The competing effects of plate stretching and transpiration on rotational stagnation-point flow

Accepted Manuscript The competing effects of plate stretching and transpiration on rotational stagnation-point flow P.D. Weidman PII: DOI: Reference:

S0997-7546(16)30505-2 http://dx.doi.org/10.1016/j.euromechflu.2017.04.003 EJMFLU 3160

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European Journal of Mechanics B/Fluids

Received date: 26 October 2016 Revised date: 21 February 2017 Accepted date: 11 April 2017 Please cite this article as: P.D. Weidman, The competing effects of plate stretching and transpiration on rotational stagnation-point flow, European Journal of Mechanics B/Fluids (2017), http://dx.doi.org/10.1016/j.euromechflu.2017.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

The competing effects of plate stretching and transpiration on rotational stagnation-point flow P. D. Weidman Department of Mechanical Engineering University of Colorado Boulder, CO 80309-0427

Abstract The simultaneous effects of transpiration through and radial extensional motion of a flat plate on Agrawal stagnation-point flow is considered. Difficulties with standard shooting techniques are overcome using Crocco variables which also serve to better elucidate the solution structure. The governing similarity equation can be mapped into that found by Weidman, Davis & Kubitschek (2008) for uniform shear flow over a stretching surface with transpiration with the proviso that the boundary conditions on the radial stretch rate and transpiration are not the identical. However, this allows us to use the stability results for multiple steady flow solutions already found. Although the two problems are qualitatively similar they are quantitatively different.

1

Introduction

Similarity solutions are often used to take a first look at difficult problems encountered in Newtonian flows, porous media flows and non-Newtonian flows, just to name a few of direct engineering interest. Many such problems are formulated using the boundary-layer approximation. Less often one finds a similarity formulation that represents an exact solution of the Navier-Stokes equations; see Drazin and Riley [1]. In either case, the boundary conditions must be consistent with the similarity ansatz. In the present study we consider the combined effects of radial extensional motion of, and transpiration through, the porous surface of a flat plate placed beneath a seldom-studied stagnation-point flow. This is the exact solution of the Navier-Stokes equations representing axisymmetric rotational stagnation-point flow impinging normal to a rigid flat surface found by Agrawal [2] using spherical coordinates. Some earlier studies of the effects of tangential wall movement and/or wall transpiration on exact stagnation-point flows will be reviewed here. Gorla [3] reported on the nonsimilar motion that ensues in axisymmetric stagnation-point flow towards a circular cylinder translating laterally at constant speed; the axisymmetric stagnation-point flow impinging on a stationary circular cylinder was discovered by Wang [4]. For this same stagnation-point flow, Paullet and Weidman [5] studied the effect of uniform transpiration through porous cylinder walls. Wang [6] investigated, inter-alia, the flow induced by a radially-shrinking sheet in both two-dimensional (Hiemenz [7]) and axisymmetric three-dimensional (Homann [8]) stagnation-point flows. Evidently Wang [6] was not aware that Riley and Weidman [9] had already reported the Himenez part of his study. Fang and Zhang [10] studied mass transfer and wall stretching of particular Falkner-Skan equations where an exact solution was available. Bhattacharyya and Layek [11] studied the effects of suction and blowing on two-dimensional stagnation-point flow and heat transfer towards a sheet with planar extensional motion using the boundary-layer approximation. Thus the Hiemenz stagnation-point flow with the combined effects of wall stretching and transpiration has been completed. Fang and Zhang [12] studied solutions for point sink flow inside a cone with mass transpiration with a moving wall. Fang and He [13] studied the steady viscous flow between two porous disks with stretching motion. Recently, Weidman and Yi Ping [14] reported on the effects of radial stretching and transpiration beneath axisymmetric Homann stagnation-point flow. While Homann [8] stagnation-point flow is irrotational in the far field, the problem studied here for Agrawal [2] stagnation-point flow is rotational in the far field. The presence of blowing or suction requires that a radial pressure gradient is available to 2

