IMECE UNSTEADY VISCOUS FLOWS AND STOKES S FIRST PROBLEM

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1 Proceedings of IMECE ASME International Mechanical Engineering Congress Chicago, Illinois, USA, November 5-0, 006 IMECE UNSTEADY VISCOUS FLOWS AND STOKES S FIRST PROBLEM Y.S. Mzychka Faclty of Engineering Applied Science Memial University of Newfondl St. John s, NL, Canada, AB 3X5 M.M. Yovanovich Department of Mechanical Engineering University of Waterloo Waterloo, ON, Canada, NL 3G ABSTRACT Unsteady viscos flows Stokes s first problem are examined. Three problems are considered: nsteady Coette flow, nsteady Poiseille flow, nsteady bondary layer flow. The relationship between these three fndamental nsteady flows Stokes first problem is illstrated. Scaling principles are sed to dedce the sht time long time characteristics of these three problems. Asymptotic analysis is sed to obtain exact sht long time characteristics to show the relationship of each problem to Stokes s first problem f sht times. Finally, compact robst models are developed f all vales of time sing the Chrchill-Usagi asymptotic crelation method to combine the sht long time characteristics. Keywds: Viscos Flow, Stokes s First Problem, Unsteady Flow, Coette Flow, Poiseille Flow, Blasis Flow, Asymptotic Modelling, Scaling NOMENCLATURE A = area, m D h = hydralic diameter, 4A/P h = characteristic channel dimension, m J 0 ( = Bessel fnction of first kind der zero J ( = Bessel fnction of first kind der one L = dct channel length, m L = arbitrary length scale, m m = series index n = asymptotic crelation parameter, Eq. (,70 n = directed nmal, m p = asymptotic crelation parameter, Eq. (7 Associate Profess, Member ASME Distingished Profess Emerits, Fellow ASME p = pressre, P a P = perimeter, m P o L = Poiseille nmber, τl/µw r = radial codinate, m s = arc length, m St = Strohal nmber, x/u t t = time, s t = dimensionless time, νt/l = local velocity, m/s = average velocity, m/s U = srface stream velocity, m/s x, y, z = Cartesian codinates, m y0 = sht time asymptote y = long time asymptote y = compact dimensionless model Greek Symbols δ = bondary layer thickness, m δ s = bondary layer thickness, m δ m = bondary layer thickness, m δ n = eigenvale η = similarity variable µ = dynamic viscosity, Ns/m ν = kinematic viscosity, m /s ρ = density, kg/m 3 τ w = wall shear stress, P a Sbscripts w = wall 0 = sht time vale = long time vale L = based on the arbitrary length L Sperscripts ( = mean vale ( = dimensionless vale

2 INTRODUCTION Stokes s first problem is a fndamental soltion in flid dynamics, which represents one of the few exact soltions to the Navier-Stokes eqations [-8]. The problem simply stated, describes the evoltion of the velocity field in the vicinity of an infinite plate which is implsively set in motion with constant velocity, U, in an infinite flid medim, as shown in Fig.. Of primary interest are the rate of penetration, δ, of the velocity field into the flid the shear stress at the srface of the plate, τ w. Fig. - Stokes s First Problem. Three additional problems can be characterized by the Stokes soltion at sht times. These are: nsteady Coette flow, nsteady Poiseille flow, nsteady bondary layer flow. In the latter case, the bondary layer near the leading edge of a semi-infinite plate will have a characteristic shear stress similar to Blasis flow ( at long times, while downstream ( at sht times the bondary layer is similar to that in Stokes flow. The three problems of interest are shown schematically in Fig.. Each of these problems has been addressed in the flid literatre sing analytical, approximate analytical, nmerical soltion methods. However, they have not all received adeqate attention from the point of view of scaling analysis asymptotic analysis, n have simple models been developed f the wall shear stress. In the case of nsteady Poiseille flow, this