BACHELOR S THESIS. Adam Janečka Flows of fluids with pressure and temperature dependent viscosity in the channel

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1 Charles Universit in Prague Facult of Mathematics and Phsics BACHELOR S THESIS Adam Janečka Flows of fluids with pressure and temperature dependent viscosit in the channel Mathematical Institute of Charles Universit Supervisor: prof. RNDr. Josef Málek, CSc., DSc. Stud program: Phsics, General Phsics 2

2 Univerzita Karlova v Praze Matematicko-fzikální fakulta BAKALÁŘSKÁ PRÁCE Adam Janečka Proudění tekutin s viskozitou závislou na tlaku a teplotě v rovinném kanále Matematický ústav Univerzit Karlov Vedoucí bakalářské práce: prof. RNDr. Josef Málek, CSc., DSc. Studijní program: Fzika, Obecná fzika 2

3 Tímto bch chtěl především poděkovat vedoucímu bakalářské práce, prof. Josefu Málkovi, za metodické vedení, trpělivost, věnovaný čas a celkovou pomoc při tvorbě této práce. Rovněž bch chtěl poděkovat konzultantovi, Vítu Průšovi, Ph.D., za cenné připomínk a ochotu při dokončování práce, a za pomoc s numerickým řešením rovnic. Prohlašuji, že jsem svou bakalářskou práci napsal samostatně a výhradně s použitím citovaných pramenů. Souhlasím se zapůjčováním práce a jejím zveřejňováním. V Praze dne 27. května 2 Adam Janečka 3

4 Contents Flows of heat-conducting incompressible fluids 6. Balance equations Constitutive equations Boundar conditions Problem geometr 2. Plane flow Boundar conditions Non-dimensional equations Solutions of selected problems 4 3. Pressure-dependent viscosities Temperature-dependent viscosities Concluding remarks 23 Bibliograph 24 4

5 Název práce: Proudění tekutin s viskozitou závislou na tlaku a teplotě v rovinném kanále Autor: Adam Janečka Katedra (ústav): Matematický ústav Univerzit Karlov Vedoucí bakalářské práce: prof. RNDr. Josef Málek, CSc., DSc. vedoucího: malek@karlin.mff.cuni.cz Abstrakt: V předložené práci studujeme ustálené proudění tepelně vodivé, homogenní, isotropní, nestlačitelné tekutin s viskozitou závislou na tlaku, teplotě a smetrickém gradientu rchlosti. Uvažujeme proudění v nekonečném rovinném kanále, které je buzeno bud tlakovým gradientem (Poiseuilleovo proudění) či pohbem jedné z desek kanálu (Couetteovo proudění), přičemž na stěnách kanálu jsou předepsán různé tp okrajových podmínek. Řídící rovnice převedeme do bezrozměrné podob a řešení hledáme ve formě paralelního proudění. Výsledné rovnice řešíme numerick a v několika případech analtick. Klíčová slova: tekutina, tok, viskozita, teplota, tlak Title: Flows of fluids with pressure and temperature dependent viscosit in the channel Author: Adam Janečka Department: Mathematical Institute of Charles Universit Supervisor: prof. RNDr. Josef Málek, CSc., DSc. Supervisor s address: malek@karlin.mff.cuni.cz Abstract: In the present work we stud stead flows of heat-conducting, homogeneous, isotropic, incompressible fluid with viscosit depending on the pressure, the temperature and the smmetric part of the velocit gradient. We stud the flow in infinite plane channel. The flow is driven either b the the pressure gradient (Poiseuille flow) or b the motion of one of the channel plates (Couette flow). Different boundar conditions are prescribed at the channel plates. We non-dimensionalize the governing equations and seek the solution in the form of parallel flow. We solve the final equations numericall and, in some cases, analticall. Kewords: fluid, flow, viscosit, temperature, pressure 5

