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Hiemenz stagnation-point flow impinging on a uniformly rotating plate
European Journal of Mechanics / B Fluids 78 (2019) 169–173
Contents lists available at ScienceDirect
European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu
Hiemenz stagnation-point flow impinging on a uniformly rotating plate Patrick Weidman Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427, United States
article
info
Article history: Received 4 March 2019 Received in revised form 7 May 2019 Accepted 13 June 2019 Available online 25 June 2019
a b s t r a c t Planar stagnation-point flow normally impinging a rotating plate is studied. Symmetries show that along a circle of radius r, the radial velocity ur can be computed from results in one quadrant and the angular velocity uθ can be computed from results in two quadrants. The results of this study for various values of the dimensionless angular velocity σ are compared with axisymmetric Homann stagnation-point flow on a rotating plate first studied by Hannah. © 2019 Elsevier Masson SAS. All rights reserved.
1. Introduction
This suggests the similarity solution of the form
Planar stagnation-point flow on a flat surface was first considered by Hiemenz [1]. Many variations on that problem have been studied including the effects of wall stretching, and transpiration through a porous wall; see the reviews by Wang [2,3]. More recent Hiemenz stagnation-point flow studies may be found in [4–6] and in [7]. In this problem we consider planar Hiemenz stagnation-point flow impinging on a uniformly rotating plate. The results presented here will be compared with the results of Hannah [8] who considered axisymmetric [9] stagnation-point flow impinging on a rotating plate. Our result as well as that of Hannah [8] represent exact solutions of the Navier–Stokes equation as defined in [10]. The presentation is as follows. The exact similarity reduction of the Navier–Stokes equation is given in Section 2. Results of numerical calculations are given in Section 3 and a formulation of Hannah’s problem is given in Section 5. Further discussion of results and concluding remarks are given in Section 6.
u(x, y, η) = c [x f ′ (η) + y g(η)],
η=
√
c
ν
z
(2.3a)
where a prime denotes differentiation with respect to η. The continuity equation for incompressible flow is satisfied by
√ w(η) = − c ν[f (η) + q(η)].
(2.3b)
Inserting this ansatz into the Navier–Stokes equations furnishes the coupled equations for η ≥ 0 as f ′′′ + (f + q)f ′′ + g 2 − f ′2 = 0,
(2.4a)
g + (f + q)g − (f + q )g = 0,
(2.4b)
q′′′ + (f + q)q′′ + 1 − q′2 + g 2 = 0,
(2.4c)
′′
′
′
′
to be solved with impermeable wall and far-field conditions f (0) = 0,
f ′ (0) = 0,
g(0) = −σ ,
2. Problem formulation
v (x, y, η) = c [yq′ (η) − x g(η)],
q(0) = 0,
f ′ (∞) = 0
(2.4d)
g(∞) = 0 q (0) = 0, ′
(2.4e) q (∞) = 1. ′
(2.4f)
Using Cartesian coordinates (x, y, z) with corresponding velocities (u, v , w ) we stipulate that there is a Hiemenz stagnation point flow aligned with the x-axis whose far-field behavior is given as
where σ = ω/c is the dimensionless angular velocity. Note that for f = g = ω = 0 one recovers from the above equations the classical problem of Hiemenz [1], viz.
u = 0,
q′′′ + qq′′ + 1 − q′2 = 0,
v = cy,
w = −cz .
(2.1)
At the surface of impingement of the plate, which is uniformly rotating with angular velocity ω, the velocity components are given by u(x, y, 0) = −ω y,
v (x, y, 0) = ω x.
E-mail address: [email protected]. https://doi.org/10.1016/j.euromechflu.2019.06.008 0997-7546/© 2019 Elsevier Masson SAS. All rights reserved.
(2.2)
q(0) = 0,
q′ (0) = 0,
q′ (∞) = 1. (2.5)
3. Presentation of results Numerical calculations were performed using the integrator package ODEINT in [11]. Integration lengths were varied to ensure solution independence of integration length.
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P. Weidman / European Journal of Mechanics / B Fluids 78 (2019) 169–173
and also the transformation (x, y) = r(cos θ, sin θ ). In this manner the horizontal velocities given in (2.3a) are rendered cylindrical using ur = c r [f ′ (η) cos2 θ + q′ (η) sin2 θ ]
(3.2a)
uθ = c r [(q′ (η) − f ′ (η)) sin θ cos θ − g(η)].
