Flows induced by a plate moving normal to stagnation-point flow

Transcription

1 Acta Mech DOI.07/s P. D. Weidman M. A. Sprague Flows induced by a plate moving normal to stagnation-point flow Received: 0 September 0 Springer-Verlag 011 Abstract The flow generated by an infinite flat plate advancing toward or receding from a normal stagnationpoint flow is obtained as an exact reduction of the Navier Stokes equations for the case when the plate moves at constant velocity V. Both Hiemenz (planar) and Homann (axisymmetric) stagnation flows are considered. In each case, the problem is governed by a Reynolds number R proportional to V. Small and large R behaviors of the shear stress parameters are found for both advancing and receding plates. Numerical solutions determined over an intermediate range of R accurately match onto the small and large R asymptotic behaviors. As a side note, we report an interesting exact solution for plates advancing toward or receding from an exact rotational stagnation-point flow discovered by Agrawal (1957). 1 Introduction A self-similar solution of the Navier Stokes equations for the squeezing motion between two plates was first reported by Wang[1]. For axisymmetric flow between two disks, this is possible when their relative distance varies as 1 αt, while their relative normal velocity varies as α/ 1 αt where α is a constant. A second exact Navier Stokes solution for the squeezing motion between rotating disks was reported by Hamza and MacDonald []. In this case, self-similarity is achieved for the same conditions given by Wang [1] if, in addition, the rotation rate of the disks varies as (1 αt) 1. In both of these problems, the relative normal motion produces a local stagnation-point flow on each plate, the former without, and the latter with, a rotational flow component. By contrast, we study the fluid motion produced by an isolated infinite flat plate moving normal to planar and axisymmetric irrotational stagnation-point flows. Here, there is no fluid squeezing motion between plates, and a self-similar reduction of the Navier Stokes equations is found only if the plate moves at constant speed toward or away from the oncoming stagnation-point flow. Apart from providing an exact solution to the Navier Stokes equations, the results provided here may have relevance to the problem of directional solidification of an interface where the material thickness of a flat plate grows linearly in time. A literature review shows that the governing equation we obtain for motion P. D. Weidman (B) Department of Mechanical Engineering, University of Colorado, Boulder, CO , USA weidman@colorado.edu M. A. Sprague School of Natural Sciences, University of California, Merced, CA 95344, USA Present Address: M. A. Sprague National Renewable Energy Laboratory, Golden, CO 80401, USA michael.a.sprague@nrel.gov

2 P. D. Weidman, M. A. Sprague toward a planar stagnation-point flow was originally derived by Brattkus and Davis [3]. In their application to directional solidification, the inverse of αs, whereα is the ratio of diffusive to convective boundary-layer thicknesses and S is the Schmidt number, corresponds to our Reynolds number R. Brattkus and Davis [3] do not solve the boundary-value problem exactly; instead, they invoke certain assumptions about the relationship between α and S to perform an approximate stability analysis of the solidifying interface in the presence of a linear temperature gradient in the solute above the interface. We note that this paper by Brattkus and Davis [3] was highlighted by Neitzel [4]inhisJournal of Fluid Mechanics article commemorating the 70th birthday of Stephen H. Davis. For further reading on the subject of directional solidification, the reader is directed to the review by Davis and Schultze [5]. Of primary consideration here are plates moving normal to two-dimensional Hiemenz [6] and axisymmetric three-dimensional Homann [7] stagnation-point flows. These exterior flows are irrotational and we find that a similarity reduction may be obtained only if the plate moves at constant speed toward or away from the oncoming stagnation flow. Another less familiar external flow considered here is the exact rotational Navier Stokes stagnation-point flow reported by Agrawal [8]. The presentation is as follows. The problem formulation for the classic stagnation-point flows is given in Sect.. In Sect. 3, we present small Reynolds number solution behaviors, and in Sect. 4, large Reynolds number asymptotics are given. The numerical results presented in Sect. 5 are followed in Sect. 6 by a self-similar stability analysis of the solutions. An exact solution for a plate moving at arbitrary speed against the rotational stagnation-point flow is given in Sect. 7, and concluding remarks are given in Sect. 8. Problem formulation Using Cartesian coordinates (x, z) with associated velocities (u,w), we assume constant-property incompressible flow and look for a similarity reduction of the unsteady Navier Stokes equations u x + w z = 0 (.1) u t + uu x + wu z = 1 ρ p x + ν(u xx + u zz ), w t + uw x + ww z = 1 ρ p z + ν(w xx + w zz ) (.a) (.b) for a plate moving normal to planar (Hiemenz) stagnation-point flow of strain rate a. Here, t is time, p is the pressure, and ν is the kinematic viscosity. Positing the solution form u = ax f (η), w = a aν f (η), η = [z S(t)], (.3) ν the continuity equation (.1) is satisfied identically and the Navier Stokes equations reduce to f + ff f + 1 = Ṡ f. (.4) aν Clearly, a reduction to an ordinary differential equation is achieved only when the plate moves at constant velocity Ṡ = V or S(t) = Vt. Thus, the similarity variable represents a Galilean transformation that makes the flow steady in a reference frame moving with the plate. In this frame, impermeability and no-slip require that u and w vanish at η = 0 and the flow approaches planar stagnation flow in the far field. Thus, the boundary-value problem is f + ff f Rf = 0, f (0) = f (0) = 0, f ( ) = 1, (.5a) in which the Reynolds number measuring the speed of the plate relative to the stagnation-point flow is R = Vl/ν, (.5b)

