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Chemically-reacting fluids with variable transport properties
Applied Mathematics and Computation 219 (2012) 1761–1775
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Chemically-reacting fluids with variable transport properties Mehrdad Massoudi a,⇑, A.K. Uguz b a b
U.S. Department of Energy, National Energy Technology Laboratory (NETL), P.O. Box 10940, 626 Cochrans Mill Road, Pittsburgh, PA 15236, USA _ Bog˘aziçi University, Department of Chemical Engineering, Bebek, 34342 Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Non-linear fluids Chemically-reacting fluids Variable viscosity Variable thermal conductivity Variable diffusion coefficient Viscous dissipation
a b s t r a c t In this paper we study the flow and heat transfer in a chemically-reacting fluid, whose transport properties, i.e., viscosity, thermal conductivity and the diffusion coefficients, are assumed to be a function of the concentration. The equations for the shearing flow of such a fluid between two long horizontal parallel plates, which are at different temperatures, are made dimensionless and the boundary value problem is solved numerically; the velocity, concentration and temperature profiles are obtained for various dimensionless numbers. Published by Elsevier Inc.
1. Introduction Studying mechanics of complex fluids, from the early days of polymer rheology to the present day nano-fluids, present a special challenge to both theoreticians and experimentalists. The main difficulty is the inherent non-linearities in the constitutive relations, thus making the numerical (or the analytical) study of the governing equations more difficult, while creating a demand for accurate measurements of the constitutive parameters. A sub-class of the complex fluids is the chemically-reacting fluids which are used in many industries and technologies such as combustion, catalysis [28], biological systems, etc. For example, coal is often used as the primary fuel in high temperature and high pressure environments such as the open-cycle MHD generators, where temperatures and pressures are of the order of 3000 K and 10 atm, respectively. Under these extreme conditions, the rheological and thermal properties of coal change and as a result the transport properties are not constant. This is not unique to coal combustion and in fact in many chemical processes viscosity or thermal conductivity of the materials under consideration are not constant and in many cases it has been shown that substantial performance benefits can be obtained by pre-heating a coal-slurry; this can affect its viscosity which in turn can affect the flow rate and pressure drop, parameters of importance in solids transport (see Gupta and Massoudi [6], and Massoudi and Christie [18]). The effects of temperature change on coal-slurry properties are controlled to a great extent by changes in the properties of the fluid. The constitutive equation used, most often, in these studies is the power-law model. Although the powerlaw model adequately fits the shear stress and shear rate measurements for many non-Newtonian fluids, it cannot always be used to accurately predict the pressure loss data measured during transport of a coal-liquid mixture in a fuel delivery system. Constitutive modeling of non-linear fluids has received much attention in the past few decades (see for example the books by Carreau et al. [4], Middleman [20], Schowalter [29], and Larson [12]), mainly due to the fact that many of the fluids used in bio-chemical industries behave as non-linear fluids, also known as non-Newtonian fluids, for example, polymer melts, suspensions, blood, coal-slurries, etc. A sub-class of these non-linear fluids which recently has been studied includes
⇑ Corresponding author. E-mail address: [email protected] (M. Massoudi). 0096-3003/$ - see front matter Published by Elsevier Inc. http://dx.doi.org/10.1016/j.amc.2012.08.015
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Nomenclature Symbol b c D f g h l K1 K L n q t T uw w x
q l h div
r
Explanation body force vector concentration symmetric part of the velocity gradient constitutive function acceleration due to gravity characteristic length identity tensor material parameter related to diffusivity thermal conductivity gradient of the velocity vector power-law index heat flux vector time Cauchy stress tensor reference velocity (chemical reaction) flux vector spatial position occupied at time t bulk density viscosity temperature divergence operator gradient symbol
chemically-reacting fluids which offer technological applications ranging from the formation of thin films for electronics (see Kee et al. [10]) to biological fluids, especially the synovial fluid (see Lai et al. [11], Bridges et al. [3], and Hron et al. [8]). Most complex fluids are multi-component mixtures. In many applications, these fluids are treated as a single continuum suspension with non-linear material properties and the techniques and models used in rheology or mechanics of non-linear fluids can generally be used to study such problems (see Oldroyd [22], Larson [12]). In this case, global or macroscopic information about the variables such as the velocity or temperature fields of the whole suspension can be obtained. In many other applications, however, there is a need to know the details of the field variables such as velocity, concentration, temperature, etc., of each component and in such cases one needs to resort to the multi-component modeling approaches (see Rajagopal and Tao [25], Massoudi [16], and Massoudi [17]). Examples of complex fluids whereby both approaches can be used are coalslurries, many of the biological fluids such as blood and the synovial fluid, and chemically-reacting fluids. Bridges and Rajagopal [2] studied the pulsatile flow of a non-linear chemically-reacting fluid. In their model, they assumed that the viscosity not only depends on the concentration of the species (or constituent whose behavior is governed by a convection-reaction– diffusion equation), but also on the velocity gradient, in a properly frame-invariant form; the fluid can be either shearthinning or shear-thickening. This approach, even though used for a single continuum complex fluid, is really based on a constrained mixture theory approach (see Humphrey and Rajagopal [9]) which implies that at each point in the complex fluid, i.e., the mixture, the host fluid coexists with the constituent, flowing in such a way that the two components are constrained to move together. In such an approach, one does not need to study the complicated coupled equations for the multi-component mixtures. Uguz and Massoudi [32] studied the heat transfer to a chemically-reacting fluid whose constitutive relation for the stress tensor was given by the model proposed by Bridges and Rajagopal [2]. Later, Hron et al. [8] developed generalized power-law models for the synovial fluid where now the exponent of this power-law model depends in various ways on the concentration of the hyaluronan [also referred to as constituent or species in the earlier studies of Bridges and Rajagopal [2] and Uguz and Massoudi [32]. In another study, Bridges et al. [3] used the three models suggested by Hron et al. [8] to study the unsteady flow of the synovial fluid between two cylinders; they also assumed the fluid to be a shear-thinning and chemically-reacting fluid. In this paper we look at chemically-reacting fluids with variable transport properties. The reaction is approximated by coupling the fluid dynamics with the concentration of the second constituent in the fluid through the change in the dynamic viscosity, thermal conductivity and the diffusion coefficients. It should be emphasized that our proposed model would reduce to the one suggested by Uguz and Massoudi [32], if we ignore the dependence of thermal conductivity and diffusion coefficients on the concentration. In Sections 2 and 3 we present the governing equations and the constitutive relations, respectively. In Section 4, we look at the flow between two parallel flat plates which are at different temperatures and present the governing equations and the boundary conditions along with our assumptions. It is followed by Section 5 where the results are analyzed. Finally, Section 6 presents our conclusions.