(3.2b)
In these coordinates the pressure is given as p = p0 −
ρ c 2 r 2 sin2 θ 2
− ρν c
(
(f + q)2 2
+ (f ′ + q′ )
) (3.3)
where p0 is the stagnation pressure. Of particular interest is the radial stress on the plate
τr = µ Fig. 1. Variation of stress parameters f ′′ (0), g ′ (0) and q′′ (0) as a function of σ . The [1] result at σ = 0 is q′′ (0) = 1.23259.
⏐ ∂ ur ⏐⏐ = ρν 1/2 c 3/2 r [cos2 θ f ′′ (0) + sin2 θ q′′ (0)] ∂ z ⏐z =0
and also the azimuthal stress on the plate given by
⏐ ∂ uθ ⏐⏐ τθ = µ = ρν 1/2 c 3/2 r [sin θ cos θ (q′′ (0) − f ′′ (0)) − g ′ (0)]. ∂ z ⏐z =0
The σ variation of the three shear stress parameters f ′′ (0), g (0) and q (0) are shown in Fig. 1 for values σ = {0, 2, 4, 6}. ′
′′
The obvious strategy is to report results in a meaningful way to compare with the results of Hannah [8] which are given in Section 4. To this end I have calculated in cylindrical coordinates (r , θ ) the radial and azimuthal velocities (ur , uθ ). This is effected using the relations ur = u cos θ + v sin θ ,
uθ = v cos θ − u sin θ
(3.1)
(3.4)
(3.5) We will compute results around the circle r = 1 for Hiemenz strain rate c = 1. There is an obvious anti-symmetry about the x-axis of the oncoming Hiemenz stagnation-point flow. Hence it is only necessary to compute results for one side of the x-axis, here chosen to be in the region 0 ≤ θ ≤ π . In this region there exist three symmetries. In the first case it is clear from Eq. (3.2a) that ur (π/2 + φ ) = ur (π/2 − φ )
(0 ≤ φ ≤ π/2).
(3.6)
Fig. 2. Radial velocity profiles ur (η) plotted for c = 1 at r = 1 for σ = {0, 2, 4, 6} respectively in panels (a), (b), (c), (d). In each figure the angles are θ = {0, π/6, π/4, π/3, π/2} with the arrow in the direction of increasing θ . The profile for θ = π/2 in panel (a) is the vertical line at the left end of the arrow.
P. Weidman / European Journal of Mechanics / B Fluids 78 (2019) 169–173
171
Fig. 3. Azimuthal velocity profiles uθ (η) plotted for c = 1 at r = 1. Results for σ = {0, 2, 4, 6} are displayed in panels (a), (b), (c), (d), respectively. The angles for each plotted profile are θ = {π/4, 5π/16, 3π/8, 7π/16, π/2, 9π/16, 5π/8, 11π/16, 3π/4} with the arrows showing the direction of increasing values of θ . The dashed lines show the profiles for θ = π/2.
Fig. 4. Radial velocity profiles ur (η) and azimuthal profiles uθ (η) for axisymmetric stagnation-point flow impinging on a rotating disk presented for σ = {0, 2, 4, 6}. The left of the arrow at uθ = 0 is the velocity profile for σ = 0.
From (3.2b) we see two symmetries, namely uθ (π/4 + φ ) = uθ (π /4 − φ )
(0 ≤ φ ≤ π/4)
(3.7a)
and uθ (3π /4 + φ ) = uθ (3π/4 − φ )
In the next section we compute the axisymmetric problem of Hannah [8] for comparison. 4. Hannah’s problem
(0 ≤ φ ≤ π/4)
(3.7b)
Thus, all radial velocity profiles ur (η) can be obtained by computation in the region 0 ≤ θ ≤ π/2 and selected profiles are shown in Fig. 2 for the values σ = {0, 2, 4, 6}. Similarly, all azimuthal velocity profiles uθ (η) can be obtained by computation in the regions π /4 ≤ θ ≤ π/2 and π/2 ≤ θ ≤ 3π/4. These profiles for σ = {0, 2, 4, 6} are displayed in Fig. 3.