3 Flows induced by a plate moving normal to stagnation-point flow where l = ν/a is an appropriate viscous length scale. Note that for R > 0, the plate moves into the stagnation flow and when R < 0, the plate recedes from it. The boundary-value problem (5.a) is that originally found by Brattkus and Davis [3] when a typographical error in their Eq. (.1) is corrected: the term FF should be replaced by FF. Integration of Eq. (.b) furnishes the pressure field [ a x ( f )] p = p 0 ρ + aν + f + Rf, (.6) where p 0 is the stagnation pressure at x = η = 0. Similarly, for a plate moving normal to axisymmetric (Homann) stagnation-point flow of strain rate a, one posits the similarity solution u = ar f (η), w = a aν f (η), η = (z Vt), (.7) ν where cylindrical coordinates (r, z) are employed with corresponding radial and axial velocities (u,w). For this case, the unsteady Navier Stokes equations reduce to f + ff f Rf = 0, f (0) = f (0) = 0, f ( ) = 1, (.8) where the pressure is now given by [ a r p = p 0 ρ aν ( f + f + Rf )]. (.9) Both stagnation flow problems may thus be encompassed in the boundary-value problem f + mff f Rf = 0, f (0) = f (0) = 0, f ( ) = 1, (.) where m = 1andm = for planar and axisymmetric stagnation-point flows, respectively. 3 Small Reynolds number behavior Assuming a regular perturbation expansion of the form f (η) = f 0 (η) + Rf 1 (η) + R f (η) + (3.1) one finds from Eq. (.) a sequence of boundary-value problems at each order of R. At O(1), we obtain f 0 0 f 0 0 f 0(0) = f 0 (0) = 0, f 0 ( ) = 1. (3.) For m = 1, this is recognized as the planar Hiemenz (1911) problem and for m =, it is the axisymmetric Homann (1936) problem. At O(R), we have the first correction for a plate moving normal to the stagnation-point flow: f 1 0 f 1 1 f 0 0 f 1 + f 0 f 1(0) = f 1 (0) = 0, f 1 ( ) = 0. (3.3) The second correction at O(R ) requires the solution of f + m ( f 0 f + f 1 f 1 + f f 0 ) f 1 f 0 f + f 1 = 0, f (0) = f (0) = 0, f ( ) = 0. (3.4) Once solved, a three-term approximation for the shear stress parameter is obtained using f (0) = f 0 (0) + Rf 1 (0) + R f (0) +. (3.5) For m = 1, integration of the lowest-order problem verifies the Hiemenz value f 0 (0) = Successive integrations of the higher-order problems give f 1 (0) = and f (0) = For m =, integration of the lowest-order problem verifies the Homann value f 0 (0) = and integrations of the higher-order problems yield f 1 (0) = and f (0) = These results are valid for both positive and negative R, i.e. for plates slowly advancing toward, or receding from, the stagnation-point flows.

