Self-similar analytic solution of the two dimensional Navier-Stokes equation with a non-Newtonian type of viscosity
aarXiv:1410.1303v3 [physics.flu-dyn] 14 Nov 2014 a nd elf-similar analytic solution of the two dimensionalNavier-Stokes equation with a non-Newtonian type ofviscosity Imre Ferenc Barna a , Gabriella Bogn´ar b a Wigner Research Center of the Hungarian Academy of Sciences Konkoly-Thege ´ut 29 -33, 1121 Budapest, Hungary b University of Miskolc, Miskolc-Egyetemv´aros 3515, Hungary
Abstract
We investigate the two dimensional incompressible Navier-Stokes(NS) andthe continuity equations in Cartesian coordinates and Eulerian descriptionfor non-Newtonian fluids. As a non-Newtonian viscosity we consider the La-dyzenskaya model with a non-linear velocity dependent stress tensor. Thekey idea is the multi-dimensional generalization of the well-known self-similarAnsatz, which was already used for non-compressible and compressible vis-cous flow studies. The geometrical interpretations of the trial function is alsodiscussed. We compare our recent results to the former Newtonian ones.
Keywords: self-similar solution, Ladyzenskaya model, non-Newtonian fluid
1. Introduction
Dynamical analysis of viscous fluids is a never-ending crucial problem. Alarge part of real fluids do not strictly follow Newtons’s law and are aptlycalled non-Newtonian fluids. In most cases, fluids can be described by morecomplicated governing rules, which means that the viscosity has some ad-ditional density, velocity or temperature dependence or even all of them.General introductions to the physics of non-Newtonian fluid can be found in[1, 2]. In the following, we will examine the properties of a Ladyzenskayatype non-Newtonian fluid [3, 4]. Additional temperature or density depen-dent viscosities will not be considered in the recent study. There are someanalytical studies available for non-Newtonian flows in connections with theboundary layer theory, which shows some similarity to our recent problem
Preprint submitted to Elsevier October 6, 2018
5, 6]. The heat transfer in the boundary layer of a non-Newtonian Ostwald-de Waele power law fluid was investigated with self-similar Ansatz from oneof us in details as well [7]. We use the two-dimensional generalization of thewell-known self-similar Ansatz [8, 9, 10], which was already used to investi-gate the three dimensional the non-compressible Newtonian NS [11] and thecompressible Newtonian NS [12] equations. We compare our results to theformer Newtonian cases. In the last part of our manuscript we present a se-ries solution for the ordinary differential equation(ODE), which was obtainedwith the help of the self-similar Ansatz.
2. Theoretical background
The Ladyzenskaya [3] model of non-Newtonian fluid dynamics can beformulated in the most general vectorial form of ρ ∂ u i ∂t + ρ u j ∂ u i ∂x j = − p∂ x i + ∂ Γ ij ∂ x j + ρ a i ∂ u j ∂ x j = 0Γ ij Def = ( µ + µ | E ( ∇ u ) | r ) E ij ( ∇ u ) E ij ( ∇ u ) Def = 12 (cid:18) ∂ u i ∂ x j + ∂ u j ∂ x i (cid:19) (1)where ρ, u i , p, a i , µ , µ , r are the density, the two dimensional velocity field,the pressure, the external force, the dynamical viscosity, the consistencyindex and the flow behavior index. The last one is a dimensionless parameterof the flow. (To avoid further misunderstanding, we use a for external fieldinstead of the letter g , which is reserved for a self-similar solution.) The E ij is the Newtonian linear stress tensor, where x (x,y) are the Cartesiancoordinates. The Einstein summation is used for the j subscript. In our nextmodel, the exponent should be r > −
1. This general description incorporatesthe following five different fluid models:Newtonian for µ > , µ = 0 , Rabinowitsch for µ , µ > , r = 2 , Ellis for µ , µ > , r > , Ostwald-de Waele for µ = 0 , µ > , r > − , Bingham for µ , µ > , r = − . (2)3or µ = 0 if r < r > µ = 0 . , r = − .
425 [2]. Parameters of a film foam withcarbon dioxides are µ = 0 . , r = − .
