Laminar flow in epicycloidal corrugated pipes

Does longitudinal surface corrugation or roughness affect viscous fluid flow? To answer this question, we study viscous flows through the entire family of pipes with epicycloidal walls, which have n cusps, n=1,2,3,…∞$n\; = \;1,2,3, \ldots \infty $ . We find that the flow becomes insensitive to the longitudinal wall corrugations due to the epicycloidal cusps as n→∞$n \to \infty $ : surface roughness of the walls does not affect the flow if it is longitudinal. Exact flows in these corrugated pipes are compared with three ‘rules of thumb’ which give general but approximate formulae for the rate of flow through a pipe of given cross‐sectional shape. Upper and lower bounds on flow, based on circumscribed and inscribed circular boundaries, give even better estimates than the three rules of thumb. All the flow formulae, exact and approximate, apply also to the twisting of solid beams.


INTRODUCTION
In the steady state of laminar flow in pipes there is no acceleration and the equation to be solved is linear [1][2][3][4][5][6][7][8]. Steady incompressible flow in a pipe of fixed cross-section satisfies It is the Poisson equation with a constant source term. The velocity along the pipe, , does not depend on the longitudianl coordinate , and the longitudinal pressure gradient is constant. A solution of (1) satisfying the no slip boundary condition that ( , ) is to be zero at the pipe wall then provides the velocity profile, and the total fluid flow rate can be found by integration. In general the flow rate is proportional to the pressure gradient | | and is inversely proportional to the viscosity .
This note discusses the fluid flow in an infinite family of pipes, with epicycloidal sections. The problem combines physics with an interesting geometry. Epicycloids are curves formed by a point on the perimeter of a circle which rolls on the outside of another circle. If the stationary inner circle has radius and is centred on the origin, the outer rolling circle has radius , and if is the azimuthal angle of the centre of the rolling circle, the cartesian coordinates of the moving point are (2) with = ∕ , Figure 1 shows sections of pipe wall for = 1, 2 and 3, Figure 2 for = 10 and = 20. Note that the minimum radius (at the cusps) is = , and the maximum radius is = (1 + 2∕ ) . The amplitude of variation in the pipe radius decreases as the number of the cusps increases.
We shall examine the flow in epicycloidal pipes. Of particular interest here is the limit of large (the number of cusps): a microscopically rough but macroscopically smooth pipe. Is the pipe flow sensitive to longitudinal microscopic roughness?

FLOW IN EPICYCLOIDAL PIPES
We can combine the parametric equations (2) for the walls of epicycloidal pipes into Hence if = , the expression − 1 +1 +1 will equal the right-hand side of (4) when = 1. For < 1 the complex variable + may be mapped onto the complex variable = by We note that 2 + 2 = 2 2 {1 − 2 +1 cos + 2 ( +1) 2 } has Laplacian equal to 4, a constant. To find the fluid flow inside the pipe we combine a function whose two-dimensional Laplacian is a constant (to satisfy (1)) with a harmonic function F I G U R E 2 The = 10 and = 20 epicycloids, in red and blue respectively. Note the increase in surface roughness and decrease in the average pipe radius and in its variation as increases.
(of zero Laplacian) to obtain zero on the walls. A possible combination for the flow speed has the form The Laplacian of the last term is zero, since it is the real part of the harmonic function , so ( 2 + 2 ) = − ∕ 2 , and hence by comparison with (1), On the wall of the pipe the flow speed is zero, which gives Hence, the required velocity profile is, with as defined above, This expression is equivalent to Equation (11) of Phan-Thien [9], (specialized from epitrochoids to epicycloids) apart from the sign of , the difference being due to our choice of orientation of the epicycloids. In order to plot the velocity F I G U R E 3 Iso-velocity contours in the cardioid, the = 1 epicycloid. The contours are at 0.1(0.1)0.9 of the maximum velocity, which occurs at ≈ −0.755 , = 0.
within a given epicycloidal pipe, we need to convert from the azimuthal angle ϕ of the centre of the rotating circle to the polar angle θ of a point [x, y] within the pipe. The relationship is found from = cos − +1 cos ( + 1) , = sin − +1 sin ( + 1) tan = = sin − sin ( + 1) cos − cos ( + 1) , The effect of the cusp is the most prominent when = 1, in the cardioid. The flow speed within the pipe (shown in Figure 3) is then The total flow through an epicycloidal pipe is found by integrating over the pipe cross-sectional area, = ∫ ∫ . It is convenient to integrate in , coordinates, for which we need the Jacobian = 2 {1 − 2 cos + 2 } . We find Following Sisavath, Jing and Zimmerman [10] we shall express the flow in terms of the 'hydraulic conductance' (we give also the value for pipes of circular cross-section, with radius , perimeter = 2 and cross-sectional area The hydraulic conductance is, from (12) and (13), The hydraulic conductance = 8 4 is for a circular pipe of radius , which is the minimum radius of the > 1 epicycloids. The flow in epicycloidal pipes is always greater than that for the inscribed circular pipe, and tends to this value for → ∞. For the cardioid