4 P. D. Weidman, M. A. Sprague 4 Large Reynolds number asymptotics Owing to the change in sign of R that fundamentally changes the governing equation (.), we consider the large-r asymptotics for advancing (R > 0) and receding (R < 0) plates separately in the following. 4.1 Large positive R Large positive R represents a plate rapidly moving toward the oncoming stagnation flow. Defining the small parameter ɛ = 1 R 1, (4.1) Eq. (.) becomes a singular perturbation problem. However, it turns out that a uniformly valid asymptotic solution may be obtained by considering the inner boundary-layer region only. Moving the plate against stagnation-point flows decreases the boundary-layer thickness, so we introduce the stretched variable ξ = η ɛ (4.) to obtain from Eq. (.) f ξξξ + ɛ(m ff ξξ fξ ) + ɛ3 + f ξξ = 0, f (0) = 0, f ξ (0) = 0, f ξ ( ) = ɛ. (4.3) The last boundary condition suggests an expansion of the form Inserting this into Eq. (4.3) yields at O(ɛ) f = ɛg 1 (ξ) + ɛ G (ξ) + ɛ 3 G 3 (ξ) +. (4.4) G 1ξξξ + G 1ξξ = 0, G 1 (0) = 0, G 1ξ (0) = 0, G 1ξ ( ) = 1 (4.5) with the solution independent of m given by G 1 (ξ) = e ξ + ξ 1. (4.6) At O(ɛ ), the problem for G is identical to that for G 1, but with homogeneous boundary conditions; hence, G (ξ) = 0. At O(ɛ 3 ), we find G 3ξξξ + G 3ξξ = G 1 ξ mg 1 G 1ξξ 1, G 3 (0) = 0, G 3ξ (0) = 0, G 3ξ ( ) = 0, (4.7) showing the first dependence on m. The solutions to these m = 1, linear equations are The shear stress parameter is then evaluated using G 3 (ξ) = 3 1 (ξ + 6ξ + 6) e ξ (m = 1), (4.8) G 3 (ξ) = 1 4 e ξ 1 (ξ + 8ξ + 9) e ξ (m = ). (4.9) f (0) = 1 ɛ G 1ξξ(0) + G ξξ (0) + ɛg 3ξξ (0) +, (4.) where we recall that ɛ = 1/R. From the preceding solutions, the two-term asymptotic behaviors f (0) R + R + (m = 1), (4.11) f (0) R + R 5 + (m = ) (4.1) are obtained. Defining the boundary-layer thickness as δ, wefindthatδ 1/R for large positive R and the dependence of the shear stress parameter on m appears only as a first correction to its leading behavior.

5 Flows induced by a plate moving normal to stagnation-point flow 4. Large negative R Large negative R represents a plate rapidly moving away fromthe stagnation flow. Defining the small parameter ɛ = 1 1, (4.13) R Eq. (.) again becomes a singular perturbation problem. Here, we are not able to obtain a uniformly valid leading-order solution, but yet are able to extract the leading behavior of the wall shear stress. When the plate recedes rapidly, our numerics show that the boundary-layer thickness increases without bound. We therefore introduce the compressed wall coordinate and new dependent variable ζ = ɛη, f (η) = μ(ɛ)f(ζ ), (4.14) where the scale factor μ(ɛ) is to be determined. With this change of variables, the boundary-value problem (.) may be written as μɛ 3 F ζζζ + μ ɛ (mff ζζ F ζ ) + 1 = μɛf ζζ, F(0) = 0, F ζ (0) = 0, F ζ ( ) = 1 μɛ. (4.15) We must balance at least the two terms in (4.15) driving the fluid motion, namely the unity term corresponding to the outer inviscid stagnation flow and the last term representing the receding plate motion; this yields With μ so determined, Eq. (4.15) now becomes μ(ɛ) = 1 ɛ. (4.16) ɛ F ζζζ + (m FF ζζ F ζ ) + 1 = F ζζ, (4.17) showing that the nonlinear terms must be included at leading order. Positing the regular perturbation expansion one obtains the leading-order boundary-value problem F(ζ ) = F 0 (ζ ) + ɛ F 1 (ζ ) +, (4.18) mf 0 F 0ζζ F 0 ζ + 1 = F 0ζζ, F 0 (0) = F 0ζ (0) = 0, F 0ζ ( ) = 1. (4.19) Clearly, these represent the inner problems that would ultimately have to be matched onto appropriate outer solutions. Explicitly, for the two stagnation-point flows, we find F 0 (ζ ) = 1 cos ζ (m = 1), (4.0a) F 0 (ζ ) = 1 ζ (m = ), (4.0b) both of which satisfy the plate conditions, but not the far field condition, in Eq. (4.19). However, these are sufficient to determine, from Eq. (4.19) evaluated at the wall, the first term for the wall shear stress parameter, viz. F (0) = F 0 (0) 1 F 0 (0) 1 = 1, (m = 1, ). (4.1) Hence, the wall shear stress parameter evaluated from f (0) = μɛ F ζζ (0) = ɛ[f 0ζζ (0) + ɛ F 1ζζ (0) + ] (4.) has the same leading behaviors for m = 1andm =, namely f (0) 1 R. (4.3)