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2. Governing equations In the absence of any electro-magnetic effects, the governing equations of motion are the conservation of mass, linear momentum, convection–diffusion, and energy equations. These are (see Slattery [30]): Conservation of mass:
@q þ divðquÞ ¼ 0 @t
ð1Þ
where q is the density of the fluid, o/ot is the partial derivative with respect to time, and u is the velocity vector. For an isochoric motion we have div u = 0. Conservation of linear momentum:
q
du ¼ div T þ qb dt
ð2Þ
where b is the body force vector, T is the Cauchy stress tensor, and d/dt is the total time derivative, given by dðÞ ¼ @ðÞ þ ½gradðÞu. The balance of moment of momentum reveals that in the absence of couple stresses, the stress tensor dt @t is symmetric. Conservation of Concentration:
@c þ div ðcuÞ ¼ f @t
ð3Þ
where c is the concentration and f is a constitutive parameter. This equation is also known as the convection-reaction–diffusion equation. Conservation of energy:
q
de ¼ T L div q þ qr dt
ð4Þ
where e is the specific internal energy, L is the gradient of velocity, q is the heat flux vector, and r is the radiant heating.1 Thermodynamical considerations require the application of the second law of thermodynamics or the entropy inequality. The local form of the entropy inequality is given by (see Liu [13]):
qg_ þ div u qs 0
ð5Þ
where g(x, t) is the specific entropy density, /(x, t) is the entropy flux, and s is the entropy supply density due to external sources, and the dot denotes the material time derivative. If it is assumed that / ¼ 1h q, and s ¼ 1h r, where h is the absolute temperature, then Eq. (5) reduces to the Clausius–Duhem inequality
q h
r h
qg_ þ div q 0
ð6Þ
Even though we do not consider the effects of the Clausius–Duhem inequality in our problem, for a complete thermomechanical study of a problem, the Second Law of Thermodynamics has to be considered (see Batra [1], Haupt [7], Müller [21], Ziegler [33], Truesdell and Noll [31], and Liu [13]). In order to ‘close’ these equations, we need to provide constitutive relations for T, q, f, e and r. In this problem, we assume that the radiation effects can be neglected. The specific internal energy, is related to the specific Helmholtz free energy [5] e ¼ W þ hg ¼ eðh; A1 Þ ¼ ^eðyÞ, where g is the specific entropy. The internal energy may in some ways depend on other parameters. Nevertheless, in the current problem, due to the nature of the kinematical assumptions for the Couette flow, e does not need to be modeled. 3. Constitutive relations We will now briefly discuss the three constitutive relations needed for the closure in this problem, namely equations for T, q, and f. (i) Stress tensor One of the main reasons for the success of the Navier–Stokes constitutive relation, in addition to its elegance and compactness, is that the stress tensor T is explicitly described in terms of D (the symmetric part of the velocity gradient). This means that once the solution to the problem, whether using numerical techniques or analytical methods represented in terms of velocity and pressure fields, is obtained, the stress at any point can be determined uniquely from the constitutive 1 Constitutive relations for complex materials can be obtained in different ways using: (a) continuum mechanics, (b) physical and experimental models, (c) numerical simulations, (d) statistical mechanics approaches, and (e) ad hoc approaches. A look at the governing Eqs. (1)–(4) reveals that constitutive relations are required for T, q, f, e, and r. Less obvious is the fact that in many practical problems involving competing effects such as temperature and concentration, the body force b, which in problems dealing with natural convection oftentimes depends on the temperature and is modeled using the Boussinesq assumption (see Rajagopal et al. [24], now might have to be modeled in such a way that it is also a function of concentration). Furthermore, in many problems involving chemical reactions, there is usually a (heat) source term, Q, in Eq. (4) which also has to be constitutively modeled (see Massoudi and Phuoc [19]).
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relation. Furthermore, from a computational point of view, it is much easier and less cumbersome to solve the equations for the explicit models.2 Perhaps the simplest model which can capture the normal stress effects is the second grade fluid, or the Rivlin–Ericksen fluid of grade two [26,31]. This model has been used and studied extensively [5] and is a special case of fluids of differential type. For a second grade fluid the Cauchy stress tensor is given by:
T ¼ p1 þ lA1 þ a1 A2 þ a2 A21
ð7Þ
where p is the indeterminate part of the stress due to the constraint of incompressibility, l is the coefficient of viscosity, a1 and a2 are material moduli which are commonly referred to as the normal stress coefficients. The kinematical tensors A1 and A2 are defined through
A1 ¼ L þ LT A2 ¼
dA1 þ A1 L þ ðLÞT A1 dt
L ¼ grad u
ð8aÞ ð8bÞ ð8cÞ
The thermodynamics and stability of fluids of second grade have been studied in detail by Dunn and Fosdick [15].3 For a chemically-reacting fluid,4 Bridges and Rajagopal [2] suggested a constitutive relation of the type:
T ¼ p1 þ lðc; A1 ÞA1
ð9Þ
where p is the constraint due to incompressibility and
c¼
qr qr þ q
ð10Þ
where q is the density of the fluid and qr denotes the density of the (coexisting) reacting fluid. Furthermore they assumed
lðc; A1 Þ ¼ l ðcÞ½1 þ atrðA21 Þn
ð11Þ
where n determines whether the fluid is shear-thinning (n < 0), or shear-thickening (n > 0). The second law of thermodynamics requires the constant a P 0 [2]. It is seen that the viscosity is assumed to depend on the concentration c whereby depending on the form of l⁄(c) the fluid can be either a chemically-thinning or chemically-thickening fluid, implying a decrease or an increase in the viscosity, respectively, as c increases. In this paper, we will generalize the results of our recently published work (see Uguz and Massoudi [32]) by assuming that in general all the transport coefficients can be expressed as:
n ðcÞ ¼ ^ng j ðcÞ with j ¼ 1; 2
ð12aÞ
g 1 ðcÞ ¼ ðx1 c2 þ x2 Þ
ð12bÞ
g 2 ðcÞ ¼ ðx3 x1 c2 Þ
ð12cÞ
where n stands for viscosity l, or diffusion j, or thermal conductivity K, and x’s are positive constants. The form given by (12b) implies chemically-thickening and (12c) chemically-thinning. In Uguz and Massoudi [32] only the viscosity was assumed to be a function of c. (ii) Concentration flux In their study of the pulsatile flow of a chemically-reacting fluid, Bridges and Rajagopal [2] suggested the following constitutive relation for f:
f ¼ div w
ð13Þ
2 There are, however, cases such as Oldroyd type fluids (see Oldroyd [22]) and other rate-dependent models whereby it is not possible to express T explicitly in terms of D and other kinematical variables. For such cases, one must resort to implicit theories, for example, of the type f(T, D, h) = 0 where h is the temperature (see Rajagopal [23]). 3 They show that if the fluid is to be thermodynamically consistent in the sense that all motions of the fluid meet the Clausius-Duhem inequality and that the l0 It is known that for many non-Newtonian fluids which are specific Helmholtz free energy of the fluid be a minimum in equilibrium, then a1 0; a1 þ a 2 ¼ 0 assumed to obey Eq. (7), the experimental values reported for a1 and a2 do not satisfy the above restriction. 4 The terms ‘chemically reactive’ and ‘chemically reacting’ as used here are not necessarily the same and they might mean different things to people in different research communities. The former is most often used in the combustion and chemical engineering traditions where the reactions are accompanied by heat transfer/release. The latter term is commonly used in fields such as Biorheology where there are many chemical reactions occurring in blood, for example, but there is no substantial heat release/generation (see Bridges and Rajagopal [2]). Thus the terminology that we have used is in line with this interpretation of the term, ‘chemically-reacting fluids.’
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where w is a flux vector, related to the chemical reactions occurring in the fluid and was assumed to be given by a constitutive relation similar to the Fick’s assumption, namely
w ¼ K 1 rc
ð14Þ
where K1 is a material parameter which in general is not constant. Bridges and Rajagopal [2] assumed that K1 is a scalar-valued function of (the first Rivlin-Ericksen tensor) A1
K 1 ¼ K 1 ðA1 Þ ¼ jA21
ð15Þ
Fig. 1. Physical configuration of the Couette flow between two parallel plates separated by a distance h. The upper wall is set into motion and held at constant speed Uw whereas the bottom plate is stationary. The concentration, c, and the temperature, h, are kept constant at the walls. The unit vectors in the x and y directions are i and j, respectively.
Fig. 2a and 2b. The effect of Re on velocity u, when d = 0.1, C = 1, c = 10, and n = 0.5, for a chemically-thickening case given by g1(c) and a chemicallythinning case given byg2(c).