THREE RULES OF THUMB FOR FLOW THROUGH PIPES
We shall relate the viscous flow through pipes with epicycloidal cross-sections to three 'rules of thumb' as discussed below. The motivation is provided by geophysical and biological applications: geophysical and biological fluid flows are in irregular crevices and ducts. Rules of thumb have therefore been developed to enable rapid estimates of flow in terms of easily measured parameters, for example the radii of two circles, one of which just fits inside the pipe, the other outside. Three such rules are (i) the Saint-Venant [11] formula, in which the flow is related to the area and the polar moment about the centroid of the pipe cross-section; (ii) the Aissen [12] approximation, in which the flow is given by the cross-sectional area and the diameters of the inscribed and circumscribed circles, and (iii) the area-perimeter or hydraulic radius formula of various authors, which states that for a given pressure gradient and fluid viscosity the total flow rate is proportional to 3 ∕ 2 , where is the perimeter length of a pipe section normal to the pipe axis. Two of the three approximate rules for total fluid flow through pipes originated in an engineering context, the torsion of beams: fluid flow and beam torsion are mathematically equivalent; see for example Section 16 of [13]. The three main approximations for the total fluid flow rate Q were discussed in [13,14], so we shall just give the formuale and brief explanatory notes. All three approximations become exact for a pipe of circular cross-section, by construction.

The Saint-Venant approximation
The de Saint-Venant formula [11,15] relating to torsion of cylindrical rods, as applied to flow in pipes, is [10] is the polar second moment about the centroid of the pipe cross-section: ( ) is the distance from the centroid to the pipe wall, at azimuthal angle of the centre of the rolling circle. (The first equality needs to be modified for the cardioid, for which the centroid does not coincide with the origin.) For a circular cross-section of radius , = 2 4 = 2 ∕2 , and so = .

The Aissen approximation
Aissen also derived an approximate formula for the torsion of a bar [12,15]. In the Aissen approximation applied to fluid flow [10], The flow is approximated in terms of the pipe cross-sectional area and the diameters of the largest inscribed and smallest circumscribed circles, and respectively.

The area-perimeter or hydraulic radius approximation
The hydraulic radius is defined [10] as ∕ . ( is the area and the perimeter length of a pipe section normal to the pipe axis.) In the area-perimeter or hydraulic radius approximation the conductance is proportional to the area times the square of the hydraulic radius. Making the rule correct for a circular pipe gives the conductance This 'rule of thumb' has been discussed several times in the literature [10,[16][17][18][19][20]. The physical basis of having dependence on the perimeter is that, other things being equal, the larger the wall perimeter the greater the viscous drag on the flow.
Sisavath, Jing and Zimmerman [10] (their Table 1) have compared the three approximations with exact flows for pipes with the following cross-sections: thin ellipse, equilateral triangle, square, rectangular slit, and semicircle. The author [14] has done the same for the family of pipe cross-sections based on ( + ) , = 3, 4, .., ∞, where is the number of vertices in the pipe cross section. Here we shall compare the three approximations for total flow with exact results for epicycloidal pipes, for number of cusps ranging to infnity.

