INFLUENCE OF WALL WAVINESS ON FRICTION AND PRESSURE DROP IN CHANNELS

An attention has been given to investigate the flow behavior of an incompressible viscous fluid confined in horizontal wavy channels and set in motion due to the movement of the upper waand the pressure differences. The governing equations have been solved analytically as well as numerically subject to the relevant boundary conditions by assuming that the solution consists of two parts: a mean part and a disturbance or perturbed part. For small and mod- erate Reynolds numbers, the analytical solution for the perturbed part has been found to be in good agreement with the numerical one. The effects of Reynolds number, the pressure gradient parameter, and the undulation wavenumber on fric- tion and pressure drop are found to be quite significant. In addition to the flow behavior for both long and short waves and for large Reynolds numbers, the effect of the wall waviness on friction and pressure drop has been examined for any arbitrary amplitude of the wavy wall.


i. INTRODUCTION.
There are many physical applications of coupled pressure and shear flows.
These include, hydrodynamic lubrication of a sleeve bearing, tube flight vehicles for the development of transportation systems.In addition to having many appli- cations, these flows are of sufficient interest in its own right.They pose no theoretical problems when the fluid motion is laminar, fully developed and con- fined between parallel flat walls.However, if the shear flows occur between wavy walls, there are certain mathematical difficulties involved with the study of such flows.Consequently, an approximate theoretical treatment may be pro- vided to gain some understanding of shear flow problems.
Literature is replete with theoretical as well as experimental problems dealing with the flow pattern of a viscous fluid confined between flat solid boundaries.Mention may be made of the studies (of these problems) by Berman [i], Balaram [2], Hahn and Kettleborough [3] and Szeri et al. [4]  However, there are many physical situations in which the surfaces of the solid boundaries are wavy in nature.For example, even the surface formed by cleavage of mica con- tains irregularities of the order of 20 A in size, and the irregularities of the surface of an ideally smooth quartz crystal can be up to I00 A in height, which have been discussed in detail by Kragelskii [5].
Although several authors have studied the problems of viscous fluid flows confined between flat boundaries, an attention has been hardly given to these flows in horizontal wavy channels with a moving wall.It seems to us that the findings of the investigations of the fluid flows confined between wavy boundaries will have a definite bearing on the hydrodynamic theory of lubrication, in particular on the bearing industry.So the main objective of this paper is to investigate the effects of wall waviness on friction and pressure drop of the generalized Couette flow.Since the theoretical treatment of the problem is complicated, it is solved by a numerical perturbation technique.The solution consists of two parts: a mean part, corresponding to the fully developed mean flow and a small disturbance.The former coincides with the well known Couette flow after modifications resulting from the different non-dimensionalizations employed in classical Couette flow and the present problem.The latter is ob- tained by the use of the method of superposition coupled with an orthonormaliza- tion procedure and a variable-step Runge-Kutta-Fehlberg integration scheme of Scott and Watts [6].A comparison is made between the numerical and the analyti- cal solutions.

FORMULATION AND SOLUTION OF THE PROBLEM.
We consider a channel with the wavy walls represented by y* d* + e* cosk*x* (YI say) and y* -d* + * cos(k'x* + 0) (Y2 say) The representation of the second wall alows us to study the flow behavior in the following four different types of channels by giving the values 0, /2, and 37/2 to 0: (i) the crest of a wall corresponds to the crest of the other wall of the channel; (ii) one of the walls considered in (i) has a phase-advance/lag; (iii) the crest of a wall corresponds to the trough of the other; and (iv) one of the walls considered in (iii) has a phase-advance/lag.
We make the following assumptions: (a) all fluid properties are constant; (b) the flow is laminar, steady and two-dimensional; and (c) the effects of elastic distortion are neglected (as in reference [3]).Under these assumptions, the equations which govern the steady, two-dimensional motion of a viscous incom- + * a--2 + a---y-)'  By the method of perturbations, given by Nayfeh [7], we take the flow field in the form u (x,y) u0(y) + u 1(x,y), P0(X) + PI(x'Y) (2.9) where the perturbations Ul, v!, and Pl are small compared with the mean or the zeroth-order quantities.With the help of (2.9), equations (2.5)   (2.(2.15) Ul -ee u, v 0 on y -I where a prime denotes differentiation with respect to y and ee (e # 0(i/e)).
(2.19) numerically by the method of superpositon coupled with an orthonormalization procedure and a variable-step Runge-Kutta-Fehlberg integration scheme.Also, an analytical solution has been obtained, within the long-wave approximation, and these results are compared with the numerical ones, for the skin friction and pressure drop (see Figure i).