52 [13].In dilatant or shear thickening fluid, the apparent viscosity increases withincreased stress. Typical examples are suspensions of corn starch in water(sometimes called oobleck) or sand in water.We set the external force to zero in our investigation a i = 0. In two dimen-sions, the absolute value of the stress tensor reads: | E | = [ u x + v y + 1 / u y + v x ) ] / , (3)where the u ( u, v ) coordinate notation is used from now on. For a bettertransparency - instead of the the usual partial derivation notation - we applysubscripts for partial derivations. (Note, that in three dimensions the ab-solute value of the stress tensor would be much complicated containing sixterms instead of three.) Introducing the following compact notation L = µ + µ | E | r , (4)our complete two dimensional NS system for incompressible fluids can beformulated much clearer: u x + v y = 0 ,u t + uu x + vu y = − p x /ρ + L x u x + Lu xx + L y u y + v x ) + L u yy + v xy ) ,v t + uv x + vv y = − p y /ρ + L y v y + Lv yy + L x u y + v x ) + L v xx + u xy ) , (5)which is our starting point for the next investigation.We apply the physically relevant self-similar Ansatz to system (5). The formof the one-dimensional self-similar Ansatz is given in [8, 9, 10] T ( x, t ) = t − α f (cid:16) xt β (cid:17) := t − α f ( ω ) , (6)where T ( x, t ) can be an arbitrary variable of a partial differential equa-tion(PDE) and t means time and x means spatial dependence. The similarity4xponents α and β are of primary physical importance since α represents therate of decay of the magnitude T ( x, t ), while β is the rate of spread (or con-traction if β < t >
0. The most powerfulresult of this Ansatz is the fundamental or Gaussian solution of the Fourierheat conduction equation (or for Fick’s diffusion equation) with α = β = 1 / t and the space coordinate x appear only in the combination of f ( x/t β ). It means that the existence ofself-similar variables implies the lack of characteristic length and time scales.These solutions are usually not unique and do not take into account the ini-tial stage of the physical expansion process. These kind of solutions describethe intermediate asymptotic of a problem: they hold when the precise ini-tial conditions are no longer important, but before the system has reachedits final steady state. They are much simpler than the full solutions and soeasier to understand and study in different regions of parameter space. Afinal reason for studying them is that they are solutions of a system of ODEand hence do not suffer the extra inherent numerical problems of the fullpartial differential equations. In some cases self-similar solutions helps tounderstand diffusion-like properties or the existence of compact supports ofthe solution.At first we introduce the two dimensional generalization of the self-similarAnsatz, (6) which might have the form of T ( x, z, t ) = t − α f (cid:18) F ( x, y ) t β (cid:19) (7)where F ( x, y ) could be understood as an implicit parametrization of a one-dimensional space curve with continuous first and second derivatives. In ourformer studies [11, 12], we explain that for the non-linearity in the Navier-Stokes equation unfortunately only the F ( x, z ) = x + y + c function is validin Cartesian coordinates. It basically comes from the symmetry properties ofthe system like u x = u y . (Other locally orthogonal coordinate systems havenot been investigated yet.)Every two dimensional flow problem can be reformulated with the help ofthe stream function Ψ via u = Ψ y and v = − Ψ x , which automatically fulfillsthe continuity equation. The system of (5) is now reduced to the following5wo PDEsΨ yt + Ψ y Ψ yx − Ψ x Ψ yy = − p x ρ + ( L Ψ yx ) x + (cid:20) L yy − Ψ xx ) (cid:21) y − Ψ xt − Ψ y Ψ xx + Ψ x Ψ xy = − p y ρ + (cid:20) L yy − Ψ xx ) (cid:21) x − ( L Ψ yx ) y (8)with L = µ + µ (cid:2) xy + (Ψ yy − Ψ xx ) (cid:3) r/ . Now, search the solution of this PDE system such as: Ψ = t − α f ( η ) , p = t − ǫ h ( η ) , η = x + yt β , where all the exponents α, β, γ are real numbers. (Solutionswith integer exponents are called self-similar solutions of the first kind andthey can be obtained from dimensional argumentation as well.)Unfortunately, the constraints, which should fix the values of the expo-nents become contradictory, therefore no unambiguous ODE can be formu-lated. This means that the PDE and the stream function does not haveself-similar solutions. In other words the stream function has no diffusiveproperty. This is a very instructive example of the applicability of the trialfunction of (6).Let’s return to the original system of (5) and try the Ansatz of u = t − α f ( η ) , v = t − δ g ( η ) , p = t − ǫ h ( η ) , η = x + yt β (9)where all the exponents are real number again and f, g, h are called the shapefunctions. The next step is to determine the exponents. From the continuityequation we simple get arbitrary β and δ = α relations. The two NS dictateadditional constraints. (We skip the trivial case of µ = 0 , µ = 0, which wasexamined in our former paper as the Newtonian fluid. [11]) Finally, we get µ = 0 , µ = 0 , α = δ = (1 + r ) / , β = (1 − r ) / , ǫ = r + 1 . (10)Note, that r remains free, which describes various fluids with diverse physicalproperties, this meets our expectations. For the Newtonian NS equation,there is no such free parameter and in our former investigation we got fixedexponents with a value of 1 /
2. For a physically relevant solutions, whichis spreading and decaying in time all the exponents Eq. (10) have to bepositive determining the − < r < µ (1 + r ) f ′′ [2 f ′ ] r/ + (1 − r )2 ηf ′ + (1 + r )2 f = 0 , (11)6here prime means derivation with respect to η . Note that for the numericalvalue r = 0 we get back the ODE of the Newtonian NS equation for twodimensions. In three dimensions the ODE reads:9 µ f ′′ − η + c ) f ′ + 32 f ( η ) − c a = 0 . (12)Its solutions are the Kummer functions [14] f = c · KummerU (cid:18) − , , ( η + c ) µ (cid:19) + c · KummerM (cid:18) − , , ( η + c ) µ (cid:19) + c − a , (13)where c and c are integration constants, c is the mass flow rate, and a is theexternal field. These functions have no compact support. The correspondingvelocity component however, decays for large time like v ∼ /t for t → ∞ ,which makes these results physically reasonable. A detailed analysis of (13)was presented in [11]. A similar investigation was performed for compress-ible Newtonian fluids in [12], where the results are given by the Whitakkerfunctions, which have strong connections to the Kummer functions as well.Unfortunately, we found no analytic solution or integrating factor for (11)at any arbitrary values of r . For sake of completeness, we mention that withthe application of the symmetry properties of the Ansatz u xx = u xy = u yy = − v xx = − v xy = − v yy the following closed form can be derived for the pressurefield h = ρ ( µ r/ f ′ r +1 + f η − ˜ c f ) + ˜ c , (14)where ˜ c and ˜ c are the usual integration constants.
3. Phase Plane Analysis
Applying the transition theorem, a second order ODE is always equivalentto a first order ODE system. Let us substitute f ′ = l, f ′′ = l ′ , then f ′ = l,l ′ = − (cid:18) (1 − r )2 ηl + (1 + r )2 f (cid:19) / (cid:0) µ (1 + r )2 r/ l r (cid:1) , (15)where prime still means derivation with respect to η . This ODE system isstill non-autonomous and there is no general theory to investigate such phase7ortraits. We can divide the second equation of (15) by the first one to geta new ODE, where the former independent variable η becomes a free realparameter dldf = − (cid:18) (1 − r )2 ηl + (1 + r )2 f (cid:19) / (cid:0) µ (1 + r )2 r/ l r +1 (cid:1) . (16)Figure 1 shows the phase portrait diagram of (16) for water pulp the materialparameters are r = − .
425 and µ = 0 .