Using cylindrical coordinates (r , θ, z) with corresponding velocities (ur , uθ , w ), we take the similarity solutions ansatz ur = crF (η), ′
uθ = crG(η),
√
w = −2 c ν F (η),
η=
√
c
ν
z. (4.1)
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P. Weidman / European Journal of Mechanics / B Fluids 78 (2019) 169–173
Inserting this ansatz into the radial and azimuthal Navier– Stokes equation gives, along with incorporation of far-field conditions, the boundary-value problem F ′′′ + 2FF ′′ − F ′2 + G2 + 1 = 0,
F (0) = 0,
F ′ (0) = 0,
F ′ (∞) = 1. G′′ + 2(FG′ − F ′ G) = 0,
(4.2a) G(0) = σ ,
G(∞) = 0
(4.2b)
where σ = ω/c. For comparison with the results for Hiemenz stagnation-point flow on a rotating disk we plot in Fig. 4 the radial and azimuthal velocities along the circle r = 1 for Homann strain rate c = 1. The results are computed for σ = {0, 2, 4, 6}. It should be noted that the results for the radial velocities are not identical since they depend on G(η) which itself depends on σ . Fig. 5. Variation of maximum radial velocities for σ = {0, 2, 4, 6} as a function of θ .
Fig. 6. Diagram in θ -σ space showing points of overshoots marked by the stars and points where overshoots do not occur marked by solid dots.
5. Analysis of results There are two competing aspects that produce radial motion. First there is the radial motion due to the Hiemenz stagnationpoint flow which for σ = 0 is the only physical mechanism for radial flow. However, when the plate rotates it centrifuges fluid radially outward and thus produces a second mechanism for radial motion. Indeed this is the physics behind von Ka´ rma´ n flow over a rotating disk beneath a quiescent fluid; see [12]. This centrifugal effect is clearly seen for the [8] problem in Fig. 4 where the radial profiles for σ = 4 and σ = 6 overshoot their far-field values ur (∞) = 1. The centrifugal effect becomes increasingly predominant with increasing σ . This is apparent for our Hiemenz flow problem in Fig. 2. We plot the maximum values of radial velocities observed in this figure as a function of θ in Fig. 5 for σ = {0, 2, 4, 6} over the range 0 ≤ θ ≤ π/2. Next we find the region in σ -θ parameter space where the radial flow overshoots its far field value ur (∞). These results are plotted in Fig. 6 which shows the predominance of overshoot with increasing σ . Clearly these overshoots are a result of the von Ka´ rma´ n component of the motion. 5.1. Comparison with the [8] problem In this Section 1 compare results of my Hiemenz stagnationpoint flow on a rotating disk with those for Homann stagnationpoint flow on a rotating disk found by Hannah [8]. First consider the far-field values ur (∞) and uθ (∞). The radial and azimuthal velocity relations in (3.2) show that the far-field values for the present problem for c = r = 1 are given as ur (∞) = sin2 θ,
uθ (∞) = sin θ cos θ.
(5.1)
These results are plotted as solid lines over the range 0 ≤ θ ≤ π in Fig. 7. In this same figure we show the constant values ur (∞) and uθ (∞) for axisymmetric stagnation-point flow studied by Hannah [8] plotted as the dashed lines. Fig. 7. Comparison of far-field velocities ur (∞) and uθ (∞) with those of the axisymmetric problem of Hannah [8].
Here c is the strain rate of the Homann stagnation-point flow. In this formulation one can only back out the limit of zero plate rotation. Hannah [8] used a more general η variable so that both the limits of zero plate rotation and zero stagnation-point flow could be recovered. However, the variable selected in (4.1) is best suited for comparison of results.
6. Summary, discussion and conclusion The problem of planar Hiemenz Stagnation-point flow impinging on a rotating disk has been studied in detail. For best comparison with Hannah’s problem of axisymmetric stagnationpoint flow impinging on a disk, we have computed radial and angular velocities around a circle of radius r = 1 for the choice of Hiemenz strain rate c = 1. The radial velocities ur (η) and angular velocities uθ (η0) depend on both the angle θ around the circle and the dimensionless rotation rate σ . We note that the choices c = r = 1 are immaterial since the radial and azimuthal
P. Weidman / European Journal of Mechanics / B Fluids 78 (2019) 169–173
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Fig. 8. Comparison of radial and azimuthal velocities at σ = 4. The solid lines are for θ = π/2 in the present study and the dashed lines are from the problem of Hannah [8].
