On the no-slip boundary conditions for squeeze flow of viscous fluids

Mechanics Research Communications 70 (2015) 1–3

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On the no-slip boundary conditions for squeeze flow of viscous fluids Lorenzo Fusi ∗ , Angiolo Farina, Fabio Rosso Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy

a r t i c l e

i n f o

Article history: Received 14 August 2015 Received in revised form 26 August 2015 Accepted 28 August 2015 Available online 22 October 2015

a b s t r a c t A typical class of boundary conditions for squeeze flow problems in lubrication approximation is the one in which the squeezing rate and the width between the squeezing plates are constant. This hypothesis is justified by claiming that the plates moves so slowly that they can be considered static. In this short note we prove that this assumption leads to a contradiction and hence cannot be used. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Squeeze flow Lubrication approximation Boundary conditions

1. Introduction We analyze a particular class of boundary conditions that are commonly prescribed when modelling the squeeze flow of a viscous fluid confined between parallel plates in lubrication regime. In particular, we show that the usual assumption of plates moving with a constant speed which is so slow that they can be considered static (quasi-steady approximation) leads to a contradiction. This assumption, which is often made in squeeze problems [1,2,4], needs to be justified by a proper non-dimensional analysis, as it always happens when considering steady and quasi-steady approximations. When dealing with squeeze flow in lubrication regime, the aspect ratio ε (i.e. the ratio between the characteristic longitudinal length and the characteristic transversal length of the fluid) is typically very small so that O(ε) and o(ε) terms can be neglected at the leading order in the governing equations and in the boundary conditions. Performing the usual lubrication expansion we show that, independently of the selected time scale, the assumption of plates moving with constant velocity and the assumption that the distance between the plates is constant are contradictory and therefore cannot be used. 2. Mathematical formulation of the problem We limit our analysis to the axisymmetric squeezing flow between parallel discs, but our results can be applied to the

∗ Corresponding author. Tel.: +39 0554237147; fax: +39 0554237133. E-mail address: [email protected]fi.it (L. Fusi). http://dx.doi.org/10.1016/j.mechrescom.2015.08.007 0093-6413/© 2015 Elsevier Ltd. All rights reserved.

planar case as well. Consider an incompressible material confined between two parallel circular plates of radius1 R* that are approaching each other with prescribed velocity. Denote h* (t) as half the distance between the plates. The upper plate moves with velocity ˙ ˙ h(t) < 0, while the lower with velocity −h(t) > 0. Assume that the Cauchy stress has the form ␴* = − p* I +  * , where p* I is the mean normal stress tensor and  * is the deviatoric stress tensor. In the case of a viscous fluid (Newtonian or non-Newtonian) we write  ∗ = ∗ (˙ ∗ )˙ ∗ , ˙ ∗

∇ u∗

(1) T + ∇ u∗

u*

= is the rate of deformation tensor, is where velocity and where * (viscosity) is a constant in case the fluid is Newtonian. Suppose that the velocity field of the type u∗ = u∗ (r ∗ , z ∗ , t ∗ )er + v∗ (r ∗ , z ∗ , t ∗ )er (Fig. 1). The governing equations of the system are

∂v∗ 1 ∂ ∗ ∗ + (r u ) = 0, ∂z ∗ r ∗ ∂r ∗

 ∗

 ∗

∂u∗ ∂u∗ ∂u∗ + u∗ + v∗ ∗ ∗ ∂t ∂r ∂z ∗ ∂ v∗ ∂v∗ ∂v∗ + u∗ + v∗ ∗ ∗ ∂t ∂r ∂z ∗

 =−

 =−

∗ ∗) ∗ ∂rz 1 ∂(r ∗ rr ∂p∗ + ∗ + −  , ∗ ∗ ∗ r r∗ ∂r ∂r ∂z

∗) ∗ ∂zz ∂p∗ 1 ∂(r ∗ rz + ∗ + , ∗ ∗ r ∂z ∂r ∂z ∗

where the first comes from the incompressibility constraint while the others come from the balance of linear momentum. Because of symmetry we confine our study only to the upper portion of

1

Starred letters indicate dimensional quantities.

2

L. Fusi et al. / Mechanics Research Communications 70 (2015) 1–3

Fig. 1. A schematic representation of the system.

the system, i.e. z* ∈ [0, h* ] and r* ∈ [0, R* ]. Assuming no-slip on the upper plate we have v∗ (r ∗ , h∗ , t ∗ ) = h˙ ∗ and u* (r* , h* , t* ) = 0. Symmetry implies (see [1])

v∗ = 0,

∗ ∗ rz = rz = 0 on z ∗ = 0,

(2)

u∗ = 0,

∗ ∗ rz = rz = 0 on r ∗ = 0.

(3)

Rescale the space variables and velocity components with r ∗ = R∗ r,

z ∗ = H ∗ z,

u∗ = U ∗ u,

v∗ = V ∗ v,

(4)

where h* (0) = H* is the initial position of the upper plate. Mass conservation becomes

 H ∗ U ∗  1 ∂(ru) R∗ V ∗

 

r

∂r

+

∂v = 0. ∂z

(5)

time taken by the plate to reach the position z = 0. Suppose now to rescale time, pressure and deviatoric stress in the following way t ∗ = T ∗ t, where P∗ =

v = v(r, t)

1

⇒

ru = f (z)

 (2) ⇒

 (3) ⇒

v ≡ 0,

=

 H∗U ∗  R∗ V ∗

= 1.

