488
Ind. Eng. Chem. Fundam. 1982, 21, 486-488
Isothermal Creeping Flow in Rectangular Channels The problem of steady isothermal creeping flow in rectangular channels is examined. An exact analytical solution to this problem was obtained earlier by Srinivasan et al. (1979) in the form of a double Fourier series. Their series converges rather slowly. We have found a solution in terms of a single Fourier series. The present solution has the advantage that it converges extremely rapidly and is much simpler to use than that of Srinivasan et al. for practical computations. Indeed, a five-term approximation of our series is sufficient for convergence (in contrast with a 50-term approximation needed in the double Fourier series of Srinivasan et al.) with an accuracy up to five decimal places. Flow in a sloping closed (and open) channel is treated first. This is followed by an analysis of the free surface flow in a horizontal open channel.
Introduction To describe experiments on the condensation of vapor inside a rotating tube, Johnston (1973) studied the isothermal creeping flow through a channel of rectangular cross section as shown in Figure 1. In his work, Johnston examined (a) the flow through a sloping closed (and open) channel and (b) the free surface flow through a horizontal open channel. The aim was to calculate the volumetric flow rate as a function of the channel depth for a closed channel and the change in liquid level for a horizontal open channel. The plane flow of a viscous incompressible fluid through a rectangular channel is governed by the equation (see Srinivasan et al., 1979, for derivation). (1)
and appropriate boundary conditions. If we define a dimensionless velocity tp as tp =
.
-./[."
&(P ax
+ pgh)
and introduce dimensionless lengths Y =y/a; Z = z / a
1
(2)
4=f++ (8) where E( Y,Z) is a particular solution of eq 4 and +(Y,Z) solves the associated Dirichlet problem. We seek a particular solution of eq 4 in the form [(Y,Z)= A P
(3)
(4) Johnston (1973) obtained a numerical solution of eq 4 subject to the no-slip boundary conditions
440,Z) = 4(1,Z) = 4(Y,O) = 4(Y,ff)= 0 (5) for a closed channel of aspect ratio CY and the conditions
m,z) = m o ) = (atp/az)(~,CY/2) =o
-$/[
a4
(9)
Substituting eq 9 in eq 4 we find
(6)
A = -72; B = 0 to satisfy eq 10 and arbitrarily take
(11)
C=E=F=O; D=Y2 As a result we obtain
(12)
E(Y,Z) = f / , Y ( l- Y)
a
- -(P ax
+ pgh)
]
as a function of CY. The details of his numerical computation are not known. Recently Srinivasan et al. (1979) derived an exact analytical solution to this problem in the form of a double Fourier series. Their solution, being analytical in nature, is more convenient to use than the numerical solution of Johnston (1973). However, the double Fourier series solution obtained by Srinivasan et
aZ+/ap+ a2+/az2= o
(144
+(O,Z) = -[(0,2) = 0
(14b)
+(1,Z) = -((l,Z) = 0
(14~)
+(Y,O)= -[(Y,O) = ' / Y ( Y - 1)
(7a)
= JaJ1tp(Y,Z) dY dZ
(13)
The problem for +(Y,Z)now becomes
for an open channel of aspect ratio a/2. Here CY = b / a . Using his numerical solution, Johnston calculated the carrying capacity of the channel under consideration and plotted the dimensionless volumetric flow rate
F(CY) =
+ Sz2 + CY2 + DY + E 2 + F
2A + 2B -1 (10) the remaining coefficients are arbitrary. We may choose
we can write eq 1 in the form
=
al. converges rather slowly. We have found an exact analytical solution in terms of a single Fourier series. Our series solution converges extremely rapidly and is much simpler to use than that of Srinivasan et al. (1979). Indeed, a five-term approximation of the series is found sufficient for convergence (in contrast to a 50-term approximation needed in the solution of Srinivasan et al.), with an accuracy up to five decimal places. We present this solution in the following sections. First we examine the flow through a sloping closed rectangular channel. This is followed by an analysis of the flow in a sloping open channel of uniform depth. Finally, we consider free surface flow (varying depth) in a horizontal open channel. Sloping Closed Rectangular Channel To determine the flow field in a sloping closed rectangular channel, we need to solve the Poisson equation (4) together with the Dirichlet boundary conditions (5). Let
