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Decay of shear layers and vortex sheets
Journal of Non -Newtonian FItlid Mechanics, 15 (1984) 199-226 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
199
DECAY OF SHEAR LAYERS AND VORTEX SHEETS
R.S. RIVLIN Lehigh Universiv,
Bethlehem, PA (U.S.A.)
(Received November 8, 1983)
Summary
An incompressible fluid is contained in the domain between two stationary infinite parallel rigid plates. It is assumed that for shear flows, the shear stress in an element of the fluid depends linearly on the history of the velocity gradient in that element. It is supposed that initially two steady shear layers exist in the fluid and are symmetrically disposed with respect to the mid-plane. The time-dependent velocity field which results from the removal of the forces maintaining this steady flow is calculated in the cases when the fluid is Newtonian and when it is Maxwellian. The limiting cases when the shear layers reside in an unbounded space of the fluid and when they further become vortex sheets are discussed.
1. Introduction
In a previous paper [l] we considered the time-dependent rectilinear flow of an incompressible viscoelastic fluid in a domain bounded by stationary infinite rigid parallel plates. It was assumed that prior to time t = 0 the fluid is at rest and at time t = 0 is subjected to a longitudinal pressure gradient which is subsequently held constant. The pressure gradient was assumed to depend in an arbitrary manner on distance from the mid-plane and an expression for the Laplace transform of the resulting velocity was obtained. It was assumed that the fluid is such that the relation between the shear stress and the history of the velocity gradient in an element of the fluid is linear. In [l] this result was applied to the problem of run-up of plane Poiseuille flow, i.e. the case when the pressure gradient is independent of position, in the particular cases when the fluid is Newtonian, or Maxwellian. The
200 dependence of the velocity on position and time was obtained by inverting its Laplace transform. This result was then used to calculate the time-dependent velocity field which results when the initial conditions are those of steady plane Poiseuille flow and the pressure gradient supporting it is removed. In the present paper, the expression for the Laplace transform of the velocity field obtained in [l] is used to calculate the velocity field which results when the pressure gradient is such that the steady velocity ultimately attained at time t = oo consists of two shear layers, symmetrically disposed with respect to the mid-plane, in which the velocity gradients are equal in magnitude but opposite in sign, but are constant in each shear layer. Results are obtained for both Newtonian and Maxwellian fluids. These results are used to calculate the time-dependent velocity fields which result when the shear layers are present initially and at time t = 0 the forces maintaining them are removed. The results for a Newtonian fluid are given in $5 and those for a Maxwellian fluid in 58. In the case when the fluid is Newtonian the velocity field spreads diffusively throughout the fluid (Fig. 2), decaying to zero at infinite time. When the fluid is Maxwellian the velocity field may be interpreted as the superposition of eight waves which originate at time t = 0 at the boundaries of the shear layers and are reflected back and forth from the rigid boundaries. Two of these eight waves originate from each boundary of the shear layers. One of these travels in the positive and the other in the negative direction (Fig. 4). The special cases when the initial shear layers exist in an unbounded space of Newtonian or Maxwellian fluid are discussed in $6 and 59 respectively. The limiting cases in which these shear layers are vortex sheets are discussed in 57 and 510 respectively. From the results in 59 and 510 for a Maxwellian fluid the velocity field associated with the decay of a single shear layer or vortex sheet in an unbounded space of Maxwellian fluid can be read off easily. This is done explicitly in $10 for the case of a single vortex sheet. From this result the time-dependent velocity field associated with the decay of an arbitrary initial longitudinal velocity distribution could be calculated by simple integration. 2. A basic result We consider an incompressible fluid to be contained between two fixed infinite parallel plates situated at x = f h in some rectangular Cartesian coordinate system x. We suppose that the fluid is initially at rest and that it is subjected at time t = 0 to a pressure gradient P(x) in a direction parallel to the plates, which is independent of t for t > 0. It is easy to show that a
201 possible resulting flow field is one in which the velocity is longitudinal at each point. Let u(x, t) be this velocity and let E(x, S) be its Laplace transform. We suppose that the shear stress o(t) in the fluid is related to the velocity gradient history u’(x, T) (- 00 -C 7 < t) by * a(t)
=Jt
f(t-CO
+‘(x,
r)dr,
(2.1)
where the prime denotes differentiation with respect to x. It was shown in [l] that if P(x) is an even function of x and we introduce the notation
s= (g$‘z>
(2.2)
then
G? 4 --L(cosh
[x[o(/3)
ssf(s)
- i’P(
sinh @dP
p) sinh @d/3 + tanh [hl’P(
- sinh lx
xP( p) cash #dP
J0
The corresponding velocity version integral, thus:
u(x, t) = &/Y_t’mes5(x,
j3) cash SpdS]
.
(2.3)
field can be obtained
s)ds,
Y 100
by means
of the in-
(24
where y is a positive constant such that U(x, S) is analytic for Re s > y. We now suppose that the fluid, instead of being initially at rest, is in a steady state of longitudinal flow with velocity u(x, cc). At time t = 0 we remove the pressure gradient which supports this flow, i.e. we apply an additional pressure gradient -P(x). The resulting velocity field U(X, t) is then given by 24(x, t) = u(x,
co) - u(x,
t).
(2.5)
* We shall interpret the relation (2.1) as u=
‘+f(t-+‘(x,7)dq Lt f-r0 I --m
r>O.
This allows us to include as special cases Newtonian exhibit instantaneous elasticity.
fluids and other fluids which do not
202 3. Uniform shear layers In this section we consider run-up in the case when the steady flow which is attained at time t = cc is one in which there are two regions of uniform (non-zero) velocity gradient, separated from each other and from the boundaries by regions in which the velocity gradient is zero. We suppose that the regions of non-zero velocity gradient are symmatrically disposed with respect to the mid-plane and occupy the domains x = [a, b] and x = [--a, -b], where h > b B a > 0. Let - Q, and K,, be the velocity gradients in these regions. Then,
u(x, co) =
K,,(b-X)
b>,X>U,
K&--U)
IXI
%b 0
-aax>,
+ b)
h a 1x12 b.
We may express the corresponding u’(x, cc)=
(3.1)
-b,
velocity
gradient
field in the form
-K,,{H(x-u)--(x-b)+ff(x+u)-H(x+b)},
(3 *2)
where H( ) is the Heaviside unit step function. The equation of motion for steady shearing flow of the fluid fluid is u”(X, X)0)= 9(x)//A, where p denotes
(3.3)
the steady-state
viscosity
of the fluid. Then, from (3.2) and
(3.3), P(X)=pK,,{ti(X-a)-6(x-b)+S(x+u)-S(x+b)},
where S( ) is the Dirac delta function. (2.3) we obtain
(3.4)
Using (3.4) to substiute
for P( /3) in
;(x,s)=~~~(x,s),
ifa
(3 -5)
;(x,s)=?@~(x,s),
ifb<]xl
(3 -6)
ti(x,s)=~$(x,.s),
ifIxI
(3.7)
where
@,(x,s)=
~~(s)rosh~I{coshSusinhI(h-Ixl)-coshSxsillhS(h-b)}, -
203 @*(x,s)=
sf(s) -
$(x,s)=
rr(~fo~ -
kosh Sh {sinh sh { sinh
From (3.8)-(3.10)
S( h - lxl)(cosh So - cash @)I y
c( h - u) - sinh S( h - b)} .
(3.9) (3.10)
it follows that as I[1 + 0
J(+%(x,
4 = (b - Ixl) + O(lS12),
f(.+D*(x,
s) =X2@
f(+&(x,
s) = (b - 4 + O(1512).
- Ma2
- b2) + 0(lS14),
(3.11)
4. Some special cases The case in which the body of fluid, in which the run-up of the shear layers takes place, occupies the whole of three-dimensional space is obtained by letting h + cc in equations (3.8)-(3.10). We note that 5, defined by (2.2), is double-valued. However, each of the expressions for is(x, S) given by (3.5)-(3.10) is unaltered if one of these values is replaced by the other. In order to obtain asymptotic expressions for C( x, S) which are valid in the limiting case h = 00, we need to choose one or other of the branches for 2. We choose the branch for which Re 3 > 0 when Re s >, 0. We make a branch cut on the negative real axis and consider the s-plane for which -r < arg s =Gr. Then for h = cc we obtain from (3.5)-(3.10) for Re s > 0 U(x, S) =_
S;:)’
e Slxlcosh{a-e-3bcoshlx), -
ifa
(4.1)
-s’“l(cosh {a - cash lb),
if b < 1x1~ cc,
(4.2)
PKo E(x, S) = pe SSJb)
E(x,s)=-!%cosh{x(e-r”-e-lb)
iflxl
dfb)
(4.3)
We now consider the further limit when b - a + 0 and K,, --) 00, while (b - U)Ko = ij,, a constant. The system then becomes one in which, in the limit t + cc, there are two parallel vortex sheets situated at x = +a, with vorticities TijO. We then obtain from (4.2) and (4.3) E(x,s)=
--
2g)ie -
3(1X1-0) _ e-S(lxl+a)}
if
a
1x1,
<
S
u(x, s) =
-_!I$!$(e-SCa-lxl)
+ e-T(O+lxl)}
if
1x1 <
a.
(4.4 (4.5)
204 5. Newtonian fluids In the case when the fluid is Newtonian f(t-7)=@--),
f(s)=q,
and has viscosity
PL=77,
77,we have (5.1)
and from (2.2)
It is then evident from (3.5)-(3.11) points for which s=O
that the poles of iJ(x, S) are located at the
orcoshSh=O.
(5 03)
With (5.2) the second of these equations s,=
-$(2n-1)2?rZTj/ph2
(n=l,2,...).
yields the solutions
s = s, where (5.4)
It follows that G(x, S) is analytic for Re s 2 0, except at s = 0 where it has a simple pole. U(x, t) is therefore given by the inversion integral (2.4), which can be evaluated by applying the Residue Theorem with the Bromwich contour of Fig. 1. There AB is the line Re s = y and lY is a circular arc of radius R centered at the origin and we take the limiting situation when R = co. We note from (3.5)-(3.10) that Lt
R-cc
Jr
e%(x,
s)ds = 0.
(5 -5)
Let K, and K,, be the residues of e’%( x, S) at s = 0 and s = s, respectively. Then, u(x,
f)=K,+
f
K,.
(5 -6)
n=l
Fig. 1. Bromwich contour for inversion of Laplace transform of u for Newtonian fluid.
205 We find from (3.5)-(3.11),
K,=
with (5.2), that
0
if b < 1x1~ h,
K,,(&-(XI)
ifa
(5 -7)
if 1x1f a,
i ‘Co@ - a> and 8K,h
K, =
cos (2n - ex 2h
(2n - 1)27r2 cos
en- w _cos on- lb
(W
2h
We now consider the decay of the velocity field (3.1) if the pressure gradient maintaining it is removed. Then, the resulting velocity field U(X, t) is given by (2.5). From (3.1) and (5.7) we have u(x,
co) = K,.
(5 -9)
Then (5.6), (5.9) and (2.5) yield 2.4(x, t)=
-
(5.10)
E K,. n=l
We can rewrite quantities X=x/h,
eqns.
A = a/h,
(5.8) and
B = b/h,
(5.10)
T= qt/ph’,
in terms
of the dimensionless
U= u/tc,,h.