velocities in Eqs. (3.2a), (3.2b) can be made dimensionless by dividing through by c r. The obvious symmetry is about the x-axis and even within the chosen region of study 0 ≤ θ ≤ π , there exists one symmetry as noted in Eq. (3.6) for ur and two symmetries as noted in Eq. (3.7) for uθ . Using these symmetries one can obtain the radial velocity profiles plotted for σ = {0, 2, 4, 6} shown in Fig. 2 and the azimuthal velocity profiles plotted for the same values of σ in Fig. 3. The chosen values of θ for the azimuthal profiles are θ = {π/4, 5π /16, 3π /8, 7π/16, π/2, 9π/16, 5π/8, 11π/16, 3π/4}. The values of maximum radial velocity (ur )max displayed in Fig. 5 are not all overshoot values. The region of radial velocity overshoots displayed over θ –σ space in Fig. 6 shows that for all σ ≥ 3 all maximum radial velocities indeed represent overshoots of the far-field values. The far-field velocities ur (∞) and uθ (∞) plotted over the region 0 ≤ θ ≤ π are compared with their respective constant values for the axisymmetric problem of Hannah [8] in Fig. 7. Observe in Fig. 7 that at θ = π/2 both the present problem and Hannah’s problem have the same maxima ur (∞) and uθ (∞). This suggests that the radial and azimuthal velocities might be similar for the two problems. As a test on this, we make a comparison in Fig. 8 between the ur (η) and the uθ (η) profiles computed for σ = 4 wherein the solid lines are from this study and the dashed lines are Hannah’s results. Though both radial profiles exhibit an overshoot, the radial and azimuthal profiles are not identical. This is not surprising since the profiles in the present problem are obtained from integration of three ODE’s while that for Hannah’s problem are derived from integration of two ODE’s and there is no apparent way that one system can be mapped into the other. A final comment is made about stagnation of flow on the plate. When the plate is stationary at σ = 0 there is a stagnation line along the x-axis on the surface of the plate. But when the plate rotates, there is stagnation only at the central point x = y = 0 on the surface of the plate. Now that axisymmetric Homann stagnation-point flow and the planar Hiemenz stagnation-point flow impinging on a rotating plate have been studied in detail, the next problem is obviously [13] stagnation-point flow impinging on a rotating plate. Since the Howarth problem has two strain rates, c and b and there is rotation, the problem will depend on three parameters, namely the angle θ , the strain rate ratio b/c, and a dimensionless rotation rate σ . Note that when b = c one has axisymmetric stagnation-point flow impinging on a rotating plate — the problem studied by Hannah [8]. Also, when b = 0 one has the problem planar stagnation-point flow impinging on a
rotating plate — the problem discussed here. The study by Abbassi and Rahimi [14] considers Howarth stagnation-point flow on a stationary plate, including the effects of heat transfer. However, I am not aware of any study of Howarth stagnation-point flow on a rotating plate. Acknowledgment The author acknowledges Andrzej Herczynski in the Department of Physics at Boston College for reading the manuscript and offering several constructive suggestions for improvement. References [1] K. Hiemenz, Die grenzschicht an einem in den gleichformigen flussigkeitsstrom eingetauchten geraden kreiszylinder, Dinglers J. 326 (1911) 321–324. [2] C.Y. Wang, Similarity stagnation point solutions of the Navier–Stokes equations – review and extension, Eur. J. Mech. B Fluids 27 (2008) 678–683. [3] C.Y. Wang, Review of similarity stretching exact solutions of the Navier–Stokes equations, Eur. J. Mech. B Fluids 30 (2011) 475–479. [4] P.D. Weidman, Obliquely-intersecting hiemenz flows: a new interpretation of howarth stagnation-point flows, Fluid Dyn. Res. 44 (2012) 065509, 15 pages. [5] P.D. Weidman, Axisymmetric stagnation-point flow on a spiraling disk, Phys. Fluids 26 (2014) 073603, 14 pages. [6] P.D. Weidman, Hiemenz stagnation-point flow impinging on a biaxially stretching surface, Meccanica (2017) http://dx.doi.org/10.1007/s11012017-0761-7. [7] P.D. Weidman, M.R. Turner, Stagnation-point flows with stretching surfaces: A unified formulation and new results, Euro. J. Mech. B/Fluids 61 (2017) 144–153. [8] D.M. Hannah, Forced flow against a rotating disk, Rep. Memor. Aerosp. Res. Count. Lond. (2772) (1947). [9] F. Homann, Der einfluss grosser zahigkeit bei stromung um zylinder, Z. Angew. Math. Mech. 16 (1936) 153–164. [10] P. Drazin, N. Riley, The Navier–Stokes Equations: A Classification of Flows and Exact Solutions, in: London Mathematical Society Lecture Note Series, vol. 334, Cambridge University Press, 2006. [11] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1989. [12] T. Von Kármán, Über Laminare und turbulent reibung, Z. angew. Math. Mech. 1 (1921) 233–251. [13] L. Howarth, The boundary layer equations in three dimensional flow - part II. The flow near a stagnation point, Philos. Mag. 42 (1951) 1433–1440. [14] A.S. Abbassi, A.B. Rahimi, Nonaxisymmetric three-dimensional stagnationpoint flow and heat transfer on a flat plate, J. Fluids Eng. 131 (2009) 074501.