18. We consider η = 0 .
03 as the”general time variable” to be positive as well. With the knowledge of theexponent range − < r < +1, two general properties of the phase space canbe understood by analyzing Eq. (16):Firstly, the derivative df /dl is zero at zero nominator values, which means l = − (1 + r ) / (1 − r ) f /η . This one is a straight line passing through theorigin with gradient of 0 < (1 + r ) / (1 − r ) < ∞ for − < r < +1. On Figure1, the numerical value of the gradient is − . /η = − . df /dl or the direction field is not defined for anynegative l values because the power function l − ( r +1) in the denominator isnot defined for negative l arguments. We cannot extract a non-integer rootfrom a negative number. The denominator is always positive.These properties explain that there are two kinds of possible trajectories orsolutions exist in the phase space. One type has a compact support, and theother has a finite range. We may consider the x axis as the velocity f ∼ v ( η )and y axis as the f ′ ∼ a ( η ) acceleration for a fixed scaled time η = const. It also means that the possible velocity and accelerations for a general timecannot be independent from each other. The factors of the second derivative f ′′ show some similarity to the porous media equation, where the diffusioncoefficient has also an exponent. This is the essential original responsibilityfor the solution with compact support [15]. This is our main result.
4. Approximate Solutions
Our objective is to show the existence of analytic solutions to the differ-ential equation (11) and to determine the approximate local solution f ( η ).We use the shooting method and give the conditions at η = 0 with initialconditions f (0) = A, f ′ (0) = B. (17)We will consider (11) as a system of certain differential equations, namely,the special Briot-Bouquet differential equations. For this type of differential8 igure 1: The phase portrait diagram of (16) for η = 0 . , r = − .
425 and µ = 0 . f ( η ) about η = 0 we apply thefollowing theorem:Briot-Bouquet Theorem [17]: Let us assume that for the system of equations ξ dz dξ = u ( ξ, z ( ξ ) , z ( ξ )) ,ξ dz dξ = u ( ξ, z ( ξ ) , z ( ξ )) , ) (18)where functions u and u are holomorphic functions of ξ, z ( ξ ) , and z ( ξ )near the origin, moreover u (0 , ,
0) = u (0 , ,
0) = 0 , (19)then a holomorphic solution of (18) satisfying the initial conditions z (0) = 0 ,z (0) = 0 exists if none of the eigenvalues of the matrix ∂u ∂z (cid:12)(cid:12)(cid:12) (0 , , ∂u ∂z (cid:12)(cid:12)(cid:12) (0 , , ∂u ∂z (cid:12)(cid:12)(cid:12) (0 , , ∂u ∂z (cid:12)(cid:12)(cid:12) (0 , , (20)is a positive integer.Briot-Bouquet theorem ensures the existence of formal solutions z = X ∞ k =0 a k ξ k , z = X ∞ k =0 b k ξ k (21)for system (18), and also the convergence of formal solutions.This theorem has been successfully applied to the determination of localanalytic solutions of different nonlinear initial value problems [18]-[20].Let us consider the initial value problem (11), (17) and take its solution inthe form f ( η ) = η α Q (cid:0) η β (cid:1) , η ∈ (0 , η c ) , (22)where function Q ∈ C (0 , η c ) (23)for some positive value η c . Substituting f ( η ) = η α Q (cid:0) η β (cid:1) (24)10nto (11) one can get K η (cid:8) β (cid:2) ( β + 1) η β Q ′ + βη β Q ′′ (cid:3)(cid:9) (cid:0) Q + βη β Q ′ (cid:1) r +1 − r η (cid:0) Q + βη β Q ′ (cid:1) + 1 + r ηQ = 0 (25)for α = 1. Let us introduce the new variable ξ such as ξ = η β (26)and function Q as follows Q ( ξ ) = a + a ξ + z ( ξ ) , (27)where a , a are real constants, and z ∈ C (0 , η βc ) , (28) z (0) = 0 , z ′ (0) = 0. We make difference between two cases: A = 0 and A = 0.If A = 0 then Q fulfills the following properties Q (0) = a , Q ′ (0) = a ,Q ′′ ( ξ ) = z ′′ ( ξ ) . From the initial