problem has recently been addressed by the aths [9]. The key reslts are inclded here f completeness. This paper addresses each of these three problems develops simple compact models f the dimensionless wall shear stress as a fnction of dimensionless time. These models are developed by sing scaling principles [0] to dedce the crect asymptotic behavi, asymptotic analysis to obtain the exact soltions to the limiting behavi [], non-linear sperposition to obtain the compact model [,3]. In all three cases, the compact models are compared with the me complex exact soltions. These models may be sed to model the transient flid behavi in micro-systems involving liqids gases providing that non-continm effects are minimal. Fig. 3 - Asymptotic Model Development. Fig. - Three Unsteady Viscos Flow Problems. Compact models are easily developed sing the asymptotic crelation method proposed by Chrchill Us-

3 agi []. Once the exact asymptotic behavi of a system of interest is known f sht long time, these limits are then combined to develop a simple compact model f the dimensionless qantity of interest. In general fm we may write: y = [(y 0 n + (y n ] /n ( The crelation parameter n can be either positive negative. In the case of n > 0 the asymptotes are combined to give a concave p fnction, while f n < 0 the asymptotes are combined to give a concave down fnction. Figre 3, illstrates the case f n < 0. Once the asymptotes are known, simple criteria f the transition point can easily be determined by finding the intersection of the asymptotes: y 0 y ( This expression is then solved in closed fm to give the nominal critical vale of the independent parameter, which is then sed as a transitional criterion. We now proceed to examine the characteristics soltions of for fndamental problems in nsteady viscos flow. FOUR FUNDAMENTAL PROBLEMS For fndamental problems in nsteady viscos flow are: Stokes s first problem, nsteady Coette flow, nsteady Poiseille flow, nsteady bondary layer flow. Each is discssed from the point of view of scaling, asymptotics, compact modelling. Stokes s First Problem One of the most imptant fndamental problems in nsteady viscos flows is that of implsively started motion of a body in an infinite flid medim. The simplest problem is that of sddenly setting an infinite flat plate in motion with velocity U. This particlar problem is often referred to as Stokes s first problem (Schlichting [], Crrie [3] is also referred to in some texts as the Rayleigh problem (Rosenhead [], Telionis [4], Panton [5]. The momentm transpt problem referred to as Stokes s first problem Stokes flow [3], reqires obtaining the soltion to: t = ν y (3 which is sbject to: y = 0 (0, t = U y (y, t = 0 t = 0 (y, 0 = 0 (4 The problem stated above describes viscos diffsion, however analogos problems in heat mass diffsion exist. The soltion is easily obtained in terms of the complementary err fnction. Befe proceeding to an exact soltion, we see that a simple scale analysis can provide mch sefl infmation. We assme that t t, U, y δ, where δ is the bondary layer thickness penetration depth of the flow field. Using these scales we arrive at U t ν U δ (5 δ νt (6 f the penetration depth. Problems in semi-infinite domains are of the penetration type, i.e., δ νt δ νx/u. The shear stress at the plate srface may be approximately obtained as: τ w µu δ, defining a dimensionless shear stress µu νt (7 τ τ wδ µu (8 The exact soltion is fond in terms of the complementary err fnction [-8]: U = erf(y/ νt = erfc(y/ νt (9 where erfc( is the complementary err fnction. The soltion is readily compted in any mathematical software package. Finally, it is of interest to determine the drag on the plate. The shear stress is defined as: τ w = µ y (0 y=0 which becomes: τ w = µu πνt ( This reslt agrees with the scaling analysis except f the fact of / π. We may now define the dimensionless shear stress from Eq. (8 as: τ = π ( The bondary layer thickness may now be obtained sing Eq. (9. The approximate vale f δ as obtained from Eq. (9 f the case where /U = 0.0 gives: δ = νt (3 We may also define two additional measres of bondary layer thickness. One is related to the shear stress throgh Eq. ( gives: 3

