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On the laminar boundary-layer flow over rotating spheroids

A. Samad a,1, S.J. Garrett b,*

a Department of Mathematics, University of Peshawar, Pakistanb Department of Mathematics, University of Leicester, UK

a r t i c l e i n f o

Article history:Received 3 September 2009Accepted 18 May 2010Available online 9 June 2010Communicated by K.R. Rajagopal

Keywords:Laminar boundary-layer flowRotating spheroidOblateProlate

a b s t r a c t

We study the laminar boundary-layer flow over a general spheroid rotating in otherwise stillfluid. In particular, we distinguish between prolate and oblate spheroids and use an appro-priate spheroidal coordinate system in each case. An eccentricity parameter e is used to dis-tinguish particular bodies within the oblate or prolate families and the laminar-flowequations are established for each family with e as a parameter. In each case, setting e = 0reduces the equations to those already established in the literature for the rotating sphere.We begin by solving the laminar-flow equations at each latitude using a series-solutionapproximation. A comparison is then made to solutions obtained from an accurate numericalmethod. The two solutions are found to agree well for a large range of latitudes and eccentric-ities, and at these locations the series solution is to be preferred due to its simplicity and easeof computation. A discussion of the resulting flows is given with particular emphasis on theimplications for their hydrodynamic stability. Their stability characteristics are expected tobe very similar to those over the rotating sphere as already studied in the literature.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The continuous development of spinning projectiles and other industrial applications has led to the need to understandthe laminar boundary-layer flow and subsequent onset of transition over the general family of rotating spheroids. Clearly amethod of accurate computation of the laminar flow at various locations over the rotating spheroid is the first stage in anysuch investigation, and this is presented here. An investigation of the convective and absolute instability properties of thelaminar flows obtained will be presented in later publications.

Garrett and Peake’s [1–3] related stability analyses of the rotating-sphere boundary layer began by first computing thelaminar flow profiles using the series-solution method due to Howarth [4], Nigham [5] and Banks [6] and then proceededby using a more accurate numerical solution. It is therefore natural that we proceed in a similar way.

To our knowledge the only published work on the laminar boundary layer over a rotating spheroid is due to Fadnis [7]who extended the Nigham series solution for the rotating sphere. However, Banks has since showed a flaw in Nigham’s solu-tion and this follows through into Fadnis’s work. Indeed, the formulation used is such that the results cannot be verifiedagainst the laminar-flow profiles already established for a rotating sphere, which is a particular case of spheroid.

In Section 2 we formulate the governing partial differential equations (PDEs) for the laminar boundary-layer flow overrotating spheroids. Distinct coordinate systems are used for each spheroidal family (prolate and oblate) and we define aneccentricity parameter to distinguish particular bodies within each family. In Section 3.1 the governing PDEs are solved usingan extension of the method originally developed by Banks for the rotating sphere. The resulting flow profiles are compared

0020-7225/$ - see front matter � 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijengsci.2010.05.001

* Corresponding author. Tel.: +44 0116 2523899.E-mail addresses: [email protected] (A. Samad), [email protected], [email protected] (S.J. Garrett).

1 Present address: Department of Mathematics, University of Leicester, UK.

International Journal of Engineering Science 48 (2010) 2015–2027

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with direct numerical solutions of the PDEs obtained using a commercially available routine in Section 3.2. A discussion ofthe resulting flows is given in Section 5 with particular emphasis on the implications for their hydrodynamic stability.

Unfortunately there are no other experimental or theoretical studies of rotating-spheroid boundary layers in the litera-ture and we are therefore unable to verify our results directly. However, our formulation is consistent with existing inves-tigations into the rotating sphere and disk boundary layers and we find favorable comparisons between those and our resultswhen appropriate parameter values are used.

2. Formulation

We formulate the steady laminar-flow equations for each spheroidal family using a distinct coordinate system. In eachcase a Cartesian frame of reference is used that is fixed in space and has origin located at the center of the body. The spheroidrotates with constant angular velocity Xw about the z-axis. The quantity gw is then the distance from the origin and normalto the body surface at a particular latitude h and azimuth /. Furthermore, dw is the distance of each focus from the origin.Note that an asterisk denotes a dimensional quantity.

For the prolate spheroid we use a prolate spheroidal coordinate system defined relative to the Cartesian coordinates as

xH ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigH2 � dH2

qsin h cos /; yH ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigH2 � dH2

qsin h sin /; zH ¼ gH cos h:

Similarly, for the oblate spheroid we use an oblate spheroidal coordinate system defined as

xH ¼ gH sin h cos /; yH ¼ gH sin h sin /; zH ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigH2 � dH2

qcos h:

In both cases we consider the hemispheroid defined by rotation of the region 0 6 h 6 p/2 and 0 6 / 6 2p about the z-axis,with surface given by gH

0 ðh;/Þ such that the spheroid nose is at gH

0 ð0;/Þ. Both coordinate systems are consistent with thosediscussed by Morse [8] and we have confirmed that each system (gw,h,/) is orthogonal and reduces to the spherical coor-dinate system as dw ? 0.