maintain the flow. A unique feature of the problem is that while λ, the stretching parameter, appears only in the boundary conditions, the transpiration parameter µ appears both in the boundary conditions and in the governing equation. Multiple solutions are found to exist in restricted regions of λ-µ parameter space with the feature that, for µ greater than a small negative number, a segment of the negative λ-axis, depending on µ, is associated with a zero wall stress solution. This leads to a complicated set of possibilities which include unique, dual and triple solutions as µ and λ are varied. Moreover, an interesting quadruple feature occurs for µ, λ both positive. We note at the outset that the present problem has close similarity to that of Weidman, Davis and Kubitschek [12], hereafter cited as WDK, who considered uniform shear flow over a stretching surface with transpiration. The similarity resides in the fact that the present problem has far-field dimensional radial velocity u∗ = a r∗ z ∗ while the analogous flow cited above has far-field uniform velocity u∗ = β z ∗ . Indeed we will show that the results of WDK may be mapped by an affine transformation into the problem studied here, though care must be taken to identify correspondence between the stretching and transpiration parameters. A sketch of the flow and boundary conditions in dimensional variables is shown in figure 1. In this figure 2w0∗ is the suction through the porous boundary and b is the radial stretch rate of the surface. The presentation is as follows. The standard similarity reduction of the problem is developed in §2 and its correspondence with that of WDK is elucidated. Difficulties encountered

while integrating the governing equation in certain regions of parameter space using a shooting method motivated a reformulation of the problem using Crocco variables. This variant

formulation and subsequent multiple steady solutions obtained for blowing (µ < 0), suction (µ > 0), and zero transpiration (µ = 0) are also given in §2. A perturbation analysis of

solutions around a focal point and the asymptotic behavior of solutions for large λ and large µ are presented in §3. The stability of multiple solutions are reported in §4 and concluding remarks are given in §5.

2

Problem Formulation and Solution

Following standard notation we denote (r∗ , z ∗ ) cylindrical coordinates, (u∗ , w∗ ) the respective coordinate velocities, and p∗ the fluid pressure. Transforming Agrawal’s [2] exact NavierStokes solution found with the aid of spherical coordinates to cylindrical coordinates gives the surprisingly simple result w∗ = −az ∗ 2

u∗ = ar∗ z ∗ ,

3

(2.1)

in which a is the strength of the flow with units 1/LT . This may be interpreted as a rotational axisymmetric stagnation-point flow on the impermeable boundary z ∗ = 0 on which the viscous no-slip boundary condition is satisfied. Introducing the dimensionless variables (r, η), (u, w) and p as  ν 1/3 (r∗ , z ∗ ) = (r, η), (u∗ , w∗ ) = a1/3 ν 2/3 (u, w), p∗ = ρν 4/3 a2/3 p a

(2.2)

the continuity and axisymmetric Navier-Stokes equations in the absence of swirl become 1 (ru)r + wη = 0 r

(2.3a)

1 u uur + wuη = −pr + urr + ur + uηη − 2 r r 1 uwr + wwη = −pη + wrr + wr + wηη . r For uniform transpiration through the porous boundary we modify (2.1) to read w∗ = −az ∗ 2 − 2w0∗

u∗ = ar∗ z ∗ ,

(2.3b) (2.3c)

(2.4)

which may be written in dimensionless coordinates as u = rη,

w = −(η 2 + 2µ),

µ=

w0∗ a1/3 ν 2/3

(2.5)

where µ is the transpiration parameter. Insertion into the governing equations (2.3) and solving for the pressure field gives 2

2

p(r, η) = p0 + µ(r − 2η ) −



η4 + 2η 2



(2.6)

where p0 is the stagnation pressure. This shows that for finite transpiration µ 6= 0 a radial

pressure gradient is necessary to maintain the rotational Agrawal stagnation-point flow. Further, the inclusion of radial plate stretching gives u∗ (r∗ , 0) = br∗ ,

→

u(r, 0) = λr,

λ=

b a2/3 ν 1/3

(2.7)

where λ is the plate stretching parameter. The boundary conditions suggest a similarity solution in the form u(r, η) = rf ′ (η),

w(η) = −2f (η)

(2.8)

which satisfies the continuity equation (2.3a) and insertion into (2.3b,c) taking into account the boundary conditions furnishes the boundary-value problem f ′′′ + 2f f ′′ − f ′2 = 2µ 4