4 δ s = π νt.77 νt (4 the other relates the total mass of flid set in motion above the srface of the plate defined as [4]: which gives Uδ m = 0 dy (5 δ m = π νt.8 νt (6 The three thicknesses relate to one another accding to: δ m < δ s < δ (7 Later, we shall see that the δ m definition plays an imptant role in the asymptotic reslts f nsteady Poiseille flow. Unsteady Coette Flow We now re-examine the problem of nsteady Coette flow. Coette flow has been sed as the fndamental method f the measrement of viscosity. Frther, it is sed in many wall driven applications as means of estimating the drag fce. The problem reqires soltion to the following viscos diffsion eqation: sbject to: t = ν y (8 y = 0 (0, t = U y = h (h, t = 0 t = 0 (y, 0 = 0 (9 Scale Analysis First, the application of scaling principles f sht times shows that the problem is accrately modelled as Stokes flow when δ νt < h F sht time, the wall shear stress scales accding to: τ 0 µu µu (0 δ νt f long time it scales accding to τ µu h We may define a dimensionless wall shear stress as: This leads to the following reslts: ( τ = τ w τ ( where τ t τ t 0 t (3 t = νt h (4 The nsteady Coette flow also has an analoge in transient heat condction f a plane wall sbject to a sdden step change in temperatre at one srface. In this case t is replaced by the Forier nmber, αt/h. Asymptotic Analysis The exact soltion may be fond in the texts of Sherman [6], White [7], Batchel [8], is often repted as: ( U = y h n= sin(nπy/h nπ exp( n π νt/h (5 The soltion f very sht times reqires many terms to achieve convergence of the velocity field. However, f very sht times, the velocity field has not penetrated far enogh into the flid to reach the stationary wall. Ths, the soltion governed by Stokes s first problem is valid soon after the flow commences. It is well known that f t 0.0, the field can be represented by Eq. (9. The shear stress is obtained from: τ w = µ y (6 y=0 which gives: τ w = µu h ( + exp( n π νt/h n= (7 defining the dimensionless shear stress τ w /τ, we obtain τ = + exp( n π t (8 n= where τ = µu/h. The above eqation contains the following asymptotes: τ = t 0 π t (9 τ = t which agree with the scaling reslts in the der of magnitde sense. Compact Model The dimensionless wall shear may be me easily modelled by considering the asymptotes 4

5 above the following eqation sing the approach of Chrchill Usagi [] Eq. (. Using Eq. ( with the asymptotes defined by Eq. (9, with at least one known point, one can solve f n. In the present case, we obtain n 5/3 to give: τ = [ ( π t 5/3 + ] 3/5 (30 which is mch me efficient than the series fm. The above eqation provides accracy of ± 0.3 percent. It is plotted in Fig. 4 along with data obtained from Eq. (8. The transition point which may be defined by the intersection of the asymptotes, shows that steady state is obtained when t > /π τ / τοο Exact Soltion, Eq. (8 Model, Eq. ( t* Fig. 4 - τ f Unsteady Coette Flow. Unsteady Poiseille Flow Unsteady Poiseille flow is described by the following nsteady momentm eqation: ν t = µ dp dz + x + y (3 which is sbject to the bondedness condition along the axis of the geometry,, homogeneos Dirichlet conditions at the bondary, = 0, the initial condition = 0, when t = 0. The system is shown in Fig.. It consists of an infinitely long dct cylinder of arbitrary bt constant cross-sectional area, A, bonded by perimeter, P. It is of interest to obtain the area mean velocity, obtained by integrating the soltion f over the crosssectional area: (t = da (3 A A Also, of interest is the perimeter averaged mean shear stress at