We introduce e ¼ dH=gH

m 2 ½0;1�, which defines the constant eccentricity of the cross-sectional ellipse. The quantity gH

m isthe length of the semi-major axis of the spheroid which is along the axis of rotation for the prolate spheroid and perpendicularto it in the oblate case. The Navier–Stokes equations are transformed to either coordinate system with some manipulationusing the transformations defined above and both sets of equations reduce to those in spherical coordinates for e = 0. We ap-ply Prandtl’s boundary-layer assumptions to obtain the dimensional boundary-layer equations that govern the laminar flow.

In order to obtain the non-dimensional boundary-layer equations we scale the velocities on the equatorial surface speedof the spheroid, as in Eq. (1). This is consistent with Garrett and Peake’s formulation of the rotating sphere

U ¼ UH

XHaH; V ¼ VH

XHaH; W ¼ WH

ðmHXHÞ1=2 : ð1Þ

Here U(g,h;e), V(g,h;e) and W(g,h;e) are the scaled velocities in the h-, /- and g-directions, respectively. Note that aw is theequatorial radius of the body defined separately for each spheroid: for the prolate case aH ¼ gH

m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

and for the oblatecase aH ¼ gH

m. Further, g is the distance in the normal direction from the surface of the spheroid, scaled on the boundary-layerthickness dw = (mw/Xw)1/2, such that g ¼ ðXH=mHÞ1=2ðgH � gH

0 Þ, where mw is the coefficient of kinematic viscosity. Since we areconsidering spheroids that rotate within otherwise still fluids, the mean pressure Pw is constant and can be neglected in thisanalysis.

For the prolate family, the resulting laminar-flow equations are

W@U@gþ U

@U@h� V2 cot h ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

1� e2 cos2 h

r@2U@g2 ; ð2Þ

W@V@gþ U

@V@hþ UV cot h ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

1� e2 cos2 h

r@2V@g2 ; ð3Þ

@W@gþ @U@hþ e2 cos h sin h

1� e2 cos2 hþ cot h

� �U ¼ 0; ð4Þ

and for the oblate family

W@U@gþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� e2p U

@U@h� V2 cot h

� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

1� e2 sin2 h

s@2U@g2 ; ð5Þ

W@V@gþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� e2p U

@V@hþ UV cot h

� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2

1� e2 sin2 h

s@2V@g2 ; ð6Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p @W

@gþ @U@hþ cot h� e2 cos h sin h

1� e2 sin2 h

� �U ¼ 0: ð7Þ

2016 A. Samad, S.J. Garrett / International Journal of Engineering Science 48 (2010) 2015–2027


We note that in both cases the limit e = 0 reduces the laminar-flow equations to those for the rotating sphere as found in theliterature (see [2,6], for example). The boundary conditions in both cases are given by

U ¼W ¼ V � sin h ¼ 0 on g ¼ 0; ð8ÞU ¼ V ¼ 0 as g!1; ð9Þ

which represent the non-slip boundary condition on the body surface and the quiescent fluid condition at the edge of theboundary layer.

3. Methods of solution

3.1. Series-solution method

In order to solve Eqs. (2)–(4) and (5)–(7) at particular latitudes for a given eccentricity, a series expansion solution in pow-ers of h is sought of the form

Uðg; eÞ ¼ hF1 þ h3F3 þ � � � ; ð10ÞVðg; eÞ ¼ hG1 þ h3G3 þ � � � ; ð11ÞWðg; eÞ ¼ H1 þ h2H3 þ � � � ð12Þ

Here Fn, Gn and Hn are functions of the non-dimensional variable g and parameter e, and n = 1,3,5, . . . This is consistent withthe series solution originally proposed by Howarth and Banks where e = 0. The boundary conditions (8) and (9) can then bewritten as

Fnð0Þ ¼ Hnð0Þ ¼ Gnð0Þ �1n!ð�1Þðn�1Þ=2 ¼ 0; ð13Þ

Fnð1Þ ¼ Gnð1Þ ¼ 0: ð14Þ

After substituting the above series expansions into Eqs. (2)–(4) we obtain a set of non-linear ODEs involving terms up toand including n = 7. These are stated in Appendix A.1 as Eqs. (15)–(26). Similar expressions are obtained for the oblate familyand these are stated in Appendix B.1 as Eqs. (27)–(38).

It is interesting to note that the leading-order equations (15), (19) and (23) in the prolate case give the von Kármán equa-tions [9] in an inertial frame of reference (but with modified boundary conditions arising from scaling on the equatorial sur-face speed). We therefore see that the flow close to the nose of the rotating prolate spheroid is very similar to that over therotating disk. This is to be expected as the spheroids are locally flat in that region. For the oblate case we see that Eqs. (27),(31) and (35) are similar, but with powers of 1 � e2 appearing.