(2.9a)

f ′ (0) = λ,

f (0) = µ,

f ′′ (∞) = 1

(2.9b, c, d )

where µ > 0 corresponds to suction and λ > 0 represents stretching away from the origin. The corresponding pressure field is given by p(r, η) = p0 + µr2 − 2(f 2 + f ′ )

(2.10)

Note that the exact solution f (η) = µ + η 2 /2, valid at λ = 1 for all transpirations µ, defines the {0, 1} focal point of the system. Of practical interest is the radial wall shear stress τr given by ∂u∗ τr = ρν ∗ = ρν 4/3 a1/3 rf ′′ (0). ∂z z∗ =0

We now show that boundary value problem (2.9) can be obtained from that found by

WDK for linear shear flow above a unilaterally stretching plate with transpiration. For that problem WDK found the boundary-value-problem 3F ′′′ + 2F F ′′ − F ′2 = 2µ F ′ (0) = λ,

F (0) = µ,

(2.11a)

F ′′ (∞) = 1

(2.11b, c, d )

in which the similarity ansatz is u(x, z) = x1/3 F ′ (ζ),

w(x, z) =

1 [2F (ζ) − ζF ′ (ζ)], 3x1/3

ζ=

z x1/3

.

(2.11e, f , g)

where µ and λ are the transpiration and stretching parameters. Introducing the affine transformation f (ζ) = αf (η), transforms (2.11a) into



3β α



η = βζ

f ′′′ + 2f f ′′ − f ′2 =

µ α2 β 2

(2.12)

.

(2.13)

To obtain (2.9) it is clear that the term in parentheses must be unity. Moreover, the righthand-side of (2.13) must equal µ and also the boundary condition F (0) = αf (0) must satisfy (2.9b). The above results then require α = 3β,

µ=

µ α2 β 2

which has solution α = 32/3 ,

β=

=

µ α

(2.14a)

.

(2.14b)

1 31/3

The correspondence is completed by identifying µ and λ as µ=

µ , α

λ = βλ 5

(2.15)

Owing to the above relation, the results calculated in WDK correspond only to special values of µ and λ in the present problem. For this reason we perform new calculations for appropriately chosen values of these parameters, though the stability calculations reported in WDK will be used to shorten the presentation of flow stability. We note the antagonistic roles of transpiration and pressure gradient on the flow. When µ > 0 there is suction which acts to accelerate the flow near the wall, but Eq. (2.10) shows that this is accompanied by an adverse pressure gradient which tends to decelerate the flow. Strong adverse pressure gradients generally lead to boundary layer separation. In the opposite case, for µ < 0 corresponding to blowing, the pressure gradient is favorable. However, blowing can precipitate boundary layer lift off. Consequently, for both blowing and suction, the streamwise pressure gradient acts in opposition to wall transpiration in their tendencies to precipitate flow separation.

2.1

Crocco variables

The standard shooting technique to obtain the steady solutions encountered convergence difficulties when completing the λ < 0 portions of the µ < 0 curves in figures 4 and 5. More and more accurate initial guesses to the wall shear stress were required which eventually became elusive. We chose to exploit the absence of η in (2.9a) in preference to a more elaborate computation. It proved both expedient and helpful to introduce the Crocco variables s = f ′ (η), g = f ′′ (η), in terms of which problem (2.9) takes the form d2 g dg + (s2 + 2µ) = 0, 2 ds ds with f (0) = µ replaced by the ‘starting’ condition   dg + 2µ = s2 + 2µ, g ds g2

g(∞) = 1

(2.16)

at s = λ.

(2.17)

The exact solution, f (η) = µ + η 2 /2, valid for all µ at λ = 0, is mapped into the semi-infinite line g = 1, s > 0 in the (s, g)-plane. For each µ, there is a family of solutions of (2.16), all √ lying in g < 1 and with g ′′ and g ′ having opposite signs except in the interval |s| < −2µ when µ < 0. By writing the differential equation in the inverted form  2 2 ds 2d s 2 g = (s + 2µ) 2 dg dg

(2.18)

it is readily shown that any g(s) curve crossing the s-axis must do so vertically; such a point is a point of inflexion of f (η). On substituting s ∼ s0 + 21 α2 g 2 + . . . in (2.18), one finds that α2 = (s20 + 2µ)−1 as required by (2.17). Thus the ‘starting’ condition (2.17) is satisfied at such a crossing point. 6