ε3 Re

u ≡ 0,

Hence, if we look for a solution with u, v ≡ / 0, then necessarily

= O(1). Since H* , R* , V* , U* are characteristic values we can safely assume (7)

With this choice we are stating that the average time needed for a particle to be squeezed out of the system is equal to the average

are given by (see [1,2]) ∗ =

U ∗ H∗

(9)

and the non-dimensional momentum balance R∗ ∂ u ∂u ∂u +u +v U ∗ T ∗ ∂t ∂r ∂z

 (6)

,

∗2

(8)

1 ∂(ru) ∂v + = 0, r ∂r ∂z

We immediately notice that ⇒

H

*

 ∗ = ∗ ,

Setting ε = H* /R* and recalling (7) we get the non-dimensional mass balance

εRe

1

and

∗ U ∗ R∗



=

P*

p∗ = P ∗ p,



R∗ ∂ v ∂v ∂v +u +v U ∗ T ∗ ∂t ∂r ∂z

=−

∂p ε + ∂r r

 =−





∂(rrr ) ∂rz −  + , ∂r ∂z

∂p ε2 ∂(rrz ) ∂zz + +ε , r ∂z ∂r ∂z

where we have defined the Reynolds number as Re =

∗ U ∗ H ∗ ∗

Remark 1. ¯ = Re

Some authors [1,3,4] define the Reynolds number as

∗ V ∗ H ∗ ∗

.

¯ = εRe. ⇒ Re

L. Fusi et al. / Mechanics Research Communications 70 (2015) 1–3

By definition Re =

¯ = Re

∗ U

The non-dimensional version of (12) is

∗2

−1 ∗ U ∗ H ∗ 2 ∗ V ∗ −1

∗ U ∗ R∗

=

=

dh = −1, dt

radial inertial forces viscous shear forces axial inertial forces extensional stress forces

¯ and Re are “sufficiently” small (creeping flow) both leads When Re to the same set of equations, since inertial effects are negligible. 3. Quasi-steady solution for lubrication approximation with constant plates velocity In the squeeze flow problem, a typical set of boundary conditions is given by (see [1,2,4]) h∗ = H ∗ ,

∗

dh ∗ ∗ = −V . dt

(10)

The non-dimensional version is h = 1,

dh = −1. dt

∗

h∗ = H ∗ − V ∗ t ∗ .

(11)

(12)

If we are interested to describe the evolution of the system from the instant in which the plates begin to move (t* = 0) to the instant in which the fluid has been completely squeezed out, then the natural choice for the characteristic time T* is T∗ =

R∗ H∗ = ∗. U∗ V

∂rz ∂p + = 0, ⎪ ∂ r ∂z ⎪ −

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ − ∂p = 0 ∂z

which yields 0 = rz

∂p0 z. ∂r

(15)

Since t can also become O(1) we cannot neglect the term −t in (15) and so (11)1 does not hold true. If, on the other hand, we consider a sufficiently small time interval, then t will be sufficiently smaller than 1, and hence negligible in (15), and so (11)1 holds true. Hence, suppose we select a time interval such that the term −V* t* is negligible in (12). Recalling that at the leading order all O(ε) terms are neglected, we must select a time scale such that at most V ∗T ∗ = O(ε) H∗ so that h = 1. Therefore let us set T∗ = ε

H∗ R∗ =ε ∗ V∗ U

(16)

⎧ 1 ∂(ru0 ) ∂v0 ⎪ ⎪ + = 0, ⎪ ⎪ r ∂r ∂z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 0 0 Re

∂u

=−

∂p

⎪ ∂t ∂r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ − ∂p = 0

+

∂rz = 0, ∂z

(17)

∂z

which reduces to (14) if Re  1. In this case h∗ − H ∗ = −V ∗ t ∗ ⇒ h = 1 − εt. ∗

dh dh V ∗T ∗ = −V ∗ ⇒ = − ∗ = −ε. ∗ H dt dt Hence, neglecting O(ε) terms, we get

(13)

Indeed, this selection implies that we are describing the system in the time scale of the whole squeezing process. Assuming Re  O(1), the problem at the leading order reduces to

⎧ 1 ∂(ru0 ) ∂v0 ⎪ ⎪ + = 0, ⎪ ⎪ r ∂r ∂z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0 0

h = 1 − t.

The problem at the leading order now becomes

Conditions (10) and (11) come from the hypothesis that the plates move so slowly that their position can be considered constant. Of course this must be justified by an appropriate non-dimensional analysis since h˙ ∗ = −V ∗ implies h* = H* − V* t* and not h* = H* . Here we prove that, when operating within the context of lubrication, assumption (10) is not consistent Suppose that the plates are moving with constant velocity, so that dh ∗ ∗ = −V , dt

3

h = 1,

dh = 0, dt

which is once again not consistent with (11). We may conclude that the lubrication approximation conditions (11) are not consistent; therefore, they cannot be used to derive a stationary solution of the problem. References

(14) [1] J. Engmann, C. Servais, A.S. Burbidge, Squeeze flow theory and applications to rheometry: a review, J. Non-Newton. Fluid Mech. 132 (2005) 1–27. [2] L. Muravleva, Squeeze plane flow of viscoplastic Bingham material, J. NonNewton. Fluid Mech. 220 (2015) 148–161. [3] P. Singh, V. Radhakrishnan, K.A. Narayan, Squeezing flow between parallel plates, Ing. Arch. 60 (1990) 274–281. [4] D.N. Smyrnaios, J.A. Tsamopoulos, Squeeze flow of Bingham plastics, J. NonNewton. Fluid Mech. 100 (2001) 165–190.