The solution of Eqs. (15)–(26) and (27)–(38) subject to conditions (13) and (14) represents a two-point boundary valueproblem which is solved using a shooting method that incorporates a fourth order Runge–Kutta integrator over a suitablylarge domain. To decide upon the domain size that accurately approximates infinity, the shooting method was used overa variety of domain sizes until the solution converged. A domain of integration between g = 0 and 20 was found to be suf-ficiently large for all e 6 0.7 considered.

If we exclude the equatorial region (close to h = 90�), we find that the series solutions are everywhere convergent for allvalues of e 6 0.7. As a check of the numerical code we note that the same values as calculated by Garrett and Banks wereobtained for the first four quantities of F 0nð0Þ; G0nð0Þ; Hnð1Þ when e = 0 (note that a prime denotes differentiation with re-spect to g). These values, together with values at e = 0–0.7, are given in Appendices A.2 and B.2 for the prolate and oblatecases, respectively. For the prolate case, we note that the leading-order (n = 1) boundary values are identical for all e. Thisreflects that e does not appear in the governing ODEs at this order; this does not happen in the oblate case.

The resulting flow profiles are discussed in Section 4 where they are compared with those arising from the numericalmethod of Section 3.2. The computational advantage of the series-solution approximation is that only one run of the codeis required for each e: storing the resulting Fn(g;e), Gn(g;e), Hn(g;e) enables the velocity components at each latitude tobe obtained from the construction in Eqs. (10)–(12). However, it is clear that the series solution will become increasinglyinaccurate as the latitude increases and this is discussed in Section 4.

3.2. Numerical solution

Manohar [10] and Banks [11] solve the special case of Eqs. (2)–(4) and (5)–(7) with e = 0 using finite difference techniquesto produce accurate basic flow profiles at each latitude. We extend these by computing solutions in the general case ofe 6 0.7 for both families. The spatial discretization is performed using the Keller box scheme [14] and the method of lines[15] is employed to reduce the PDEs to a system of ODEs in h at each mesh point. The resulting system is solved at each lat-itude by marching from a given complete solution at h = 5� towards the equator h = 90� in one degree increments. At eachlatitude a backward differentiation formula method is used over a grid of 2000 data points between g = 0 and g = 20. The

A. Samad, S.J. Garrett / International Journal of Engineering Science 48 (2010) 2015–2027 2017


initial solution at h = 5� is found using the series solutions of Section 3.1. The computational routine used is commerciallyavailable from NAG as D03PEF.

The resulting profiles for e = 0 have been compared to the results of Banks [11] and Garrett and Peake and complete agree-ment is found up to the equator. These profiles can be seen in Fig. 1 of [2], for example.

4. Results

In this section we present the flow profiles obtained from the numerical method and comparisons with the profiles ob-tained from the series-solution approximation are made. Experimental observations of the boundary-layer flow over rotatingspheres have noted that the boundary layer erupts at latitudes close to the equator (see [12,13], for example) and it is as-sumed that this will be the case for spheroids of all eccentricities. Theoretical profiles for latitudes above 80� are thereforenot considered.

4.1. Prolate spheroid

Fig. 1 demonstrates laminar-flow profiles obtained from the numerical solution with increasing eccentricity at two dif-ferent locations. At h = 10�, when increasing e from 0 (the sphere) we find that there is almost negligible effect on the velocitycomponents. Further over the body (at h = 70�, for example) the variation in flow profile is slightly more pronounced,although manifests mostly in the normal velocity component. As discussed in Section 5, this has only limited implicationsfor the stability of the flow.

In order to understand the development of the flow over prolate spheroids, Fig. 2 shows the flow components for e = 0.3and 0.7 at all latitudes up to h = 80�. Note that the latitudinal velocity is inflectional at all latitudes and eccentricities whichimplies that it is unstable to crossflow instabilities according to Rayleigh’s theorem. We also note that fluid is entrained intothe boundary layer at all latitudes through the negative W-component, but has a region of reverse flow close to the surfacewhich first appears at a particular latitude between h = 64� and 66� depending on the value of e. Fig. 1 shows that the mag-nitude of the reverse flow is decreased with e at these high latitudes.

The profiles arising from the series-solution approximation are compared to the numerical solutions at various latitudesand eccentricities in Figs. 3–5. From these a purely visual comparison can be made and we observe that the series solutionsmatch the numerical solutions very well at low latitudes, however discrepancies arise at higher latitudes which are exagger-ated for increased eccentricity. For e = 0.7 we see qualitatively different behaviour from the series solution at sufficientlyhigh latitudes. Further investigation shows that such a major discrepancy is not found at any latitude for e < 0.7.

0 0.01 0.02 0.03 0.040

5

10

15

20θ=10

U

η

0 0.05 0.1 0.15 0.20

5

10

15

20θ=70

U

η

0 0.05 0.1 0.15 0.20

5

10

15

20

V

η

0 0.2 0.4 0.6 0.8 10

5

10

15

20

V

η

−1 −0.8 −0.6 −0.4 −0.2 00

5

10

15

20

W

η

−0.8 −0.6 −0.4 −0.2 0 0.20

5

10

15

20

W

η

Fig. 1. Prolate spheroid velocity profiles at latitudes h = 10� and 70� with increasing e = 0–0.7 (left to right in each frame).