2.2

Suction solutions (µ > 0)

Numerical solutions of (2.9) for µ > 0 were performed using the standard shooting technique in which the shear stress parameter f ′′ (0) is varied to converge on the far-field condition f ′′ (ηmax ) = 1 for sufficiently large values of ηmax . Parametric solution curves in {λ, f ′′ (0)}space for µ = {0.0, 0.1, 0.25, 0.5} are displayed in figure 2. Note that the µ = 0 curve

terminates at {0,0} while curves for µ > 0 continue as lower solution branches yielding dual solutions for all λ > λc . The turning point values λc , associated with positive wall stress and listed in Table 1, were determined to an estimated accuracy ±0.0001 using numerical interpolation of data in the region of the turning point.

Examination of a Crocco solution, obtained by numerical integration of (2.16), verifies that, for points in the third quadrant of the parameter space, the ‘starting’ condition (2.17) is additionally satisfied at g = 0 and a point on the other side of the focal point {0,1} on a constant µ curve in the {λ, f ′′ (0)}-plane, for a total of three points on the corresponding

g(s) curve. Similarly, for points in the fourth quadrant, (2.17) is additionally satisfied at g = 0 and a pair of points on either side of the focal point {0,1}, for a total of four points on the corresponding g(s) curve. The zero wall stress solutions exhibit a bifurcation of each µ > 0 curve in the form of a

finite segment, depending on µ, of the negative λ-axis. As µ decreases, this segment moves towards the origin. A comparison of the ‘reversed flow’ solution I and the three embedded solutions II, III and IV for λ = 2.0 is given in figure 3. The embedded property can be verified by moving the f ′ (η) profile I downwards by successive values η0 and seeing that they coincide with profiles II, III and IV. The stability of these solutions will be discussed in §4.

2.3

Blowing solutions (µ < 0)

Calculations for µ < 0 corresponding to positive transpiration through the plate were easily carried out for all λ > 0 using the standard shooting technique. Difficulties were encountered, however, on each µ < 0 curve at some negative value of λ and overcome by use of the Crocco variables, as described above. With g ′ (s) and g ′′ (s) now having the same sign in the interval √ |s| < −2µ, there is a marked tendency for g(s) to become very ‘flat’ while remaining √ positive, as s decreases towards − −2µ.

Numerical computations based on the Crocco solution show that each level curve of µ < 0 √ in the second quadrant parameter space has (λ0 , 0) as an end point, where λ0 = − −2µ. It is found that, for values of µ near zero, the ‘starting’ condition (2.17) is satisfied at two

distinct points on each solution curve g(s) for fixed µ. These provide a pair of points, either side of the focal point {0,1}, on a constant µ curve in the {λ, f ′′ (0)}-plane and demonstrate that the velocity profile for the left-hand point includes the displaced profile corresponding 7

to the right-hand point. Results from integrations using both the f (η) formulation in (2.9) and the g(s) formulation in (2.16) are combined to obtain the wall stress solution curves displayed in figure 4. Note that unique solutions exist for µ = -0.5 and -0.25 whilst triple solutions exist for µ = -0.02 and -0.10, though the λ-range of their existence for µ = -0.02 is extremely small. We now show how results from WDK may be used to generate further solutions for µ < 0. The values µ = {−0.5, −0.25, −0.10, −0.02} are found through the transformation

to correspond to µ = {−0.240375, −0.120187, −0.0480749, −9.615 × 10−3 } in the present problem and these solutions are displayed in figure 4 as dashed lines. Figure 5 shows details of the parametric curves near the λ-axis. At these negative values of µ there is no bifurcation of the µ = 0 end point into lower branch solutions; rather the end point just slides continuously to the left as µ decreases from zero. It is clear that each wall stress curve terminates on the λ-axis, indicated by the black dots in figure 5. The values of the left turning point λc and right turning point λt (when they exist), and the end point values λ0 are listed in Table 3. Since the plate is shrinking into the origin, f ′ (η) is negative close to the wall, with a smooth transition to the displaced outer stagnation flow with increasing η. For small negative values of µ the boundary layer lifts off the wall. Details of this blow-off are not presented since they are very similar to the results reported in WDK.