6 P. D. Weidman, M. A. Sprague Though matching with the outer solution has not been accomplished, we nonetheless find that the leading behavior of the shear stress is independent of m and that the boundary-layer thickness scales as δ R for large negative R. At the request of a referee, we give here an outline of the matching procedure for m = 1. From Eq. (4.0a), one obtains the inner velocity solution f (ζ ) = sin ζ, (4.4) where ζ = ɛη. This solution fails as η η 0 = π/(ɛ), but this is sufficient to satisfy the outer boundary condition f ( ) = F 0ζ ( ) = 1, since η 0 as ɛ 0. For finite ɛ, we thus take the uniformly valid solution as { f sin ɛη 0 η η0, (η) = (4.5) 1 η η 0. Carrying out the matching for m = is more tedious and will not be presented here. 5 Numerical results Numerical solutions of Eq. (.) were obtained with a Matlab routine built on the bvp4c boundary-valueproblem solver. Solutions were calculated over the domain 0 η η max,whereη max was chosen to be sufficiently large that results were insensitive to changes in the integration length. Our numerical results for m = 1 are summarized in the shear stress distribution in Fig. 1 plotted over the range R, which is sufficient to capture the large-r asymptotic behaviors found in Sect. 4 plotted as dotted lines. The three-term small-r behavior found in Sect. 3 is plotted as the dashed line. Similar results for the shear stress parameter for m = are displayed in Fig.. Similarity velocity profiles f (η) for m = 1 are plotted using a shifted logarithmic scale for η in Fig. 3 adopted to display all profiles in the range 40 R on the same plot. One observes the rapid growth in boundary-layer thickness as R becomes increasingly negative. Velocity profiles for m = are shown in Fig. 4. Defining the boundary-layer thickness as δ = η[0.99 f ( )], we show the variation of boundary-layer thickness with R in Fig. 5. In this figure, one may see the large positive R asymptotic behaviors δ R and the large negative R behaviors δ 1/ R, in agreement with the analyses given in Sect. 4 for both the m = 1and m = stagnation-point flows. The prefactors multiplying these behaviors, however, depend on the stagnation flow in consideration. We now compare the asymptotic solution given as Eq. (4.5)form = 1 with the exact numerical similarity velocity profiles for R = 40, 0, with corresponding small parameters ɛ = 0.05, 0.050, Fig. 1 Shear stress parameter for plates advancing toward (R > 0) or receding from (R < 0) planar Hiemenz stagnation-point flow. The center dashed line is the small-r behavior given by Eq. (3.5)form = 1. The dotted lines show the large-r asymptotics for advancing plates given by Eq. (4.11) and for receding plates given by Eq. (4.1)

7 Flows induced by a plate moving normal to stagnation-point flow Fig. Shear stress parameter for plates advancing toward (R > 0) or receding from (R < 0) axisymmetric Homann stagnationpoint flow. The center dashed line is the small-r behavior given by Eq. (3.5) form =. The dotted lines show the large-r asymptotics for advancing plates given by Eq. (4.1) and for receding plates given by Eq. (4.1) Fig. 3 Velocity profiles for plates advancing toward (R > 0) or receding from (R < 0) planar Hiemenz stagnation-point flow; the dashed line is for R = 0 representing a stationary plate. Note the shifted ordinate for the logarithmic scale The results previously exhibited in Fig. 3 on a shifted logarithmic ordinate are now plotted in Fig. 6 using an unshifted linear ordinate to clearly display sine-function behavior of the asymptotic results. Note that a deviation between the asymptotics and the numerical profiles can only be seen at R =. 6 Flow stability It is of interest to determine the stability of these flows. In particular, one may wonder about the stability of solutions for receding plates in these stagnation-point flows. To test the stability of our solutions steady in the similarity variable η, we now admit explicit unsteady behavior of the form f (η, τ) in solution ansatz (3), where τ = at is dimensionless time. The resulting equations for two-dimensional and axisymmetric three-dimensional motions are f + mff f Rf = f τ. (6.1) Following Merkin [9], small disturbances of growth rate α are posited in the form f (η, τ) = f 0 (η) + h(η)e ατ, (6.)