(5 .ll)
We obtain
u= -
eii,,
(5.12)
n-1
0.2
0.6
0.4
0.8
1.0
X
Fig. 2. U vs. X for Newtonian fluid for various values of T with A = 0.2, B = 0.5. I. T = 0; II. T= 0.05; III. T= 0.2; IV. ‘l- = 0.5.
206 where 8 (2n - l)*a* X{cos[(n-f)rB]
cos[(n -f)7rX] -cos[(n-f)?rA]}
exp[-(n-+)*a*T].
(5.13)
As an illustration of this result, in Fig. 2 U is plotted against X, for the case when A = 0.2 and 3 = 0.5, for various values of T and we note that the decay of the shear layer takes place smoothly in a diffusive manner. 6. Newtonian fluids (h = 00) With (5.1), eqns. (4.1)-(4.3)
E(x, s) =-eKo+‘(cosh d
yield
la - cash lb),
b < 1x1~ h;
(6-2)
Ix/< a,
u(x, s) =
where l is given by (5.2). Each of the expressions (6.1)-(6.3) has a discontinuity on the negative real axis. Therefore, we cannot evaluate the corresponding inversion integrals by using the Residue Theorem, as was done in the case when U(x, s) is given by the expressions (3.5)-(3.10). We note also that each of the expressions (6.1)-(6.3) has a pole at s = 0 and is analytic and single-valued everywhere else for -7r < arg s < 7r. We can therefore evaluate the inversion integrals by using Cauchy’s theorem with the modified Bromwich contour shown in Fig. 3. There C and c are arcs of circles, centered at s = 0, with radii R and E respectively and we take the limiting case R --) 00 and E + 0. We easily see, from (6.1)-(6.3), that
et\
jest6 (x, s)ds = -2i%,(b-1-x))
c
Lt e%(x,s)ds=O e-0 / c
ek\/e’s(x,s)ds=
c
(aglxlgb),
(6.5)
(b
-2imc,(b-a)
(6.4)
(IxI
(6.6)
207
Fig. 3. Modified Bromwich contour for inversion of Laplace transform of v for unbounded space of Newtonian fluid.
Also, for each of the expressions for E(x, S) in (6.1)-(6.3) Lt
Jc
R+cc
e%(x,
s)ds = 0.
(6.7)
In order to calculate the contributions of the integrals on I and i to the inversion integral, we write a = - Re s and note that on 1
i vcu’/2
I=
i -
irxP2
(6.8)
on 5,
where
( p/7Jp2.
v=
(6.9)
Then we obtain from each of the expressions (6.1)-(6.3)
JI+i
e%(x,
2iK0 s)ds =--v-1(x,
t),
for C(x, S) (6.10)
where
I(
X,
t) = ~~~~~~
vxa”‘(cos vaa”’
- cos vba’/2)da.
(6.11)
The expression (6.11) for 1(x, t) may be rewritten as
1(x, t) = 2/* -cos e-cr2’ 0 a2
vxa(cos vua - cos vba)da.
(6.12)
208 From (6.12) we obtain 3%
(see eqn. (7.4.6) in [2])
t) = - 2 OOOe-a” cos VX(Y(COS YU~ - cos vba)da /
at
= -~/gme~a2r[cosva(x+n)+cosva(x-a) -cos
va(x
+ b) - cos Y~(X - b)]da
0 i ?T
c-3
-
-exp
t
-
exp -
2(x + u)’ 4t
2(x + b)* 4t
i
1 i
+exp -
- exp i
i
u(x,
t)=
U(X,
t)=K&-U)--1(x,
u)’
4t i
V”( x - b)* 4t
From Cauchy’s Theorem with the contour and (6.10), the inversion integral (2.4) yields U(X,f)=K&‘-IXI)+(X,t)
V’(X -
)I ’
in Fig. 3 and eqns. (6.4)-(6.7)
(6.14)
(U
-21(x,
t)
(6.15)
(b
t)
(6.13)
(Ixl
(6.16)
We now consider the decay of the velocity field (3.1) with h = co if, at time t = 0, the pressure gradient maintaining it is reduced to zero. Then, the resulting velocity field u(x, t) is given, from (2.5), (3.1) and (6.16) by
24(x,t)
=21(x,
(6.17)
t).
7. Vortex sheets in a Newtonian fluid We have seen in 94 that in the case in which, in the limit t + 00, there are two vortex sheets situated at x = + a, with vorticities T ij,, G( x, S) is given by (4.4) and (4.5). If the fluid is Newtonian we obtain, with (5.1), qx,
s)
=
E(x,s)=z{e
_
$J {e-Swl-aL -ATa-1.4)
e-S(14+4},
+ e-sc~+l~l)},
u
<
(7.1)
IxI,
Ix1 < a,
where 5 is given by (5.2). Taking the inverse Laplace
P-2) transforms
of (7.1) and
209 (7.2) we obtain, u(x,
t)=
with the notation
-+tij,{erfc[jv(Ixl-LZ)~-“~] -erfc[$v(lxl+
u(x,
(6.9),
t) = +L&{erfc[$v(a +erfc[$v(a
a)t-“2]},
1x1> a,
(7.3)
Ix)< a.
(7.4)
- Ixl)t-1’2] + l~l)i-‘/~]},
From (7.3) and (7.4)
u(x, oo)=
I-4 > a 1x1-=a-
0,
l
_ ql,
(7.5)
With (2.5) and (7.5) it follows that the velocity the decay of a pair of vortex sheets, situated vorticities f Go, is given by 24(x, t) = $ijo{erf[+v(lxl
+ CZ)~-“~]
IxI> a,
-erf[$Y(IxI-a)t-“2]}, u(x,
field u(x, t) associated with at +a and having initially
(7.6)
t) = +GO{erf[+Y(IxI + LZ)~-‘/~] +erf[+Y(IxI
- ~2)t-“~]},
IxI<
a.
Since erf(0) = 0, it follows from (7.6) that U(X, cc) = 0, as it should. Also, since erf(cc) = 1, it follows from (7.6) that U(X, 0) = 0 for (xl > a and ijo for 1x1~ a, as it should. From (7.6) we obtain al.4
at=
1 f(lxI--a)exp -[ v’( 4ru)’ II, -
w 4&3/Z
( Ix1 + u) exp
i
-
“1’1”4:
a)’
[
1x1 -
(7.7)
the upper (lower) sign applying if Ix/> a (-c a). This result can also be obtained from (6.17) and (6.13) in the limiting case when b - a + 0, ~~ + 00, with IC~(~ - a) = 3,. 8. Maxwellian fluids We now suppose that the fluid is Maxwellian f(t
- T) = ;e+Q/A,
where 17and X are positive constants. f(s)
and that f( t - T) is given by
=&
5 = Y[S(hS + l)]“‘,
(8.1) Then, with (2.2), /J = q,
(8.2)
210 where y = (P/11Y2 V(x, S) is given by (3.5)-3.10). poles at the points at which s=O,
Each of these expressions
(8.4
cosh{h=O.
The last of these equations r2(2n
yields, with (8.2) 2,
- 1)2
(n = 1,2,...). 4v2h2 Thus the poles of i( x, S) occur at s(Xs+l)=
-
(8.3) for 6(x, S) has
(8.5)
s = 0,
s=,ps-
n
(8.6)
=- 2LJ-lf[1-a%(2n-1)2]1’2],
where
x
L=-
(8.7) v2h2 ’ We note that all of these poles lie in the half of the s-plane for which Re s G 0 and that, except at the poles, C(x, S) is single-valued and analytic. Let m be the largest value of n for which 1 - 7PL(2n
- Q2 >, 0.
(8.8) Then, for n < m, sz, s; lie on the negative real axis. For n > m, s,’ and s; are complex conjugates lying on the line Re s = - 1/2X. As in the case when the fluid is Newtonian, u(x, t) is given by the inversion integral (2.4), which can, in turn, be evaluated by using the Residue Theorem with the Bromwich contour of Fig. 1. We denote the residues at s = 0, s,’ , s; by K,, Kz, K; respectively. Then, u(x,
t)=K,+
f (K,+ +K,). n=l We find, from (3.5)-(3.11), with (8.2), that
K,=
JG,(b-~xI),
ablxl
0,
b < 1x1
i Ko(b - a),
(8 -9)
(8.10)
1x1=Ga
and K+
_
n-
%l(W + 1) hv2s,+(2Xs,f
cos[(2n
- l)#os[(2n
- l)E]
+ 1)
I}
esir,
(8.11)
2K,(hS,
K,- =
+ 1)
hv2s,(2hs,
cos [ (2n
+ 1)
- l)E
I}
(2n - 1) g
-cos
- 1)3cos[(2n
es;‘.
We now consider the decay of the velocity field (3.1) if the pressure gradient maintaining it is removed. Then, the resulting velocity field U(X, t) is given by (2.5). From (8.6), (8.9) and (8.11) u(x,
(8.12)
00) = K,
and from (2.5), (8.9) amd (8.12) 24(x, Q=
-
g
(K;
(8.13)
t-K,-).
We define the dimensionless A=a/h, q+,s,-
B=b/h, =Xs,+,hs,,
quantities
X=x/h, u=u
T=t/X, (8.14)
K,hL ’
Then, from (8.6) *, S+,S;=~(-1+[1-a2L(2n-1)2]1/2),
(8.15)
and (8.13) can be rewritten U(X,
T)=
-
as (8.16)
$ (R,‘+R,), n=l
where k+
n-
k-
_
= n
2(%++ 1) s,+(2s,+
+ 1)
-COs[(n
-
+ 1)
-cos[(n -
-t)vX]{cos[(n-+rA]
(8.17)
f)~7B]}e~~~,
2(&i-+ 1) s,-(2s,-
co&l
cos[(?z -+X]
{cos[(n
- +A]
+)vB]}esLT.
We note that the expression (5.10) for U(X, t) when the fluid is Newtonian can be obtained from (8.6), (8.11) and (8.13) by taking X = 0. We then have, with (8.7) and (8.3), s,+=s,,
s,=-00,
(8.18)
212 where s, is given by (5.4). From (8.11) we obtain K; =K,,
K,
(8.19)
=O,
where K,, is given by (5.8) and hence U(X, t) is given by (5.10). It is seen in the Appendix (eqns. (11.6)-(11.8)) that the velocity field u(x, t) can be regarded as the superposition of a number of waves. These originate at x = + a and x = +b at time t = 0 and are reflected back and forth at the boundaries. From each of these positions waves travel in both directions with speeds l/E, where C = h‘/‘Y. At the front of each wave there is a discontinuity in the velocity u( x, t) the magnitude of which can be easily obtained from (11.6)-(11.9). If the time t is sufficiently small and the point x is sufficiently close to a so that only the front leaving x = a has had time to reach x, then the velocity field u(x, t) at x is given by (11.12). When the front is at x, the velocity u(x, t) immediately behind the front is given by (11.13), and its spatial gradient u’(x, t) by (11.13),. In the case when only the fronts leaving x = a, or x = b, or both, have had time to reach x, the velocity u(x, t) is given by (11.14). From these results, corresponding results for u(x, t) can be obtained by using (2.5) and (3.1). Since U(X, 0) = u( x, cc) is continuous, it follows that the jump at a front of the velocity U(X, t) is the negative of that at the corresponding front of the velocity u(x, t). The results obtained can be written in dimensionless form by using (8.14). We find that for the waves originating at x = a at time t = 0 the jump in the dimensionless velocity U( X, T) (defined as the velocity immediately behind the wave-front minus the velocity immediately ahead of it) is given, from (2.5), (8.14) and (11.13), by 1
- -exp
2 Lii2
[
- ---&IX-J].
