ТЕОРЕТИЧЕСКИЕ ОСНОВЫ ХИМИЧЕСКОЙ ТЕХНОЛОГИИ, 2011, том 45, № 1, с. 54-67
ANALYTICAL INVESTIGATION OF VISCOELASTIC CREEPING FLOW AND HEAT TRANSFER INSIDE A CURVED RECTANGULAR DUCT © 2011 M. Norouzia, M. H. Kayhania, M. R. H. Nobarib, F. Talebic
aShahrood University of Technology, Shahrood, Iran bAmirkabir University of Technology, Tehran, Iran cSemnan University, Semnan, Iran
email@example.com Поступила в редакцию 16.09.2009 г.
An exact solution for creeping viscoelastic flow and heat transfer in a curved rectangular duct is presented based on the order of magnitude technique. Here, the Criminale—Eriksen—Filbey model is used as constitutive equation. The closed form of axial velocity, flow rate, flow resistance ratio, pressure distribution, stress field, temperature and Nusselt number are presented, and the effect of aspect ratio, curvature ratio and both of the first and the second normal stress differences on the flow and heat transfer are investigated. One of the noticeable results of current research is confirming the independency of flow resistance and Nusselt number of creeping flow inside the curved duct to curvature ratio at aspect ratio 0.89077.
Flow in a curved duct is one of the fundamental fields in fluid mechanic which has a wide range of applications in industries. This reason motivated many researchers to study this flow using analytical, experimental and numerical approaches. The most of previous studies are restricted to Newtonian fluid flow and few researches have been done about non-Newtonian fluids. It is important to know that there are many families of non-Newtonian fluids such as dilatant, pseudoplastic, Bingham, Hershel-Buckley, thixotropic, rheopectic and viscoelastic fluids which their flows in curved ducts have a lot of applications in chemical and petrochemical industries, injection of polymeric liquids, food and drug production, medical matter, biotechnology, etc. For example, due to the higher risk of apoplexy in curved streaks and capillaries, investigating the blood flow (as a non-Newtonian fluid) in curved channels with circular or elliptical cross section is excessively essential for medical aims. The biological flows are of increasing importance in microfluidics, where lithographic methods typically produce channels of square or rectangular cross-section. These channels are widely used in biologic kits (such as the kits for extraction of the DNA, detection of cancers cells and bacteria, blood sample preparation, glucose monitoring, etc.), fuel cells, cooling systems for small spaces, etc. The viscoelas-tic flow inside the closed channels is also applied for determining the rheological properties ofviscoelastic fluids such as viscosity and normal stress differences.
Using the perturbation method is common to finding the approximate analytical solution for Newtonian fluid flow inside the curved pipes that has been used at first times by Dean [1, 2]. In this method, the ratio of radius of pipe to radius of curvature is called curvature ratio which is used as a perturbation parameter. Therefore, this
solution is valid only for small curvature ratios which include loose curved ducts with the radius of curvature larger than seven times of the duct hydraulic diameter. Similar to Dean's work, many researchers used the perturbation method for solving the flow ofnon-Newtonian fluids inside the curved pipes [3—12]. A summary of these studies is presented in Table. The effect ofWeissenberg number, Reynolds number and curvature ratio on axial velocity distribution, shape and intensity of secondary flows and flow rate of non-Newtonian fluids is investigated in these researches. In spite of the advantages of the perturbation method to obtaining the approximating analytical solution, this method is restricted to flow inside the curved pipes with circular and annular cross section that is related to the too complicated form of perturbation series for other geometries. Also, the results of this method are only valid at low Dean numbers. Robertson and Muller  confirmed that the result of perturbation method for Newtonian fluid flow inside the curved pipe is not valid at Dean numbers larger than 30. Depending on the nonlinearity form of constitutive equation, the singularity situation is one of the main problems that resists to find the analytical solution with perturbation method. For example, Jitchote and Robertson  showed that by considering the second normal stress difference on constitutive equation, the singularity situation is caused and it is not possible to find a unique solution for this problem. Zhang et al.  introduced another series solution based on the Galerkin method to obtain the analytical results for high Dean number and Weissenberg number conditions.