6 Chapter Flows of heat-conducting incompressible fluids We will stud flows of heat-conducting, homogeneous, isotropic, incompressible fluids with viscosit depending on the pressure and temperature. Mathematical description of the problem requires knowledge of the balance equations (for mass, linear momentum and energ), constitutive equations (for the Cauch stress and the heat flux) and boundar conditions. Let us briefl comment all these three ke components of the mathematical approach.. Balance equations We shall assume that the fluid occupies a open connected set Ω R 3 with the boundar Ω. Motion of such a fluid is well described through the velocit field v = (v, v 2, v 3 ) : (, ) Ω R 3, the densit ρ : (, ) Ω R +, the pressure p : (, ) Ω R + and the internal energ e : (, ) Ω R +. Let us recall the standard notation. For an scalar field ϕ, an vector field (e.g. velocit) v and an tensor field S we have ϕ,t := ϕ t, div v = i v i x i, ( ϕ) i = ϕ x i, ( v) ij = v i, x j (div S) i = S ij. x j j (.) We shall describe the behavior of the fluid through the balance equations expressed in their Eulerian form. The balance of mass is in the form ρ,t + div(ρv) =. (.2) Balance of linear momentum is a generalization of Newton s second law in classical mechanics ρ (v,t + [ v]v) = div T T + ρb, (.3) where T denotes the Cauch stress tensor and b denotes the specific bod force. In the absence of internal couples (moments per unit volume), the balance of angular momentum implies that the Cauch stress is smmetric, i.e., T = T T. (.4) 6

7 The balance of energ is given b ρ (E,t + E v) = div(tv q) + ρb v, (.5) where E = v e is the sum of kinetic energ and internal energ and q is the heat flux. B subtracting the scalar product of (.3) and v from (.5) we obtain the the balance of internal energ in the form ρ (e,t + e v) = T v div q. (.6) The constraint of incompressibilit in fluid mechanics is usuall in the form div v = tr D =, (.7) where D is the smmetric part of the velocit gradient, i.e., D = 2 ( v+ vt ) [5]. We will suppose even stronger condition, namel ρ = ρ, where ρ (, + ), which ensures the validit of (.2). If we identif T := T and q := q, we can re-write the sstem of balance ρ ρ equations as: div v =, (.8) v,t + div(v v) = div T + b, (.9) e,t + div(ev) = T D div q. (.) More details concerning balance equations can be found, for example, in []..2 Constitutive equations The constitutive equation for internal energ is specified b simple temperature dependent model e(θ) = c V θ, (.) where θ is temperature and c V is the heat capacit at a constant volume [6]. The heat flux q is related to the variation of temperature through the fluid [4]. The relevant constitutive equation takes the form q = q (p, θ, D(v), θ) = k(p, θ, D(v) 2 ) θ, (.2) where k denotes the thermal conductivit and we will assume it to be constant. Relation (.2) is called the Fourier s law. It has been know since the time of Newton, that T and v are related [3]. This relationship changes with the variation of θ. A general algebraic relation describing this fact can be written as g(θ, T, v) =, (.3) where g is isotropic tensor function of the second order. From the principle of material frame-indifference follows that (.3) must satisf, for all orthogonal tensors Q, g(θ, QTQ T, Q vq T ) = Qg(θ, T, v)q T. (.4) 7

8 From (.4) follows that we ma use onl the smmetric part of the velocit gradient. Using the representation theorems for isotropic functions of the form (.4) we obtain α I + α T + α 2 D + α 3 T 2 + α 4 D 2 + α 5 (TD + DT) + α 6 (T 2 D + DT 2 ) + α 7 (TD 2 + D 2 T) + α 8 (T 2 D 2 + D 2 T 2 ) = (.5) where the material moduli α i, i =,..., 8, depend on θ, tr T, tr D, tr T 2, tr D 2, tr T 3, tr D 3, tr(td), tr(t 2 D), tr(d 2 T), tr(t 2 D 2 ). To obtain a more specific subclass of fluids we shall first consider α i, i = , to be zero. Thus (.5) simplifies to α I + α T + α 2 D =. (.6) If we take the trace of the previous equation and satisf the incompressibilit constraint, we conclude that or 3α + α tr T = (.7) α = tr T 3 α (.8) and we define the pressure (the mean normal stress) as p := tr T. B substituting the relation for p into (.6) we 3 obtain T = pi + α 2 α D. (.9) B defining viscosit as µ(θ, p, D 2 ) := α 2 2 α we finall get the Cauch stress tensor of the form T = pi + 2µ(θ, p, D 2 )D, tr D =, (.2) where we define the deviatoric (or viscous) part of the stress tensor S := 2µ(θ, p, D 2 )D. (.2) Our subclass subsumes several fluids including the following fluids we shall stud later:. Navier-Stokes fluid and its generalizations S = 2µ(θ, p)d(v) (.22) Navier-Stokes fluid is the one with constant viscosit. 2. Power-law fluid S = 2µ(θ, p) D(v) r 2 D(v), < r < + (.23) 8