8 P. D. Weidman, M. A. Sprague Fig. 4 Velocity profiles for plates advancing toward (R > 0) or receding from (R < 0) axisymmetric Homann stagnation-point flow; the dashed line is for R = 0 representing a stationary plate. Note the shifted ordinate for the logarithmic scale Fig. 5 Boundary-layer thicknesses for plates advancing toward (R > 0) or receding from (R < 0) planar Hiemenz (solid circles) and axisymmetric Homann (open circles) stagnation-point flows where f 0 (η) satisfies the steady boundary-value problem (.). Inserting Eq. (6.) into(6.1) and linearizing furnishes the eigenvalue problem h + m ( f 0 h + f 0 h) f 0 h + Rh = αh, h(0) = h (0) = h ( ) = 0. (6.3) Note that since the problem is linear, eigenfunctions and eigenvalues may be found by shooting from η = 0 using h (0) = 1 for convenience. Solutions of (6.3) give an infinite set of eigenvalues α 1 <α <α 3 < ; if the smallest eigenvalue α 1 is negative, there is an initial growth of disturbances and the flow is unstable to self-similar disturbances; when α 1 is positive, there is an initial decay and the flow is stable to self-similar disturbances. Integration of boundary-value problem (6.3) to determine lowest eigenvalues α 1 for both m = 1andm = was found to become increasingly difficult with two different numerical codes as R increases for R < 0. We are able to obtain converged solutions up to R = 3.0form = 1andR = 3.5form =. The results given in Fig. 7 show that a minimum occurs near R = 1.5 form = 1andR =.0 form =. In this figure, we have defined the shorthand notation α 11 = α 1 (m = 1) and α 1 = α 1 (m = ). Note that the ordinate for α 1 is shifted two units upward to display the observed minima clearly. We infer from the upward trend beyond these minima that all lowest eigenvalues satisfy α 1 >.0, and thus, the solutions are self-similarly stable. Generally,

9 Flows induced by a plate moving normal to stagnation-point flow Fig. 6 Comparison of large negative R asymptotics (dashed lines) for similarity velocity profiles (solid lines) atr = 40, 0, with the numerically calculated profiles Fig. 7 Values of α 11 = α 1 (m = 1) and of α 1 = α 1 (m = ). The values of α 1 are shifted upward by two units to clearly exhibit the minima in each distribution of the lowest eigenvalues this analysis of self-similar stability discerns stable from unstable solution branches when dual solutions exist. Since in the present case no dual solutions exist for either m = 1orm =, it is not surprising that all solutions are self-similarly stable. The problem of instability at high values of R would have to be determined by some other more exacting analysis; see for example Drazin and Reid []. 7 A rotational stagnation-point flow Using spherical polar coordinates, Agrawal [8] found a new exact axisymmetric Navier Stokes stagnationpoint flow solution that, in its ultimate simplicity, satisfies both impermeability and no-slip at the stationary flat plate. It is much easier to derive this result using axisymmetric cylindrical coordinates (r, z) and make use of the Stokes streamfunction ψ(r, z) from which the respective radial and axial velocities (r 1 ψ z, r 1 ψ r ) are obtained. In this format, the exact solution found by Agrawal is simply ψ(r, z) = kr z, (7.1) where k is a dimensional constant measuring the strength of the rotational stagnation-point flow. We now consider the problem of a plate moving normal to this flow.