(8.20)
Similarly for waves originating at x = b the jump in U( X, T) is given by 1 -exp 2 L’12
[
- &IX-
811.
(8.21)
We note that the dimensionless speed with which the wave-fronts propagate is L112. As an example, in Figs. 4(a)-(d) the calculated values of U are plotted against X, in the range (0, l), for T = 0.05, 0.1, 0.2, 0.4 and L = 0.25, i.e. the dimensionless speed is 0.5. In each figure the U vs. X curve for T = 0 is shown for purposes of comparison. For all of the values of T for which the U vs. X curves are plotted in Fig. 4, the waves emanating from x = -a and x = -b have not had time to reach positions in the range X = (0, l), nor have the waves emanating from x = a and x = b had time to reach the
213 1.5
-
I.5 T-0.1
T=.05 1.0
1.0
-
b.5
L_._--“0 0.6
-0.5
X
‘-F 0.2
1.0
0.8
- 0.5
\
-1.0
-
X
1.0
0.8
(b)
-1.0
(a)
0.6
I.5 T -
0.2
T-0.4
I I.0 -
0.5
\
U
0
I 0.2
-1.0
L
0.6
(C)
Fig. 4. U vs. X for Maxwellian
x
0.8
1 I .o
u
I 0.6
0
-1.0~
X
0.8
, 1.0
(d)
fluid for various values of T with L = 0.25, A = 0.4, B = 0.5.
boundaries. Fig. 4(a) shows the situation when the fronts of the waves emanating from x = a and x = b have not yet met. Fig. 4(b) shows the situation when they have just met at x = +(a + b) and Figs. 4(c) and (d) show situations when they have passed each other. The directions of the jumps in the velocity U, which occur immediately after removal of the forces maintaining the initial shear flow in the region X= (0.4, OS), may at first sight seem curious. This is less so when one realizes that in order to maintain the initial flow, the force system which must act on the fluid in the region X= (0.4, 0.5) must consist not only of a uniform pressure gradient in the region X= (0.4, 0.5), but also of surface tractions acting tangentially on the fluid in the region. The traction acting on the surface X= 0.4 is in the positive direction and that acting on the surface X= 0.5 is in the negative direction. The tractions which act on the surfaces X = 0.4 and X = 0.5 of the fluid in the regions X = (0, 0.4) and (0.5, 1) respectively are zero.
214 This situation, in which there are discontinuities in the tangential component of the stress at the surfaces X = 0.4 and 0.5, is admittedly artificial. It reflects an idealization of the initial conditions similar to that which was made by Lamb [3] in discussing the decay of a vortex sheet in a Newtonian fluid. 9. Maxwellian fluid (R = co) In the case when the shear layers exist in an infinite space of Maxwellian fluid, we obtain an expression for u(, t) by letting h + 00 in eqns. (11.6)-(11.8) of the Appendix. Then for all finite times H(t - CA,,,) = 0 unless M = 0. Accordingly u(x, t ) is given by
u(x, t) = ‘I&,(x, t -
:A& +‘$,,&, t - %,,,)W - F&J -‘K,,(x, t - :&)H(t - fiA& -\ko,&, t - &,,)H(t - fA,,,) a < 1x1~ b, u(x, t) = \k,,,(x, t - %,,,)H(t - E&) + %,,,(x,t - %,,)H( t - fk,,,) -\ka,s(x,t - fA,,)H(t - G,,s) -%,,(x, t - %,,,)H(t - &,7) b< 1x1, ub, t) = %,z(x, t - A&f(t - %J FA,.,)H(t
-
+\ko,&, t - fA,,dH(t - &,,,) -\k,,&, t - W,,6)H(t - &,6) -\k,,+, t - $,,,)H(t - CA,,,) 1x1~a.
P-1)
P-2)
(9.3)
In (9.1)-(9.3) \ko,,(x, t) is the inverse Laplace transform for ;ji;,,,(x, S) defined by (cf. (11.2))
%&, 4 =
Kg(As + 1)
24
ed-l+
+Ao,,l
P-4)
and the A’s are defined in (11.1). The results (9.1)-(9.4) can, of course, also be obtained from (4-l)-(4.3) by a procedure similar to that used in the Appendix. It is evident that To Jx, S) is analytic for Re s > 0. Its inverse Laplace transform is therefore given by the inversion integral, thus:
\Irg,a(x,t)
=&/‘_yi”estTo+(x, Y ‘00
s)ds,
P-5)
where y is a positive constant. In order to evaluate this, we first write
q)&,
s) = ;i;O*,a(X, s) -I- G(s),
(9.6)
where
~~,(x,s)=e(s){exp[(-r+fs)A,,,] G(s) =
--I},
(9.7)
+ 1)
K&S
2s[
*
Then,
%,a(x, t>= %gx,0 + w,
(9.8)
where q&J X, t) and e(t) are the inverse Laplace transforms of 3&(x, and e(s) respectively. We note from (8.2) that As + -=
ST
x
1
F/m
+
s)
(9-9)
It follows (see, for example, eqns. (29.3.49) and (29.3.51) in [2]) that (9.10) where I,, and Ii are zeroth and first order modified Bessel functions of the first kind, q,&(x, s), defined in (9.7), has a simple pole at s = 0, a discontinuity on the segment ( - l/h, 0) of the negative real axis, and is single-valued and analytic elsewhere for -T c arg s < ?r. Accordingly, its inverse Laplace transform is given by the inversion integral (9.11)
Fig. 5. Modified Bromwich contour for inversion of Laplace transforms for Maxwellian fluid.
216 where y is a positive constant. This can be evaluated by applying Cauchy’s Theorem with the modified Bromwich contour shown in Fig. 5. There TA is the line Re s = y and I, r, c0 are circular arcs centered at s = 0; c and C are circular arcs centered at s = - l/X. I’ and F have radii R and q,, c, Z have radii E. We take the limiting situation when R + 00and E -, 0. We obtain ,k\ / esfT,&( x, s)ds = i7rK0A,,a,
co eS’T&(x, s)ds = 0, Lt c-0 J c-k.? (9.12)
eS1%&( x, s)ds = 0, Lt R-m / r+r e”q&(
x,
s)ds = 0,
In deriving the last of these results we use the notation /I = - Re s and note that iv/_
s_
on 1
(9.13)
i
-iv/_
on 7.
It follows from the inversion integral (9.11), by applying Cauchy’s theorem with the contour shown in Fig. 5 and writing (Y= X/3, that
*o+, 1)= - tW&,,a-
KOX G
1 j 0
(1 -
(y)l”
a3/2
exp - a(t+
x cos c*Ao,u - exp( $A,.,)]da,
cAo,a)
1 (9.14)
where (9.15)
5* =;/m. Then from (9.14) ‘I%%,
KJ t - ;A,,,) = - ?‘%Ao,~ - &&(x,
t),
(9.16)
217 where J&(X,
t) =il
(’ $“2
eVat/‘[cos l*A,,= - exp( iuAO,,)]dor.
(9.17)
Also, from (9.10),
From (9.8) ‘&,,(x,
t -
FA,,,) = !&‘$x, t - CA&++,
t - ;A&.
(9.19)
In the limit as t --) co, Jo+(x, t) = 0, so that, from (9.16), *,&(x,
t - ;A,,,)
(9.20)
= - OKRA,,+.
Also, from (9.18) it follows that as t --j co
0(x, t - CA,+,) -~(~)l”[l++)]. Then, from (9.19)-(9.21)
(9.21)
and (9.1)-(9.3)
we obtain
u(x, cc) = - &(A~,I
+ A,,, - A,,, - A,,,)
(a G 1x1Q b),
u(x, m) = - &(Ao,I
- A,,, - Ao,s - Ao,,)
(b G IN),
v(x, m)=
-~KO(AO,Z+A~,~-A~,~-A~,,)
(9.22)
(IN-).
With (11.1) we see that these results agree with (3.1), as they should. We now consider that the velocity field (3.1) with h = co is maintained in the fluid by an appropriate pressure gradient and that at time t = 0 this pressure gradient is removed. The resulting velocity field U(X, t) can be easily calculated from (2.5) with (9.1)-(9.3), (9.16)-(9.19) and (3.1). Since in this section h = co, we cannot write the expressions obtained for the velocity field in terms of the dimensionless quantities defined in (8.14). We note that the speed with which the wave-fronts are propagated is l/g. The distance travelled by a wave-front in time X is then X/E. We accordingly define the dimensionless quantities
(4
B, X,
v=
V;/K&
a,,) u=
=;(a,b, x, &,a), T= t/x,
(9.23)
U~/K&
We then obtain from (9.15)-(9.19) %X,(X,
T-
A,,,) = ‘@&(X,
T-
A,,,) + d( X, T - &J,
(9.24)
where
%&x(X, T- &,) = 6 ( X,
$,_(X T),
fil,,, -
A,,,) = 4e-@o..)/2
T -
(1 +
T-
6,,,)r,jT-;“-)
&,)l,(T-;ova)},
+(T-
(9.25)
with
-&,,(X9 T) =il (l-$“2 eeaT(cos[ &/m]
-
exp( &p)}da.(9.26)
Then, the dimensionless expressions for V( 2, F) can be easily read off from (9.1)-(9.3) by replacing \k,,,, x, t, ;A,, by Go,,, X, T, A,,, respectively. U( X, T) is then given by U(X,
T)=
V(X,
co)-
V(X,
T).
(9.27)
From (9.26) it follows that Go,JX, aT
1 (1 - (Y)1’2
T) =
p
-o J
emar(cos[ &,/m]
- exp( d,,,cu))dcu. (9.28)
Now, we can easily show by writing (Y= sin2$+ that 1 (1 - q2
J/2
J0 =
e
-‘+-&,)da
+e -u---i\,.,)/2
= $re
J0
“(1 + cos ‘p) exp[f(T-
-v-&..v2(
Io(
T-+9
+,,(
a,,,)
cos cp]d+ (9.29)
T-$..)).
Again, from (9.25), it follows that &(X,
TaT
ii,,,) =
+e*7-A0.0)/lj Io( T-P,*a)
Then, from (9.24), (9.25), and (9.28)-(9.30),
+I,(
T-2i\07”)}_
(9.30)
we obtain
%$F( X, T - Ao,,)= $ l1 (’a,$)“2cos[ &,{~~] epsT
da.
(9.31)
From (9.4) and the Initial Value Theorem it follows that
(
exp - &ho_
1
.
(9.32)
219 It follows from (9.1)-(9.3), (2.5) and (9.32) that at the front of each of the waves emanating from x = a there is a jump in u( x, t) given by
KCJ
-Texp
(
-&x-al).
At the front of each of the waves emanating u(x, t) given by
-KtJexp 2t
(
from x = b there is a jump
in
(9.34)
-&lx-bl).