the srface: τ w (t = µ ds (33 P n The dimensionless mean velocity may be defined as: = L dp µ dz (34 while the dimensionless shear stress at the srface is defined with respect to the steady state vale: τ = τ w τ (35 where τ = A dp P dz (36 is obtained from the steady state fce balance. Scale Analysis We now examine what infmation scale analysis provides. Eqation (3 represents a balance of three qantities: stage, generation, diffsion. There are three distinct flow regions in this problem that may be considered. Flly developed flow exists after a very long time, at sht times, there exists a potential ce a very thin bondary layer region. Each of these regions may be analyzed sing scale analysis. The following scales will be sed:, t t, /L f long time /δ f sht time. Here, L is as yet ndetermined characteristic length scale related to the geometry δ is the bondary layer thickness penetration scale associated with early times. First, f long times t, flly developed flow, the balance between generation diffsion leads to: L G (37 GL (38 where G = dp. The dimensionless mean velocity becomes: µ dz = as t (39 GL Next, f sht times t 0, the balance is between stage generation, considering the potential ce, this leads to ν t G (40 Gνt (4 νt L t as t 0 (4 5

6 Next, in the bondary layer region, the balance between stage diffsion leads to: ν t δ (43 δ νt (44 which is the intrinsic penetration depth. The flow becomes flly developed when δ L, sch that νt L (45 t νt L (46 Finally, we wish to develop expressions f the mean shear stress at the srface defined as: τ w = µ (47 n We mst consider the two limiting cases of sht time long time. F long time, t, the shear stress becomes: τ w µ (48 L We may also relate the shear stress to the sorce G f flly developed flows, where τ = A µg (49 P Ths, if we define τ = τ w /τ, we obtain τ = ( as t (50 A P GL Finally, f sht times, t 0, the shear stress becomes τ w µ (5 δ Also, from the momentm transpt eqation we see that, after combining Eqs. (5 (5: δ G (5 τ w µgδ (53 Finally defining τ as befe, we obtain τ δ A/P t L as t 0 (54 A/P It is clear from scaling analysis, that two distinct characteristics are present. These are the dimensionless mean velocity, dimensionless mean shear stress, τ. Each has the following asymptotic behavior: t t 0 (55 t τ t t 0 τ t (56 Asymptotic Analysis The exact asymptotic behavior f small large times may now be examined f each dimensionless qantity of interest. The aths recently addressed these isses in [9], only a brief smmary is given below. F long time, t, the flow is characterized by a balance between generation diffsion. This problem has been analyzed extensively f momentm transpt soltions to many configrations may be fond in the literatre reference books. The reslts are sally presented in the fm of the dimensionless grop f Re, the Fanning friction fact Reynolds nmber prodct defined as: fre L = τ L µ = P o L (57 where P o is referred to in the flids literatre as the Poiseille nmber [7]. We can frther introdce the sorce term throgh the flly developed flow balance Eq. (36 obtain: P o L = τ L µ = A P GL (58 Rearranging, f sing the definition of the dimensionless mean velocity, Eq. (34, we obtain = A/P P o L L (59 Finally, by virte of the definition of the dimensionless shear stress, Eq. (35 the vale f τ, the dimensionless asymptotic limit f τ is: τ = (60 F sht times, t 0, the transpt is characterized by a balance between generation stage. The eqation of transpt which may be solved in the potential ce when δ is small is: ν t = G (6 6