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Fig. 2c and 2d. The effect of Re on concentration c, when d = 0.1, C = 1, c = 10, and n = 0.5, for a chemically-thickening case given by g1(c) and a chemicallythinning case given by g2(c).
where j was assumed to be constant and kk denotes the trace-norm. This form was also used by Uguz and Massoudi [32]. In this problem we generalize the form of j by assuming
jðcÞ ¼ j g j ðcÞ with j ¼ 1; 2 g 1 ðcÞ
ð16Þ
g 2 ðcÞ
where and are given by Eq. (12b and c) above. (iii) Heat flux vector The most popular and widely used form of the constitutive relation for the heat flux vector is given by the Fourier’s assumption where
q ¼ Krh
ð17Þ
where K is the thermal conductivity. In general, K, for anisotropic materials is a fourth order tensor, which can also depend on concentration, temperature, etc. There have been many experimental and theoretical studies related to this issue (see Massoudi [14], and Massoudi [15]). In fact, the general forms of the constitutive relations for the q and w could also depend on temperature and concentration gradients and include terms often referred to as the Dufour and the Soret effects. In this problem, we assume that K is also given a similar structure to the viscosity and diffusion coefficients, that is
KðcÞ ¼ K g j ðcÞ with j ¼ 1; 2
ð18Þ
where g 1 ðcÞ and g 2 ðcÞ are given by Eq. (12b and c) above. If we assume that q can depend explicitly on the temperature gradient, concentration gradient, etc., then the problem would become highly non-linear. An example of this type of constitutive relation is a power-law type model suggested by Rodrigues and Urbano [27], where
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Fig. 2e and 2f. The effect of Re on temperature h, when d = 0.1, C = 1, c = 10, and n = 0.5, for a chemically-thickening case given by g1(c) and a chemicallythinning case given by g2(c).
q ¼ Krh ¼ jrhjp2 rh where 1 < p < 1
ð19Þ
When p = 2, the above equation reduces to the usual Fourier’s law. Eq. (17) is by no means the most general one, especially for cases involving density variation (see Massoudi [14], and Massoudi [15] for a review of this). In the next section, we will look at the heat transfer and flow of the non-linear fluid model described in this section. 4. Flow due to shear Whenever non-linear constitutive relations are studied, the solution procedures, for solving the governing equations, whether analytical or computational, are more complicated. Exact solutions are very rare in heat transfer studies of non-linear materials or multiphase flows. Next to the exact solutions, finding solutions to simple boundary value problems are extremely desirable. In this section, we will study the flow and heat transfer to a non-linear chemically-reacting fluid between two long parallel plates where the lower plate is fixed and heated and the upper plate is set into motion and is at a lower temperature than the lower plate (see Fig. 1). For the problem under consideration, we make the following assumptions: (i) the motion is steady, (ii) the radiant heating r is ignored, (iii) the constitutive equation for the stress tensor is given by Eq. (9), the constitutive relation for the diffusion term is given by Eq. (14), and the constitutive equation for the heat flux vector is that of Fourier’s law, given by Eq. (17);
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(iv) the concentration, velocity and temperature fields are of the form
8 9 > < c ¼ cðyÞ > = u ¼ uðyÞi > > : ; h ¼ hðyÞ
ð20Þ
with (20) the conservation of mass is automatically satisfied, and by substituting Eqs. (9)–(11) into Eq. (2), the equations of linear momentum reduce to
@p d 0¼ þ @x dy
"
l
2 !n # du du ðcÞ 1 þ 2a dy dy
@p @p ¼ ¼0 @y @z
ð21Þ
Substituting (13)–(16) into (3), the diffusion-convection equation reduces to
" 2 # d du dc ¼0 jðcÞ dy dy dy
ð22Þ
And substituting (9) and (17) into (4), and assuming r = 0, the energy equation becomes:
Fig. 3a and 3b. The effect of n on velocity u, when d = 0.1, C = 1, c = 10, and Re = 1, for a chemically-thickening case given by g1(c) and a chemically-thinning case given by g2(c).
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Fig. 3c and 3d. The effect of n on concentration c, when d = 0.1, C = 1, c = 10, and Re = 1, for a chemically-thickening case given by g1(c) and a chemicallythinning case given by g2(c).
l
2 !n 2 du du d dh ¼0 K 2 ðcÞ ðcÞ 1 þ 2a þ dy dy dy dy
ð23Þ
where due to the kinematical assumptions made for the velocity and temperature fields, the term corresponding to the derivative of disappears from the energy equation. That is ddte ¼ 0 and a simplified form of the energy equation is obtained. The above equations are subject to the boundary conditions:
at y ¼ 0 :
8 > : c ¼ c0
ð24Þ
at y ¼ h :
8 > < u ¼ uw h ¼ hh > : c ¼ ch
ð25Þ
Clearly other types of boundary conditions related to the gradients of velocity, temperature, and concentration are also possible. For the present problem, we have assumed the no-slip condition for the velocity, and we have also assumed that the values of the concentration and temperature at the two plates are known, i.e., are prescribed. Eqs. (20)–(23) along with the boundary conditions (24) and (25) are to be made dimensionless and solved numerically. Let us define the dimensionless , the velocity u , the concentration c and the temperature distance y h by the following equations
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Fig. 3e and 3f. The effect of n on temperature h, when d = 0.1, C = 1, c = 10, and Re = 1, for a chemically-thickening fluid given by g1(c) and a chemicallythinning case given by g2(c).
y ¼ ; y h
¼ u
u ; uw
h ¼ h h0 ; Dh
c ¼
c c0 Dc
ð26Þ
where uw is a shearing velocity at the top plate, Dh = hhh, Dc = chc0, and h is the distance between the two plates. The dependence of the physical parameters on the concentration, i.e., Eqs (12a), (16), and (18) are now expressed as
l ðcÞ ¼ l^ g j ðcÞ
ð27Þ
jðcÞ ¼ j^ g j ðcÞ
ð28Þ
^ ðcÞ KðcÞ ¼ Kg j
ð29Þ
respectively. With these, the dimensionless forms of the governing equations become:
" 2 !n # 1 d du du ¼ c g j ðcÞ 1 þ d Re dy dy dy
/
( 2 ) d du dc g j ðcÞ ¼0 dy dy dy
ð30Þ
ð31Þ
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Fig. 4a and 4b. The effects of d on u, for C = 1, c = 10, Re = 1, and n = 0.5, for chemically-thickening case g1(c) and a chemically-thinning case given by g2(c).
2 !n 2 du du d dh ¼0 g j ðcÞ þ g j ðcÞ 1 þ d dy dy dy dy Re
C
ð32Þ
where c is related to the pressure gradient term, and the bars have been dropped for simplicity. The dimensionless numbers that arise are
Re ¼
qhuw 2au2 j^ Dcu2w qu3 h ; d ¼ 2w ; / ¼ ; C¼ w 4 ^ Dh l^ K h h
ð33Þ
where Re is the Reynolds number, d is a dimensionless number signifying the dependence of the shear viscosity on the shear rate (with a = 0 implying no dependence of l on the shear rate), / is a dimensionless number related to the chemically-reacting nature of the fluid (related to the diffusion coefficient K1), and C is a measure of viscous dissipation (see Massoudi and Christie [18]). The dimensionless forms of the boundary conditions become:
at y ¼ 0 :
8 > : c¼0
ð34Þ
at y ¼ 1 :
8 > : c¼1
ð35Þ
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Fig. 4c and 4d. The effects of d on c, for C = 1, c = 10, Re = 1, and n = 0.5, for chemically-thickening case g1(c) and a chemically-thinning case given by g2(c).
In the next section we solve numerically Eqs. (30)–(32) subject to boundary conditions (34), and (35). This will be a parametric study where different values for the dimensionless numbers listed in Eq (33) except / as it does not affect the fields, along with different values for n will be used. 5. Numerical solutions Recall that the form of the dependence of the physical parameters on the concentration is represented by Eq. (12). We use the following simplified expressions:
g 1 ðcÞ ¼ 100ð4c2 þ 1Þ
ð36Þ
g 2 ðcÞ ¼ 100ð5 4c2 Þ
ð37Þ
where g1(c) represents the chemically-thickening contribution, and g2(c) is for chemically-thinning fluids. These expressions were suggested by Bridges and Rajagopal [2]. Eq. (30)–(32) are solved using the Spectral Method with Chebyshev grid points which are defined as yðjÞ ¼ cos ðjp=NÞ where j = 0, 1, . . . , N, where N denotes the cut-off frequency. These non-linear governing equations are recast into
2 0 ( ðpÞ )2 1n ðpþ1Þ 3 1 d 4 du ðpÞ @ 5 ¼ c A du g ðc Þ 1 þ d Re dy j dy dy
ð38Þ
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Fig. 4e and 4f. The effects of d on h, for C = 1, c = 10, Re = 1, and n = 0.5, for chemically-thickening case g1(c) and a chemically-thinning case given by g2(c).