10 P. D. Weidman, M. A. Sprague The unsteady Navier Stokes equations for axisymmetric flow without swirl given by Goldstein [11]is t (D ψ) + 1 [ ψ r r z (D ψ) r (D ψ) ψ ] + ψ z r z (D ψ) = ν D 4 ψ, (7.a) where D = r 1 r r z z. (7.b) For a plate moving normal to this unique stagnation-point flow, we posit a solution of the form ψ = kr η, η = z S(t) (7.3) and observe that all differentiations with respect to z may be replaced by differentiations with respect to η. Therefore, the steady part of the Navier Stokes equation (7.a,b) is again identically zero, and one is left with the unsteady component t (D ψ) = t ( kr ) 0. (7.4) Thus, Eq. (7.3) represents the exact unsteady Navier Stokes solution valid for arbitrary plate motion S(t). This rather amazing result is seen to pivot on the fact that Agrawal s steady solution (7.1) satisfies all boundary conditions at the plate. Note, however, that since the flow is rotational, it is not a viscous potential-flow solution of the Navier Stokes equations as discussed in the book by Joseph, et al. [1]. In Agrawal s solution, no adjustment is required to match the outer rotational flow to the no-slip boundary condition at the plate. For the Hiemenz [6] and Homann [7] stagnation-point flow problems, on the other hand, an inner boundary-layer solution is required to match the outer irrotational flow to the no-slip boundary condition at the plate. 8 Summary and conclusion We have presented exact Navier Stokes solutions for a plate moving normal to two irrotational stagnationpoint flows the planar m = 1Hiemenz[6] flow and the axisymmetric m = Homann[7] flow. Moreover, an exact Navier Stokes solution is found for a plate moving normal to the rotational stagnation-point flow found by Agrawal [9]. In each case, the similarity variable normal to the plate is η = a/ν[z S(t)].Forthe irrotational stagnation-point flows, a similarity reduction is found only for a plate moving at constant speed V, Fig. 8 Reynolds number variation of the planar (m = 1) shear stress parameter plotted on log log scale. The solid circles are data for advancing plates (R > 0), and the open circles are data for receding plates (R < 0). The dashed lines are the leading-order asymptotic behaviors R for R > 0and1/ R for R < 0. The horizontal dotted line is the Hiemenz (R = 0) shear stress value f (0) = 1.36

11 Flows induced by a plate moving normal to stagnation-point flow i.e. for S(t) = Vt. For the rotational stagnation flow, on the other hand, the exact solution admits arbitrary plate motions z = S(t), a feature attributed to the fact that Agrawal s stagnation-point flow needs no adjustment to satisfy no-slip at the surface of the plate. Argawal s solution must be considered a mathematical curiosity since it is our understanding that, to date, there is no experimental verification of the existence of this flow in Nature or in a laboratory setting. However, that does not preclude the possibility that it will be observed in the future. Both irrotational stagnation-point flow problems exhibit interesting reciprocal asymptotic relations for the shear stress parameters f (0) and the boundary-layer thicknesses δ for advancing and receding plates. The asymptotical analyses given in Sect. 4 for both positiveand negativereynoldsnumbers possess the following features for both m = 1andm = type stagnation-point flows: f (0) R, δ 1 (R > 0), (8.1) R f (0) 1, δ R (R < 0). (8.) R This reciprocal feature for the shear stress parameter is displayed in Fig. 8 in a log log format for planar (m = 1) stagnation-point flow; very similar results are obtained for axisymmetric (m = ) stagnation-point flow. In closing, we note that it is easy to incorporate uniform rotation of a flat plate moving against Homann stagnation-point flow. For a plate rotating at angular velocity ω, a similarity reduction of the Navier Stokes equations yields a coupled pair of ordinary differential equations governed by two dimensionless parameters, our Reynolds number R = V/ aν and a swirl parameter S = ω/a. This provides work for future study. References 1. Wang, C.-Y.: The squeezing of a fluid between two plates. J. Appl. Mech. 43, (1976). Hamza, E.A., MacDonald, D.A.: A similar flow between two rotating disks. Quart. Appl. Math. 41, (1984) 3. Brattkus, K., Davis, S.H.: Flow induced morphological instabilities: stagnation-point flows. J. Crystal Growth 89, (1988) 4. Neitzel, G.P.: Stephen H. Davis-70, and counting. J. Fluid Mech. 647, 3 1 (0) 5. Davis, S.H., Schultze, T.P.: Effects of flow on morphological stability during directional solidification. Metal. Mater. Trans. 7A, (1996) 6. Hiemenz, K.: Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers J. 36, (1911) 7. Homann, F.: Der Einfluss grosser Zähigkeit bei Strömung um Zylinder. ZAMM 16, (1936) 8. Agrawal, H.L.: A new exact solution of the equations of motion with axial symmetry. Quart. J. Mech. Appl. Math., 4 44 (1957) 9. Merkin, J.H.: On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math. 0, (1985). Drazin, P.G., Reid, W.H.: Hydrodynamic Stability, nd edn. Cambridge University Press, Cambridge (004) 11. Goldstein, S.: Modern Developments in Fluid Dynamics, Vol 1. pp Dover, New York (1965) 1. Joseph, D., Funada, T., Wang, J.: Potential Flows of Viscous and Viscoelastic Fluids. Cambridge University Press, Cambridge (007)