3

Focal point analysis and large λ and µ asymptotics

In this section the parametric solution behavior near the {0,1} focal point is analyzed and asymptotic results are provided for large λ and also for large µ.

3.1

The neighborhood of the focal point {0,1}

The nature of the parametric solution curves near the focal point {0,1} may be analyzed by

introducing, for any µ, the perturbation expansion solution developed in powers of λ ≪ 1 1 f (η) = (η + λ)2 + µ − λ2 f2 (η) + · · · 2

(3.1)

and its derivatives into (1.9). This yields, at O(λ2 ), the boundary-value problem f2′′′ + (η 2 + 2µ)f2′′ − 2ηf2′ + 2f2 = 0 1 f2′ (0) = 0, f2′′ (∞) = 0. f2 (0) = , 2 By the simple expedient of differentiation, one finds the separable equation f2iv + (η 2 + 2µ)f2′′′ = 0 8

(3.2a) (3.2b, c, d )

(3.3)

two integrations of which yield f2′′

=A

Z

∞

e−(t

3 /3+2µt)

dt

(3.4)

η

satisfying f2′′ (∞) = 0. Since we need f2′′ (η) only near η = 0, write   Z η −(t3 /3+2µt) ′′ e dt f2 (η) = A I(µ) −

(3.5)

0

where the definite integral I(µ) is given in terms of the related Airy function Hi(x) defined in Abramowitz and Stegun [19], namely Z ∞ 3 I(µ) ≡ e−(t /3+2µt) dt = π Hi(−2µ).

(3.6)

0

Now expand the integrand of the definite integral in (3.5) to sufficient order and integrate twice to obtain



 η3 η2 4 f2 (η) = C + Bη + A I(µ) − + O(η ) . 2 6

(3.7)

Boundary conditions (3.2b,c) require B = 0 and C = 1/2. The last arbitrary constant is determined by evaluating (3.2a) at η = 0 in order to recover the constant sacrificed in the above differentiation; this confirms A=

1 . 1 − 2µI(µ)

(3.8)

Evaluation of (3.5) at η = 0, and use of the relation f ′′ (0) = 1 − λ2 f2′′ (0), yields the desired result

f ′′ (0) = 1 − λ2

I(µ) . [1 − 2µI(µ)]

(3.9)

Since by inspection I(µ) < (1/2µ) for µ > 0, this confirms the downward concavity of the numerically determined solution curves at the {0,1} focal point.

3.2

Large λ asymptotics

Inspection of equations (2.9), with λ ≫ 1, suggests the rescaling η = λ−α ξ,

f (η) = λα F (ξ)

(3.10)

which maintains the form of the differential equation and yields F ′′′ + 2F F ′′ − F ′2 = 2µλ−4α F (0) = λ−α µ,

F ′ (0) = λ1−2α , 9

F ′′ (∞) = λ−3α

(3.11a) (3.11b, c, d )

where a prime now denotes differentiation with respect to ξ. On setting α = 1/2 to obtain F ′ (0) = 1, a suitable solution ansatz is F (ξ) = F0 (ξ) + µλ−1/2 F1 (ξ) + λ−1 F2 (ξ) + . . .

(3.12)

in which, according to (3.11), the first two functions are governed by F0′′′ + 2F0 F0′′ − F0′2 = 0 F0 (0) = 0,

F0′ (0) = 1,

F0′′ (∞) = 0

and F1′′′ + 2(F0 F1′′ + F0′′ F1 ) − 2F0′ F1′ = 0 F1 (0) = 1,

F1′ (0) = 0,

F1′′ (∞) = 0.

The factor µ in (3.12) was anticipated by noting that the problem for F1 is linear and forced solely by its initial value. Then the asymptotic form of a level µ curve in the {f ′ (0), f ′′ (0)}plane is given, according to (3.10) and (3.12), by f ′′ (0) = λ3/2 F ′′ (0) ∼ λ3/2 F0′′ (0) + λµF1′′ (0).

(3.13)

Numerical integrations give F0′′ (0) = −1.17372 and F1′′ (0) = −1.06643. A comparison of

the two-term large-λ asymptotic results with the numerically determined values of f ′′ (0) for µ = {−2, −1, 0, 1, 2, 3} is shown in figure 6.