Analogous results apply for the waves emanating from x = - a and x = - b. In each case the jump is defined as the value of U(X, t) immediately behind the front minus the value immediately ahead of the front. In terms of the dimensionless quantities defined in (9.23) we see from (9.33) and (9.34) that the jump in U( X, T) at the fronts of the waves emanating from X = A and X = B are given by - iexp( - 51X - Al) and - $ exp( - f )X - BI) respectively,. In the numerical example, the results of which are shown in Fig. 4, the wave-fronts have not yet reached the boundaries. Consequently, if appropriate allowance is made for the differences in the normalizations adopted in 58 and this section, the same results can be obtained by using the formulae for U derived in this section. 10. Vortex sheets in a Maxwellian fluid We have seen in 54 that in the case in which, in the limit t + co, there are two vortex sheets situated at x = + a, with vorticities T ijo, ti( x, S) is given by (4.4) and (4.5). If the fluid is Maxwellian then, with (8.2) i(x,s)=
-
c&(hs+ 1) 2s
_ {e I(lxlka) - e-3(lxl+a)} if a < 1x1, (10.1)
G&s+l) E(x, S) = 2s
_
_
{e Ra Ix’)+ e-S(a+lxl)} if 1x1< a,
where ~=Y\/I~.
(10.2)
We can rewrite (10.1) as E(x, S) = -O,(
x, S) exp[ - cs( 1x1 - a)]
+&(x,s) 5(x, s) = 0,(x, +0,(x,
exp[-Cs(lxl+a)],a
s) exp[ -Zs(lxI
- a)]
s) exp[ -Bs( IxI+ a)], IxI< a,
(10.3)
220 where
G,( x, s) = 62,~exp[(-3+ts)(lxl-u)], G,(x, s) =
Gj,
It follows from the Translation u(x,
t) = -0,(x, +0,(x,
u(x,
(10.4)
Fexp[(-5+cs)(lxl+n)].
t) = 0,(x,
t - iqlxj
- a))H(t
t - iqIxI+ t - Y(lxl-
Theorem
a))H(t a))H(t
that u( x, t) is given by
- iqlxl-
a))
- F(lxl+
a)),
- iqlxl-
(u < 1x1)
(10.5)
u))
+o,(x,t-E(lxl+ul)H(t-t(lxI+u)),
(Ixlu),
where O,( x, t) and O,( x, t) are the inverse Laplace transforms of 0,(x, s) and o,( x1 s) respectively. on the segment Since 0, and e, have poles at s = 0, are discontinuous ( - l/h, 0) of the negative real axis, and are analytic elsewhere, their inverse Laplace transforms can be calculated by means of the inversion integrals. These, in turn, can be evaluated by using Cauchy’s Theorem with the modified Bromwich contour shown in Fig. 5. We thus obtain, with (9.13),
&x, t), a)) = +ijo - $J*(x,t),
0,(x,
t - iqlxl -a)) = +Ljo-
0,(x,
t - iqlxl+
(10.6)
where
(10.7)
With these expressions u(x, t) is given by (10.5). From (lOS)-(10.7) we obtain
4%
00)’ i
0 I Go
(a+I)
(I-4 -4.
(10.8)
The velocity u(x, t) associated with the decay of a pair of vortex sheets can easily be obtained from (2.5) with (10.5)-(10.8). If x > 0, the first (second) term in (10.5), can be regarded as a wave which emanated from x = a (-a) at time t = 0 and travels in the positive direction of the x-axis. The first (second) term in (10.5), can be regarded as a wave
221 which emanated from x = a ( -a) at time t = 0 and travels in the negative (positive) direction of the x-axis. (Analogous interpretations of (10.5) can be made in the case when x < 0.) At the front of each of these waves there is a velocity discontinuity. The magnitudes of these discontinuities (value immediately behind the front minus value immediately ahead of the front) can be calculated by using the Initial Value Theorem. However, in order to do so we recognize that each of the waves emanating from x = a, say, represents the superposition of two waves, which emanate from x = a and x = b, in the limiting case when b - a + 0. The jumps in u(x, t) at the fronts of the waves emanating from x = a and x = b are given by (9.33) and (9.34) respectively, where - K~ is the initial velocity gradient in the region a < x < b. It follows that the jumps in U(X, t) at the fronts of the waves which emanate from x = a in the limiting case when b + a, K,, + co and (b - a)Ko = Go are given by I_t{k$[exp(--&lx--bl)-exp(-&1.x-al)]) = *
$ijo exp( - &lx
- al),
(10.9)
where the upper (lower) sign applies if x > a ( -c a). In terms of the dimensionless quantities defined in (9.23) we obtain from (10.9) the result that the jump in U( X, T) at the fronts of the waves emanating from X = A are given by ++exp(-+1X-Al).
(10.10)
It is evident, from the interpretation of u(x, t) in (10.5) as the superposition of waves, that, for a single vortex sheet of strength -ij, located at x = a, u(x, t) is given by u(x,
t)=
u(x,
t)=@,[x,
-O,[x,
t-i;(x-a)]H[t-C(x-a)] t-E(a-x)]H[t-F(a-x)]
(x7a), (x
(10.11)
Corresponding results for the velocity field U(X, t) associated with the decay of a single vortex sheet of strength - &, located at x = a can be easily obtained from (2.5). From (10.6),, (10.7), and (10.11) we obtain -‘ij
u(x, co)= i
2 O (x ’
+ijo
(x
a>
It then follows from (2.5) (lO.ll), i.e. for t > SIX - al,
u(x,
t) = T
21,(x, t),
(10.12)
(10.12) and (10.6) that behind
the fronts,
(10.13)
222 where Jr (x, t) is given by (10.7),, and the upper (lower) sign applies if x > a ( -C a). Equation (10.13) can be rewritten in terms of the dimensionless quantities defined in (9.23) as U(X,
T)=
+r(x,
(10.14)
T),
where J,( X, 7’) is given, from (10.7),, by j,(X,
T)=
~‘~e-“Tsin[(~X~-
(10.15)
l)/m]dol.
11. Appendix We now introduce
the notation
A m.1 = 2mh + 1x1- a,
Am,2 = 2mh + a - 1x1,
A m.3 = 2mh + a + 1x1,
A,,, = 2mh - a - 1x1,
A ms = 2mh + 1x1 - b, A m.7 = 2mh + b + 1x1,
A,,, = 2mh + b -‘/xl,
(11.1)
A,,, = 2mh - b - 1x1,
F = p’/*y 7 and K&b \k,,Jx,
s) =
+ 1) exP[(-l+
2.4
(11.2)
iW,,,j.
Then taking Re { > 0 for Re s > 0, it follows from (3.5)-(3.10), that for Re s 2 0 8(x, s)=
g (-l)“{!Fn,, n=O -\kn+1,4
+K+r,s
exp( - CsAn,t) + k,,3 exp( - CsA,,,)
exP(-~~An+,,.J
- \kn,6 exp(
with (8.2),
-K+1,2
exP(-wz+,.2)
- “sA~,~) - G,,, exp( - ZsA,*,)
exp(-~~A,+I,~)+~,,+~,~ex~(-~~A,+I,J} u
E(x,s)=
E (-l)“{!&,r n=O -%+1,4
(11.3)
exp( - CsAn,r) + q”,3 exp( - FsA~,~)
exp(-~~An+lA)-~~+1,2ex~(-~~A,+,,2)
- T”,5 exp( - VSA~,~) - \k,,, exp( - EsAn,,) ++n+r,*
exP(-YSA,+1,8)+\kn+1,6
exP(-6sA,+,,,)} b i 1x1G h,
(11.4)
223 E(x,s)=
2 (-l)“{\JI,,,exp(-CsA,,,)+\k,,,exp(-FsA,,,) n=O -JI.+,,,
exp(-vsA,+,,,)-\Ir,+l,l
ex~k~sA,+,,,)
-\kn,6 exp( - i;sA,,,) - ;2;,,, exp( - 5sA,,7) +%,8
exp(-+A,+,,d
+ %+l,5 exp(-isA,+,,,)} IXI< a.
We denote the inverse Laplace transform of k,Jx, the unit step function by H( ). It then follows from the Translation Theorem that U(V)=
S) by \~,Jx,
f) and
a
(11.6)
b < 1x1~ h
(11.7)
2 (-l)“{\Ir,,,(x,t-~A,,,)H(t-tA,,,) n=O +\kn,&,
u(v)=
(11.5)
t - FA,.,)H(t
- FAn.3)
- %+1,4 (x> t - EAn+&H(t
- FA,,,,,)
- *PI+,,2 (XT t-
- EA,+,z)
cA,+*,M(t
- *,Jx,
t - fA,,@(t
- VA,,,)
-‘I’,&,
t - fA,,,)H(t
- CA,,,)
+ %+I,8 (XT t - fA,+,,,Mt
- fan+*,*)
+ %I+*,5 (XT t - ~An+l,S)Hf
- ~A,+,,,))
9
E (-1>“{~~.~tx,t-~A,,~)~tt-~A,,,) n=O +
\kn,&, t - EA,,#(t - gA,,,3)
(x9t - ~An+l,dWf- ~An+1,.4) - %+1,4 -%+1,2(XTt - %+*,*)~(f - %+1,2) -*&, t - %,#tt - &,s) -*&,
t - %,,)Htt
- %,,,)
tx,t- %,+&H(t - %r+u) + \kn+*3 + %+I,6(x, t -
;A n+l.CM(f
- ~An+1,6)),
224
Each of the terms in (11.6)-(11.8) represents a disturbance which propagates with speed l/5. We consider the disturbance given by a term !Pm,,( x, t - FA,,,)H( t - ;A,,,). This is zero for t -Z EAm,a and has the value at the \k,,,,(x, t - CA,,,) for t a FA,,,+. The magnitude of the discontinuity value of x given by A,,_ = t/E can be calculated ‘by using the Initial Value Theorem. We have from (11.2) and (8.2) (11.9) We note from (11.1) and (11.2) that q&,s)=
the upper (lower) sign being taken if x is positive (11.9) and (8.2) the Initial Value Theorem yields *;,Jx,
(11 JO)
r(-l)“(-{+5s)k,,,(x,s),
o)=
+(-l)“!$ exp( -=$L+
(negative).
From (ll.lO),
(11.ll)
We now consider x > 0. The terms in (11.6)-(11.8) involving qO,, and \k0,2 can be interpreted as waves emanating from x = a at time t = 0 and travelling in the positive and negative directions of x respectively. Those involving \Ib,5 and \k0,6 can be interpreted as waves emanating from x = b at time t = 0 and travelling in the positive and negative directions of x as waves respectively. The terms involving *0,3 and \ko4 can be interpreted emanating from x = - CI at time t = 0 and travelling in the positive and negative directions of x respectively. Those involving \k0,7 and q,,* can be interpreted as waves emanating from x = -b at time t = 0 and travelling in the positive and negative directions of x respectively. The terms involving \k,,, with m # 0 represent the contributions of these waves to the resultant velocity field after they have undergone reflections at the boundaries at x= fh.
226 where the upper (lower) sign applies to the wave travelling in the positive (negative) direction of x. Acknowledgement The results in this paper were obtained in the course of research sponsored by the U.S. Army Research Office under Contract No. DAAG 29-82-K-0026 with Lehigh University. References 1 R.S. Rivlin, J. Non-Newtonian Fluid Mech., 14 (1984) 203-217. 2 M. Abramowitz and LA. Stegun @is.), Handbook of Mathematical Bureau of Standards, 1964. 3 H. Lamb, Hydrodynamics, Dover, New York, 1945.