Contents lists available at ScienceDirect
European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu
Hiemenz stagnation-point flow impinging on a uniformly rotating plate Patrick Weidman Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427, United States
article
info
Article history: Received 4 March 2019 Received in revised form 7 May 2019 Accepted 13 June 2019 Available online 25 June 2019
a b s t r a c t Planar stagnation-point flow normally impinging a rotating plate is studied. Symmetries show that along a circle of radius r, the radial velocity ur can be computed from results in one quadrant and the angular velocity uθ can be computed from results in two quadrants. The results of this study for various values of the dimensionless angular velocity σ are compared with axisymmetric Homann stagnation-point flow on a rotating plate first studied by Hannah. © 2019 Elsevier Masson SAS. All rights reserved.
1. Introduction
This suggests the similarity solution of the form
Planar stagnation-point flow on a flat surface was first considered by Hiemenz [1]. Many variations on that problem have been studied including the effects of wall stretching, and transpiration through a porous wall; see the reviews by Wang [2,3]. More recent Hiemenz stagnation-point flow studies may be found in [4–6] and in [7]. In this problem we consider planar Hiemenz stagnation-point flow impinging on a uniformly rotating plate. The results presented here will be compared with the results of Hannah [8] who considered axisymmetric [9] stagnation-point flow impinging on a rotating plate. Our result as well as that of Hannah [8] represent exact solutions of the Navier–Stokes equation as defined in [10]. The presentation is as follows. The exact similarity reduction of the Navier–Stokes equation is given in Section 2. Results of numerical calculations are given in Section 3 and a formulation of Hannah’s problem is given in Section 5. Further discussion of results and concluding remarks are given in Section 6.
u(x, y, η) = c [x f ′ (η) + y g(η)],
η=
√
c
ν
z
(2.3a)
where a prime denotes differentiation with respect to η. The continuity equation for incompressible flow is satisfied by
√ w(η) = − c ν[f (η) + q(η)].
(2.3b)
Inserting this ansatz into the Navier–Stokes equations furnishes the coupled equations for η ≥ 0 as f ′′′ + (f + q)f ′′ + g 2 − f ′2 = 0,
(2.4a)
g + (f + q)g − (f + q )g = 0,
(2.4b)
q′′′ + (f + q)q′′ + 1 − q′2 + g 2 = 0,
(2.4c)
′′
′
′
′
to be solved with impermeable wall and far-field conditions f (0) = 0,
f ′ (0) = 0,
g(0) = −σ ,
2. Problem formulation
v (x, y, η) = c [yq′ (η) − x g(η)],
q(0) = 0,
f ′ (∞) = 0
(2.4d)
g(∞) = 0 q (0) = 0, ′
(2.4e) q (∞) = 1. ′
(2.4f)
Using Cartesian coordinates (x, y, z) with corresponding velocities (u, v , w ) we stipulate that there is a Hiemenz stagnation point flow aligned with the x-axis whose far-field behavior is given as
where σ = ω/c is the dimensionless angular velocity. Note that for f = g = ω = 0 one recovers from the above equations the classical problem of Hiemenz [1], viz.
u = 0,
q′′′ + qq′′ + 1 − q′2 = 0,
v = cy,
w = −cz .
(2.1)
At the surface of impingement of the plate, which is uniformly rotating with angular velocity ω, the velocity components are given by u(x, y, 0) = −ω y,
v (x, y, 0) = ω x.
E-mail address: [email protected]. https://doi.org/10.1016/j.euromechflu.2019.06.008 0997-7546/© 2019 Elsevier Masson SAS. All rights reserved.
(2.2)
q(0) = 0,
q′ (0) = 0,
q′ (∞) = 1. (2.5)
3. Presentation of results Numerical calculations were performed using the integrator package ODEINT in [11]. Integration lengths were varied to ensure solution independence of integration length.
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P. Weidman / European Journal of Mechanics / B Fluids 78 (2019) 169–173
and also the transformation (x, y) = r(cos θ, sin θ ). In this manner the horizontal velocities given in (2.3a) are rendered cylindrical using ur = c r [f ′ (η) cos2 θ + q′ (η) sin2 θ ]
(3.2a)
uθ = c r [(q′ (η) − f ′ (η)) sin θ cos θ − g(η)].