COMPARISON OF THREE RULES OF THUMB WITH EXACT FLOWS
We shall need the diameters and of the inscribed and circumscribed circles, the perimeter length , the pipe crosssectional area , and the polar second moment about the centroid of the pipe, . From (3) we have, for > 1, the diameters of the largest inscribed and smallest circumscribed circles, = 2 , = 2 (1 + 2∕ ). The perimeter length is found by integrating the length element = Note that the perimeter does not tend to 2 at large , but to 8 , 4∕ times larger. This is in contrast to the area and the polar second moment of epicycloids, which tend to those of a circle of radius as → ∞.
The cross-sectional area is, on using the Jacobian above (12), The polar second moment about the centroid is, for > 1, We are now able to compare the approximate and exact total flows. The hydraulic conductance in the Saint-Venant approximation is, from (16), (21) and (22), for all epicycloidal pipes from = 2 to infinity, and with the exact given by (14), The hydraulic conductance according to the Aissen rule of thumb is, from (18) and = 2 , = 2 (1 + 2∕ ), again for all epicycloidal pipes from = 2 to infinity, Finally, the hydraulic approximation, from (19), (20) and (21) The hydraulic rule of thumb estimates the flow in epicycloidal pipes as always smaller than the exact flow, although it is close for = 1 and = 2. At large it underestimates the flow, by the factor 2 ∕16 ≈ 0.617.
It remains to evaluate the Aissen flow value (18) for the cardioid. The diameter of the largest inscribed circle is = 2 . The diameter of the smallest circumscribed circle is = 3 √ 3 ≈ 5.196 , as shown in the Appendix. The ratio of the Aissen estimate to the exact total flow within the = 1 epicycloidal pipe is 216 √ 3∕527 ≈ 0.71. The ratios of approximate total flows to exact total flows ∕ are shown in Figure 4.

DISCUSSION
We have compared three rules of thumb for total fluid flow through pipes with exact values for the complete set of epicycloidally corrugated pipe shapes. Formulae were given for the flows and related to the pipe properties for corrugation number ranging to infinity. Of particular interest is the behavious at large where there are many cusped corrugations, but with small variation in pipe radius. We have found that the Saint-Venant and Aissen total flows are correct in the limit → ∞, whereas the area-perimeter rule fails by the factor 2 ∕16 ≈ 0.617. A similar behaviour was seen for another pipe family [14] based on ( + ) , again with angularities increasing in number as increases, but variation in radius decreasing with . The conclusion we draw is that the Saint-Venant and Aissen approximate total flows cope well with many longitudinal corrugations in pipes, but the area-perimeter rule does not. The flow speed goes to zero at the pipe wall; pipe 'roughness', in the sense of there being many small longitudinal indentations, does not impede flow. But such roughness can increase the pipe perimeter; in the case discussed here the perimeter does not tend to that that of a circular pipe, but to value a larger by 4∕ . Hence the failure of the 3 ∕ 2 rule, which overestimates the effect of small variations in pipe radius.
The remarks above apply also to the torsion of beams, since fluid flow and beam torsion are mathematically equivalent, as discussed in Section 16 of [13]. The total flow rate is in one-to-one correspondence with the torsional rigidity. See also [21,22] for general theorems on functions with constant Laplacian, with application to torsion problems.

NOTE ADDED JUNE 2022
Professor Zimmerman of Imperial College, London, has pointed out that as well as the flow being bounded below by that through the largest inscribed circle, the flow is also bounded above by that through the smallest circumscribed circle, as proved rigorously in [23]. Excluding the special case of = 1 (the cardioid), we have and (1 + 2∕ ) for the radii of the largest inscribed and smallest circumscribed circles. The normalized hydraulic conductance thus has the bounds The exact ratio, from (14), is = 1 + 4 + 10 2 + 13 3 + 6 4 , clearly within the bounds given in (28). Professor Zimmerman noted that the arithmetic mean of the lower and upper bounds is close to (and lies above) the exact value: The arithmetic mean (29) of the upper and lower bounds, is better than the Saint-Venant and Aissen approximations when > 3 (and much better than the area-perimeter rule), for the family of epicycloidal pipes.
One may also consider the geometric mean, (1 + 2∕ ) 2 = 1 + 4 + 4 2 . Both the arithmetic and geometric means are correct to first order in the −1 expansion; the arithmetic mean is better in the higher-order terms. The Saint-Venant and Aissen approximations are not correct to order −1 , and the area-perimeter rule fails badly as → ∞ for the corrugated pipes considered here.
We may conclude that, for families of longitudinally corrugated pipes such as considered here and in [14], in which the corrugations decrease in radial amplitude as their number increases, the mean of the upper and lower bounds proposed by the Reviewer is better than the other rules of thumb considered here. It is also simpler to calculate.