9 3. Generalized power-law fluid S = 2µ(θ, p) ( r 2 κ + D(v) 2) 2 D(v), < r < +, κ R. (.24) All the above mentioned models can be generalized as S = 2µ(θ, p)β( D 2 )D. (.25) Studing the power-law fluid or the generalized power-law fluid, we will focus on two important values of r, r = { 3, } The value r = 4 is used in glacier 3 dnamics for models based on Glen s flow law [6]. We shall consider the viscosit to be dependent either on pressure or temperature. We will stud two pressure dependent viscosit models µ(p) = αp γ, (.26) µ(p) = µ e αp, (.27) where α is a constant, and the following three temperature dependent viscosit models:. Constant viscosit model 2. Renolds model 3. Vogel s model µ(θ) = µ, (.28) µ(θ) = µ exp( mθ), (.29) ( ) a µ(θ) = µ exp, (.3) b + θ where µ, m, a and b are constants. Using the mentioned constitutive equations (.), (.2) and (.2), the sstem of balance equations can be written as: where denotes the Laplace operator. div v =, (.3) v,t + div(v v) = p + div S + b, (.32) c V θ,t + c V div(θv) = S D + k θ, (.33).3 Boundar conditions We assume that the boundar Ω is not permeable v n = on [, ) Ω, (.34) where n is the unit outer normal to the boundar. For the velocit field we will consider the Navier s slip boundar condition λv t + ( λ)sn t = on [, ) Ω, (.35) 9

10 where t is an tangent vector at the boundar, i.e., t n = and the parameter λ meets λ [, ] including two limiting cases. If λ =, (.35) reduces to slip boundar condition. On the contrar, λ = means no slip boundar condition [2]. Concerning the temperature, we prescribe the temperature values on the boundar θ Ω = θ on [, ) Ω. (.36) For fluids with pressure-dependent viscosit, we need to prescribe the pressure in some point p(x) = p at an x Ω. (.37) In the specific case of plane Poiseuille flow, when the flow is driven b a pressure potential we need to specif the driving force. This can be done b fixing the pressure gradient along the main flow direction (e.g., x-direction) at some point x Ω p x = C, (.38) where C < is a constant (a datum of the problem).

11 Chapter 2 Problem geometr 2. Plane flow We will stud stead full developed flow of an incompressible fluid between two infinite parallel plates located in = ±h of an orthogonal Cartesian coordinate sstem (see figure 2.). We will seek the velocit field of the form v = u()e x, (2.) which means that (.7) is automaticall satisfied. It follows from (2.) that u v =, D(v) = u u, D(v) = u, (2.2) 2 2 where the prime smbol denotes the coordinate derivative, i.e., u := du. In d order to eliminate (from aesthetic reasons) the factor 2 in case of the power-law fluid and the generalized power-law fluid, we redefine the generalized viscosit as µ := ( 2) r 2 µ. From now on, we will use these rescaled viscosities. From our assumptions, it also immediatel follows that so that (.32) reduces to v,t + div(v v) =, (2.3) p = div S + b. (2.4) In most cases, we will also neglect external bod forces, i.e., b =, so that we can assume p =, (2.5) z which implies that p is independent of z, i.e., p = p(x, ). Furthermore, we shall assume the temperature to satisf meaning that Thus (.33) simplifies to or, using (.25), to θ = θ(), (2.6) e,t + div(ev) =. (2.7) S 2 u + kθ =, (2.8) µ(θ, p)β( u 2 )[u ] 2 + kθ =, (2.9)