Functions,
National
199
DECAY OF SHEAR LAYERS AND VORTEX SHEETS
R.S. RIVLIN Lehigh Universiv,
Bethlehem, PA (U.S.A.)
(Received November 8, 1983)
Summary
An incompressible fluid is contained in the domain between two stationary infinite parallel rigid plates. It is assumed that for shear flows, the shear stress in an element of the fluid depends linearly on the history of the velocity gradient in that element. It is supposed that initially two steady shear layers exist in the fluid and are symmetrically disposed with respect to the mid-plane. The time-dependent velocity field which results from the removal of the forces maintaining this steady flow is calculated in the cases when the fluid is Newtonian and when it is Maxwellian. The limiting cases when the shear layers reside in an unbounded space of the fluid and when they further become vortex sheets are discussed.
1. Introduction
In a previous paper [l] we considered the time-dependent rectilinear flow of an incompressible viscoelastic fluid in a domain bounded by stationary infinite rigid parallel plates. It was assumed that prior to time t = 0 the fluid is at rest and at time t = 0 is subjected to a longitudinal pressure gradient which is subsequently held constant. The pressure gradient was assumed to depend in an arbitrary manner on distance from the mid-plane and an expression for the Laplace transform of the resulting velocity was obtained. It was assumed that the fluid is such that the relation between the shear stress and the history of the velocity gradient in an element of the fluid is linear. In [l] this result was applied to the problem of run-up of plane Poiseuille flow, i.e. the case when the pressure gradient is independent of position, in the particular cases when the fluid is Newtonian, or Maxwellian. The
200 dependence of the velocity on position and time was obtained by inverting its Laplace transform. This result was then used to calculate the time-dependent velocity field which results when the initial conditions are those of steady plane Poiseuille flow and the pressure gradient supporting it is removed. In the present paper, the expression for the Laplace transform of the velocity field obtained in [l] is used to calculate the velocity field which results when the pressure gradient is such that the steady velocity ultimately attained at time t = oo consists of two shear layers, symmetrically disposed with respect to the mid-plane, in which the velocity gradients are equal in magnitude but opposite in sign, but are constant in each shear layer. Results are obtained for both Newtonian and Maxwellian fluids. These results are used to calculate the time-dependent velocity fields which result when the shear layers are present initially and at time t = 0 the forces maintaining them are removed. The results for a Newtonian fluid are given in $5 and those for a Maxwellian fluid in 58. In the case when the fluid is Newtonian the velocity field spreads diffusively throughout the fluid (Fig. 2), decaying to zero at infinite time. When the fluid is Maxwellian the velocity field may be interpreted as the superposition of eight waves which originate at time t = 0 at the boundaries of the shear layers and are reflected back and forth from the rigid boundaries. Two of these eight waves originate from each boundary of the shear layers. One of these travels in the positive and the other in the negative direction (Fig. 4). The special cases when the initial shear layers exist in an unbounded space of Newtonian or Maxwellian fluid are discussed in $6 and 59 respectively. The limiting cases in which these shear layers are vortex sheets are discussed in 57 and 510 respectively. From the results in 59 and 510 for a Maxwellian fluid the velocity field associated with the decay of a single shear layer or vortex sheet in an unbounded space of Maxwellian fluid can be read off easily. This is done explicitly in $10 for the case of a single vortex sheet. From this result the time-dependent velocity field associated with the decay of an arbitrary initial longitudinal velocity distribution could be calculated by simple integration. 2. A basic result We consider an incompressible fluid to be contained between two fixed infinite parallel plates situated at x = f h in some rectangular Cartesian coordinate system x. We suppose that the fluid is initially at rest and that it is subjected at time t = 0 to a pressure gradient P(x) in a direction parallel to the plates, which is independent of t for t > 0. It is easy to show that a
201 possible resulting flow field is one in which the velocity is longitudinal at each point. Let u(x, t) be this velocity and let E(x, S) be its Laplace transform. We suppose that the shear stress o(t) in the fluid is related to the velocity gradient history u’(x, T) (- 00 -C 7 < t) by * a(t)
=Jt
f(t-CO
+‘(x,
r)dr,
(2.1)
where the prime denotes differentiation with respect to x. It was shown in [l] that if P(x) is an even function of x and we introduce the notation
s= (g$‘z>
(2.2)
then
G? 4 --L(cosh
[x[o(/3)
ssf(s)
- i’P(
sinh @dP
p) sinh @d/3 + tanh [hl’P(
- sinh lx
xP( p) cash #dP
J0
The corresponding velocity version integral, thus:
u(x, t) = &/Y_t’mes5(x,
j3) cash SpdS]
.
(2.3)
field can be obtained
s)ds,
Y 100
by means
of the in-
(24
where y is a positive constant such that U(x, S) is analytic for Re s > y. We now suppose that the fluid, instead of being initially at rest, is in a steady state of longitudinal flow with velocity u(x, cc). At time t = 0 we remove the pressure gradient which supports this flow, i.e. we apply an additional pressure gradient -P(x). The resulting velocity field U(X, t) is then given by 24(x, t) = u(x,
co) - u(x,
t).
(2.5)
* We shall interpret the relation (2.1) as u=
‘+f(t-+‘(x,7)dq Lt f-r0 I --m
r>O.
This allows us to include as special cases Newtonian exhibit instantaneous elasticity.
fluids and other fluids which do not
202 3. Uniform shear layers In this section we consider run-up in the case when the steady flow which is attained at time t = cc is one in which there are two regions of uniform (non-zero) velocity gradient, separated from each other and from the boundaries by regions in which the velocity gradient is zero. We suppose that the regions of non-zero velocity gradient are symmatrically disposed with respect to the mid-plane and occupy the domains x = [a, b] and x = [--a, -b], where h > b B a > 0. Let - Q, and K,, be the velocity gradients in these regions. Then,
u(x, co) =
K,,(b-X)
b>,X>U,
K&--U)
IXI
%b 0
-aax>,
+ b)
h a 1x12 b.
We may express the corresponding u’(x, cc)=
(3.1)
-b,
velocity
gradient
field in the form
-K,,{H(x-u)--(x-b)+ff(x+u)-H(x+b)},
(3 *2)
where H( ) is the Heaviside unit step function. The equation of motion for steady shearing flow of the fluid fluid is u”(X, X)0)= 9(x)//A, where p denotes
(3.3)
the steady-state
viscosity
of the fluid. Then, from (3.2) and
(3.3), P(X)=pK,,{ti(X-a)-6(x-b)+S(x+u)-S(x+b)},
where S( ) is the Dirac delta function. (2.3) we obtain
(3.4)
Using (3.4) to substiute
for P( /3) in
;(x,s)=~~~(x,s),
ifa
(3 -5)
;(x,s)=?@~(x,s),
ifb<]xl
(3 -6)
ti(x,s)=~$(x,.s),
ifIxI
(3.7)
where
@,(x,s)=
~~(s)rosh~I{coshSusinhI(h-Ixl)-coshSxsillhS(h-b)}, -
203 @*(x,s)=
sf(s) -
$(x,s)=
rr(~fo~ -
kosh Sh {sinh sh { sinh
From (3.8)-(3.10)
S( h - lxl)(cosh So - cash @)I y
c( h - u) - sinh S( h - b)} .
(3.9) (3.10)
it follows that as I[1 + 0
J(+%(x,
4 = (b - Ixl) + O(lS12),
f(.+D*(x,
s) =X2@
f(+&(x,
s) = (b - 4 + O(1512).
- Ma2
- b2) + 0(lS14),
(3.11)
4. Some special cases The case in which the body of fluid, in which the run-up of the shear layers takes place, occupies the whole of three-dimensional space is obtained by letting h + cc in equations (3.8)-(3.10). We note that 5, defined by (2.2), is double-valued. However, each of the expressions for is(x, S) given by (3.5)-(3.10) is unaltered if one of these values is replaced by the other. In order to obtain asymptotic expressions for C( x, S) which are valid in the limiting case h = 00, we need to choose one or other of the branches for 2. We choose the branch for which Re 3 > 0 when Re s >, 0. We make a branch cut on the negative real axis and consider the s-plane for which -r < arg s =Gr. Then for h = cc we obtain from (3.5)-(3.10) for Re s > 0 U(x, S) =_
S;:)’
e Slxlcosh{a-e-3bcoshlx), -
ifa
(4.1)
-s’“l(cosh {a - cash lb),
if b < 1x1~ cc,
(4.2)
PKo E(x, S) = pe SSJb)
E(x,s)=-!%cosh{x(e-r”-e-lb)
iflxl
dfb)
(4.3)
We now consider the further limit when b - a + 0 and K,, --) 00, while (b - U)Ko = ij,, a constant. The system then becomes one in which, in the limit t + cc, there are two parallel vortex sheets situated at x = +a, with vorticities TijO. We then obtain from (4.2) and (4.3) E(x,s)=
--
2g)ie -
3(1X1-0) _ e-S(lxl+a)}
if
a
1x1,
<
S
u(x, s) =
-_!I$!$(e-SCa-lxl)
+ e-T(O+lxl)}
if
1x1 <
a.
(4.4 (4.5)
204 5. Newtonian fluids In the case when the fluid is Newtonian f(t-7)=@--),
f(s)=q,
and has viscosity
PL=77,
77,we have (5.1)
and from (2.2)
It is then evident from (3.5)-(3.11) points for which s=O
that the poles of iJ(x, S) are located at the
orcoshSh=O.
(5 03)
With (5.2) the second of these equations s,=
-$(2n-1)2?rZTj/ph2
(n=l,2,...).
yields the solutions
s = s, where (5.4)
It follows that G(x, S) is analytic for Re s 2 0, except at s = 0 where it has a simple pole. U(x, t) is therefore given by the inversion integral (2.4), which can be evaluated by applying the Residue Theorem with the Bromwich contour of Fig. 1. There AB is the line Re s = y and lY is a circular arc of radius R centered at the origin and we take the limiting situation when R = co. We note from (3.5)-(3.10) that Lt
R-cc
Jr
e%(x,
s)ds = 0.
(5 -5)
Let K, and K,, be the residues of e’%( x, S) at s = 0 and s = s, respectively. Then, u(x,
f)=K,+
f
K,.
(5 -6)
n=l
Fig. 1. Bromwich contour for inversion of Laplace transform of u for Newtonian fluid.
205 We find from (3.5)-(3.11),
K,=
with (5.2), that
0
if b < 1x1~ h,
K,,(&-(XI)
ifa
(5 -7)
if 1x1f a,
i ‘Co@ - a> and 8K,h
K, =
cos (2n - ex 2h
(2n - 1)27r2 cos
en- w _cos on- lb
(W
2h
We now consider the decay of the velocity field (3.1) if the pressure gradient maintaining it is removed. Then, the resulting velocity field U(X, t) is given by (2.5). From (3.1) and (5.7) we have u(x,
co) = K,.
(5 -9)
Then (5.6), (5.9) and (2.5) yield 2.4(x, t)=
-
(5.10)
E K,. n=l
We can rewrite quantities X=x/h,
eqns.
A = a/h,
(5.8) and
B = b/h,
(5.10)
T= qt/ph’,
in terms
of the dimensionless
U= u/tc,,h.
