Squeezing flow between parallel plates

Wear, 45 (1977) 177 - 185 0 Elsevier Sequoia S.A., Lausanne

SQUEEZING

177 - Printed

FLOW BETWEEN

in the Netherlands

PARALLEL

PLATES

P. S. GUPTA and A. S. GUPTA Department

(Received

of Mathematics,

June

Indian

Institute

of Technology,

14, 1976; in final form December

Kharagpur

(India)

14, 1976)

Summary A similarity solution for the full Navier-Stokes equations is presented for the unsteady flow between two plates approaching or receding from each other symmetrically. The similarity solution exists when the distance between the plates varies as the square root of a linear function of time. In the case of axisymmetric squeezing flow the total resistance to the upper circular disk has been calculated.

1. Introduction Most investigations of unsteady flow in channels and of fluid flow near an oscillating plate are confined to velocity or pressure fluctuations imposed axially along the channel or to longitudinal oscillations of the plate [ 11. Other problems in channel flows exist when the confining walls have a transverse motion. One example of such flow of significance in lubrication is the squeezing flow between two parallel plates. Jackson [2] has presented an iterative solution and has evaluated the relative magnitudes of the inertial and viscous effects. The tackiness of liquid adhesives is also a squeeze film effect [ 31. Squeeze film theory may also be applied to the rolling of a pneumatic tyre on a slick pavement where the important factor is pavement texture [ 41. Langlois [ 51 and Salbu [ 61 analysed isothermal compressible squeeze films neglecting inertial effects. An explicit solution of the squeeze flow problem taking inertial terms into account was attempted by Thorpe [7] , but his solution fails to satisfy the requisite boundary conditions. His perturbation solution when the plates approach (or recede) with constant velocity is erroneous. The present paper presents a correct solution to the rectilinear squeezing flow of an incompressible viscous fluid between two infinite parallel plates studied by Thorpe. The radial squeeze film problem was investigated and the resistance to the motion of the upper plate was calculated.

2. Rectilinear

flow

Consider the rectilinear squeezing flow of an incompressible viscous fluid between two infinite parallel plates (Fig. 1). The plates move symmetrically with respect to the central axis of the channel which is taken as the x axis, the y axis being perpendicular to it.

---I=---t+

it__;_---___-_-_--__~__

w

Fig. 1. Rectilinear

flow.

If it is assumed that the flow is two dimensional, equations are

au

_+U;+,;=_:J

at

av -_+u;;+,_;. at

=..__i

and the equation

of continuity

the Navier-Stokes

g+V($+E!)

(1)

g+v(;$+;;)

(2)

is

au av --- + - = 0 ax ay

(3)

If the dimensionless variable 77= y/u(t) is introduced, where 2a(t) is the distance between the plates at any time t, eqns. (1) - (3) become

au au _++-++_~=-__++ at ax

au

1 ap

aav

P ax

av au vav -++_++-____-_--_.__-_++ at ax a a77

i

ap

pa al7

a2u 1 a2u ____++-_-( ax2 a2 av2 1 a2v i a2v __-+---__._ i ax2 a2 aq2

au 1 av +_--_-co is a a77 The equation

u = ;;q

of continuity

v,(t)f’(rl)

(eqn. (6)) is satisfied by

v = v,(t)f(l))

where C is a constant related to the inlet condition of the channel and v,(t) (= da/dt) is the velocity of the upper plate. The flow is clearly symmetrical about y = 0. The boundary conditions on u(x, Q, t) and v(x, 77,t) are

(4)

(5)

179

24(x, 1, t) = 0 u(x, 0, t)

+G 1, t)’ u,(t)

aw, 0, t)

= 0

=

o

a77

The first two conditions at 77= 1 reflect the no-slip condition at the upper wall and the remaining two follow from the symmetry of the flow at 77= 0. In addition to the above boundary conditions, Thorpe assumed a rigid wall at x = 0, and to make u = 0 at x = 0 he used ~=__.__--_XUt)f’(?)) a instead of eqn. (7). However, at this rigid wall u = 0 also and this is not satisfied by the form he assumed for u, namely u = u,(t)f(r)) In the present analysis, no rigid wall at x = 0 is assumed, so that u and u are not required to vanish at x = 0 for C # 0. If eqn. (7) is substituted in eqns. (4) and (5) and the pressure is eliminated

(9) where a prime denotes differentiation with respect to 77.The boundary tions are determined from eqns. (7) and (8) to be f(1)

= 1

f(0) = 0 Equation solution

au, --=R

f’(1)

condi-

= 0

(10)

f”(O)= 0

(9) is the same as that derived by Thorpe. a2dv,/dt .____

V

=

Thus for a similarity

RQ

(11)

VU,

where R and Q are constants. integrates as

Since u, = da/dt, the first equation

u(t) = (Lt + Mp2

of (11) (12)

where L and M areconstants. When L > 0 and M > 0,the plates move apart symmetrically with respect to q= 0. In contrast, when M > 0 and L < 0,the plates approach each other and squeezing flow exists with similar velocity profiles as long as M + Lt > 0.From eqns. (11) and (12), it follows that Q = -1. Equation (9) then becomes