(14d)
+(Y,CY)= -E(Y,CY) = ' / Y ( Y - 1) (14e) The problem (14) constitutes a Dirichlet problem for (Y,Z) in a rectangle, with nonhomogeneous boundary conditions. A solution of this problem is quite straightforward and can be derived using separation of variables. We easily find
+-
+(Y,Z)=
2 (a, cosh naZ + b, sinh naZ) sin naY
n=l
where
0196-4313/82/1021-0486$01.25/0@ 1982 American Chemical Society
(15)
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 1
a, =
'I
Y(Y - 1) sin n r Y dY
= 0 if n is even; = -4/n3r3 if n is odd
487
(16)
and b, = a, (1- cosh nra)/sinh nra
(17)
As a consequence, the solution for 4(Y,Z) is given by 4(Y,Z) = 4
'/zY (1 - Y) - -
"
r 3 n=l
/
[([sinh((2n - l)r(cr - 2))+ sinh
(2n - l)aZ] sin (2n-l)aY)/((2n - 1)3sinh (2n - 1)ra)I (18) The dimensionless volumetric flow rate F ( a ) is then determined by substituting eq 18 in eq 7 and integrating the result. We have 1 F(a) = z a -
D/RECT/W OF FLO w
Figure 1. Cross section of the channel.
0.J 0'6
E2 r5 n=l
[coth (2n - 1)aa - cosech (2n - l)ra]/(2n - 1)5
1
(19) Using the series in eq 19, the flow function F ( a ) is computed for various values of a. This series is found to converge extremely rapidly. In fact, a five-term approximation of the series is found sufficient for convergence, with an accuracy up to five decimal places. In view of this rapid convergence, computation of F(a) can be carried out on a hand calculator, if preferred. Table I shows a typical convergence pattern for a = 3. A comparison of the results of Table I with those of Srinivasan et al. (see Table I of their paper) shows that the series given in eq 19 converges much more rapidly than the double Fourier series for F(a) obtained by these authors. The variation of the flow rate F with the aspect ratio a is plotted in Figures 2 and 3. The results are in good agreement with those obtained by Johnston (1973) and by Srinivasan et al. (1979). Sloping Open Channel We now consider the steady flow down a sloping open channel of width a and uniform depth b f 2. In this case dP/dx = 0 and we need to solve eq 4 subject to the no-slip boundary conditions at the walls 4(0,Z) = d1,Z) = 4(Y,O) = 0
n
0.OY
4
t
t0.031 /
c
0.01
0 01
0
0.2
0.Y
(20)
and the condition of zero shear at the surface
e
Figure 2. Variation of dimensionless flow rate with aspect ratio.
0.6
0.0
I.0
d-
Figure 3. Variation of dimensionless flow rate with aspect ratio.
a4 -(Y, a/2) = 0
Table I. Typical Convergence Pattern for (Y = 3
az
It is clear that solution (18) of the previous section automatically satisfies the boundary conditions (20) and (21). As a result, we may conclude that the volumetric flow rate for a sloping open channel of width a and uniform depth b f 2 is half the flow rate for a sloping closed channel of dimensions a and b. Horizontal Open Channel Here we consider free surface flow (varying depth) in a horizontal rectangular channel. The slope of the free surface is small in practice so that the volumetric flow rate, to a good approximation, can be given by eq 7a with Q = 2Q0 and dP f ax = 0, where Qois the volumetric flow rate for a rectangular open channel of width a and uniform depth b f 2. Thus the dimensionless flow rate is
n 1 2 2 4 5
F 0.19772 0.19750 0.19749 0.19748 0.19748
A
0.22968 0.22905 0.22900 0.22899 0.22899
h being the local depth of the liquid in the open channel. Following Srinivasan et al. (1979), we define a depth ratio H as
H = h/a
= 0.5a
(23)
and introduce a new coordinate x* in the direction opposite to the flow. Then eq 22 can be written in the form 4rQ0 dX F(a)d a = -
c
a4Pg
(22) with
488
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
X = x*/a
(25)
6.0
.