A better measure of the accuracy of the series solution with respect to the numerical solution can be obtained from theroot mean square error

EX;e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNj¼1jXNumðgj; eÞ � XSerðgj; eÞj2

N

s:

Here X = U, V, W indicates a velocity component and gi is a discretized point on g 2 [0,20]. The subscripts Num and Ser indi-cate the numerical or series-solution velocity profiles, respectively.

Tables 1–3 give the values of EX,e computed at h = 10�, 30� and 70� at a range of eccentricities, with N = 2000. From thesewe see that the accuracy of the series solution reduces with increased latitude and eccentricity, as expected from the visualinspections. The implications of these error values is discussed in Section 5.

4.2. Oblate spheroid

The development of flow over oblate spheroids for fixed e is found to be very similar to that over prolate spheroids andthe equivalent plot to Fig. 2 is not shown here. The latitudinal velocity is again inflectional at all latitudes and eccentricities.We also note that fluid is entrained into the boundary layer at all latitudes through a negative W-component and has a regionof reverse flow close to the surface for sufficiently high latitudes.

Fig. 6 demonstrates laminar-flow profiles obtained from the numerical solution with increasing eccentricity at two dif-ferent locations. At h = 10�, when increasing e from 0 (the sphere) we find that there is almost negligible effect on the velocitycomponents. Further over the body (at h = 70�, for example) the variation in flow profile is slightly more pronounced,although manifests mostly in the normal velocity component. This is as found for the prolate case and indeed the U andV profiles behave in exactly the same way. However, the effect of increasing the eccentricity at any particular location is seento have the opposite effect on the normal velocity by moving the profiles in the opposite direction, entraining more fluid intothe boundary layer. It is also seen to increase the magnitude of the reserve flow that appears close to the surface at suffi-ciently high latitudes. This is more clearly seen in Fig. 7 where the flow profiles for both cases at e = 0.7 are compared againstthose for the rotating sphere (e = 0) at low and high latitudes. At each latitude we see that the W-profile limits to an equaldistance either side of the sphere limit. The small region of reverse flow in the normal direction is seen to be greater for theoblate case. We also see that the latitudinal and azimuthal flows in the oblate case are more sensitive to eccentricity at lowlatitudes, this is reflected in e appearing at the leading order in the series solution ODEs (27)–(38). However, as the latitudeincreases, the U- and V-components for prolate and oblate spheroids become indistinguishable at any particular e.

0 0.05 0.1 0.15 0.20

5

10

15

20 e=0.3

U

η

0 0.05 0.1 0.15 0.20

5

10

15

20 e=0.7

U

η

0 0.2 0.4 0.6 0.8 10

5

10

15

20

V

η

0 0.2 0.4 0.6 0.8 10

5

10

15

20

V

η

−1 −0.5 0 0.50

5

10

15

20

W

η

−1 −0.5 0 0.50

5

10

15

20

W

η

Fig. 2. Prolate spheroid velocity profiles at h = 10�–80� in increments of 10� (left to right in each frame) for e = 0.3 and 0.7.



0 0.02 0.040

5

10

15

20θ=10

U

η

0 0.1 0.20

5

10

15

20θ=50

U

η

0 0.1 0.20

5

10

15

20θ=70

U

η

0 0.1 0.20

5

10

15

20

V0 0.5 10

5

10

15

20

V0 0.5 10

5

10

15

20

V

−1 −0.5 00

5

10

15

20

W−1 −0.5 00

5

10

15

20

W−1 −0.5 0 0.50

5

10

15

20

W

Fig. 3. Comparison of the numerical (solid line) and series solutions (cross points) at h = 10�, 50�, 70� for e = 0.3.

0 0.02 0.040

5

10

15

20θ=10

U

η

0 0.1 0.20

5

10

15

20θ=50

U

η

0 0.1 0.20

5

10

15

20θ=70

U

η

0 0.1 0.20

5

10

15

20

V0 0.5 10

5

10

15

20

V0 0.5 10

5

10

15

20

V

−1 −0.5 00

5

10

15

20

W−1 −0.5 00

5

10

15

20

W−1 −0.5 0 0.50

5

10

15

20

W




0 0.02 0.040

5

10

15

20θ=10

U

η

0 0.1 0.20

5

10

15

20θ=50

U

η

0 0.1 0.20

5

10

15

20θ=70

U

η

0 0.1 0.20

5

10

15

20

V0 0.2 0.4 0.6

0

5

10

15

20

V0 0.5 1

0

5

10

15

20

V

−1 −0.5 00

5

10

15

20

W−1 −0.5 00

5

10

15

20

W−2 −1 0 10

5

10

15

20

W


Table 1RMS error of the series-solution approximation at h = 10� on the prolate spheroid.

e EU,e EV,e EW,e

0.0 8.32299839e�005 4.82027924e�005 1.47195175e�0030.1 8.32627790e�005 4.91963666e�005 1.47029442e�0030.3 7.91565682e�005 5.10325181e�005 1.44650613e�0030.5 8.99574949e�005 5.34531952e�005 1.54027200e�0030.7 9.07023453e�005 5.39351851e�005 1.54630408e�003


e EU,e EV,e EW,e



e EU,e EV,e EW,e




RMS errors have been calculated for the oblate solutions and the results show similar orders of magnitude and behaviourto those reported for the prolate case in Tables 1–3, although not shown here. Again, the accuracy of the series solution re-duces with increased latitude and eccentricity.