(3.2b)
In these coordinates the pressure is given as p = p0 −
ρ c 2 r 2 sin2 θ 2
− ρν c
(
(f + q)2 2
+ (f ′ + q′ )
) (3.3)
where p0 is the stagnation pressure. Of particular interest is the radial stress on the plate
τr = µ Fig. 1. Variation of stress parameters f ′′ (0), g ′ (0) and q′′ (0) as a function of σ . The [1] result at σ = 0 is q′′ (0) = 1.23259.
⏐ ∂ ur ⏐⏐ = ρν 1/2 c 3/2 r [cos2 θ f ′′ (0) + sin2 θ q′′ (0)] ∂ z ⏐z =0
and also the azimuthal stress on the plate given by
⏐ ∂ uθ ⏐⏐ τθ = µ = ρν 1/2 c 3/2 r [sin θ cos θ (q′′ (0) − f ′′ (0)) − g ′ (0)]. ∂ z ⏐z =0
The σ variation of the three shear stress parameters f ′′ (0), g (0) and q (0) are shown in Fig. 1 for values σ = {0, 2, 4, 6}. ′
′′
The obvious strategy is to report results in a meaningful way to compare with the results of Hannah [8] which are given in Section 4. To this end I have calculated in cylindrical coordinates (r , θ ) the radial and azimuthal velocities (ur , uθ ). This is effected using the relations ur = u cos θ + v sin θ ,
uθ = v cos θ − u sin θ
(3.1)
(3.4)
(3.5) We will compute results around the circle r = 1 for Hiemenz strain rate c = 1. There is an obvious anti-symmetry about the x-axis of the oncoming Hiemenz stagnation-point flow. Hence it is only necessary to compute results for one side of the x-axis, here chosen to be in the region 0 ≤ θ ≤ π . In this region there exist three symmetries. In the first case it is clear from Eq. (3.2a) that ur (π/2 + φ ) = ur (π/2 − φ )
(0 ≤ φ ≤ π/2).
(3.6)
Fig. 2. Radial velocity profiles ur (η) plotted for c = 1 at r = 1 for σ = {0, 2, 4, 6} respectively in panels (a), (b), (c), (d). In each figure the angles are θ = {0, π/6, π/4, π/3, π/2} with the arrow in the direction of increasing θ . The profile for θ = π/2 in panel (a) is the vertical line at the left end of the arrow.
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171
Fig. 3. Azimuthal velocity profiles uθ (η) plotted for c = 1 at r = 1. Results for σ = {0, 2, 4, 6} are displayed in panels (a), (b), (c), (d), respectively. The angles for each plotted profile are θ = {π/4, 5π/16, 3π/8, 7π/16, π/2, 9π/16, 5π/8, 11π/16, 3π/4} with the arrows showing the direction of increasing values of θ . The dashed lines show the profiles for θ = π/2.
Fig. 4. Radial velocity profiles ur (η) and azimuthal profiles uθ (η) for axisymmetric stagnation-point flow impinging on a rotating disk presented for σ = {0, 2, 4, 6}. The left of the arrow at uθ = 0 is the velocity profile for σ = 0.
From (3.2b) we see two symmetries, namely uθ (π/4 + φ ) = uθ (π /4 − φ )
(0 ≤ φ ≤ π/4)
(3.7a)
and uθ (3π /4 + φ ) = uθ (3π/4 − φ )
In the next section we compute the axisymmetric problem of Hannah [8] for comparison. 4. Hannah’s problem
(0 ≤ φ ≤ π/4)
(3.7b)
Thus, all radial velocity profiles ur (η) can be obtained by computation in the region 0 ≤ θ ≤ π/2 and selected profiles are shown in Fig. 2 for the values σ = {0, 2, 4, 6}. Similarly, all azimuthal velocity profiles uθ (η) can be obtained by computation in the regions π /4 ≤ θ ≤ π/2 and π/2 ≤ θ ≤ 3π/4. These profiles for σ = {0, 2, 4, 6} are displayed in Fig. 3.
Using cylindrical coordinates (r , θ, z) with corresponding velocities (ur , uθ , w ), we take the similarity solutions ansatz ur = crF (η), ′
uθ = crG(η),
√
w = −2 c ν F (η),
η=
√
c
ν
z. (4.1)
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P. Weidman / European Journal of Mechanics / B Fluids 78 (2019) 169–173
Inserting this ansatz into the radial and azimuthal Navier– Stokes equation gives, along with incorporation of far-field conditions, the boundary-value problem F ′′′ + 2FF ′′ − F ′2 + G2 + 1 = 0,
F (0) = 0,
F ′ (0) = 0,
F ′ (∞) = 1. G′′ + 2(FG′ − F ′ G) = 0,
(4.2a) G(0) = σ ,
G(∞) = 0
(4.2b)
where σ = ω/c. For comparison with the results for Hiemenz stagnation-point flow on a rotating disk we plot in Fig. 4 the radial and azimuthal velocities along the circle r = 1 for Homann strain rate c = 1. The results are computed for σ = {0, 2, 4, 6}. It should be noted that the results for the radial velocities are not identical since they depend on G(η) which itself depends on σ . Fig. 5. Variation of maximum radial velocities for σ = {0, 2, 4, 6} as a function of θ .