8 ! !9 ðpÞ 2 ðpþ1Þ = d < du dc ðpÞ ¼0 g ðc Þ / ; dy : j dy dy ( ) 1n ! ! ! ðpÞ 2 ðpÞ ðpþ1Þ ðpþ1Þ du du du d p dh @ A þ ¼0 g ðc Þ g ðc Þ 1 þ d dy j Re j dy dy dy dy
C
ð39Þ
0
ðpÞ
ð40Þ
Here, the superscript (p) denotes the guessed quantity and (p + 1) is the computed one. This computation step is preferred as linearization around a known state was not possible. The initial guess is taken as unity for both the velocity and the concentration. Then, the convergence is checked with the norm of the difference of the present value and the previous value. The code is also ensured to independent of the initial guess. In our previous paper [32] we only considered the case of a chemically-thickening fluid. In the current study, we look at both cases, i.e., we will also look at a chemically-thinning fluid. It should be noted that in our previous study [32] the value for the pressure drop term c was assumed to be 10 (but it was mistakenly written as +10). In the current study, for the effects of Re, we have corrected this minor numerical issue; this changed the values slightly but did not affect the qualitative response as observed in the figures. In the present study, the effect of the different parameters, namely, Re, n, d, and C on the velocity, concentration, and temperature fields are investigated. In our parametric study, we first look at the effects of varying Re, for the case of d = 0.1, C = 1, c = 10, and n = 0.5, for two different types of chemically-reacting fluids, namely a chemically-thickening case given by g1(c) and a chemically-thinning case given by g2(c). The results for the velocity, concentration and temperature are given in Fig. 2a–f. The interesting observation is the reversal of the curves. There does not seem to be a great deal of influence on the velocity profiles for both cases of g1(c) and g2(c) (as seen in Fig. 2a and 2b). However, the impact
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Fig. 5a and 5b. The effect of C on h, when d = 0.1, c = 10, Re = 1, and n = 0.5, for a chemically-thickening fluid g1(c) and a chemically-thinning g2(c).
of these two parameters on the concentration and temperature profiles is more pronounced for the case of chemically-thickening fluid, as seen in Fig. 2c and 2d. It is also noticed that as Re increases, at any given point Y in the flow, the value of c decreases while the value of h increases. Fig. 3a–f show the influence of exponent n on u, c, and h, for the selected parameters d = 0.1, C = 1, c = 10, and Re = 1. Again, for the case of chemically-thickening, g1(c), as shown in Fig. 3a, there is very little difference between the three cases of n, 0.0, 0.5 (shear-thinning fluid), and +0.5 (shear-thickening fluid). More noticeable differences are observed in all cases for the chemically-thinning case given by g2(c) . Fig. 4a–f show the influence of d (the dimensionless number signifying the dependence of the viscosity on the shear rate with a = 0 implying no dependence of l on the shear rate, where a P 0) on u, c, and h, for the selected parameters C = 1, c = 10, Re = 1, and n = 0.5, for two different types of chemically-reacting fluids, namely a chemically-thickening case given by g1(c) and a chemically-thinning case given by g2(c). Again, it is observed that the impact of the chemically-thinning fluid given by g2(c) is more vivid on the velocity, concentration and temperature profiles. Fig. 5a and b show the influence of C (a measure of viscous dissipation) on h, for the selected parameters d = 0.1, c = 10, Re = 1, and n = 0.5, for two different types of chemically-reacting fluids, namely a chemically-thickening case given by g1(c) and a chemically-thinning case given by g2(c). As we can see from Eqs (30)–(32), C appears only in the energy equation and therefore, it influences the temperature h . It is seen that not only C but also g1(c) and g2(c) have significant impact on the temperature distribution. Clearly as C increases (a greater degree of viscous dissipation), the value of the maximum temperature within the flow increases; interestingly, when the fluid is chemically-thickening, the maximum temperature occurs below the centerline, whereas when the fluid is chemically-thinning, this maximum temperature point shifts and moves closer to the top plate. Clearly, this should be further investigated by changing the boundary conditions and or geometry, for example solving flow in a pipe and seeing if the curvature has any effect on this observation.
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6. Conclusions We have studied the flow and heat transfer in a non-linear reactive fluid, with variable transport properties, between two parallel plates. The flow is assumed to be steady state and the velocity, concentration, and temperature fields are computed accordingly. In the present problem, we have performed a parametric study for a limited range of the dimensionless numbers corresponding to viscous effect (l), the non-linear nature of the fluid (d and the exponent n), and viscous dissipation (C). It is observed that in general, the impact of the chemically-thinning fluid given by g2(c) is more vivid on the velocity, concentration and temperature profiles. Perhaps the most noted impact of all the dimensionless numbers is the effect of C where it is observed that as C increases (a greater degree of viscous dissipation), the value of the maximum temperature within the flow increases; interestingly, when the fluid is chemically-thickening, the maximum temperature occurs below the centerline, whereas when the fluid is chemically-thinning, this maximum temperature point shifts and moves closer to the top plate. It might be suggested that in Eq. (31), C and Re could be combined to give a new dimensionless variable. Although this is true, we prefer to keep Re as it is and examine the effect of C separately. This way, it would be easier to attribute the results observed to either Re appearing in the momentum and energy equations or the C appearing only in the energy equation. Furthermore, it needs to be mentioned that had we assumed more general constitutive relations for w and q, such as w ¼ K1 rc K2 rh or q ¼ Krh K3 rc, then Eqs (22) and (23) would have been more non-linear and also coupled. As it stands, due to the simple forms of constitutive relations for w (chemical reaction flux vector) given by Eq (14), Eq (31) is not coupled with Eq (32) and the term u (a dimensionless number related to the chemically-reacting nature of the fluid, related to the diffusion coefficient K1) simply drops off the equations. Acknowledgements Kerem Uguz acknowledges financial support from Bogazici University Research Fund Project 6007. References [1] R.C. Batra, Elements of Continuum Mechanics, American Institute of Aeronautics and Astronautics (AIAA) Inc., Reston, VA, 2006. [2] C. Bridges, K.R. Rajagopal, Pulsatile flow of a chemically-reacting nonlinear fluid, Comput. Math. Appl. 52 (2006) 1131–1144. [3] C. Bridges, S. Karra, K.R. Rajagopal, On modeling of the synovial fluids: unsteady flow of a shear-thinning, chemically-reacting fluid mixture, Comput. Math. Appl. 60 (2010) 2333–2349. [4] P.J. Carreau, D. De Kee, R.J. Chhabra, Rheology of Polymeric Systems, Hanser/Gardner Publications, Cincinnati, OH, 1997. [5] J.E. Dunn, R.L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Arch. Rat. Mech. Anal. 56 (1974) 191–252. [6] G. Gupta, M. Massoudi, Flow of a generalized second grade fluid between heated plates, Acta Mech. 99 (1993) 21–33. [7] P. Haupt, Continuum Mechanics and Theory of Materials, second ed., Springer, Berlin, 2002. [8] J. Hron, J. Malek, P. Pustejovska, K.R. Rajagopal, On the modeling of the synovial fluid, Advances in Tribology, vol. 2010, article ID 104957, pp. 12. [9] J.D. Humphrey, K.R. Rajagopal, A constrained mixture model for growth and remodeling of soft tissues, Math. Models Methods Appl. Sci. 12 (2002) 407–430. [10] R.J. Kee, M.E. Coltrin, P. Glarborg, Chemically Reacting Flow: Theory and Practice, John Wiley and Sons, 2003. [11] W.M. Lai, S.C. Kuei, V.C. Mow, Rheological equations for synovial fluids, ASME J. Biomech. Eng. 100 (1978) 169–186. [12] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, USA, 1999. [13] I.S. Liu, Continuum Mechanics, Springer-Verlag, Berlin, 2002. [14] M. Massoudi, On the heat flux vector for flowing granular materials, part I: effective thermal conductivity and background, Math. Methods Appl. Sci. 29 (2006) 1585–1598. [15] M. Massoudi, On the heat flux vector for flowing granular materials, part II: derivation and special cases, Math. Methods Appl. Sci. 29 (2006) 1599– 1613. [16] M. Massoudi, A note on the meaning of mixture viscosity using the classical continuum theories of mixtures, Int. J. Eng. Sci. 46 (2008) 677–689. [17] M. Massoudi, A mixture theory formulation for hydraulic or pneumatic transport of solid particles, Int. J. Eng. Sci. 48 (2010) 1440–1461. [18] M. Massoudi, I. Christie, Effects of variable viscosity and viscous dissipation on the flow of a third grade fluid, Int. J. Nonlinear Mech. 30 (1995) 687– 699. [19] M. Massoudi, T.X. Phuoc, Flow of a non-linear (density-gradient-dependent) viscous fluid with heat generation viscous dissipation and radiation, Mathematical Methods in the Applied Sciences 31 (2008) 1685–1703. [20] S. Middleman, Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977. [21] I. Müller, On the entropy inequality, Arch. Rat. Mech. Anal. 26 (1967) 118–141. [22] J.G. Oldroyd, An approach to non-Newtonian fluid mechanics, J. Non-Newtonian Fluid Mech. 14 (1984) 9–46. [23] K.R. Rajagopal, On implicit constitutive theories for fluids, J. Fluid Mech. 550 (2006) 243–249. [24] K.R. Rajagopal, G. Saccomandi, L. Vergori, On the Oberbeck–Boussinesq approximation for fluids with pressure dependent viscosities, Nonlinear Anal. Real World Appl. 10 (2009) 1139–1150. [25] K.R. Rajagopal, L. Tao, Mechanics of Mixtures, World Scientific Publishing, New Jersey, 1995. [26] R.S. Rivlin, J.L. Ericksen, Stress-deformation relations for isotropic materials, J. Ration. Mech. Anal. 4 (3) (1955) 323–425. [27] J.F. Rodrigues, J.M. Urbano, On the stationary Boussinesq–Stefan problem with constitutive power laws, Int. J. Non-Linear Mech. 33 (1998) 555–566. [28] D.R. Rosner, Transport Processes in Chemically Reacting Flow Systems, Butterworths, Boston, 1986. [29] E.R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, New York, 1978. [30] J.C. Slattery, Advanced Transport Phenomena, Cambridge University Press, 1999. [31] C. Truesdell, W. Noll, The Non-Linear Field Theories of Mechanics, Springer-Verlag, New York, 1992. [32] A.K. Uguz, M. Massoudi, Heat transfer and Couette flow of a chemically reacting non-linear fluid, Math. Methods Appl. Sci. 33 (2010) 1331–1341. [33] H. Ziegler, An Introduction to Thermomechanics, North-Holland Publishing Company, Amsterdam, 1983. Second revised edition.