12 h x h Figure 2.: Flow between two plates 2.2 Boundar conditions In our geometr, the tangent vector and the outer normal on the boundar take the form t = (,, ), n = (, ±, ), (2.) where + holds at the upper plate and at the lower one. The choice of velocit field of the form (2.) automaticall satisfies the permeabilit constaint (.34). We will deal with the two following tpes of flow with boundar conditions derived from (.35):. Plane Poiseuille flow with boundar conditions λ u( h) ( λ )µ(θ, p)β( u )u ( h) = at the lower plate (2.) λ 2 u(h) + ( λ 2 )µ(θ, p)β( u )u (h) = at the upper plate, (2.2) where λ λ Plane Couette flow when the upper plate moves with a prescribed velocit (V,, ). The velocit has to satisf (2.) at the lower plate and λ 2 (u(h) V ) + ( λ 2 )µ(θ, p)β( u )u (h) = at the upper plate. (2.3) The prescribed temperatures at the lower and the upper plate are 2.3 Non-dimensional equations θ = θ( h), θ 2 = θ(h). (2.4) We shall introduce the following variables and parameter ȳ = h, ū = u V, µ = µ µ, T = θ θ θ 2 θ, Γ = µ V 2 k(θ 2 θ ), (2.5) where V is the characteristic velocit. In case of plane Couette flow, V corresponds to the velocit of the upper plate. Concerning the plane Poiseuille flow, V is given b V = h2 dp 2µ dx, (2.6) 2

13 which is the velocit for plane Poiseuille flow for Navier-Stokes fluid. In case of the power-law fluid and the generalized power-law fluid, we redefine the parameter Γ as Γ := [ h V ]r 2 Γ. Then, (2.9) in its the non-dimensional form is subject to the boundar conditions T + Γ µ(t, p)β( ū 2 )[ū ] 2 = (2.7) T () =, T () =. (2.8) We can also re-write the temperature dependent viscosit models:. Constant viscosit model 2. Renolds model where M = m(θ 2 θ ), µ(t ) =, (2.9) µ(t ) = exp( MT ), (2.2) 3. Vogel s model where θ = A, A = B+T µ(t ) = exp[θ(t ) θ()], (2.2) a θ 2 θ and B = (b+θ ) [7]. (θ 2 θ ) We shall also introduce the non-dimensional form of (2.4). The equation is alread divided b the pressure ρ as mentioned in section.. Thus we have ( ) ( ) p µ = div ρ ρ β( D 2 )D + b. (2.22) If we assume b of the form b = (, g, ) and use (2.5) we obtain ( ) ( ) p µ = div ρ V 2 ρ V L µ(t, p)β( D 2 ) D + Lg V b, (2.23) 2 where L is the characteristic linear dimension. B defining an analogue of the Renolds number Re = ρ V L µ, (2.24) and the Froude number Fr = V Lg, (2.25) we finall obtain the sought non-dimensional form ( ) p = div Re µβ( D 2 ) D + b. Fr 2 (2.26) In the following, we will omit the bars above the non-dimensional variables. 3

14 Chapter 3 Solutions of selected problems 3. Pressure-dependent viscosities We will consider the flow of a fluid modelled b (.22) and seek the pressure field in the form p(x, ) = F (x)g(). Considering the plane Poiseuille flow and on neglecting the bod forces, onl trivial solutions exist for viscosit of the form (.26) and for γ. For viscosit of the form (.27), solution is not possible [3]. In the following, some of the results and methods are adpoted from [3]. (i) µ(p) = αp γ If γ = and we assume u () > on (, ), from (2.4) follows which is equivalent to p x = α p u + αpu, p = α p x u, (3.a) (3.b) p x ( α2 [u ] 2 ) = αpu, (3.2a) p ( α2 [u ] 2 ) = α 2 pu u. (3.2b) The first equation can be written as ( x ln p(x, ) = ln + αu /2) =: C αu (), (3.3) which implies that p(x, ) = C 2 ()e C ()x, (3.4) where C 2 is either non-negative or non-positive. Substituting (3.4) into (3.2b) leads to C 2() + C 2 ()C ()x = αu ()C 2 ()C (), (3.5) which implies that C () C = const., (, ). Denoting C 3 () b L() we then have p(x, ) = ±L()e C x, L, (3.6) 4