4

Flow Stability

Following Merkin [20], the linear temporal stability of steady similarity solutions have been conducted by WDK for selected values of µ. This results in the eigenvalue equation and boundary conditions given in equation (26) of WDK. Solutions give an infinite set of eigenvalues α1 < α2 < α3 < · · · . If the smallest eigenvalue α1 is negative, there is an initial

growth of disturbances and the flow is unstable; when α1 is positive, there is an initial decay and the flow is stable. We take the opportunity to correct a misprint in WDK: the last term in the linear eigenvalue equation (26a) of WDK needs to be written αh′ instead of αh. Calculations have been carried out in WDK for selected values of µ. For those values of µ exhibiting turning points, local theory of the saddle-node bifurcation is sufficient to ensure that generically there is a transition of stability across each turning point; see, for example, Kuznetsov [21]. Consequently, since WDK has shown that the upper branches of the µ curves are always stable, one readily concludes that the branch below λc is unstable whilst the branch below λt is again stable. 10

Consider first the dual steady solutions found at µ = {0, 0.1, 0.25, 0.5} shown in figure 2.

Since by WDK the upper branches are stable, the lower branches below the critical points λc are unstable. Results are summarized in the similarity velocity plots f ′ (η) displayed in figure 3 which shows that only the profile IV marked by the solid line is stable while the remaining three profiles marked by the dashed lines are unstable. Consider now the various steady flow multiplicities found for blowing in figures 4 and 5. Since there are no turning points for the curves µ = −0.24037, −0.25 and -0.5, they are

each everywhere stable upper branch solutions. For the remaining values, on the other hand, there are left turning points λc and right turning points λt as listed in Table 3, which give

rise to triple solutions. Since the upper branches are all stable, it follows that the branch below λc is unstable and that the flow restabilizes on the branch below λt . It is clear that a transition, at some value of µ in the interval (−0.120187, −0.24037), between unique and triple solutions must exist. An example of the triple solution behavior for µ = −0.12019 at λ = −0.46456 is given in figure 7 in which solid lines are stable similarity profiles and the dashed line is the intermediate unstable profile. These profiles were transformed from the µ = −0.25, λ = 0.67 calculations of WDK which give the respective values µ = −0.12019 and λ = −0.45456. We conclude this section by displaying sample stable similarity velocity profiles at λ = 1.0

and 2.5 in figure 8. Results for an impermeable wall µ = 0 are shown by the solid lines and the effects of suction at µ = 0.25, 0.5 and blowing at µ = −0.25, −0.5 are shown by the

various dashed lines.

5

Summary and conclusion

The existence, multiplicity, and linear stability of self-similar solutions for Agrawal [2] stagnation-point flow of strength a over an infinite flat plate are determined. Results are provided in (plate stretching, wall stress) {λ, f ′′ (0)}-space for selected values of the transpiration parameter µ. With only plate stretching the external radial pressure gradient is zero, but when there is blowing or suction through the porous plate, a pressure gradient is required to maintain the flow. A stationary plate focal point {0,1} exists about which

solutions are regular and easily analyzed in terms of an Airy function. Numerical integration difficulties encountered using standard shooting techniques to track parametric solution curves at negative values of µ are overcome by reformulating the problem using Crocco variables which additionally provides a basis for analyzing solution behaviors not obvious using the original variables. The parametric curve of dual solutions for µ = 0 terminates an end point {0,0}. At positive values of λ, velocity profiles on the lower branch

exhibit reverse flow near the wall while those on the upper branch do not. Within the 11