(5 .ll)
We obtain
u= -
eii,,
(5.12)
n-1
0.2
0.6
0.4
0.8
1.0
X
Fig. 2. U vs. X for Newtonian fluid for various values of T with A = 0.2, B = 0.5. I. T = 0; II. T= 0.05; III. T= 0.2; IV. ‘l- = 0.5.
206 where 8 (2n - l)*a* X{cos[(n-f)rB]
cos[(n -f)7rX] -cos[(n-f)?rA]}
exp[-(n-+)*a*T].
(5.13)
As an illustration of this result, in Fig. 2 U is plotted against X, for the case when A = 0.2 and 3 = 0.5, for various values of T and we note that the decay of the shear layer takes place smoothly in a diffusive manner. 6. Newtonian fluids (h = 00) With (5.1), eqns. (4.1)-(4.3)
E(x, s) =-eKo+‘(cosh d
yield
la - cash lb),
b < 1x1~ h;
(6-2)
Ix/< a,
u(x, s) =
where l is given by (5.2). Each of the expressions (6.1)-(6.3) has a discontinuity on the negative real axis. Therefore, we cannot evaluate the corresponding inversion integrals by using the Residue Theorem, as was done in the case when U(x, s) is given by the expressions (3.5)-(3.10). We note also that each of the expressions (6.1)-(6.3) has a pole at s = 0 and is analytic and single-valued everywhere else for -7r < arg s < 7r. We can therefore evaluate the inversion integrals by using Cauchy’s theorem with the modified Bromwich contour shown in Fig. 3. There C and c are arcs of circles, centered at s = 0, with radii R and E respectively and we take the limiting case R --) 00 and E + 0. We easily see, from (6.1)-(6.3), that
et\
jest6 (x, s)ds = -2i%,(b-1-x))
c
Lt e%(x,s)ds=O e-0 / c
ek\/e’s(x,s)ds=
c
(aglxlgb),
(6.5)
(b
-2imc,(b-a)
(6.4)
(IxI
(6.6)
207
Fig. 3. Modified Bromwich contour for inversion of Laplace transform of v for unbounded space of Newtonian fluid.
Also, for each of the expressions for E(x, S) in (6.1)-(6.3) Lt
Jc
R+cc
e%(x,
s)ds = 0.
(6.7)
In order to calculate the contributions of the integrals on I and i to the inversion integral, we write a = - Re s and note that on 1
i vcu’/2
I=
i -
irxP2
(6.8)
on 5,
where
( p/7Jp2.
v=
(6.9)
Then we obtain from each of the expressions (6.1)-(6.3)
JI+i
e%(x,
2iK0 s)ds =--v-1(x,
t),
for C(x, S) (6.10)
where
I(
X,
t) = ~~~~~~
vxa”‘(cos vaa”’
- cos vba’/2)da.
(6.11)
The expression (6.11) for 1(x, t) may be rewritten as
1(x, t) = 2/* -cos e-cr2’ 0 a2
vxa(cos vua - cos vba)da.
(6.12)
208 From (6.12) we obtain 3%
(see eqn. (7.4.6) in [2])
t) = - 2 OOOe-a” cos VX(Y(COS YU~ - cos vba)da /
at
= -~/gme~a2r[cosva(x+n)+cosva(x-a) -cos
va(x
+ b) - cos Y~(X - b)]da
0 i ?T
c-3
-
-exp
t
-
exp -
2(x + u)’ 4t
2(x + b)* 4t
i
1 i
+exp -
- exp i
i
u(x,
t)=
U(X,
t)=K&-U)--1(x,
u)’
4t i
V”( x - b)* 4t
From Cauchy’s Theorem with the contour and (6.10), the inversion integral (2.4) yields U(X,f)=K&‘-IXI)+(X,t)
V’(X -
)I ’
in Fig. 3 and eqns. (6.4)-(6.7)
(6.14)
(U
-21(x,
t)
(6.15)
(b
t)
(6.13)
(Ixl
(6.16)
We now consider the decay of the velocity field (3.1) with h = co if, at time t = 0, the pressure gradient maintaining it is reduced to zero. Then, the resulting velocity field u(x, t) is given, from (2.5), (3.1) and (6.16) by
24(x,t)
=21(x,
(6.17)
t).
7. Vortex sheets in a Newtonian fluid We have seen in 94 that in the case in which, in the limit t + 00, there are two vortex sheets situated at x = + a, with vorticities T ij,, G( x, S) is given by (4.4) and (4.5). If the fluid is Newtonian we obtain, with (5.1), qx,
s)
=
E(x,s)=z{e
_
$J {e-Swl-aL -ATa-1.4)
e-S(14+4},
+ e-sc~+l~l)},
u
<
(7.1)
IxI,
Ix1 < a,
where 5 is given by (5.2). Taking the inverse Laplace
P-2) transforms
of (7.1) and
209 (7.2) we obtain, u(x,
t)=
with the notation
-+tij,{erfc[jv(Ixl-LZ)~-“~] -erfc[$v(lxl+
u(x,
(6.9),
t) = +L&{erfc[$v(a +erfc[$v(a
a)t-“2]},
1x1> a,
(7.3)
Ix)< a.
(7.4)
- Ixl)t-1’2] + l~l)i-‘/~]},
From (7.3) and (7.4)
u(x, oo)=
I-4 > a 1x1-=a-
0,
l
_ ql,
(7.5)
With (2.5) and (7.5) it follows that the velocity the decay of a pair of vortex sheets, situated vorticities f Go, is given by 24(x, t) = $ijo{erf[+v(lxl
+ CZ)~-“~]
IxI> a,
-erf[$Y(IxI-a)t-“2]}, u(x,
field u(x, t) associated with at +a and having initially
(7.6)
t) = +GO{erf[+Y(IxI + LZ)~-‘/~] +erf[+Y(IxI
- ~2)t-“~]},
IxI<
a.
Since erf(0) = 0, it follows from (7.6) that U(X, cc) = 0, as it should. Also, since erf(cc) = 1, it follows from (7.6) that U(X, 0) = 0 for (xl > a and ijo for 1x1~ a, as it should. From (7.6) we obtain al.4
at=
1 f(lxI--a)exp -[ v’( 4ru)’ II, -
w 4&3/Z
( Ix1 + u) exp
i
-
“1’1”4:
a)’
[
1x1 -
(7.7)
the upper (lower) sign applying if Ix/> a (-c a). This result can also be obtained from (6.17) and (6.13) in the limiting case when b - a + 0, ~~ + 00, with IC~(~ - a) = 3,. 8. Maxwellian fluids We now suppose that the fluid is Maxwellian f(t
- T) = ;e+Q/A,
where 17and X are positive constants. f(s)
and that f( t - T) is given by
=&
5 = Y[S(hS + l)]“‘,
(8.1) Then, with (2.2), /J = q,
(8.2)
210 where y = (P/11Y2 V(x, S) is given by (3.5)-3.10). poles at the points at which s=O,
Each of these expressions
(8.4
cosh{h=O.
The last of these equations r2(2n
yields, with (8.2) 2,
- 1)2
(n = 1,2,...). 4v2h2 Thus the poles of i( x, S) occur at s(Xs+l)=
-
(8.3) for 6(x, S) has
(8.5)
s = 0,
s=,ps-
n
(8.6)
=- 2LJ-lf[1-a%(2n-1)2]1’2],
where
x
L=-
(8.7) v2h2 ’ We note that all of these poles lie in the half of the s-plane for which Re s G 0 and that, except at the poles, C(x, S) is single-valued and analytic. Let m be the largest value of n for which 1 - 7PL(2n
- Q2 >, 0.
(8.8) Then, for n < m, sz, s; lie on the negative real axis. For n > m, s,’ and s; are complex conjugates lying on the line Re s = - 1/2X. As in the case when the fluid is Newtonian, u(x, t) is given by the inversion integral (2.4), which can, in turn, be evaluated by using the Residue Theorem with the Bromwich contour of Fig. 1. We denote the residues at s = 0, s,’ , s; by K,, Kz, K; respectively. Then, u(x,
t)=K,+
f (K,+ +K,). n=l We find, from (3.5)-(3.11), with (8.2), that
K,=
JG,(b-~xI),
ablxl
0,
b < 1x1
i Ko(b - a),
(8 -9)
(8.10)
1x1=Ga
and K+
_
n-
%l(W + 1) hv2s,+(2Xs,f
cos[(2n
- l)#os[(2n
- l)E]
+ 1)
I}
esir,
(8.11)
2K,(hS,
K,- =
+ 1)
hv2s,(2hs,
cos [ (2n
+ 1)
- l)E
I}
(2n - 1) g
-cos
- 1)3cos[(2n
es;‘.
We now consider the decay of the velocity field (3.1) if the pressure gradient maintaining it is removed. Then, the resulting velocity field U(X, t) is given by (2.5). From (8.6), (8.9) and (8.11) u(x,
(8.12)
00) = K,
and from (2.5), (8.9) amd (8.12) 24(x, Q=
-
g
(K;
(8.13)
t-K,-).
We define the dimensionless A=a/h, q+,s,-
B=b/h, =Xs,+,hs,,
quantities
X=x/h, u=u
T=t/X, (8.14)
K,hL ’
Then, from (8.6) *, S+,S;=~(-1+[1-a2L(2n-1)2]1/2),
(8.15)
and (8.13) can be rewritten U(X,
T)=
-
as (8.16)
$ (R,‘+R,), n=l
where k+
n-
k-
_
= n
2(%++ 1) s,+(2s,+
+ 1)
-COs[(n
-
+ 1)
-cos[(n -
-t)vX]{cos[(n-+rA]
(8.17)
f)~7B]}e~~~,
2(&i-+ 1) s,-(2s,-
co&l
cos[(?z -+X]
{cos[(n
- +A]
+)vB]}esLT.
We note that the expression (5.10) for U(X, t) when the fluid is Newtonian can be obtained from (8.6), (8.11) and (8.13) by taking X = 0. We then have, with (8.7) and (8.3), s,+=s,,
s,=-00,
(8.18)
212 where s, is given by (5.4). From (8.11) we obtain K; =K,,
K,
(8.19)
=O,
where K,, is given by (5.8) and hence U(X, t) is given by (5.10). It is seen in the Appendix (eqns. (11.6)-(11.8)) that the velocity field u(x, t) can be regarded as the superposition of a number of waves. These originate at x = + a and x = +b at time t = 0 and are reflected back and forth at the boundaries. From each of these positions waves travel in both directions with speeds l/E, where C = h‘/‘Y. At the front of each wave there is a discontinuity in the velocity u( x, t) the magnitude of which can be easily obtained from (11.6)-(11.9). If the time t is sufficiently small and the point x is sufficiently close to a so that only the front leaving x = a has had time to reach x, then the velocity field u(x, t) at x is given by (11.12). When the front is at x, the velocity u(x, t) immediately behind the front is given by (11.13), and its spatial gradient u’(x, t) by (11.13),. In the case when only the fronts leaving x = a, or x = b, or both, have had time to reach x, the velocity u(x, t) is given by (11.14). From these results, corresponding results for u(x, t) can be obtained by using (2.5) and (3.1). Since U(X, 0) = u( x, cc) is continuous, it follows that the jump at a front of the velocity U(X, t) is the negative of that at the corresponding front of the velocity u(x, t). The results obtained can be written in dimensionless form by using (8.14). We find that for the waves originating at x = a at time t = 0 the jump in the dimensionless velocity U( X, T) (defined as the velocity immediately behind the wave-front minus the velocity immediately ahead of it) is given, from (2.5), (8.14) and (11.13), by 1
- -exp
2 Lii2
[
- ---&IX-J].