15 and + αu = ±Me 2C, M >. (3.7) αu Thus u () = Me 2C, M R. (3.8) α Me 2C + We assume that λ = λ 2, so we consider the problem to be smmetric. Hence, we require that u () = which leads to M =. So that u () = e 2C α e 2C + = α sinh C cosh C, C <, (3.9) and the constant C is related to the pressure gradient along the x-direction. B integrating and satisfing the boundar condition for λ = λ 2 = we can find the explicit formula u() = αc ln ( cosh C cosh C ). (3.) Different boundar conditions (λ, λ 2 ) were also tried but led to a sulution which did not fulfilled assumption (2.) (due to the form of p). In figure 3., there are shown the velocit profiles for certain values C. The profiles are scaled so that u() d =. (3.) Pressure decline in the middle of the channel is depicted in figure C =. C = C = C = u () Figure 3.: Poiseuille flow - velocit profiles for different values of C (α = ) (ii) µ(p) = e αp Since the solution for plane Poiseuille flow does not exist, we will stud the plane Couette flow with an external bod force, e.g., gravit, in the form of b = (, g, ). (3.2) 5

16 5 C =. C = C = C = p (x, ) x Figure 3.2: Poiseuille flow - pressure along the channel for different values of C (L = ) In [3] is shown that p has to be in the form of p(x, ) = R(x) + S() (3.3) and, in addition, R(x) has to be a constant function. Thus viscosit is in the form of µ(p) = µ(s()) = Ce αs(), (3.4) where C = const. Then, it immediatel follows from (2.4) that = αµ(s)s u + µ(s)u, (3.5a) S = g. (3.5b) From the second equation, we obtain the form of pressure in the channel p = g + D, (3.6) D being a constant. B substituting the second equation into the first, we obtain a ordinar differential equation of the second order µ(s) (u αgu ) =. (3.7) B solving it and satisfing the boundar conditions (2.) and (2.3) we get the solution ( e αg e αg u() = C + λ ), (3.8) αg λ where / ( 2 sinh αg C = V αg + λ + λ ) 2, (3.9) λ λ 2 and λ, λ 2. The velocit profiles for certain values of λ, λ 2, V and g (with α = ) are depicted in figures 3.3, 3.4 and

17 V = V = 3 V = u () Figure 3.3: Couette flow - velocit profiles for different values of V (g =, λ = λ 2 = ) V = V = 3 V = u () Figure 3.4: Couette flow - velocit profiles for different values of V (g = 5, λ = λ 2 = ) V = V = 3 V = u () Figure 3.5: Couette flow - velocit profiles for different values of V (g =, λ =.2, λ 2 =.6) 7

18 3.2 Temperature-dependent viscosities In this section, we shall stud the plane Poisseuille flow with no-slip boundar conditions at the plates u(±h) =, (3.2) i.e., λ = λ 2 =. The external bod forces will be neglected and all the velocit profiles will be scaled according to (3.). Firstl, we focused on the generalized Navier-Stokes fluid (.22). We were not able to find the analtical solutions, therefore, we had to solve the problems numericall. Renolds viscosit model represents oils well [7]. The velocit profiles in the absence of viscous heating, i.e., Γ =, are shown in figure 3.6 and the temperature distributions with Γ = are depicted in figure 3.7. In figures are displaed solutions for the Vogel s model which describes lubricating oils [7]. Again, the velocit profiles are in the absence of viscous heating and the temperature distributions are with Γ =. The constant viscosit model describing the behaviour of the Navier-Stokes fluid is a special subclass of both Renolds and Vogel s models with M = and A = B = respectivel M =.8 M = M = u () Figure 3.6: Generalized Navier-Stokes fluid with Renolds viscosit model - velocit profiles for different values of M (Γ = ) For the power-law fluid model (.23), we were not able to find neither the analtical nor the numerical solution (using the standard ODE BVP solvers) for an but the constant viscosit model which can be solved analtical as u() = { L r ( ) r r + Lr on (, ), L r r r + Lr on (, ), (3.2) where L r = r r ( C ) r. (3.22) µ 8

