mass M of foam which remains in the device. However, since M depends upon the yield stress of the foam, the test nevertheless leads to an estimate for the foam’s yield stress. If no material were to leave the gap between the plates, and if the original plate separation had been h0 , the pressure within the plates would be p0 h0 / h and the force F = πR2 p0 h0 / h would vary, like (51) and (55), as h−1 . It is therefore important to verify that the material behaves plastically, rather than elastically, by observing that material is indeed squeezed out from the gap between the plates. When the foam has a yield stress, stresses will not drop to zero when the plate velocity U = 0 at the end of the test, and foam will not be subsequently extruded. It should be possible to measure the mass of foam remaining at the end of the test, though an accurate determination of this mass may be difficult when h is small.
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