0 0.01 0.02 0.03 0.040

5

10

15

20 θ=10

U

η

0 0.05 0.1 0.15 0.20

5

10

15

20 θ=70

U

η

0 0.05 0.1 0.15 0.20

5

10

15

20

V

η

0 0.2 0.4 0.6 0.8 10

5

10

15

20

V

η

−1 −0.8 −0.6 −0.4 −0.2 00

5

10

15

20

W

η

−0.8 −0.6 −0.4 −0.2 0 0.20

5

10

15

20

W

η

Fig. 6. Oblate spheroid velocity profiles at latitudes h = 10� and 70� with increasing e = 0–0.7 in 0.1 increments (left to right for U and V, right to left for W).

0 0.05 0.1 0.15 0.20

5

10

15

20 θ=70

U

η

0 0.01 0.02 0.03 0.040

5

10

15

20 θ=10

U

η

0 0.2 0.4 0.6 0.8 10

5

10

15

20

V

η

0 0.05 0.1 0.15 0.20

5

10

15

20

V

η

−0.8 −0.6 −0.4 −0.2 0 0.20

5

10

15

20

W

η

−1 −0.8 −0.6 −0.4 −0.2 00

5

10

15

20

W

η

Fig. 7. A comparison of flow profiles at h = 10� and 70� for the rotating sphere (—), prolate spheroid with e = 0.7 (-�-) and oblate spheroid with e = 0.7 (-�-).



5. Conclusions

In this paper we have derived the governing PDEs for the laminar flow within the boundary layer over both familiesof rotating spheroid. Each system is seen to limit to the known equations for the rotating sphere as e ? 0. Two methodsare used to solve the two sets of governing equations for general e: a series-solution approximation and an accuratenumerical solution. For both families of spheroid the flow is related to that over a rotating disk as the latitude is reducedto the pole. The series-solution method can therefore be thought of as successive modifications to von-Kármán-type pro-files with latitude. Indeed versions of the von Kármán equations appear at leading order in both cases. The implication ofthis for the numerical solution is that profiles similar to von Kármán could be used as the initial profile if the integrationwere to start at a sufficiently low latitude, say, h = 1�. These would be obtained from the leading-order series-solutionODEs. This avoids the need to run the higher-order series approximations at, say, h = 5� as was done here.

From visual inspections and calculation of RMS errors of the resulting series-solution profiles with respect to the numer-ical solutions we see that the series solution is very accurate at low latitudes for all e 6 0.7. However, as the latitude is in-creased the discrepancy between the two increases and this is exaggerated with increasing eccentricity. At h = 70�significantly different qualitative behaviour for the flow is found with large eccentricity.

Now that the governing ODEs for the series solutions of both families are available in Sections A.1 and B.1, the meth-od is considerably easier to use in engineering applications than the numerical method on grounds of cost and requiredexpertise. However, the decision of which method to use should be taken with the required accuracy levels kept inmind.

Ultimately our aim is to conduct stability analyses on the resulting flow profiles (in a similar manner to [1–3]) and weneed to understand the extent of the usefulness of the series solution method in terms of latitude and eccentricity for thisapplication.

First of all we note that the normal-velocity profile, W, occurs at O(1/R) in the equations that govern the stability of theflow over the rotating sphere (see Eqs. (2.13)–(2.18) of [2]). Here R is the Reynolds number that can be interpreted as ameasure of the rotation rate of the body in local analyses conducted at fixed latitudes; typically R = O(102) and above. Pre-liminary derivations of the equations for e – 0 (not shown here) confirm this for both spheroid families. The accuracy ofthe normal-velocity profile is therefore to be considered secondary to the accuracy of the latitudinal and azimuthal veloc-ities. That the W-profiles have errors 2 orders of magnitude greater than the U- and V-profiles at h = 10� in the prolate caseis therefore not a concern (see Table 1). Experience of stability analyses of flows over related geometries shows that theresulting neutral curves of convective and absolute instabilities are typically insensitive to modifications of O(10�4) to theU- and V-profiles. For this reason, we expect that the series-solution approximation would be sufficiently accurate to cor-rectly capture the stability characteristics and predict critical parameters with reasonable accuracy up to latitudes ofh = 50� for e 6 0.5 and latitudes of h = 30� for e > 0.5 in the prolate case. This range is also expected to be suitable inthe oblate case.