reverse flow there is a point of inflexion in the velocity profile which is found to furnish a zero wall stress solution. Consequently, the two solution branches are augmented by a segment, depending on µ, of the negative λ-axis, which feature persists down to a small negative value of µ. These zero wall shear stress solutions, however, are found to be linearly unstable. For example see velocity profile II in figure 3. We have appealed to the analogous problem of uniform shear flow on a semi-infinite plate with stretching and transpiration to discern the stability of flows encountered. Indeed, the problem found in WDK may be transformed directly to the present boundary-value problem, albeit for different values of µ and λ. At negative values of µ corresponding to blowing, the flows are rather complicated. In this case the µ = 0 end point slides along the negative λ-axis to predetermined values √ λ0 = − −2µ. Though these curves do not cross the λ-axis, there exist unique solutions for sufficiently negative values of µ and triple solutions up to µ = 0− . The triple solutions exhibit the interesting feature that the upper and lower branch flows are both linearly stable, while flows on the intermediate branch lying between two turning points are unstable. We are now in a position to estimate the transition point between unique and triple solutions. The rate of increase in slope steepness from µ = −0.25 to µ = −0.24037 in figure 4 suggests that the demarcation from unique to triple solutions will take place very near µ = −0.230. The stable zero shear stress solutions found at negative values of µ on the negative λ axis are not without precedent. Apart from the analogous problem of uniform shear flow over a stretching surface with transpiration studied by WDK, Wang [22] has shown that a local deposition of floating solid material on a liquid surface, spreading horizontally with velocity inversely proportional to the radius, drives a flow in the liquid characterized by zero shear stress at the interface.

12

References [1] P. Drazin, N. Riley, The Navier-Stokes equations: a classification of flows and exact solutions. London Mathematical Society Lecture Note Series 334. Cambridge University Press, Cambridge, 2006. [2] H. L. Agrawal, A new exact solution of the equations of viscour motion with axial symmetry, Q. J. Mech. Math., 10 (1957) 42-44. [3] R. S. Gorla, Nonsimilar axisymmetric stagnation flow on a moving cylinder, Int. J. Engrg. Sci., 32 (1978) 541-553. [4] C. Y. Wang, Axisymmetric stagnation flow on a cylinder, Quart. Appl. Math. 32 (1974) 207-213. [5] J. E. Paullet, P. D. Weidman, Analytical results for a BVP describing radial stagnation flow with transpiration, J. Math. Anal. Appl. 247 (2000) 246-254. [6] C. Y. Wang, Stagnation flow towards a shrinking sheet. Int. J. Non-linear Mech. 43 (2008) 377-382. [7] K. Hiemenz, Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder, Dinglers Polytech. J. 326 (1911) 321-410. [8] F. Homann, Der Einfluss grosser Zahigkeit bei der Stromung um den Zylinder und um de Kugel, Z. Angew. Math. Mech. 16 (1936) 153-164. [9] N. Riley, P. D. Weidman, Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary, SIAM J. Appl. Math. 49 (1989) 1350-1358. [10] Fang, T. and J. Zhang, An exact analytical solution of the Falkner-Skan equation with mass transfer and wall stretching, Int. J. Non-Linear Mech. 43 (2008) 1000-1006. [11] K. Bhattacharyya, G. C. Layek, Effects of suction/blowing on steady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation, Int. J. Heat Mass Transf. 54 (2011) 302-307. [12] Fang, T. and X. He, Boundary layer solutions to a point sink flow inside a cone with mass transpiration and moving wall, Eur. Phys. J. Plus 130 (2015) 21. [13] Fang, T. and X. He, Steady viscous flow between two porous disks with stretching motion, ASME J. Fluids Engrg., 138 (2016) 011201.

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[14] P. D. Weidman, Y.-P. Ma, The competing effects of wall transpiration and stretching on Homann stagnation-point flow, Euro. J. Mech. B/Fluids 60 (2016) 237-241. [15] P. D. Weidman, A. M. J. Davis, D. Kubitschek, Crocco variable formulation for uniform shear flow over a stretching surface with transpiration: Multiple solutions and stability. Z. Angew. Math. Phys. 58 (2008) 313-332. [16] P. D. Weidman, Axisymmetric rotational stagnation-point flow impinging on a rotating disk, Z. Angew. Math. Phys. 66 (2015) 3425-3431. [17] P. D. Weidman, Axisymmetric rotational stagnation-point flow on a flat liquid surface. Euro. J. Mech. B/Fluids 56 (2016) 188-191. [18] P. D. Weidman, Axisymmetric rotational stagnation-point flow impinging on a radially stretching sheet. Int. J. Non-linear Mech. 82 (2016) 1-5. [19] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, U. S. Government Printing Office, Washington, 1972. [20] J. H. Merkin, On dual solutions occurring in mixed convection in a porous medium, J. Engng. Math. 2 (1985) 171-179. [21] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Applied Mathematical Sciences (Springer, New York, 2004). [22] C. Y. Wang, Effect of spreading material on the surface of a fluid — An exact solution, Int. J. Non-Linear Mech. 6 (1971) 255-262.