R(ff”’ - f’f” -,,f”’

_ 3f”) = f”

(13) subject to conditions (10). This result is at variance with the conclusion of Thorpe who reported that a similarity solution always exists for the velocity profile and that flow is characterized by two dimensionless parameters. The

180

present analysis reveals that similarity exists only for plate separation varying as eqn. (12), the flow being characterized by the single dimensionless parameter R. Thorpe failed to recognize that in general the parameters R and Q appearing in eqn. (9) are not constants but are functions of time. For constant u, he dropped the term involving du,/dt in eqn. (9) and obtained a perturbation solution of the resulting equation for small values of R (= au,/v). This procedure is erroneous since, for constant u_,, a(t) = u,t + constant and R reduces to a linear function of time. In this case eqn. (9) may be written as R

.-.-___f’“____ -77fl” __ 2f”

=

ff”’ - f’f”

and such a relation is not valid since the left hand side is a function of time only and the right hand side is a function of n only. This discrepancy stems from the fact that no similarity solution is possible for constant u, . A regular perturbation solution of eqn. (13) exists for small values of R in the form f = fo(v)

+ Rf,(r,)

+ @fi(v)

If eqn. (14) is substituted

+ . . .

(14)

in eqn. (13) and the coefficients

of R are equated,

f’o” =()

(15)

subject to fo(0)

= 0

f;(o)

fo(l)

= 1

f; (1) = 0

Similarly,

= 0

if the coefficients

f;”

=fof;-f;,f;;-sf;‘-3f;

fp

= -3f;’

of R and R2 are equated (17)

+ f. f;,’ + fl fJ’ -fA

with the boundary

f;’ -f;

f(y - Qf;l)

(18)

conditions

fi(0) = O

f;‘(o)

fi(l)

ff(1) = 0

= O

(16)

= 0

(19)

(i = 1,2)

This process can be repeated for any number of approximations. tions of eqns. (15), (17) and (18) satisfying the above boundary are fo(n)=$

-_;

f1(77)

2; --!l~oq3+-7)l3

=g

(29)

(21)

140

f2(q) = E,q + 5;

The soluconditions

+ i&

n5 - 5-

v’ + &

+so

(22)

181

where E, and E, are given by E, = 0.0196946

E, = ---0.2892701

(23)

correct to seven decimal places. The function f’ is plotted for R = 0.05 in Fig. 2 which shows that at a given value of x the velocity component u monotonically decreases from 77= 0 to rl = 1. From the definition of R in eqn. (ll), it follows that positive values of R correspond to the case in which the two plates move apart symmetrically and vice versa. Table 1 gives the values of f uersus 77for several values of R. 1.6 -

0.6 0.4-

0

I 0.1

I 0.2

I 0.3

I 0.4

’ 0.5

’ 0.6

’ 0.7

’ 0.0

8 0.9

1.0

“1-

Fig. 2. Variation

TABLE

off’

with 17 for R = 0.05.

1

Values of

f( 7))

R

0.01 0.03 0.05

f(v) q = 0.1

q = 0.3

7) = 0.5

77 = 0.7

rj = 0.9

0.149591 0.149775 0.149960

0.436730 0.437194 0.437661

0.687759 0.688280 0.688806

0.878666 0.879000 0.879336

0.986629 0.985588 0.985647

Although the solution for small R is presented, it is possible to integrate eqn. (13) subject to eqn. (10) numerically for any value of R. Table 1 shows that at a given time and for a fixed positive value of R the normal velocity increases monotonically from Q = 0 to n = 1.

3. Radial flow Consider the squeezing flow of a viscous liquid between two circular plates which move symmetrically with respect to the central region y = 0 (Fig. 3). The flow is clearly axisymmetric about r = 0. The analysis is similar to that in Section 2. If u and u are the velocity components along the radial

t_--2c-4

Fig. 3. Radial

flow.

and axial directions coordinates are

au

au

____+u

respectively,

the equations

au ay

+v(v%-,“i)

: ;;

__+~.__=-‘--.._-

at

ar

a~

a~ +

---+l&---

at

au u

ar

The equation

.-

ay

=

--.-

of motion

in cylindrical

polar

(24)

f --aP +vo=u P

(25)

ay

of continuity + -$(ru)

a

= 0

(26)

is satisfied by taking 2.4= -

I”-__ u,(t)f’(r))

2@(f)

u = %tt)ft77)

where ri = y/a(t). As in Section 2,2a(t) and u,(t) denote the distance between the plates and the velocity of the upper plate at any time t respectively. The fact that the radial velocity u vanishes at r = 0 follows from symmetry. If p is eliminated from eqns. (24) and (25) and eqn. (27) is used,

R(QfJ” + 2f” - ff 8’1)+ f’” = Sf 16 where a prime denotes given by R=%

S=_ Y

differentiation

(28) with respect to 77,and R and S are

a2dv, jdt (29) y%v

Following the ar~ments of Section 2, a similarity solution obeys eqn. (12) with S = -R. Thus eqn. (28) reduces to

R(ff”’ _ rlfttr __3f I’) = fiv

exists when a(t) (30)

and the flow is characterized again by a single dimensionless parameter R, which contradicts the conclusion of Thorpe. Notice that eqn. (30) differs is absent in eqn. (30). The from eqn. (13) in so far as the term -Rf’f” boundary conditions for eqn. (30) are the same as those given for eqn. (10). A regular perLurbation solution of eqn. (30) in the form

183

f=fo+Rf,+R2f2

+...