7 This may be integrated solved with the initial condition (0 = 0 to give: (t = νtg (t (6 When non-dimensionalized, the soltion f sht time in the potential ce is: = νt L = t (63 The sht time shear stress may be fond by considering the classic Stokes soltion f momentm transpt in an infinite medim. The soltion f the field reslting from a step change at the srface is: y = Uerfc( νt (64 The reslt f the bondary layer thickness which acconts f the mean penetration of the field from the srface to the potential ce, i.e. the area nder the crve defined by Eq. (64, may be written as [9]: δ m U = νt where η = y/ νt, which gives 0 U erfc(η dη (65 δ m = π νt.8 νt (66 Eqation (66 acconts f the mean depth of penetration of the field to the potential region. Althogh the potential ce is in a state of change, the process is still applicable, since we are interested in the characteristics of the bondary layer which is bonded by the srface the potential ce. In the case of momentm transpt, this bondary layer defines the total mass flow, Uδ m, at any time which reslts from the implsive motion of an infinite flat plate. Using the above reslt in Eq. (53 gives: τ = P L t (67 π A In smmary, the asymptotic analysis yields: = t t 0 = A/P t LP o L τ = P L πa t t 0 τ = t (68 (69 Compact Models These exact limits may now be combined sing the asymptotic crelation method []. The models of interest may now be written in the following fms: = τ = [ ( n ] /n A/P (t n + (70 LP o L [( P L πa t p + ] /p (7 The vales of n p may now be determined from comparisons with data obtained from the exact soltions f a nmber of geometries. The fitting parameters may be obtained by either applying Eqs. (70 (7 at a single known point in the transition region by sing mltiple points minimizing the root mean sqare err. In the present wk, the latter method is sed to determine the fitting parameters. When the hydralic diameter is sed as a characteristic length scale Eqs. (70 (7 become: = τ = [ ( n ] /n (t n + (7 4P o Dh [( 8 π t p + ] /p (73 Comparisons were made in [9] with for known analytical soltions: the plane channel, the circlar tbe, the rectangle, the circlar annls. Each of these for geometries are closely related. The annls contains as special limits the tbe the channel reslts, the rectangle also contains the channel limit. We only consider the tbe channel reslts in this paper. The soltion f the plane channel [5] of width h is: (y, t = [( Gh y h ] sin(δ n 4 δn 3 cos(δ n y/h exp( δnνt/h n= (74 where (n π δ n = (75 The exact soltions f mean wall shear mean velocity are: ( = Gh 3 exp( δnνt/h δn 4 (76 n= the shear stress may be calclated from: ( exp( δ τ w = µgh nνt/h δn n= (77 Eqations (76 (77 have been sed to generate data f comparisons to the proposed models. Over the 7

8 range of < t < 0, terms were sed in the series as reqired to achieve convergence. The optimal fitting parameter f was fond to be n =.3 with a root mean sqare err (RMS of 6.03 percent. While f τ the vale was fond to be p = 6 with a 0.6 percent RMS. Graphical reslts are shown in Figs The soltion f the circlar tbe of diameter a was obtained by Szymanski [6] sing the separation of variables method. The soltion is widely discssed in many advanced level flid texts [,,4,7]. The soltion is: * Exact, Eq. (76 Model, Eq. (73 (r, t = [( Ga r 4 a 8 n= J o (δ n r/a δ 3 nj (δ n exp( δ nνt/a ] (78 E t* Fig. 5 - f the Plane Channel. where δ n are the positive roots of J 0 (δ n = 0 (79 The mean velocity may be fond by integrating over the cross-sectional area = Ga ( 8 4 n= the shear stress is fond to be: exp( δnνt/a δn 3 (80 τ* t* Fig. 6 - τ f the Plane Channel. Exact, Eq. (77 Model, Eq. (73 τ w = µga ( n= exp( δnνt/a δn ( Eqations (80 (8 have been sed to generate data f comparisons to the proposed models. Over the range of < t < 0, terms were sed in the series as reqired to achieve convergence. The optimal fitting parameter f was fond to be n =. with an RMS err of 6.94 percent. While f τ the vale was fond to be p =.8 with an RMS err of.7 percent. Graphical reslts are shown in Figs The transition point f the dimensionless shear stress may be fond from the intersection of the asymptotes in Eq. (73. This leads to t ( π/ Other soltions f the rectanglar dct [9] the annls [7] are examined in [9]. * Exact, Eq. (80 Model, Eq. (7 E t* Fig. 7 - f the Circlar Tbe. 8