condition f ′ (0) = Q (0) and we have a = B. Applying the Briot-Bouquet theorem, (25) yields ξz ′′ ( ξ ) = − β " ( β + 1) Q ′ + ξ β − Q + − r βξQ ′ Kβ ( Q + βξQ ′ ) r , (29)with K = µ (1 + r )2 r . Therefore, β = 2. We restate the second orderdifferential equation in (25) as a system of two equations u ( ξ, z ( ξ ) , z ( ξ )) = ξ z ′ ( ξ ) u ( ξ, z ( ξ ) , z ( ξ )) = ξ z ′ ( ξ ) (cid:27) (30)with choosing z ( ξ ) = z ( ξ ) z ( ξ ) = z ′ ( ξ ) (cid:27) with z (0) = 0 z (0) = 0 (cid:27) (31)and u ( ξ, z ( ξ ) , z ( ξ )) = ξ z , (32)11 ( ξ, z ( ξ ) , z ( ξ )) = ξ z ′ = −
12 [3 ( a + z ) +] − " a + (2 − r ) a ξ + z + (1 − r ) ξz K ( a + 3 a ξ + z + 2 ξz ) r . (33)In order to satisfy the conditions u (0 , , ,
0) = u (0 , , ,
0) = 0 (34)in the Briot-Boquet theorem the following connection yields a = − a − r K . (35)Therefore, the eigenvalues of matrix (20) at (0 , ,
0) are 0. Since all eigen-values are non-positive, referring to the Briot-Bouquet theorem we obtain theexistence of unique analytic solutions z and z near zero. Thus, there existsa formal solution f ( η ) = η ∞ X k =0 a k η k , (36)where the first two coefficients are already known.For the determination of coefficients a k , k >
2, we shall use the J.C.P. Millerformula [21]: " L X k =0 c k x k r/ = r L X k =0 d k ( r ) x k , (37)where d ( r ) = 1 for c = 1 , and d k ( r ) = 1 k k − X j =0 [ r k − j ) − j ] d j ( r ) c k − j , ( k ≥ . (38)Applying (38) we get recursion formula for the determination of a k . Com-paring the proper coefficients of η , one can have the values of a k for some k ;e.g., a = 2 − r K a − r − K a − r . (39)The coefficients are obtained by this method for the power series approxima-tion as f ( η ) = a η + a η + a η + . . . . (40)12imilarly, we can obtain series approximate solution for A = 0. Here weobtain that α = 0 and Q (0) = A = a , Q ′ (0) = B = a for f ( η ) = Q ( η β ).The differential equation (11) yields Kβη γ (cid:2) ( β − Q ′ + βη β Q ′′ (cid:3) β r Q ′ r + 1 − r βη β Q ′ + 1 + r Q = 0 , (41)where γ = β − β − r . The system of Briot-Bouquet equations isformulated as follows u ( ξ, z ( ξ ) , z ( ξ )) = ξ z ′ ( ξ ) u ( ξ, z ( ξ ) , z ( ξ )) = ξ z ′ ( ξ ) (cid:27) (42)where ξz ′′ ( ξ ) = 1 − ββ Q ′ − − r K ξ − γβ Q ′ − r β r − r K Qβ r Q ′ r ξ − γβ . (43)The conditions in the Briot-Bouquet theorem can be satisfied for arbitrary A and B if γ = − β, i.e., β = 1 . Then we get that u ( ξ, z ( ξ ) , z ( ξ )) = − − r K ξ ( a + z ) − r − r K ξ a + a ξ + z ( a + z ) r . (44)The method of calculation of the coefficients in the power series is similar tocase A = 0 . Equating the like powers one gets the further coefficients that a = − r K a a r , a = − (1 + r ) r K a a r − a − r K , . . . . (45)Figure 2 presents the series solution of (11) for different values of r when A = 0 and B = 1. Figure 3 presents the series solution of (11) for differentvalues of r when A = 1 and B = 0. Note, the red curve on both figurespresents the solution for r = − .
5. Conclusions
We applied a two-dimensional generalization of the self-similar Anstaz forthe Ladyzhenskaya-type non-Newtonian NS equation. We analyzed the final13
Figure 2: Approximate series solutions of f ( η ) to Eq. (11) and for different values of r when A = 0 and B = 1. The red curve is for r = − . Figure 3: Approximate series solutions of f ( η ) to Eq. (11) and for different values of r when A = 1 and B = 0. Note the red curve is for r = − .
6. Acknowledgement
This work was supported by the Hungarian OTKA NK 101438 Grant.This research was (partially) carried out in the framework of the Center ofExcellence of Innovative Engineering Design and Technologies at the Univer-sity of Miskolc. We thank for Prof. Robert Kersner for useful discussionsand comments. We dedicate this paper to our spouses.