f
If eq 24 is integrated and the condition a = o (X=O)
is used, the result is 1 24
X=-a2--
16
(26)
c- In [2 cosh2 (2n - l)aa/2] (2n -
r6 n=l
(27)
where
Eguation 27 can now be used to compute the dimensionless group X for various values of a. The results are plotted in Figure 4 and are in good agreement with those of Johnston (1973) and Srinivasan et al. (1979). Here again, a five-term approximation of the series in eq 27 is found sufficient for convergence. In this regard our results are superior to those obtained by Srinivasan et al. and are much more convenient to use. A typical convergence pattern for the computation of X is presented in Table I for a = 3. It is of interest to determine the change in the liquid level in an open channel of given length, say 1A.Q This can be done by adopting a procedure similar to that described by Srinivasan et al. (1979). We use the following steps. 1. For a initial level ao,compute the initial value of A, say A,, using eq 27. 2. Find the final value of X from Xf = Xo - [AX1 with
3. Compute afcorresponding to hf using the equation
f(4 = 0
(30)
where 1 16 f ( a ) = - a2 - -
24
(2n - l)aa/2] c In [2 cosh2 (2n -
n=l
Xf
(31) This last equation can be solved numerically using, for example, a Newton-Raphson algorithm. We may write
where the prime denotes differentiation with respect to a. Using eq 27, formula 32 can be written in the form r
In [2 cosh2 (2n - l)aaj/2] (2n 16
-E
-
r5 n=l
tanh (2n - l)7raj/2 (2n -
]
0 2
04
0 6
08
/ O
/ 2
A-
Figure 4. Dimensionless free surface profile for horizontal open channel.
hence the change in the liquid level in the channel. As pointed out by Srinivasan et al., the graphical results of Johnston (1973) are difficult to use for small values of IAXl and may lead to considerable error. The present solution being analytical as that of Srinivasan et al. is very convenient to use even for small values of lAA1. Moreover, the solution given here converges much more rapidly than that obtained by Srinivasan et al. (1979). Concluding Remarks The steady isothermal creeping flow in (a) a sloping rectangular (open and closed) channel and (b) a horizontal open channel has been analyzed. An exact analytical solution to the problem is derived in the form of a Fourier series. This solution is shown to converge extremely rapidly and is very convenient to use for practical computations. Nomenclature a = width of channel b = depth of channel F = dimensionless volumetric flow rate g = acceleration due to gravity h = local height of free surface H = h/a n = number of terms in the series lhLl = length of channel P = pressure in the liquid Q = volumetric flow rate in closed channel Qo = volumetric flow rate in open channel u = velocity in the direction of flow x = direction of flow x* = direction opposite to flow X = x*/a y = coordinate along the width of channel Y =y/a z = coordinate along the depth of channel Z = z/a cy = aspect ratio b / a X = dimensionless group defined by eq 28 IAXl = change in X 4 = dimensionless velocity p = density of liquid Literature Cited Johnston, A. K. Ind. Eng. Ctmm. Fundam. 1973. 12, 482. Srinivasan, S.; Bobba, K. M.; Stenger, L. A. Ind. Eng. Chem. Fundam. 1979, 18. 130.
Department of Mathematical Sciences (33)
Starting with the initial value ao,eq 33 can be used iteratively to compute successive approximations for cy and
Asok K. Sen
Purdue School of Science at Indianapolis Indianapolis, Indiana 46205 Received for reuiew December 17, 1981 Accepted July 23, 1982