The formulation used here has led to flow profiles in the latitudinal and azimuthal directions that are similar in boththe prolate and oblate cases at a particular eccentricity. However, a significant difference in the normal component isfound in that the profiles asymptote to equal distances either side of the rotating-sphere profiles. As discussed above,the stability analyses are expected to have the W-component appearing at O(1/R) which implies that the high Reynoldsnumber limit of the neutral curves (where the governing equations limit to the inviscid Rayleigh equation) will be verysimilar. Exactly how the critical Reynolds numbers (and other critical parameters) of the prolate and oblate flows will com-pare at particular e will be determined by the structure of the two sets of governing stability equations and how geometryeffects occur at O(1/R). However, it is likely that they will only differ slightly and that the normal component of the flowprofile may be key in distinguishing any slight differences in stability characteristics between prolate and oblate spheroidshas implications for the use of the series solution. The series solution leads to inaccurate approximations for the normalcomponent and these may dominate the very effects that we are interested in. In situations where the full perturbationsystem is to be solved with a view to distinguishing prolate and oblate spheroids the numerical solution is therefore tobe preferred.

In the above discussion we have been concerned with the use of the flow profiles within a stability analysis,however, other applications may require different conclusions about the applicability of the series-solution approximations.

Acknowledgment

A.S. would like to thank the University of Peshawar, Pakistan for financial support during this ongoing project.

Appendix A. Details of the series solution for the prolate family

A.1. Ordinary differential equations

F21 þ H1F 01 � G2

1 ¼ F 001; ð15Þ



4F1F3 þ H1F 03 þ H3F 01 � 2G1G3 þ13� e2

2ð1� e2Þ

� �G2

1 þe2

2ð1� e2Þ F21 þ H1F 01

� �¼ F 003; ð16Þ

6F1F5 þ 3F23 þ H1F 05 þ H3F 03 þ H5F 01 � 2G1G5 � G2

3 þ23� e2

ð1� e2Þ

� �G1G3

þ 145þ e2ð8� 5e2Þ

24ð1� e2Þ2

!G2

1 �e2ð4� e2Þ

24ð1� e2Þ2F2

1 þ H1F 01� �

þ e2

2ð1� e2Þ 4F1F3 þ H1F 03 þ H3F 01� �

¼ F 005; ð17Þ

8F1F7 þ 8F3F5 þ H1F 07 þ H3F 05 þ H5F 03 þ H7F 01 � 2G1G7 � 2G3G5 þ�e2

2ð1� e2Þ þ13

� �G2

3

þ 23� e2

1� e2

� �G1G5 þ

245þ ð8� 5e2Þe2

12ð1� e2Þ2

!G1G3 þ

2945� e2ð16� 2e2 þ e4Þ

240ð1� e2Þ3

!G2

1

þ e2ð16þ 28e2 þ e4Þ720ð1� e2Þ3

F21 þ H1F 01

� �� e2ð4� e2Þ

24ð1� e2Þ24F1F3 þ H1F 03 þ H3F 01� �

þ e2

2ð1� e2Þ 6F1F5 þ 3F23 þ H1F 05 þ H3F 03 þ H5F 01

� �¼ F 007; ð18Þ

2F1G1 þ H1G01 ¼ G001; ð19Þ

4F1G3 þ 2F3G1 þ H1G03 þ H3G01 þe2

1� e2 �13

� �F1G1 þ

e2

2ð1� e2ÞH1G01 ¼ G003; ð20Þ

6F1G5 þ 4F3G3 þ 2F5G1 þ H1G05 þ H3G03 þ H5G01 þ2e2

ð1� e2Þ �13

� �F1G3 þ

e2

ð1� e2Þ �13

� �F3G1 �

e2ð2� e2Þ4ð1� e2Þ2

þ 145

!F1G1

þ e2

2ð1� e2Þ ðH3G01 þ H1G03Þ �e2ð4� e2Þð24ð1� e2Þ2Þ

H1G001 ¼ G005; ð21Þ

8F1G7 þ 6F3G5 þ 4F5G3 þ 2F7G1 þ H1G07 þ H3G05 þ H5G03 þ H7G01 þ3e2

1� e2 �13

� �F1G5 þ

2e2

1� e2 �13

� �F3G3

þ e2

1� e2 �13

� �F5G1 �

e2ð5� 2e2Þ6ð1� e2Þ2

þ 145

!F1G3 �

e2ð2� e2Þ4ð1� e2Þ2

þ 145

!F3G1 þ

e2ð32þ 11e2 þ 2e4Þ360ð1� e2Þ3

� 2945

!F1G1

þ e2

2ð1� e2Þ H1G05 þ H5G01� �

þ e2ð16þ 28e2 þ e4Þ720ð1� e2Þ3

!H1G01 þ

e2

2ð1� e2ÞH3G03 �e2ð4� e2Þ

24ð1� e2Þ2H3G01 þ H1G03� �

¼ G007; ð22Þ

2F1 þ H01 ¼ 0; ð23Þ

4F3 þ H03 þe2

1� e2 �13

� �F1 ¼ 0; ð24Þ

6F5 þ H05 �e2ð2þ e2Þ3ð1� e2Þ2

þ 145

!F1 þ

e2

1� e2 �13

� �F3 ¼ 0; ð25Þ

8F7 þ H07 þe2ð2þ 11e2 þ 2e4Þ

15ð1� e2Þ3� 2

945

!F1 �

e2ð2þ e2Þ3ð1� e2Þ2

þ 145

!F3 þ

e2

1� e2 �13

� �F5 ¼ 0: ð26Þ



A.2. Computed boundary values

e n Fn(0;e) F 0nð0; eÞ Gn(0;e) G0nð0; eÞ Hn(1;e)