(31)

may be found for small R as in Section given by fo(rl) = T

2. The functions

fo, fi and f2 are

_ $

(32)

(33) f=(v) =

El ~537) + F

v3

+ &

11

q5 -

(34)

where El = --0.1348985

E, = 0.0098599

(35)

correct to seven decimal places. The function f’ is plotted for R = 0.05 in Fig. 4 which shows that at a certain radial distance r and at a given time the radial velocity monotonically decreases from Q = 0 to 77= 1. Table 2 gives the values of f uersus 17for several values of R.

f.4 t

f’

-

1.0 f.2

0.80.6 0.4

-

0.20

0.1’

0.2I

0.3I

’ 0.4

’ 0.5

0.6e

0.7’

0.6’

0.9’

1.0

“1-

Fig. 4. Variation off’

with q for R = 0.05.

TABLE 2 Values of f(v) R

0.01 0.03 0.05

f(q) q = 0.1

q = 0.3

q = 0.5

q=o.7

q=o.9

0.149565 0.149695 0.149826

0.436665 0.436996 0.437329

0.687687 0.688064 0.688442

0.878622 0.878867 0.879113

0.985522 0.985566 0.985610

184

Comparison of these values with the corresponding values in Table 1, at a certain time for a given value of R, shows that the normal velocity for the axisymmetric case is slightly less than the corresponding velocity in the rectilinear case. If C is the radius of one of the plates, the resistance to the motion of this plate (say the upper one) may be determined as F = Jc(p -po)*2nr

dr

(36)

0 where p. is the atmospheric pressure at r = C and @ -po)* at the upper plate when n = 1. Substitution of eqn. (27) in eqn. (24) and integration using the boundary condition p = p. at r = C gives

is the value of

p -p.

p_po=~(r2-c2)

it Vi 2a2

--

To evaluate p -p. @ -po)*

Vf” -

dvvJdt-5 2a

v: 4a2

r&2

f’2 + 2s

1 f’-

ff” - 3

at n = 1, eqn. (10) is substituted

-!Z$_

=

f”‘(l)(r2

with respect to r

f”’ I

(37)

in eqn. (37) and then

- C2)

When R is small, f”‘(l) may be approximated by f;(l) + Rf;“(l) + with fo, fl and f2 given by eqns. (32) - (34). Substitution of eqn. (38) in eqn. (36) and integration gives the resistance to the upper plate for small R as

R2fz’(l)

FE--

7Tpc%J, 8a3

(-3

+ 4.12500R

+ 0.0963516R2

+ O(R3)}

(39)

For squeezing flow v, < 0 and R < 0 so that eqn. (39) gives F > 0. In this case F is a measure of the load-supporting capacity of the liquid film between the plates. The load-supporting capacity for a circular plate approaching another plate with constant velocity was first calculated by Reynolds [8] by neglecting the inertial terms and assuming the motion to be quasi-steady. While the neglect of inertial terms is justified for small Reynolds numbers, the assumption of quasi-steadiness may introduce large errors unless the acceleration or deceleration of the plates is very small.

4. Summarizing

remarks

A similarity solution of the full Navier-Stokes equations for the unsteady flow between two plates approaching or receding from each other symmetrically has been presented. It is shown that a similarity solution exists

185

only when the distance between the plates varies as (Lt + M)1’2, and squeezing flow takes place for L < 0 and M > 0 as long as M + Lt> 0.

Acknowledgment One of us (P.S.G.) thanks the Council of Scientific Research for financial support.

and Industrial

Nomenclature a(t) F

L M P ;? t U u

uvf(t) 77

V P

half the distance between two plates load supporting capacity of the liquid film constants liquid pressure radial distance Reynolds number time velocity in the x direction velocity in the y direction velocity of the upper plate dimensionless variable y/a(t) kinematic viscosity density of the liquid

References 1 2 3 4 5 6 7

H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1960. J. D. Jackson, Appl. Sci. Res., Sect. A, 11 (1962) 148. J. J. Bikerman, Surface Chemistry, Academic Press, New York, 1958. D. F. Moore, Wear, 8 (1965) 245. W. E. Langlois, Q. Appl. Math., 20 (1962) 131. E. 0. Salbu, J. Basic Eng., 86 (1964) 355. J. F. Thorpe, in W. A. Shaw (ed.), Developments in Theoretical and Applied Mechanics, Vol. 3, Pergamon Press, Oxford, 1967. 8 L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1959.