(8.20)
Similarly for waves originating at x = b the jump in U( X, T) is given by 1 -exp 2 L’12
[
- &IX-
811.
(8.21)
We note that the dimensionless speed with which the wave-fronts propagate is L112. As an example, in Figs. 4(a)-(d) the calculated values of U are plotted against X, in the range (0, l), for T = 0.05, 0.1, 0.2, 0.4 and L = 0.25, i.e. the dimensionless speed is 0.5. In each figure the U vs. X curve for T = 0 is shown for purposes of comparison. For all of the values of T for which the U vs. X curves are plotted in Fig. 4, the waves emanating from x = -a and x = -b have not had time to reach positions in the range X = (0, l), nor have the waves emanating from x = a and x = b had time to reach the
213 1.5
-
I.5 T-0.1
T=.05 1.0
1.0
-
b.5
L_._--“0 0.6
-0.5
X
‘-F 0.2
1.0
0.8
- 0.5
\
-1.0
-
X
1.0
0.8
(b)
-1.0
(a)
0.6
I.5 T -
0.2
T-0.4
I I.0 -
0.5
\
U
0
I 0.2
-1.0
L
0.6
(C)
Fig. 4. U vs. X for Maxwellian
x
0.8
1 I .o
u
I 0.6
0
-1.0~
X
0.8
, 1.0
(d)
fluid for various values of T with L = 0.25, A = 0.4, B = 0.5.
boundaries. Fig. 4(a) shows the situation when the fronts of the waves emanating from x = a and x = b have not yet met. Fig. 4(b) shows the situation when they have just met at x = +(a + b) and Figs. 4(c) and (d) show situations when they have passed each other. The directions of the jumps in the velocity U, which occur immediately after removal of the forces maintaining the initial shear flow in the region X= (0.4, OS), may at first sight seem curious. This is less so when one realizes that in order to maintain the initial flow, the force system which must act on the fluid in the region X= (0.4, 0.5) must consist not only of a uniform pressure gradient in the region X= (0.4, 0.5), but also of surface tractions acting tangentially on the fluid in the region. The traction acting on the surface X= 0.4 is in the positive direction and that acting on the surface X= 0.5 is in the negative direction. The tractions which act on the surfaces X = 0.4 and X = 0.5 of the fluid in the regions X = (0, 0.4) and (0.5, 1) respectively are zero.
214 This situation, in which there are discontinuities in the tangential component of the stress at the surfaces X = 0.4 and 0.5, is admittedly artificial. It reflects an idealization of the initial conditions similar to that which was made by Lamb [3] in discussing the decay of a vortex sheet in a Newtonian fluid. 9. Maxwellian fluid (R = co) In the case when the shear layers exist in an infinite space of Maxwellian fluid, we obtain an expression for u(, t) by letting h + 00 in eqns. (11.6)-(11.8) of the Appendix. Then for all finite times H(t - CA,,,) = 0 unless M = 0. Accordingly u(x, t ) is given by
u(x, t) = ‘I&,(x, t -
:A& +‘$,,&, t - %,,,)W - F&J -‘K,,(x, t - :&)H(t - fiA& -\ko,&, t - &,,)H(t - fA,,,) a < 1x1~ b, u(x, t) = \k,,,(x, t - %,,,)H(t - E&) + %,,,(x,t - %,,)H( t - fk,,,) -\ka,s(x,t - fA,,)H(t - G,,s) -%,,(x, t - %,,,)H(t - &,7) b< 1x1, ub, t) = %,z(x, t - A&f(t - %J FA,.,)H(t
-
+\ko,&, t - fA,,dH(t - &,,,) -\k,,&, t - W,,6)H(t - &,6) -\k,,+, t - $,,,)H(t - CA,,,) 1x1~a.
P-1)
P-2)
(9.3)
In (9.1)-(9.3) \ko,,(x, t) is the inverse Laplace transform for ;ji;,,,(x, S) defined by (cf. (11.2))
%&, 4 =
Kg(As + 1)
24
ed-l+
+Ao,,l
P-4)
and the A’s are defined in (11.1). The results (9.1)-(9.4) can, of course, also be obtained from (4-l)-(4.3) by a procedure similar to that used in the Appendix. It is evident that To Jx, S) is analytic for Re s > 0. Its inverse Laplace transform is therefore given by the inversion integral, thus:
\Irg,a(x,t)
=&/‘_yi”estTo+(x, Y ‘00
s)ds,
P-5)
where y is a positive constant. In order to evaluate this, we first write
q)&,
s) = ;i;O*,a(X, s) -I- G(s),
(9.6)
where
~~,(x,s)=e(s){exp[(-r+fs)A,,,] G(s) =
--I},
(9.7)
+ 1)
K&S
2s[
*
Then,
%,a(x, t>= %gx,0 + w,
(9.8)
where q&J X, t) and e(t) are the inverse Laplace transforms of 3&(x, and e(s) respectively. We note from (8.2) that As + -=
ST
x
1
F/m
+
s)
(9-9)
It follows (see, for example, eqns. (29.3.49) and (29.3.51) in [2]) that (9.10) where I,, and Ii are zeroth and first order modified Bessel functions of the first kind, q,&(x, s), defined in (9.7), has a simple pole at s = 0, a discontinuity on the segment ( - l/h, 0) of the negative real axis, and is single-valued and analytic elsewhere for -T c arg s < ?r. Accordingly, its inverse Laplace transform is given by the inversion integral (9.11)
Fig. 5. Modified Bromwich contour for inversion of Laplace transforms for Maxwellian fluid.
216 where y is a positive constant. This can be evaluated by applying Cauchy’s Theorem with the modified Bromwich contour shown in Fig. 5. There TA is the line Re s = y and I, r, c0 are circular arcs centered at s = 0; c and C are circular arcs centered at s = - l/X. I’ and F have radii R and q,, c, Z have radii E. We take the limiting situation when R + 00and E -, 0. We obtain ,k\ / esfT,&( x, s)ds = i7rK0A,,a,
co eS’T&(x, s)ds = 0, Lt c-0 J c-k.? (9.12)
eS1%&( x, s)ds = 0, Lt R-m / r+r e”q&(
x,
s)ds = 0,
In deriving the last of these results we use the notation /I = - Re s and note that iv/_
s_
on 1
(9.13)
i
-iv/_
on 7.
It follows from the inversion integral (9.11), by applying Cauchy’s theorem with the contour shown in Fig. 5 and writing (Y= X/3, that
*o+, 1)= - tW&,,a-
KOX G
1 j 0
(1 -
(y)l”
a3/2
exp - a(t+
x cos c*Ao,u - exp( $A,.,)]da,
cAo,a)
1 (9.14)
where (9.15)
5* =;/m. Then from (9.14) ‘I%%,
KJ t - ;A,,,) = - ?‘%Ao,~ - &&(x,
t),
(9.16)
217 where J&(X,
t) =il
(’ $“2
eVat/‘[cos l*A,,= - exp( iuAO,,)]dor.
(9.17)
Also, from (9.10),
From (9.8) ‘&,,(x,
t -
FA,,,) = !&‘$x, t - CA&++,
t - ;A&.
(9.19)
In the limit as t --) co, Jo+(x, t) = 0, so that, from (9.16), *,&(x,
t - ;A,,,)
(9.20)
= - OKRA,,+.
Also, from (9.18) it follows that as t --j co
0(x, t - CA,+,) -~(~)l”[l++)]. Then, from (9.19)-(9.21)
(9.21)
and (9.1)-(9.3)
we obtain
u(x, cc) = - &(A~,I
+ A,,, - A,,, - A,,,)
(a G 1x1Q b),
u(x, m) = - &(Ao,I
- A,,, - Ao,s - Ao,,)
(b G IN),
v(x, m)=
-~KO(AO,Z+A~,~-A~,~-A~,,)
(9.22)
(IN-).
With (11.1) we see that these results agree with (3.1), as they should. We now consider that the velocity field (3.1) with h = co is maintained in the fluid by an appropriate pressure gradient and that at time t = 0 this pressure gradient is removed. The resulting velocity field U(X, t) can be easily calculated from (2.5) with (9.1)-(9.3), (9.16)-(9.19) and (3.1). Since in this section h = co, we cannot write the expressions obtained for the velocity field in terms of the dimensionless quantities defined in (8.14). We note that the speed with which the wave-fronts are propagated is l/g. The distance travelled by a wave-front in time X is then X/E. We accordingly define the dimensionless quantities
(4
B, X,
v=
V;/K&
a,,) u=
=;(a,b, x, &,a), T= t/x,
(9.23)
U~/K&
We then obtain from (9.15)-(9.19) %X,(X,
T-
A,,,) = ‘@&(X,
T-
A,,,) + d( X, T - &J,
(9.24)
where
%&x(X, T- &,) = 6 ( X,
$,_(X T),
fil,,, -
A,,,) = 4e-@o..)/2
T -
(1 +
T-
6,,,)r,jT-;“-)
&,)l,(T-;ova)},
+(T-
(9.25)
with
-&,,(X9 T) =il (l-$“2 eeaT(cos[ &/m]
-
exp( &p)}da.(9.26)
Then, the dimensionless expressions for V( 2, F) can be easily read off from (9.1)-(9.3) by replacing \k,,,, x, t, ;A,, by Go,,, X, T, A,,, respectively. U( X, T) is then given by U(X,
T)=
V(X,
co)-
V(X,
T).
(9.27)
From (9.26) it follows that Go,JX, aT
1 (1 - (Y)1’2
T) =
p
-o J
emar(cos[ &,/m]
- exp( d,,,cu))dcu. (9.28)
Now, we can easily show by writing (Y= sin2$+ that 1 (1 - q2
J/2
J0 =
e
-‘+-&,)da
+e -u---i\,.,)/2
= $re
J0
“(1 + cos ‘p) exp[f(T-
-v-&..v2(
Io(
T-+9
+,,(
a,,,)
cos cp]d+ (9.29)
T-$..)).
Again, from (9.25), it follows that &(X,
TaT
ii,,,) =
+e*7-A0.0)/lj Io( T-P,*a)
Then, from (9.24), (9.25), and (9.28)-(9.30),
+I,(
T-2i\07”)}_
(9.30)
we obtain
%$F( X, T - Ao,,)= $ l1 (’a,$)“2cos[ &,{~~] epsT
da.
(9.31)
From (9.4) and the Initial Value Theorem it follows that
(
exp - &ho_
1
.
(9.32)
219 It follows from (9.1)-(9.3), (2.5) and (9.32) that at the front of each of the waves emanating from x = a there is a jump in u( x, t) given by
KCJ
-Texp
(
-&x-al).
At the front of each of the waves emanating u(x, t) given by
-KtJexp 2t
(
from x = b there is a jump
in
(9.34)
-&lx-bl).
Analogous results apply for the waves emanating from x = - a and x = - b. In each case the jump is defined as the value of U(X, t) immediately behind the front minus the value immediately ahead of the front. In terms of the dimensionless quantities defined in (9.23) we see from (9.33) and (9.34) that the jump in U( X, T) at the fronts of the waves emanating from X = A and X = B are given by - iexp( - 51X - Al) and - $ exp( - f )X - BI) respectively,. In the numerical example, the results of which are shown in Fig. 4, the wave-fronts have not yet reached the boundaries. Consequently, if appropriate allowance is made for the differences in the normalizations adopted in 58 and this section, the same results can be obtained by using the formulae for U derived in this section. 10. Vortex sheets in a Maxwellian fluid We have seen in 54 that in the case in which, in the limit t + co, there are two vortex sheets situated at x = + a, with vorticities T ijo, ti( x, S) is given by (4.4) and (4.5). If the fluid is Maxwellian then, with (8.2) i(x,s)=
-
c&(hs+ 1) 2s
_ {e I(lxlka) - e-3(lxl+a)} if a < 1x1, (10.1)
G&s+l) E(x, S) = 2s
_
_
{e Ra Ix’)+ e-S(a+lxl)} if 1x1< a,
where ~=Y\/I~.
(10.2)
We can rewrite (10.1) as E(x, S) = -O,(
x, S) exp[ - cs( 1x1 - a)]
+&(x,s) 5(x, s) = 0,(x, +0,(x,
exp[-Cs(lxl+a)],a
s) exp[ -Zs(lxI
- a)]
s) exp[ -Bs( IxI+ a)], IxI< a,
(10.3)
220 where
G,( x, s) = 62,~exp[(-3+ts)(lxl-u)], G,(x, s) =
Gj,
It follows from the Translation u(x,
t) = -0,(x, +0,(x,
u(x,
(10.4)
Fexp[(-5+cs)(lxl+n)].
t) = 0,(x,
t - iqlxj
- a))H(t
t - iqIxI+ t - Y(lxl-
Theorem
a))H(t a))H(t
that u( x, t) is given by
- iqlxl-
a))
- F(lxl+
a)),
- iqlxl-
(u < 1x1)
(10.5)
u))
+o,(x,t-E(lxl+ul)H(t-t(lxI+u)),
(Ixlu),
where O,( x, t) and O,( x, t) are the inverse Laplace transforms of 0,(x, s) and o,( x1 s) respectively. on the segment Since 0, and e, have poles at s = 0, are discontinuous ( - l/h, 0) of the negative real axis, and are analytic elsewhere, their inverse Laplace transforms can be calculated by means of the inversion integrals. These, in turn, can be evaluated by using Cauchy’s Theorem with the modified Bromwich contour shown in Fig. 5. We thus obtain, with (9.13),
&x, t), a)) = +ijo - $J*(x,t),
0,(x,
t - iqlxl -a)) = +Ljo-
0,(x,
t - iqlxl+
(10.6)
where
(10.7)
With these expressions u(x, t) is given by (10.5). From (lOS)-(10.7) we obtain
4%
00)’ i
0 I Go
(a+I)
(I-4 -4.
(10.8)
The velocity u(x, t) associated with the decay of a pair of vortex sheets can easily be obtained from (2.5) with (10.5)-(10.8). If x > 0, the first (second) term in (10.5), can be regarded as a wave which emanated from x = a (-a) at time t = 0 and travels in the positive direction of the x-axis. The first (second) term in (10.5), can be regarded as a wave
221 which emanated from x = a ( -a) at time t = 0 and travels in the negative (positive) direction of the x-axis. (Analogous interpretations of (10.5) can be made in the case when x < 0.) At the front of each of these waves there is a velocity discontinuity. The magnitudes of these discontinuities (value immediately behind the front minus value immediately ahead of the front) can be calculated by using the Initial Value Theorem. However, in order to do so we recognize that each of the waves emanating from x = a, say, represents the superposition of two waves, which emanate from x = a and x = b, in the limiting case when b - a + 0. The jumps in u(x, t) at the fronts of the waves emanating from x = a and x = b are given by (9.33) and (9.34) respectively, where - K~ is the initial velocity gradient in the region a < x < b. It follows that the jumps in U(X, t) at the fronts of the waves which emanate from x = a in the limiting case when b + a, K,, + co and (b - a)Ko = Go are given by I_t{k$[exp(--&lx--bl)-exp(-&1.x-al)]) = *
$ijo exp( - &lx
- al),
(10.9)
where the upper (lower) sign applies if x > a ( -c a). In terms of the dimensionless quantities defined in (9.23) we obtain from (10.9) the result that the jump in U( X, T) at the fronts of the waves emanating from X = A are given by ++exp(-+1X-Al).
(10.10)
It is evident, from the interpretation of u(x, t) in (10.5) as the superposition of waves, that, for a single vortex sheet of strength -ij, located at x = a, u(x, t) is given by u(x,
t)=
u(x,
t)=@,[x,
-O,[x,
t-i;(x-a)]H[t-C(x-a)] t-E(a-x)]H[t-F(a-x)]
(x7a), (x
(10.11)
Corresponding results for the velocity field U(X, t) associated with the decay of a single vortex sheet of strength - &, located at x = a can be easily obtained from (2.5). From (10.6),, (10.7), and (10.11) we obtain -‘ij
u(x, co)= i
2 O (x ’
+ijo
(x
a>
It then follows from (2.5) (lO.ll), i.e. for t > SIX - al,
u(x,
t) = T
21,(x, t),
(10.12)
(10.12) and (10.6) that behind
the fronts,
(10.13)
222 where Jr (x, t) is given by (10.7),, and the upper (lower) sign applies if x > a ( -C a). Equation (10.13) can be rewritten in terms of the dimensionless quantities defined in (9.23) as U(X,
T)=
+r(x,
(10.14)
T),
where J,( X, 7’) is given, from (10.7),, by j,(X,
T)=
~‘~e-“Tsin[(~X~-
(10.15)
l)/m]dol.
11. Appendix We now introduce
the notation
A m.1 = 2mh + 1x1- a,
Am,2 = 2mh + a - 1x1,
A m.3 = 2mh + a + 1x1,
A,,, = 2mh - a - 1x1,
A ms = 2mh + 1x1 - b, A m.7 = 2mh + b + 1x1,
A,,, = 2mh + b -‘/xl,
(11.1)
A,,, = 2mh - b - 1x1,
F = p’/*y 7 and K&b \k,,Jx,
s) =
+ 1) exP[(-l+
2.4
(11.2)
iW,,,j.
Then taking Re { > 0 for Re s > 0, it follows from (3.5)-(3.10), that for Re s 2 0 8(x, s)=
g (-l)“{!Fn,, n=O -\kn+1,4
+K+r,s
exp( - CsAn,t) + k,,3 exp( - CsA,,,)
exP(-~~An+,,.J
- \kn,6 exp(
with (8.2),
-K+1,2
exP(-wz+,.2)
- “sA~,~) - G,,, exp( - ZsA,*,)
exp(-~~A,+I,~)+~,,+~,~ex~(-~~A,+I,J} u
E(x,s)=
E (-l)“{!&,r n=O -%+1,4
(11.3)
exp( - CsAn,r) + q”,3 exp( - FsA~,~)
exp(-~~An+lA)-~~+1,2ex~(-~~A,+,,2)
- T”,5 exp( - VSA~,~) - \k,,, exp( - EsAn,,) ++n+r,*
exP(-YSA,+1,8)+\kn+1,6
exP(-6sA,+,,,)} b i 1x1G h,
(11.4)
223 E(x,s)=
2 (-l)“{\JI,,,exp(-CsA,,,)+\k,,,exp(-FsA,,,) n=O -JI.+,,,
exp(-vsA,+,,,)-\Ir,+l,l
ex~k~sA,+,,,)
-\kn,6 exp( - i;sA,,,) - ;2;,,, exp( - 5sA,,7) +%,8
exp(-+A,+,,d
+ %+l,5 exp(-isA,+,,,)} IXI< a.
We denote the inverse Laplace transform of k,Jx, the unit step function by H( ). It then follows from the Translation Theorem that U(V)=
S) by \~,Jx,
f) and
a
(11.6)
b < 1x1~ h
(11.7)
2 (-l)“{\Ir,,,(x,t-~A,,,)H(t-tA,,,) n=O +\kn,&,
u(v)=
(11.5)
t - FA,.,)H(t
- FAn.3)
- %+1,4 (x> t - EAn+&H(t
- FA,,,,,)
- *PI+,,2 (XT t-
- EA,+,z)
cA,+*,M(t
- *,Jx,
t - fA,,@(t
- VA,,,)
-‘I’,&,
t - fA,,,)H(t
- CA,,,)
+ %+I,8 (XT t - fA,+,,,Mt
- fan+*,*)
+ %I+*,5 (XT t - ~An+l,S)Hf
- ~A,+,,,))
9
E (-1>“{~~.~tx,t-~A,,~)~tt-~A,,,) n=O +
\kn,&, t - EA,,#(t - gA,,,3)
(x9t - ~An+l,dWf- ~An+1,.4) - %+1,4 -%+1,2(XTt - %+*,*)~(f - %+1,2) -*&, t - %,#tt - &,s) -*&,
t - %,,)Htt
- %,,,)
tx,t- %,+&H(t - %r+u) + \kn+*3 + %+I,6(x, t -
;A n+l.CM(f
- ~An+1,6)),
224
Each of the terms in (11.6)-(11.8) represents a disturbance which propagates with speed l/5. We consider the disturbance given by a term !Pm,,( x, t - FA,,,)H( t - ;A,,,). This is zero for t -Z EAm,a and has the value at the \k,,,,(x, t - CA,,,) for t a FA,,,+. The magnitude of the discontinuity value of x given by A,,_ = t/E can be calculated ‘by using the Initial Value Theorem. We have from (11.2) and (8.2) (11.9) We note from (11.1) and (11.2) that q&,s)=
the upper (lower) sign being taken if x is positive (11.9) and (8.2) the Initial Value Theorem yields *;,Jx,
(11 JO)
r(-l)“(-{+5s)k,,,(x,s),
o)=
+(-l)“!$ exp( -=$L+
(negative).
From (ll.lO),
(11.ll)
We now consider x > 0. The terms in (11.6)-(11.8) involving qO,, and \k0,2 can be interpreted as waves emanating from x = a at time t = 0 and travelling in the positive and negative directions of x respectively. Those involving \Ib,5 and \k0,6 can be interpreted as waves emanating from x = b at time t = 0 and travelling in the positive and negative directions of x as waves respectively. The terms involving *0,3 and \ko4 can be interpreted emanating from x = - CI at time t = 0 and travelling in the positive and negative directions of x respectively. Those involving \k0,7 and q,,* can be interpreted as waves emanating from x = -b at time t = 0 and travelling in the positive and negative directions of x respectively. The terms involving \k,,, with m # 0 represent the contributions of these waves to the resultant velocity field after they have undergone reflections at the boundaries at x= fh.
226 where the upper (lower) sign applies to the wave travelling in the positive (negative) direction of x. Acknowledgement The results in this paper were obtained in the course of research sponsored by the U.S. Army Research Office under Contract No. DAAG 29-82-K-0026 with Lehigh University. References 1 R.S. Rivlin, J. Non-Newtonian Fluid Mech., 14 (1984) 203-217. 2 M. Abramowitz and LA. Stegun @is.), Handbook of Mathematical Bureau of Standards, 1964. 3 H. Lamb, Hydrodynamics, Dover, New York, 1945.
Functions,
National