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Chemically-reacting fluids with variable transport properties Mehrdad Massoudi a,⇑, A.K. Uguz b a b
U.S. Department of Energy, National Energy Technology Laboratory (NETL), P.O. Box 10940, 626 Cochrans Mill Road, Pittsburgh, PA 15236, USA _ Bog˘aziçi University, Department of Chemical Engineering, Bebek, 34342 Istanbul, Turkey
a r t i c l e
i n f o
Keywords: Non-linear fluids Chemically-reacting fluids Variable viscosity Variable thermal conductivity Variable diffusion coefficient Viscous dissipation
a b s t r a c t In this paper we study the flow and heat transfer in a chemically-reacting fluid, whose transport properties, i.e., viscosity, thermal conductivity and the diffusion coefficients, are assumed to be a function of the concentration. The equations for the shearing flow of such a fluid between two long horizontal parallel plates, which are at different temperatures, are made dimensionless and the boundary value problem is solved numerically; the velocity, concentration and temperature profiles are obtained for various dimensionless numbers. Published by Elsevier Inc.
1. Introduction Studying mechanics of complex fluids, from the early days of polymer rheology to the present day nano-fluids, present a special challenge to both theoreticians and experimentalists. The main difficulty is the inherent non-linearities in the constitutive relations, thus making the numerical (or the analytical) study of the governing equations more difficult, while creating a demand for accurate measurements of the constitutive parameters. A sub-class of the complex fluids is the chemically-reacting fluids which are used in many industries and technologies such as combustion, catalysis [28], biological systems, etc. For example, coal is often used as the primary fuel in high temperature and high pressure environments such as the open-cycle MHD generators, where temperatures and pressures are of the order of 3000 K and 10 atm, respectively. Under these extreme conditions, the rheological and thermal properties of coal change and as a result the transport properties are not constant. This is not unique to coal combustion and in fact in many chemical processes viscosity or thermal conductivity of the materials under consideration are not constant and in many cases it has been shown that substantial performance benefits can be obtained by pre-heating a coal-slurry; this can affect its viscosity which in turn can affect the flow rate and pressure drop, parameters of importance in solids transport (see Gupta and Massoudi [6], and Massoudi and Christie [18]). The effects of temperature change on coal-slurry properties are controlled to a great extent by changes in the properties of the fluid. The constitutive equation used, most often, in these studies is the power-law model. Although the powerlaw model adequately fits the shear stress and shear rate measurements for many non-Newtonian fluids, it cannot always be used to accurately predict the pressure loss data measured during transport of a coal-liquid mixture in a fuel delivery system. Constitutive modeling of non-linear fluids has received much attention in the past few decades (see for example the books by Carreau et al. [4], Middleman [20], Schowalter [29], and Larson [12]), mainly due to the fact that many of the fluids used in bio-chemical industries behave as non-linear fluids, also known as non-Newtonian fluids, for example, polymer melts, suspensions, blood, coal-slurries, etc. A sub-class of these non-linear fluids which recently has been studied includes
⇑ Corresponding author. E-mail address: [email protected] (M. Massoudi). 0096-3003/$ - see front matter Published by Elsevier Inc. http://dx.doi.org/10.1016/j.amc.2012.08.015
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Nomenclature Symbol b c D f g h l K1 K L n q t T uw w x
q l h div
r
Explanation body force vector concentration symmetric part of the velocity gradient constitutive function acceleration due to gravity characteristic length identity tensor material parameter related to diffusivity thermal conductivity gradient of the velocity vector power-law index heat flux vector time Cauchy stress tensor reference velocity (chemical reaction) flux vector spatial position occupied at time t bulk density viscosity temperature divergence operator gradient symbol
chemically-reacting fluids which offer technological applications ranging from the formation of thin films for electronics (see Kee et al. [10]) to biological fluids, especially the synovial fluid (see Lai et al. [11], Bridges et al. [3], and Hron et al. [8]). Most complex fluids are multi-component mixtures. In many applications, these fluids are treated as a single continuum suspension with non-linear material properties and the techniques and models used in rheology or mechanics of non-linear fluids can generally be used to study such problems (see Oldroyd [22], Larson [12]). In this case, global or macroscopic information about the variables such as the velocity or temperature fields of the whole suspension can be obtained. In many other applications, however, there is a need to know the details of the field variables such as velocity, concentration, temperature, etc., of each component and in such cases one needs to resort to the multi-component modeling approaches (see Rajagopal and Tao [25], Massoudi [16], and Massoudi [17]). Examples of complex fluids whereby both approaches can be used are coalslurries, many of the biological fluids such as blood and the synovial fluid, and chemically-reacting fluids. Bridges and Rajagopal [2] studied the pulsatile flow of a non-linear chemically-reacting fluid. In their model, they assumed that the viscosity not only depends on the concentration of the species (or constituent whose behavior is governed by a convection-reaction– diffusion equation), but also on the velocity gradient, in a properly frame-invariant form; the fluid can be either shearthinning or shear-thickening. This approach, even though used for a single continuum complex fluid, is really based on a constrained mixture theory approach (see Humphrey and Rajagopal [9]) which implies that at each point in the complex fluid, i.e., the mixture, the host fluid coexists with the constituent, flowing in such a way that the two components are constrained to move together. In such an approach, one does not need to study the complicated coupled equations for the multi-component mixtures. Uguz and Massoudi [32] studied the heat transfer to a chemically-reacting fluid whose constitutive relation for the stress tensor was given by the model proposed by Bridges and Rajagopal [2]. Later, Hron et al. [8] developed generalized power-law models for the synovial fluid where now the exponent of this power-law model depends in various ways on the concentration of the hyaluronan [also referred to as constituent or species in the earlier studies of Bridges and Rajagopal [2] and Uguz and Massoudi [32]. In another study, Bridges et al. [3] used the three models suggested by Hron et al. [8] to study the unsteady flow of the synovial fluid between two cylinders; they also assumed the fluid to be a shear-thinning and chemically-reacting fluid. In this paper we look at chemically-reacting fluids with variable transport properties. The reaction is approximated by coupling the fluid dynamics with the concentration of the second constituent in the fluid through the change in the dynamic viscosity, thermal conductivity and the diffusion coefficients. It should be emphasized that our proposed model would reduce to the one suggested by Uguz and Massoudi [32], if we ignore the dependence of thermal conductivity and diffusion coefficients on the concentration. In Sections 2 and 3 we present the governing equations and the constitutive relations, respectively. In Section 4, we look at the flow between two parallel flat plates which are at different temperatures and present the governing equations and the boundary conditions along with our assumptions. It is followed by Section 5 where the results are analyzed. Finally, Section 6 presents our conclusions.