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Table 1 µ 0.50 0.25 0.10 0.00

λc -0.2700 -0.3271 -0.3741 -0.4130

Table 1. Variation of the turning points λc for µ ≥ 0.

Table 2 λ -0.450 -0.321 0.409

f ′′ (0) 0.0 0.367 0.789

η0 1.648 2.336 3.568

Table 2. Values of λ, f ′′ (0) and shifts η0 for solutions embedded in the λ = 2 solution at µ = 0.25

Table 3 µ −9.615 × 10−3 -0.02 -0.048075 -0.10 -0.120187 -0.24037 -0.25 -0.50

λc -0.4169 -0.4227 -0.4347 -0.4628 -0.4745 — — —

λt -0.1377 -0.1971 -0.2999 -0.4173 -0.4508 — — —

λ0 -0.1387 -0.2000 -0.3101 -0.4472 -0.4903 -0.6934 -0.7071 -1.0000

Table 3. Variation of first turning point λc , the second turning point λt (when it exists), and the end point λ0 with µ. The values displayed in boldface are those calculated here and the remaining values are those carried over from WDK.

15

z* w∗ u∗

r∗ br∗ 2w0∗ Figure 1. Schematic of a meridional cut through the axisymmetric rotational stagnation-point flow impinging on a porous plate. The plate stretches radially as br∗ and there is uniform suction 2w0∗ through the plate.

1 0.5

0.00

0 ′′

f (0) -0.5 -1 -1.5 -2 -0.5

µ = 0.50

0

0.25

0.10

0.5

1

1.5

2

λ Figure 2. Parametric solution curves of f ′′ (0) as a function of λ for selected values of µ as indicated.

16

8 7 I

6 5

II

η

III

4 3

IV

2 1 0 -1

0

1

f ′(η)

2

3

4

Figure 3. Comparison of similarity velocity profiles II, III and IV embedded in similarity velocity profile I at λ = 2.0 as indicated in Figure 2b for µ = 0.25. The dashed curves are unstable profiles and the single solid curve is a stable profile.

1 0.8 0.6 ′′

f (0) 0.4 -0.50

0.2

-0.25 -0.10

-0.02

0 -1

-0.5

0

0.5

1

λ

Figure 4. Solution curves of f ′′ (0) (wall stress) as a function of λ (plate stretching) showing the effect of blowing as indicated by the negative values of µ. The solid lines are results calculated here and the dashed lines are results carried over from WDK for µ = {−0.240375, −0.120187, −0.0480749, −9.615 × 10−3 }. 17

0.3 0.25 0.2 ′′

f (0) 0.15 0.1 0.05 -0.50

-0.25

-0.10

-0.02

0 -0.05 -1

-0.8

-0.6

-0.4

-0.2

0

λ

Figure 5. Details of the parametric solution curves as a function of λ near the λ-axis for five values of µ indicated. The solid lines are results calculated here and the dashed lines are results carried over from WDK. The zero stress end point values λ0 are marked with solid dots and open circles with values given in Table 3.

0 -50 -100 -150

f ′′ (0)

-2

-200

-1

-250

0 1

-300

2

-350

µ=3

-400 0

5

10

15

20

25

30

35

40

λ

Figure 6. Comparison of large-λ asymptotics (dashed lines) with numerical calculations (solid lines) for the indicated values of µ. 18

16 14

f ′′ (0) = 0.1022

12 10

η 8 6 0.2595

4

0.5049

2 0 -1

0

1

f ′ (η)

2

3

4

Figure 7. Three similarity velocity profiles obtained for µ = −0.12019 at λ = −0.46456. The two solid curves are stable profiles and the single dashed curve is an unstable profile.

2

1.5

η 1

blowing blowing

suction

0.5 suction

0 1

1.5

2

2.5

3

′

f (η) Figure 8. Sample radial velocity profiles at λ = 1.0 on the left and λ = 2.5 on the right showing the effects of suction (µ = 0.25, 0.5) and blowing (µ = −0.25, −0.5) where the solid lines are for an impermeable wall (µ = 0). 19