19 The velocit profiles for different values of r are depicted in figure 3.. More details concerning the power-law fluids with constant viscosit can be found in [2] M = M = T () Figure 3.7: Generalized Navier-Stokes fluid with Renolds viscosit model - temperature distributions for different values of M (Γ = ) A = 47.5, B = 6.25 A = B = u () Figure 3.8: Generalized Navier-Stokes fluid with Vogel s viscosit model - velocit profiles for different values of A and B (Γ = ) 9

20 A = 47.5, B = 6.25 A = B = T () Figure 3.9: Generalized Navier-Stokes fluid with Vogel s viscosit model - temperature distributions for different values of M (Γ = ).8 r = 4/3 r = 3/2 r = u () Figure 3.: Power-law fluid with constant viscosit model - velocit profiles for different values of r 2

21 Concerning the generalized power-law fluids (.24), with some effort, we were able to find the numerical solution but onl for some parameters values. The velocit profiles and the temperature distributions for Renolds and Vogel s model are displaed in figures The temperature distributions are not mentioned for Vogel s model because we were not able to obtain the solution with viscous heating, i.e., Γ >. Once again, the constant viscosit model is a special subclass of, e.g., Renolds model with M = M = M = M = u () Figure 3.: Generalized power-law fluid with Renolds viscosit model - velocit profiles for different values of M (r = 3, Γ =, κ = ) M = M = M = u () Figure 3.2: Generalized power-law fluid with Renolds viscosit model - velocit profiles for different values of M (r = 4, Γ =, κ = ) 3 2

22 κ =.8 κ = κ = T () Figure 3.3: Generalized power-law fluid with Renolds viscosit model - temperature distributions for different values of κ (r = 3, M =, Γ = ) κ =.8 κ = κ = T () Figure 3.4: Generalized power-law fluid with Renolds viscosit model - temperature distributions for different values of κ (r = 4, M =, Γ = ) r = 3/2 r = 4/ u () Figure 3.5: Generalized power-law fluid with Vogel s viscosit model - velocit profiles for different values of r (κ =, Γ =, A = 47.5, B = 6.25) 22

23 Chapter 4 Concluding remarks In chapter 3, we have summed up so far known results. To date, the mechanic (the pressure and the smmetric part of the velocit gradient) and the temperature effects were studied separatel and in our work, we tr to combine both of these effects even with respect to different boundar conditions. The flow between two parallel plates was chosen as a simple problem geometr in which we can easil compare the mechanic and the temperature effects. From the qualitative point of view, it is obvious that the temperature gradient shifts the maximum of the velocit profile towards the warmer plate. This effect cannot be achieved b a pure mechanic model. We were not able to obtain the numerical solution directl in case of the powerlaw fluid and the generalized power-law fluid (solution onl for some parameters values). These problems would need a deeper analsis. In the future, we can further stud the fluids with pressure and temperature dependent viscosities. We could also tr to optimize the flow b setting up the temperature at the channel plates. 23

24 Bibliograph [] Gurtin M. E.: An Introduction to Continuum Mechanics, Academic Press, 98. [2] Hron J., Le Roux C., Málek J., Rajagopal K. R.: Flows of Incompressible Fluids subject to Navier s slip on the boundar, Comput. Math. Appl. 56 (28), [3] Hron J., Málek J., Rajagopal K. R.: Simple flows of fluids with pressuredependent viscosities, Proc. Ro. Soc. London Ser. A 457 (2), [4] Landau L. D., Lifshitz E. M.: Fluid mechanics, Vol. 6 (2nd ed.), Pergamon Press, 987. [5] Málek J., Rajagopal K. R.: Mathematical issues concerning the Navier- Stokes equations and some of its generalization, Handb. Differ. Equ., Evolutionar Equations, Volume 2 (25), Elsevier, Amsterdam, [6] Souček O.: Numerical modeling of ice sheet dnamics, Ph.D. Thesis, Department of Geophsics, Charles Universit in Prague, Prague, 2. [7] Szeri A. Z., Rajagopal K. R.: Flow of a non-newtonian fluid between heated parallel plates, Internat. J. Non-Linear Mech. 2 (985), 9. 24