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2. Governing equations In the absence of any electro-magnetic effects, the governing equations of motion are the conservation of mass, linear momentum, convection–diffusion, and energy equations. These are (see Slattery [30]): Conservation of mass:
@q þ divðquÞ ¼ 0 @t
ð1Þ
where q is the density of the fluid, o/ot is the partial derivative with respect to time, and u is the velocity vector. For an isochoric motion we have div u = 0. Conservation of linear momentum:
q
du ¼ div T þ qb dt
ð2Þ
where b is the body force vector, T is the Cauchy stress tensor, and d/dt is the total time derivative, given by dðÞ ¼ @ðÞ þ ½gradðÞu. The balance of moment of momentum reveals that in the absence of couple stresses, the stress tensor dt @t is symmetric. Conservation of Concentration:
@c þ div ðcuÞ ¼ f @t
ð3Þ
where c is the concentration and f is a constitutive parameter. This equation is also known as the convection-reaction–diffusion equation. Conservation of energy:
q
de ¼ T L div q þ qr dt
ð4Þ
where e is the specific internal energy, L is the gradient of velocity, q is the heat flux vector, and r is the radiant heating.1 Thermodynamical considerations require the application of the second law of thermodynamics or the entropy inequality. The local form of the entropy inequality is given by (see Liu [13]):
qg_ þ div u qs 0
ð5Þ
where g(x, t) is the specific entropy density, /(x, t) is the entropy flux, and s is the entropy supply density due to external sources, and the dot denotes the material time derivative. If it is assumed that / ¼ 1h q, and s ¼ 1h r, where h is the absolute temperature, then Eq. (5) reduces to the Clausius–Duhem inequality
q h
r h
qg_ þ div q 0
ð6Þ
Even though we do not consider the effects of the Clausius–Duhem inequality in our problem, for a complete thermomechanical study of a problem, the Second Law of Thermodynamics has to be considered (see Batra [1], Haupt [7], Müller [21], Ziegler [33], Truesdell and Noll [31], and Liu [13]). In order to ‘close’ these equations, we need to provide constitutive relations for T, q, f, e and r. In this problem, we assume that the radiation effects can be neglected. The specific internal energy, is related to the specific Helmholtz free energy [5] e ¼ W þ hg ¼ eðh; A1 Þ ¼ ^eðyÞ, where g is the specific entropy. The internal energy may in some ways depend on other parameters. Nevertheless, in the current problem, due to the nature of the kinematical assumptions for the Couette flow, e does not need to be modeled. 3. Constitutive relations We will now briefly discuss the three constitutive relations needed for the closure in this problem, namely equations for T, q, and f. (i) Stress tensor One of the main reasons for the success of the Navier–Stokes constitutive relation, in addition to its elegance and compactness, is that the stress tensor T is explicitly described in terms of D (the symmetric part of the velocity gradient). This means that once the solution to the problem, whether using numerical techniques or analytical methods represented in terms of velocity and pressure fields, is obtained, the stress at any point can be determined uniquely from the constitutive 1 Constitutive relations for complex materials can be obtained in different ways using: (a) continuum mechanics, (b) physical and experimental models, (c) numerical simulations, (d) statistical mechanics approaches, and (e) ad hoc approaches. A look at the governing Eqs. (1)–(4) reveals that constitutive relations are required for T, q, f, e, and r. Less obvious is the fact that in many practical problems involving competing effects such as temperature and concentration, the body force b, which in problems dealing with natural convection oftentimes depends on the temperature and is modeled using the Boussinesq assumption (see Rajagopal et al. [24], now might have to be modeled in such a way that it is also a function of concentration). Furthermore, in many problems involving chemical reactions, there is usually a (heat) source term, Q, in Eq. (4) which also has to be constitutively modeled (see Massoudi and Phuoc [19]).
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relation. Furthermore, from a computational point of view, it is much easier and less cumbersome to solve the equations for the explicit models.2 Perhaps the simplest model which can capture the normal stress effects is the second grade fluid, or the Rivlin–Ericksen fluid of grade two [26,31]. This model has been used and studied extensively [5] and is a special case of fluids of differential type. For a second grade fluid the Cauchy stress tensor is given by:
T ¼ p1 þ lA1 þ a1 A2 þ a2 A21
ð7Þ
where p is the indeterminate part of the stress due to the constraint of incompressibility, l is the coefficient of viscosity, a1 and a2 are material moduli which are commonly referred to as the normal stress coefficients. The kinematical tensors A1 and A2 are defined through
A1 ¼ L þ LT A2 ¼
dA1 þ A1 L þ ðLÞT A1 dt
L ¼ grad u
ð8aÞ ð8bÞ ð8cÞ
The thermodynamics and stability of fluids of second grade have been studied in detail by Dunn and Fosdick [15].3 For a chemically-reacting fluid,4 Bridges and Rajagopal [2] suggested a constitutive relation of the type:
T ¼ p1 þ lðc; A1 ÞA1
ð9Þ
where p is the constraint due to incompressibility and
c¼
qr qr þ q
ð10Þ
where q is the density of the fluid and qr denotes the density of the (coexisting) reacting fluid. Furthermore they assumed
lðc; A1 Þ ¼ l ðcÞ½1 þ atrðA21 Þn
ð11Þ
where n determines whether the fluid is shear-thinning (n < 0), or shear-thickening (n > 0). The second law of thermodynamics requires the constant a P 0 [2]. It is seen that the viscosity is assumed to depend on the concentration c whereby depending on the form of l⁄(c) the fluid can be either a chemically-thinning or chemically-thickening fluid, implying a decrease or an increase in the viscosity, respectively, as c increases. In this paper, we will generalize the results of our recently published work (see Uguz and Massoudi [32]) by assuming that in general all the transport coefficients can be expressed as:
n ðcÞ ¼ ^ng j ðcÞ with j ¼ 1; 2
ð12aÞ
g 1 ðcÞ ¼ ðx1 c2 þ x2 Þ
ð12bÞ
g 2 ðcÞ ¼ ðx3 x1 c2 Þ
ð12cÞ
where n stands for viscosity l, or diffusion j, or thermal conductivity K, and x’s are positive constants. The form given by (12b) implies chemically-thickening and (12c) chemically-thinning. In Uguz and Massoudi [32] only the viscosity was assumed to be a function of c. (ii) Concentration flux In their study of the pulsatile flow of a chemically-reacting fluid, Bridges and Rajagopal [2] suggested the following constitutive relation for f:
f ¼ div w
ð13Þ
2 There are, however, cases such as Oldroyd type fluids (see Oldroyd [22]) and other rate-dependent models whereby it is not possible to express T explicitly in terms of D and other kinematical variables. For such cases, one must resort to implicit theories, for example, of the type f(T, D, h) = 0 where h is the temperature (see Rajagopal [23]). 3 They show that if the fluid is to be thermodynamically consistent in the sense that all motions of the fluid meet the Clausius-Duhem inequality and that the l0 It is known that for many non-Newtonian fluids which are specific Helmholtz free energy of the fluid be a minimum in equilibrium, then a1 0; a1 þ a 2 ¼ 0 assumed to obey Eq. (7), the experimental values reported for a1 and a2 do not satisfy the above restriction. 4 The terms ‘chemically reactive’ and ‘chemically reacting’ as used here are not necessarily the same and they might mean different things to people in different research communities. The former is most often used in the combustion and chemical engineering traditions where the reactions are accompanied by heat transfer/release. The latter term is commonly used in fields such as Biorheology where there are many chemical reactions occurring in blood, for example, but there is no substantial heat release/generation (see Bridges and Rajagopal [2]). Thus the terminology that we have used is in line with this interpretation of the term, ‘chemically-reacting fluids.’