Fig. 6. Diagram in θ -σ space showing points of overshoots marked by the stars and points where overshoots do not occur marked by solid dots.
5. Analysis of results There are two competing aspects that produce radial motion. First there is the radial motion due to the Hiemenz stagnationpoint flow which for σ = 0 is the only physical mechanism for radial flow. However, when the plate rotates it centrifuges fluid radially outward and thus produces a second mechanism for radial motion. Indeed this is the physics behind von Ka´ rma´ n flow over a rotating disk beneath a quiescent fluid; see [12]. This centrifugal effect is clearly seen for the [8] problem in Fig. 4 where the radial profiles for σ = 4 and σ = 6 overshoot their far-field values ur (∞) = 1. The centrifugal effect becomes increasingly predominant with increasing σ . This is apparent for our Hiemenz flow problem in Fig. 2. We plot the maximum values of radial velocities observed in this figure as a function of θ in Fig. 5 for σ = {0, 2, 4, 6} over the range 0 ≤ θ ≤ π/2. Next we find the region in σ -θ parameter space where the radial flow overshoots its far field value ur (∞). These results are plotted in Fig. 6 which shows the predominance of overshoot with increasing σ . Clearly these overshoots are a result of the von Ka´ rma´ n component of the motion. 5.1. Comparison with the [8] problem In this Section 1 compare results of my Hiemenz stagnationpoint flow on a rotating disk with those for Homann stagnationpoint flow on a rotating disk found by Hannah [8]. First consider the far-field values ur (∞) and uθ (∞). The radial and azimuthal velocity relations in (3.2) show that the far-field values for the present problem for c = r = 1 are given as ur (∞) = sin2 θ,
uθ (∞) = sin θ cos θ.
(5.1)
These results are plotted as solid lines over the range 0 ≤ θ ≤ π in Fig. 7. In this same figure we show the constant values ur (∞) and uθ (∞) for axisymmetric stagnation-point flow studied by Hannah [8] plotted as the dashed lines. Fig. 7. Comparison of far-field velocities ur (∞) and uθ (∞) with those of the axisymmetric problem of Hannah [8].
Here c is the strain rate of the Homann stagnation-point flow. In this formulation one can only back out the limit of zero plate rotation. Hannah [8] used a more general η variable so that both the limits of zero plate rotation and zero stagnation-point flow could be recovered. However, the variable selected in (4.1) is best suited for comparison of results.
6. Summary, discussion and conclusion The problem of planar Hiemenz Stagnation-point flow impinging on a rotating disk has been studied in detail. For best comparison with Hannah’s problem of axisymmetric stagnationpoint flow impinging on a disk, we have computed radial and angular velocities around a circle of radius r = 1 for the choice of Hiemenz strain rate c = 1. The radial velocities ur (η) and angular velocities uθ (η0) depend on both the angle θ around the circle and the dimensionless rotation rate σ . We note that the choices c = r = 1 are immaterial since the radial and azimuthal
P. Weidman / European Journal of Mechanics / B Fluids 78 (2019) 169–173
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Fig. 8. Comparison of radial and azimuthal velocities at σ = 4. The solid lines are for θ = π/2 in the present study and the dashed lines are from the problem of Hannah [8].