9 τ* Exact, Eq. (8 Model, Eq. (73 is governed by Blasis flow. F sht times, /x <<, the flow is nsteady wall shear is obtained from Stokes flow. Conversely, similar characteristics are obtained when x is small, i.e. near the leading edge, /x >>, when x is large, i.e. downstream from the leading edge, /x <<. A connection between Stokes flow Blasis flow was first illstrated by Rayleigh []. Rayleigh [] sggested that the Blasis problem cold be modeled sing an Oseen approximation whereby f steady flows, Eq. (83 cold be replaced by t* Fig. 8 - τ f the Circlar Tbe. Unsteady Bondary Layer Flow The nsteady development of laminar bondary layers has been considered by several researchers, see Rosenhead [], Telionis [4] Sherman [6]. This particlar problem has been tackled sing both approximate analytical nmerical methods. The problem nder consideration is that of implsively started stream over a semi-infinite flat plate. In time space the flow progresses from a Stokes flow field to a Blasis flow field, as shown in Fig.. The governing eqations are continity the nsteady x-momentm eqation: x + v y = 0 (8 t + x + v y = ν y (83 which are sbject to the following bondary initial conditions: y = 0, v = 0 y = U (84 x = 0 = U t = 0 = 0 As we shall see, an imptant parameter in the soltion of this problem is the groping: x = St (85 where St is one fm of the Strohal nmber [4]. The characteristics of this problem are sch that at long times, /x >>, the flow is steady the wall shear U x = ν y (86 which may be written as Eq. (3 if t = x/u. This leads to the soltion f linearized Blasis problem of the fm ( (η U = erf y (87 νx/u If we consider the effective residence time defined by t = x/u, the drag co-efficient becomes: C f,x = π µ ρux =.8 Rex (88 which over predicts laminar bondary layer they, which yields a constant Rayleigh sggested that t = x/0.346u with no proof. This vale of the effective time gives a constant A me general approach can be taken by linearizing the laminar bondary layer eqations by fmally defining an effective velocity [8]. This approach gives a similar vale f the effective residence time as obtained by Rayleigh, bt with a me fmal approach. Using the eqations f nsteady bondary layer flow, the crect asymptotic characteristics will be obtained, a simple model developed f the shear stress. Scale Analysis Application of scaling analysis to Eqs. (8 (83 yields several imptant characteristics. First, the continity eqation gives the transverse velocity scale: v Uδ (89 x F sht times, we may neglect the inertia terms in Eq. (83 we obtain: δ 0 νt (90 τ 0 µu (9 δ 0 while f long times, we neglect the transient term, obtain: νx δ (9 U 9

10 τ 0 µu (93 δ Finally, we see that the ratio of the sht time long time bondary layer thicknesses yields: δ 0 (94 δ x If we define a dimensionless shear stress accding to τ/τ 0 we then obtain the following dimensionless scales: τ 0 τ x x 0 x (95 Asymptotic Analysis Stewartson [9] obtained an approximate soltion to the nsteady bondary layer flow by linearizing the momentm eqation sing an Oseen approximation. The reslts were given as piecewise soltion: ( y = Uerf νt x < = Uerf ( U y νx x > (96 These approximate reslts illstrate how the flow transitions from a Stokes flow to a Blasis flow. However, this approximation is only valid near the leading edge of the plate. Stewartson [9] also obtained soltions sing the momentm integral eqation. Stewartson s [9] reslts f the wall shear were fond to be in good agreement with the limits of Stokes flow Blasis flow: τ w = 0.534ρU ν/t τ w = 0.38ρU Uν/x x.65 x.65 (97 These reslts show abrpt transition from one type of flow to the other. In fact this transition is not as abrpt generally occrs in the range of < /x < 4. We may now examine the exact asymptotes f the wall shear sing: τ 0 = 0.565ρU ν/t τ = 0.33ρU Uν/x x.65 x.65 (98 The approximate transition point may be obtained by eqating the two asymptotes, τ 0 τ, to obtain U t/x We may also define a dimensionless wall shear τ = τ/τ 0. This leads to the following dimensionless asymptotes: τ 0 = τ = x x 3 x 3 (99 Compact Models These asymptotes may be combined sing Eq. ( with n, in the fm τ = + ( x / (00 The accompanying plot in Fig. 9, shows the natre of the transition from Stokes flow to Blasis flow. The data points were obtained sing the procedre otlined in Sherman [6] Dwyer [0]. Comparison of the above model with the nmerical data yields n. This simple robst model provides a means to establish the transient response f a semi-infinite plate sbjected to an implsively started stream. τ / τ s Nmerical Soltion Model Eq. ( / x Fig. 9 - τ f Unsteady Bondary Layer Flow. SUMMARY AND CONCLUSIONS The present wk re-examined three fndamental nsteady viscos flow problems: nsteady Coette flow, nsteady Poiseille flow, nsteady bondary layer flow. Their link to Stokes s first problem was established by means of scaling asymptotic analysis. Asymptotic reslts were combined sing the Chrchill-Usagi [] 0

11 method, to develop robst compact models f the dimensionless mean wall shear stress. These new models provide accrate reslts an alternative to sing me complex series soltions graphical reslts. ACKNOWLEDGMENTS The aths acknowledge the financial sppt of the Natral Sciences Engineering Research Concil of Canada (NSERC nder the Discovery grants program. The first ath also acknowledges the assistance of Karl Lawrence, who re-solved the nsteady bondary layer flow problem nmerically to provide the data in Fig. 9. REFERENCES [] Rosenhead, L., Laminar Bondary Layers, Dover, Pblishing, pp ,959, New Yk, NY. [] Schlichting, H., Bondary Layer They, McGraw- Hill, pp , 979, New Yk, NY. [3] Crrie, I.G., Fndamental Mechanics of Flids, McGraw-Hill, pp. 4-8, 993, New Yk, NY. [4] Telionis, D.P., Unsteady Viscos Flows, Springer- Verlag, pp. 79-0, 98. [5] Panton, R.L., Incompressible Flow, Wiley, pp. 66-7, 995. [6] Sherman, F., Viscos Flow, McGraw-Hill, pp , 99, New Yk, NY. [7] White, F.M.,Viscos Flid Flow, McGraw-Hill, pp. 3-43, 99, New Yk, NY. [8] Batchel, G.K., Introdction to Flid Dynamics, Cambridge University Press, pp , 970. [9] Mzychka, Y.S. Yovanovich, M.M., Compact Models f Transient Condction Viscos Transpt in Non-Circlar Geometries with a Unifm Sorce, International Jornal of Thermal Sciences, In Press, 007. [0] Bejan, A. Convection Heat Transfer, Wiley, 995. [] Leal, L.G., Laminar Flow Convective Transpt Processes, Btterwth-Heinemann, pp. 9, 99, Boston, MA. [] Chrchill, S. W. Usagi, R., A General Expression f the Crelation of Rates of Transfer Other Phenomena, American Institte of Chemical Engineers, Vol. 8, pp. -8, 97. [3] Yovanovich, M.M., Asymptotes Asymptotic Analysis f Development of Compact Models f Microelectronics Cooling, Keynote Address, Semitherm 003, San Jose, CA, March 003. [4] Yan, S.W., Fondations of Flid Mechanics, Prentice-Hall, pp. 9-93, 967. [5] Lamb, H., A Paradox in Flid Motion, Jornal of the London Mathematical Society, Vol., pp. 09-, 97. [6] Szymanski, P., Qelqes Soltions Exactes des Eqations de l Hydrodynamiqe d Flide Visqex dans le Cas d n Tbe Cylindriqe, Jornal Mathematiqe Pre et Appliqee, Vol., pp , 93. [7] Müller, W., Zm Problem der Anlafstromng einer Flssigkeit im geraden Rohr mit Kreisring nd Kreisqerschnitt, ZAMM, Vol. 6, pp. 7-38, 936. [8] Yovanovich, M.M., Lee, S. Gayowsky, T., Approximate Analytic Soltion of Laminar Fced Convection From an Isothermal Plate, AIAA Paper 9-048, 30th Aerospace Sciences Meeting Exhibit, Janary 6-9, Reno, NV, 99. [9] Stewartson, K., On the Implsive Motion of a Flate Plate in a Viscos Flid, Qarterly Jornal of Mechanics, Vol. 4, pp. 8-98, 95. [0] Dwyer, H.A., Calclation of Unsteady Leading Edge Bondary Layers, AIAA Jornal, Vol. 6, pp , 968.