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where w is a flux vector, related to the chemical reactions occurring in the fluid and was assumed to be given by a constitutive relation similar to the Fick’s assumption, namely
w ¼ K 1 rc
ð14Þ
where K1 is a material parameter which in general is not constant. Bridges and Rajagopal [2] assumed that K1 is a scalar-valued function of (the first Rivlin-Ericksen tensor) A1
K 1 ¼ K 1 ðA1 Þ ¼ jA21
ð15Þ
Fig. 1. Physical configuration of the Couette flow between two parallel plates separated by a distance h. The upper wall is set into motion and held at constant speed Uw whereas the bottom plate is stationary. The concentration, c, and the temperature, h, are kept constant at the walls. The unit vectors in the x and y directions are i and j, respectively.
Fig. 2a and 2b. The effect of Re on velocity u, when d = 0.1, C = 1, c = 10, and n = 0.5, for a chemically-thickening case given by g1(c) and a chemicallythinning case given byg2(c).
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Fig. 2c and 2d. The effect of Re on concentration c, when d = 0.1, C = 1, c = 10, and n = 0.5, for a chemically-thickening case given by g1(c) and a chemicallythinning case given by g2(c).
where j was assumed to be constant and kk denotes the trace-norm. This form was also used by Uguz and Massoudi [32]. In this problem we generalize the form of j by assuming
jðcÞ ¼ j g j ðcÞ with j ¼ 1; 2 g 1 ðcÞ
ð16Þ
g 2 ðcÞ
where and are given by Eq. (12b and c) above. (iii) Heat flux vector The most popular and widely used form of the constitutive relation for the heat flux vector is given by the Fourier’s assumption where
q ¼ Krh
ð17Þ
where K is the thermal conductivity. In general, K, for anisotropic materials is a fourth order tensor, which can also depend on concentration, temperature, etc. There have been many experimental and theoretical studies related to this issue (see Massoudi [14], and Massoudi [15]). In fact, the general forms of the constitutive relations for the q and w could also depend on temperature and concentration gradients and include terms often referred to as the Dufour and the Soret effects. In this problem, we assume that K is also given a similar structure to the viscosity and diffusion coefficients, that is
KðcÞ ¼ K g j ðcÞ with j ¼ 1; 2
ð18Þ
where g 1 ðcÞ and g 2 ðcÞ are given by Eq. (12b and c) above. If we assume that q can depend explicitly on the temperature gradient, concentration gradient, etc., then the problem would become highly non-linear. An example of this type of constitutive relation is a power-law type model suggested by Rodrigues and Urbano [27], where
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Fig. 2e and 2f. The effect of Re on temperature h, when d = 0.1, C = 1, c = 10, and n = 0.5, for a chemically-thickening case given by g1(c) and a chemicallythinning case given by g2(c).
q ¼ Krh ¼ jrhjp2 rh where 1 < p < 1
ð19Þ
When p = 2, the above equation reduces to the usual Fourier’s law. Eq. (17) is by no means the most general one, especially for cases involving density variation (see Massoudi [14], and Massoudi [15] for a review of this). In the next section, we will look at the heat transfer and flow of the non-linear fluid model described in this section. 4. Flow due to shear Whenever non-linear constitutive relations are studied, the solution procedures, for solving the governing equations, whether analytical or computational, are more complicated. Exact solutions are very rare in heat transfer studies of non-linear materials or multiphase flows. Next to the exact solutions, finding solutions to simple boundary value problems are extremely desirable. In this section, we will study the flow and heat transfer to a non-linear chemically-reacting fluid between two long parallel plates where the lower plate is fixed and heated and the upper plate is set into motion and is at a lower temperature than the lower plate (see Fig. 1). For the problem under consideration, we make the following assumptions: (i) the motion is steady, (ii) the radiant heating r is ignored, (iii) the constitutive equation for the stress tensor is given by Eq. (9), the constitutive relation for the diffusion term is given by Eq. (14), and the constitutive equation for the heat flux vector is that of Fourier’s law, given by Eq. (17);
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(iv) the concentration, velocity and temperature fields are of the form
8 9 > < c ¼ cðyÞ > = u ¼ uðyÞi > > : ; h ¼ hðyÞ
ð20Þ
with (20) the conservation of mass is automatically satisfied, and by substituting Eqs. (9)–(11) into Eq. (2), the equations of linear momentum reduce to
@p d 0¼ þ @x dy
"
l
2 !n # du du ðcÞ 1 þ 2a dy dy
@p @p ¼ ¼0 @y @z
ð21Þ
Substituting (13)–(16) into (3), the diffusion-convection equation reduces to
" 2 # d du dc ¼0 jðcÞ dy dy dy
ð22Þ
And substituting (9) and (17) into (4), and assuming r = 0, the energy equation becomes:
Fig. 3a and 3b. The effect of n on velocity u, when d = 0.1, C = 1, c = 10, and Re = 1, for a chemically-thickening case given by g1(c) and a chemically-thinning case given by g2(c).
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Fig. 3c and 3d. The effect of n on concentration c, when d = 0.1, C = 1, c = 10, and Re = 1, for a chemically-thickening case given by g1(c) and a chemicallythinning case given by g2(c).
l
2 !n 2 du du d dh ¼0 K 2 ðcÞ ðcÞ 1 þ 2a þ dy dy dy dy
ð23Þ
where due to the kinematical assumptions made for the velocity and temperature fields, the term corresponding to the derivative of disappears from the energy equation. That is ddte ¼ 0 and a simplified form of the energy equation is obtained. The above equations are subject to the boundary conditions:
at y ¼ 0 :
8 >
ð24Þ
at y ¼ h :
8 > < u ¼ uw h ¼ hh > : c ¼ ch
ð25Þ
Clearly other types of boundary conditions related to the gradients of velocity, temperature, and concentration are also possible. For the present problem, we have assumed the no-slip condition for the velocity, and we have also assumed that the values of the concentration and temperature at the two plates are known, i.e., are prescribed. Eqs. (20)–(23) along with the boundary conditions (24) and (25) are to be made dimensionless and solved numerically. Let us define the dimensionless , the velocity u , the concentration c and the temperature distance y h by the following equations
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Fig. 3e and 3f. The effect of n on temperature h, when d = 0.1, C = 1, c = 10, and Re = 1, for a chemically-thickening fluid given by g1(c) and a chemicallythinning case given by g2(c).
y ¼ ; y h
¼ u
u ; uw
h ¼ h h0 ; Dh
c ¼
c c0 Dc
ð26Þ
where uw is a shearing velocity at the top plate, Dh = hhh, Dc = chc0, and h is the distance between the two plates. The dependence of the physical parameters on the concentration, i.e., Eqs (12a), (16), and (18) are now expressed as
l ðcÞ ¼ l^ g j ðcÞ
ð27Þ
jðcÞ ¼ j^ g j ðcÞ
ð28Þ
^ ðcÞ KðcÞ ¼ Kg j
ð29Þ
respectively. With these, the dimensionless forms of the governing equations become:
" 2 !n # 1 d du du ¼ c g j ðcÞ 1 þ d Re dy dy dy
/
( 2 ) d du dc g j ðcÞ ¼0 dy dy dy
ð30Þ
ð31Þ
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Fig. 4a and 4b. The effects of d on u, for C = 1, c = 10, Re = 1, and n = 0.5, for chemically-thickening case g1(c) and a chemically-thinning case given by g2(c).