velocities in Eqs. (3.2a), (3.2b) can be made dimensionless by dividing through by c r. The obvious symmetry is about the x-axis and even within the chosen region of study 0 ≤ θ ≤ π , there exists one symmetry as noted in Eq. (3.6) for ur and two symmetries as noted in Eq. (3.7) for uθ . Using these symmetries one can obtain the radial velocity profiles plotted for σ = {0, 2, 4, 6} shown in Fig. 2 and the azimuthal velocity profiles plotted for the same values of σ in Fig. 3. The chosen values of θ for the azimuthal profiles are θ = {π/4, 5π /16, 3π /8, 7π/16, π/2, 9π/16, 5π/8, 11π/16, 3π/4}. The values of maximum radial velocity (ur )max displayed in Fig. 5 are not all overshoot values. The region of radial velocity overshoots displayed over θ –σ space in Fig. 6 shows that for all σ ≥ 3 all maximum radial velocities indeed represent overshoots of the far-field values. The far-field velocities ur (∞) and uθ (∞) plotted over the region 0 ≤ θ ≤ π are compared with their respective constant values for the axisymmetric problem of Hannah [8] in Fig. 7. Observe in Fig. 7 that at θ = π/2 both the present problem and Hannah’s problem have the same maxima ur (∞) and uθ (∞). This suggests that the radial and azimuthal velocities might be similar for the two problems. As a test on this, we make a comparison in Fig. 8 between the ur (η) and the uθ (η) profiles computed for σ = 4 wherein the solid lines are from this study and the dashed lines are Hannah’s results. Though both radial profiles exhibit an overshoot, the radial and azimuthal profiles are not identical. This is not surprising since the profiles in the present problem are obtained from integration of three ODE’s while that for Hannah’s problem are derived from integration of two ODE’s and there is no apparent way that one system can be mapped into the other. A final comment is made about stagnation of flow on the plate. When the plate is stationary at σ = 0 there is a stagnation line along the x-axis on the surface of the plate. But when the plate rotates, there is stagnation only at the central point x = y = 0 on the surface of the plate. Now that axisymmetric Homann stagnation-point flow and the planar Hiemenz stagnation-point flow impinging on a rotating plate have been studied in detail, the next problem is obviously [13] stagnation-point flow impinging on a rotating plate. Since the Howarth problem has two strain rates, c and b and there is rotation, the problem will depend on three parameters, namely the angle θ , the strain rate ratio b/c, and a dimensionless rotation rate σ . Note that when b = c one has axisymmetric stagnation-point flow impinging on a rotating plate — the problem studied by Hannah [8]. Also, when b = 0 one has the problem planar stagnation-point flow impinging on a
rotating plate — the problem discussed here. The study by Abbassi and Rahimi [14] considers Howarth stagnation-point flow on a stationary plate, including the effects of heat transfer. However, I am not aware of any study of Howarth stagnation-point flow on a rotating plate. Acknowledgment The author acknowledges Andrzej Herczynski in the Department of Physics at Boston College for reading the manuscript and offering several constructive suggestions for improvement. References [1] K. Hiemenz, Die grenzschicht an einem in den gleichformigen flussigkeitsstrom eingetauchten geraden kreiszylinder, Dinglers J. 326 (1911) 321–324. [2] C.Y. Wang, Similarity stagnation point solutions of the Navier–Stokes equations – review and extension, Eur. J. Mech. B Fluids 27 (2008) 678–683. [3] C.Y. Wang, Review of similarity stretching exact solutions of the Navier–Stokes equations, Eur. J. Mech. B Fluids 30 (2011) 475–479. [4] P.D. Weidman, Obliquely-intersecting hiemenz flows: a new interpretation of howarth stagnation-point flows, Fluid Dyn. Res. 44 (2012) 065509, 15 pages. [5] P.D. Weidman, Axisymmetric stagnation-point flow on a spiraling disk, Phys. Fluids 26 (2014) 073603, 14 pages. [6] P.D. Weidman, Hiemenz stagnation-point flow impinging on a biaxially stretching surface, Meccanica (2017) http://dx.doi.org/10.1007/s11012017-0761-7. [7] P.D. Weidman, M.R. Turner, Stagnation-point flows with stretching surfaces: A unified formulation and new results, Euro. J. Mech. B/Fluids 61 (2017) 144–153. [8] D.M. Hannah, Forced flow against a rotating disk, Rep. Memor. Aerosp. Res. Count. Lond. (2772) (1947). [9] F. Homann, Der einfluss grosser zahigkeit bei stromung um zylinder, Z. Angew. Math. Mech. 16 (1936) 153–164. [10] P. Drazin, N. Riley, The Navier–Stokes Equations: A Classification of Flows and Exact Solutions, in: London Mathematical Society Lecture Note Series, vol. 334, Cambridge University Press, 2006. [11] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1989. [12] T. Von Kármán, Über Laminare und turbulent reibung, Z. angew. Math. Mech. 1 (1921) 233–251. [13] L. Howarth, The boundary layer equations in three dimensional flow - part II. The flow near a stagnation point, Philos. Mag. 42 (1951) 1433–1440. [14] A.S. Abbassi, A.B. Rahimi, Nonaxisymmetric three-dimensional stagnationpoint flow and heat transfer on a flat plate, J. Fluids Eng. 131 (2009) 074501.