ZEROTH-ORDER SOLUTION (MEAN PART).
The solution for the zeroth-order velocity u0, satisfying the differen- tial equation (2.10) and the boundary conditions (2.14), has been obtained but is not presented here.The expression for u 0 at various values of y has been evaluated numerically for several sets of values of the parameters R and C.
These results were found to be in good agreement with those of the generalized Couette flow. 2 20 + eeikx [eiOu"(-l) "(-i)] To study the effects of wall waviness on friction, in general, we can rewrite the expressions (4.1) and (4.2) as ikx ei i r + ee IF1] ikx ei Y2 r20 + e IF2[ 2 and, in particular, hope to bring out the salient features of IF I, IF21, l' and 2 in this paper (see Figures 2, 3, 5, 6, 8 and 9).
The expressions for r and 02 have been obtained from the zeroth-order solution and have been evaluated numerically for several sets of values of the parameters R and C. It is clear that expression (4.3) is the zeroth-order skin friction at the walls and that its numerical values correspond physically to the behavior of the flow at the walls in the case of a channel whose walls are flat (generalized Couette flow).

PRESSURE-DROP.
We refer to (2.5) and (2.6) and obtain the fluid pressure where L is an arbitrary constant and Z(y) ("' k2 ') ik (noV' '9) u Equation (5.1) can be written as p (x,-l)p (x,l) =----(-i) -Z (i) ikx (5.2) where has been referred to as the pressure drop since it indicates the dif- ference in the pressure on the wavy walls, with x fixed.The amplitude and the phase B were evaluated numerically for several sets of values of the dimensionless parameters entering the problem and are presented in Figures 4, 7, and i0.In what follows, we record the qualitative differences in the behavior of the flow characteristics that show clearly the effects of the wavy walls of the channels.

RESULTS AND DISCUSSION.
As mentioned in the introduction, the analytical solution of IFil i (i 1,2), QI, and B are compared with the numerical ones in Figure I.
From Figure 1 it is evident that the analytical and numerical solutions are in good agreement, for small and moderate Reynolds numbers.However, at high Rey- nolds numbers, the analytical solution differs very much from the numerical solu- tion.In Figures 2-10 only the numerical solution results for the amplitudes and phases (arising from wall waviness) are plotted for various values of R, C, and k. (IQI, B) are presented in Figures 3 and 4, respectively.From Figure 2, it follows that the amplitude IFII increases with R: physically it means that The amplitudes IFII, IF21, and IQI are enhanced significantly with an increase in either the Reynolds number R or the pressure gradient parameter C or the wavenumber k, in almost all the channels considered.Of all the para- meters, the Reynolds number and the pressure-gradient parameter have the strong- est influences on the amplitudes.However, the phases I' 2' and B were found to have different trends in different channels, depending on the parameters R, C and k; they are very much affected by the parameters R and C. Finally, it may be concluded that the present analysis is capable of providing us with information relating to the effects of wall-waviness on friction and pressure drop for long as well as short waves at small, moderate, and large Reynolds numbers.
pressible fluid in the channel are 8u* u*.p__* p* (u* x* + v* y, 3) x---* dy- where u* and v* are the velocity components, p* is the pressure, and the other symbols have their usual meanings.The boundary conditions relevant to the problem are taken as u* U, v* 0 on y* d* + * cosk*x*, 1 u* 0, v* 0 on y* -d* + * cos(k'x* / (2.4)We define dimensionless variables as x order where P0 C Also with the help of (2 9) the boundary con-

Figure 2
Figure 2 shows the behavior of IF if and