2 !n 2 du du d dh ¼0 g j ðcÞ þ g j ðcÞ 1 þ d dy dy dy dy Re
C
ð32Þ
where c is related to the pressure gradient term, and the bars have been dropped for simplicity. The dimensionless numbers that arise are
Re ¼
qhuw 2au2 j^ Dcu2w qu3 h ; d ¼ 2w ; / ¼ ; C¼ w 4 ^ Dh l^ K h h
ð33Þ
where Re is the Reynolds number, d is a dimensionless number signifying the dependence of the shear viscosity on the shear rate (with a = 0 implying no dependence of l on the shear rate), / is a dimensionless number related to the chemically-reacting nature of the fluid (related to the diffusion coefficient K1), and C is a measure of viscous dissipation (see Massoudi and Christie [18]). The dimensionless forms of the boundary conditions become:
at y ¼ 0 :
8 > : c¼0
ð34Þ
at y ¼ 1 :
8 > : c¼1
ð35Þ
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Fig. 4c and 4d. The effects of d on c, for C = 1, c = 10, Re = 1, and n = 0.5, for chemically-thickening case g1(c) and a chemically-thinning case given by g2(c).
In the next section we solve numerically Eqs. (30)–(32) subject to boundary conditions (34), and (35). This will be a parametric study where different values for the dimensionless numbers listed in Eq (33) except / as it does not affect the fields, along with different values for n will be used. 5. Numerical solutions Recall that the form of the dependence of the physical parameters on the concentration is represented by Eq. (12). We use the following simplified expressions:
g 1 ðcÞ ¼ 100ð4c2 þ 1Þ
ð36Þ
g 2 ðcÞ ¼ 100ð5 4c2 Þ
ð37Þ
where g1(c) represents the chemically-thickening contribution, and g2(c) is for chemically-thinning fluids. These expressions were suggested by Bridges and Rajagopal [2]. Eq. (30)–(32) are solved using the Spectral Method with Chebyshev grid points which are defined as yðjÞ ¼ cos ðjp=NÞ where j = 0, 1, . . . , N, where N denotes the cut-off frequency. These non-linear governing equations are recast into
2 0 ( ðpÞ )2 1n ðpþ1Þ 3 1 d 4 du ðpÞ @ 5 ¼ c A du g ðc Þ 1 þ d Re dy j dy dy
ð38Þ
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Fig. 4e and 4f. The effects of d on h, for C = 1, c = 10, Re = 1, and n = 0.5, for chemically-thickening case g1(c) and a chemically-thinning case given by g2(c).
8 ! !9 ðpÞ 2 ðpþ1Þ = d < du dc ðpÞ ¼0 g ðc Þ / ; dy : j dy dy ( ) 1n ! ! ! ðpÞ 2 ðpÞ ðpþ1Þ ðpþ1Þ du du du d p dh @ A þ ¼0 g ðc Þ g ðc Þ 1 þ d dy j Re j dy dy dy dy
C
ð39Þ
0
ðpÞ
ð40Þ
Here, the superscript (p) denotes the guessed quantity and (p + 1) is the computed one. This computation step is preferred as linearization around a known state was not possible. The initial guess is taken as unity for both the velocity and the concentration. Then, the convergence is checked with the norm of the difference of the present value and the previous value. The code is also ensured to independent of the initial guess. In our previous paper [32] we only considered the case of a chemically-thickening fluid. In the current study, we look at both cases, i.e., we will also look at a chemically-thinning fluid. It should be noted that in our previous study [32] the value for the pressure drop term c was assumed to be 10 (but it was mistakenly written as +10). In the current study, for the effects of Re, we have corrected this minor numerical issue; this changed the values slightly but did not affect the qualitative response as observed in the figures. In the present study, the effect of the different parameters, namely, Re, n, d, and C on the velocity, concentration, and temperature fields are investigated. In our parametric study, we first look at the effects of varying Re, for the case of d = 0.1, C = 1, c = 10, and n = 0.5, for two different types of chemically-reacting fluids, namely a chemically-thickening case given by g1(c) and a chemically-thinning case given by g2(c). The results for the velocity, concentration and temperature are given in Fig. 2a–f. The interesting observation is the reversal of the curves. There does not seem to be a great deal of influence on the velocity profiles for both cases of g1(c) and g2(c) (as seen in Fig. 2a and 2b). However, the impact
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Fig. 5a and 5b. The effect of C on h, when d = 0.1, c = 10, Re = 1, and n = 0.5, for a chemically-thickening fluid g1(c) and a chemically-thinning g2(c).
of these two parameters on the concentration and temperature profiles is more pronounced for the case of chemically-thickening fluid, as seen in Fig. 2c and 2d. It is also noticed that as Re increases, at any given point Y in the flow, the value of c decreases while the value of h increases. Fig. 3a–f show the influence of exponent n on u, c, and h, for the selected parameters d = 0.1, C = 1, c = 10, and Re = 1. Again, for the case of chemically-thickening, g1(c), as shown in Fig. 3a, there is very little difference between the three cases of n, 0.0, 0.5 (shear-thinning fluid), and +0.5 (shear-thickening fluid). More noticeable differences are observed in all cases for the chemically-thinning case given by g2(c) . Fig. 4a–f show the influence of d (the dimensionless number signifying the dependence of the viscosity on the shear rate with a = 0 implying no dependence of l on the shear rate, where a P 0) on u, c, and h, for the selected parameters C = 1, c = 10, Re = 1, and n = 0.5, for two different types of chemically-reacting fluids, namely a chemically-thickening case given by g1(c) and a chemically-thinning case given by g2(c). Again, it is observed that the impact of the chemically-thinning fluid given by g2(c) is more vivid on the velocity, concentration and temperature profiles. Fig. 5a and b show the influence of C (a measure of viscous dissipation) on h, for the selected parameters d = 0.1, c = 10, Re = 1, and n = 0.5, for two different types of chemically-reacting fluids, namely a chemically-thickening case given by g1(c) and a chemically-thinning case given by g2(c). As we can see from Eqs (30)–(32), C appears only in the energy equation and therefore, it influences the temperature h . It is seen that not only C but also g1(c) and g2(c) have significant impact on the temperature distribution. Clearly as C increases (a greater degree of viscous dissipation), the value of the maximum temperature within the flow increases; interestingly, when the fluid is chemically-thickening, the maximum temperature occurs below the centerline, whereas when the fluid is chemically-thinning, this maximum temperature point shifts and moves closer to the top plate. Clearly, this should be further investigated by changing the boundary conditions and or geometry, for example solving flow in a pipe and seeing if the curvature has any effect on this observation.
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6. Conclusions We have studied the flow and heat transfer in a non-linear reactive fluid, with variable transport properties, between two parallel plates. The flow is assumed to be steady state and the velocity, concentration, and temperature fields are computed accordingly. In the present problem, we have performed a parametric study for a limited range of the dimensionless numbers corresponding to viscous effect (l), the non-linear nature of the fluid (d and the exponent n), and viscous dissipation (C). It is observed that in general, the impact of the chemically-thinning fluid given by g2(c) is more vivid on the velocity, concentration and temperature profiles. Perhaps the most noted impact of all the dimensionless numbers is the effect of C where it is observed that as C increases (a greater degree of viscous dissipation), the value of the maximum temperature within the flow increases; interestingly, when the fluid is chemically-thickening, the maximum temperature occurs below the centerline, whereas when the fluid is chemically-thinning, this maximum temperature point shifts and moves closer to the top plate. It might be suggested that in Eq. (31), C and Re could be combined to give a new dimensionless variable. Although this is true, we prefer to keep Re as it is and examine the effect of C separately. This way, it would be easier to attribute the results observed to either Re appearing in the momentum and energy equations or the C appearing only in the energy equation. Furthermore, it needs to be mentioned that had we assumed more general constitutive relations for w and q, such as w ¼ K1 rc K2 rh or q ¼ Krh K3 rc, then Eqs (22) and (23) would have been more non-linear and also coupled. As it stands, due to the simple forms of constitutive relations for w (chemical reaction flux vector) given by Eq (14), Eq (31) is not coupled with Eq (32) and the term u (a dimensionless number related to the chemically-reacting nature of the fluid, related to the diffusion coefficient K1) simply drops off the equations. Acknowledgements Kerem Uguz acknowledges financial support from Bogazici University Research Fund Project 6007. References [1] R.C. Batra, Elements of Continuum Mechanics, American Institute of Aeronautics and Astronautics (AIAA) Inc., Reston, VA, 2006. [2] C. Bridges, K.R. Rajagopal, Pulsatile flow of a chemically-reacting nonlinear fluid, Comput. Math. Appl. 52 (2006) 1131–1144. [3] C. Bridges, S. Karra, K.R. 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