0.0 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.22129 �0.16667 0.24764 0.160745 0.00000 0.02071 0.00833 �0.02569 0.000847 0.00000 �0.00189 �0.00020 0.00181 0.00085

0.1 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.22009 �0.16667 0.24600 0.162365 0.00000 0.01979 0.00833 �0.02447 0.000077 0.00000 �0.00160 �0.00020 0.00147 0.00103

0.2 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.21633 �0.16667 0.24085 0.167435 0.00000 0.01684 0.00833 �0.02057 �0.002437 0.00000 �0.00064 �0.00020 0.00031 0.00168

0.3 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.20953 �0.16667 0.23153 0.176635 0.00000 0.01126 0.00833 �0.01318 �0.007377 0.00000 0.00138 �0.00020 �0.00219 0.00313

0.4 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.19864 �0.16667 0.21660 0.191355 0.00000 0.00175 0.00833 �0.00048 �0.016307 0.00000 0.00537 �0.00020 �0.00735 0.00633

0.5 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.18166 �0.16667 0.19332 0.214315 0.00000 �0.01455 0.00833 0.02147 �0.032787 0.00000 0.01373 �0.00020 �0.01858 0.01392

0.6 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.15442 �0.16667 0.15597 0.251145 0.00000 �0.04445 0.00833 0.06211 �0.065737 0.00000 0.03360 �0.00020 �0.04619 0.03436

0.7 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.10708 �0.16667 0.09107 0.315165 0.00000 �0.10737 0.00833 0.14869 �0.142047 0.00000 0.09208 �0.00020 �0.12947 0.10254

Appendix B. Details of the series solution for the oblate family

B.1. Ordinary differential equations

F21 þ


H1F 01 � G21 ¼ ð1� e2ÞF 001; ð27Þ

4F1F3 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H1F 03 þ H3F 01� �

� 2G1G3 þ13þ e2

2

� �G2

1 �e2

2F2

1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H1F 01� �

¼ ð1� e2ÞF 003; ð28Þ

6F1F5 þ 3F23 þ


ðH1F 05 þ H3F 03 þ H5F 01Þ � 2G1G5 � G23 þ

23þ e2

� �G1G3 þ

145� e2

3þ e4

8

� �G2

1

þ e2

6� e4

8

� �F2


H1F 01� �

� e2

24F1F3 þ


H1F 03 þ H3F 01� ��

¼ ð1� e2ÞF 005; ð29Þ



8F1F7 þ 8F3F5 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H1F 07 þ H3F 05 þ H5F 03 þ H7F 01� �

� 2G1G7 � 2G3G5 þe2

2þ 1

3

� �G2

3 þ23þ e2

� �G1G5

þ 245� 2e2

3þ e4

4

� �G1G3 þ

2945þ e2

15� e4

8þ e6

16

� �G2

1 þ�e2

45þ e4

12� e6

16

� �F2


H1F 01� �

þ e2

6� e4

8

� �4F1F3 þ


H1F 03 þ H3F 01� ��

� e2

26F1F5 þ 3F2


H1F 05 þ H3F 03 þ H5F 01� ��

¼ ð1� e2ÞF 007;

ð30Þ

2F1G1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H1G01 ¼ ð1� e2ÞG001; ð31Þ

4F1G3 þ 2F3G1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H1G03 þ H3G01� �

� e2 þ 13

� �F1G1 �

e2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

2H1G01 ¼ ð1� e2ÞG003; ð32Þ

6F1G5 þ 4F3G3 þ 2F5G1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H1G05 þ H3G03 þ H5G01� �

� 2e2 þ 13

� �F1G3 � e2 þ 1

3

� �F3G1

þ e2

2� e4

4� 1

45

� �F1G1 �


2H3G01 þ H1G03� �

þ e2

6� e4

8

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H1G01 ¼ ð1� e2ÞG005; ð33Þ

8F1G7 þ 6F3G5 þ 4F5G3 þ 2F7G1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H1G07 þ H3G05 þ H5G03 þ H7G01� �

� 3e2 þ 13

� �F1G5 � 2e2 þ 1

3

� �F3G3

� e2 þ 13

� �F5G1 �

e4

2� 5e2

6þ 1

45

� �F1G3 �

e4

4� e2

2þ 1

45

� �F3G1 �

2945þ 4e2

45� 5e4

24þ e6

8

� �F1G1

� e2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

2H1G05 þ H5G01� �

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p e2

45� e4

12þ e6

16

� �H1G01 �


2H3G03

þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p e2

6� e4

8

� �H3G01 þ H1G03� �

¼ ð1� e2ÞG007; ð34Þ

2F1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2p

H01 ¼ 0; ð35Þ


H03 �13þ e2

� �F1 ¼ 0; ð36Þ


H05 þ2e2

3� e4 � 1

45

� �F1 �

13þ e2

� �F3 ¼ 0; ð37Þ


H07 �2

945þ 2e2

15� e4 þ e6

� �F1 �

145� 2e2

3þ e4

� �F3 �

13þ e2

� �F5 ¼ 0: ð38Þ

B.2. Computed boundary values


0.0 1 0.00000 0.51023 1.00000 �0.61592 �0.884473 0.00000 �0.22129 �0.16667 0.24764 0.160745 0.00000 0.02071 0.00833 �0.02569 0.000847 0.00000 �0.00189 �0.00020 0.00181 0.00085

0.1 1 0.00000 0.51280 1.00000 �0.61902 �0.884473 0.00000 �0.22363 �0.16667 0.25050 0.158925 0.00000 0.02173 0.00833 �0.02702 0.001667 0.00000 �0.00217 �0.00020 0.00218 0.00071

0.2 1 0.00000 0.52075 1.00000 �0.62862 �0.884473 0.00000 �0.23084 �0.16667 0.25927 0.153395 0.00000 0.02481 0.00833 �0.03104 0.004067 0.00000 �0.00296 �0.00020 0.00321 0.00035




0.3 1 0.00000 0.53487 1.00000 �0.64566 �0.884473 0.00000 �0.24354 �0.16667 0.27461 0.143915 0.00000 0.03001 0.00833 �0.03777 0.007877 0.00000 �0.00415 �0.00020 0.00474 �0.00004

0.4 1 0.00000 0.55671 1.00000 �0.67203 �0.884473 0.00000 �0.26298 �0.16667 0.29785 0.130045 0.00000 0.03745 0.00833 �0.04731 0.012857 0.00000 �0.00557 �0.00020 0.00652 �0.00026

0.5 1 0.00000 0.58917 1.00000 �0.71121 �0.884473 0.00000 �0.29140 �0.16667 0.33138 0.111115 0.00000 0.04743 0.00833 �0.05991 0.018687 0.00000 �0.00700 �0.00020 0.00824 �0.00008

0.6 1 0.00000 0.63779 1.00000 �0.76990 �0.884473 0.00000 �0.33311 �0.16667 0.37978 0.086125 0.00000 0.06059 0.00833 �0.07611 0.025087 0.00000 �0.00828 �0.00020 0.00962 0.00061

0.7 1 0.00000 0.71447 1.00000 �0.86246 �0.884473 0.00000 �0.39715 �0.16667 0.45267 0.053555 0.00000 0.07847 0.00833 �0.09736 0.031817 0.00000 �0.00933 �0.00020 0.01057 0.00169

References

[1] S.J. Garrett, The Stability and Transition of the Boundary Layer on Rotating Bodies, Ph.D. Thesis, Cambridge University, 2002.[2] S.J. Garrett, N. Peake, The stability and transition of the boundary layer on a rotating sphere, J. Fluid Mech. 456 (2002) 199–217.[3] S.J. Garrett, N. Peake, The stability of the boundary layer on a rotating sphere in a uniform axial flow, Eur. J. Mech., B 23 (2004) 241–253.[4] L. Howarth, Note on the boundary layer on a rotating sphere, Phil. Mag. Ser. 7 42 (1951) 1308–1315.[5] S.D. Nigam, Note on the boundary layer on a rotating sphere, Z. Angew. Math. Phys. 5 (1954) 151–155.[6] W.H.H. Banks, The boundary layer on a rotating sphere, Quart. J. Mech. Appl. Math. 48 (1965) 443–454.[7] B.S. Fadnis, Boundary layer on rotating spheroids, ZAMP 5 (1954) 156–163.[8] P.M. Morse, Methods of Theoretical Physics, McGraw-Hill, 1953.[9] T. Von Kármán, Uber laminare und turbulente Reiburg, Z. Angew. Math. Mech. 1 (1921) 233–252.

[10] R. Manohar, The boundary on a rotating sphere, Z. Angew. Math. Phys. 18 (1967) 320.[11] W.H.H. Banks, The laminar boundary layer on a rotating sphere, Acta Mech. 24 (1976) 273–287.[12] O. Sawatzki, Das Strömungsfeld un eine rotiernde kugel, Acta Mech. 9 (1970) 159–214.[13] Y. Kohama, R. Kobayashi, Boundary-layer transition and the behaviour of spiral vortices on rotating spheres, J. Fluid Mech. 137 (1983) 153–164.[14] H.B. Keller, A new difference scheme for parabolic problems, in: J. Bramble (Ed.), Numerical Solution of Partial Differential Equations, vol. 22, Academic

Press, 1970, pp. 273–287.[15] M. Berzins, P.M. Dew, R.M. Furzeland, Developing software for time-dependent problems using the method of lines and differential-algebraic

integrators, Appl. Numer. Math. 5 (1989) 375–397.

Appendix B.2 (continued)


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