Regarding to the constraints of perturbation method, some of researchers used the numerical methods to study this problem such as Phan-Thien and Zhang  and Fan et al.  about Oldroyd-B fluid and Helin et al. 
Summary list of researches which used the perturbation method
Researchers Geometry Constitutive Equation Major Result
Thomas and Walters  curved pipe Oldroyd-B Drag reduction of curved pipe in small elastic property
Sharma and Prakash  curved pipe SOF Increasing the secondary flows intensity by raising the first normal stress difference of fluid
Iemoto [5, 6] curved pipe with different kind of curvatures Power Law & Oldroyd-B Faster flow adaptation with curvature by increasing the elastic property
Bowen  curved pipe UCM & SOF Studying the effect of relaxation and retardation time of material on drag reduction/enhancement of creeping flow
Sarin [8, 9] curved circular  and elliptical  pipe with slowly varying curvature Oldroyd-B Investigating the effect of Deborah number on shape of secondary flows and axial velocity
Das  curved pipe Bingham Investigation of the influence of yield number on velocity components and frictional resistance
Robertson and Muller  curved circular and annular pipe Oldroyd-B Investigation of the effect ofWeissenberg number and viscosity ratio on flow rate
Jitchote and Robertson  curved pipe SOF Finding the solution by considering the effect of the second normal stress difference
and Boutabba et al.  about PTT fluid. Norouzi et al.  investigated the flow of second order fluid inside the curved rectangular duct via both numerical and analytical methods. They focused on the inverse effect of the normal stress differences on the flow field and obtained the forced balance relations in the core region of flow field. The heat transfer of viscoelastic flow has been also studied by Norouzi et al. . They showed that raising the first normal stress difference causes to increase the heat transfer of flow while the negative second normal stress difference has completely reverse effect.
Also, the effect of duct's rotation on the flow of Old-royd-B fluid in curved ducts has been investigated in researches of Chen et al  and Zhang et al. .
In this paper, creeping flow and heat transfer of viscoelastic fluid inside the curved rectangular duct is investigated analytically. The Criminale—Eriksen—Filbey (CEF) model is used as constitutive equation and both effects of the first and the second normal stress differences on flow field have been considered. Here, we try to complete the analytical solution of Norouzi et al.  about creeping flow via presenting more proofs and clarifications about this flow and completing the solution for pressure distribution, stress field and convective heat transfer. In creeping flow inside the curved duct, the amount of secondary flows velocity is smaller than the axial velocity. Regarding to this supposition, the order of magnitude technique is used in current research to find the exact solution for this problem. This method is used for the first time by Fan et al.  for studying the Old-royd-B fluid in curved circular pipes. They presented the force balance relation for core region which is useful for studying the numerical results but due to the complication form of governing equations in toroidal coordinate system, they did not reported any solution for creeping
flow inside the curved circular pipes. Here, the effect of aspect ratio, curvature ratio, Weissenberg number and normal stress differences on axial velocity, flow rate, flow resistance ratio, pressure distribution, stress components, temperature and Nusselt number is investigated in detail.
Geometry. In this paper, creeping flow ofCEF fluid in a curved rectangular duct is investigated analytically. Figure 1 shows the geometry ofduct in current research. Unlike the previous researches which used the curvilinear coordinate systems especially toroidal coordinate system, the orthogonal cylindrical coordinate system is used in current investigation. Here, it is assumed that the pitch curvature radius of duct (R) is constant. Regarding to this
figure, a and b are the lengths of duct cross section in radial (r) and lateral (z) direction respectively and 8 is the angle of curvature (direction of main flow).
Non-Dimensional Parameters. The following non-dimensional parameters are used in current investigation:
x¡ v x¡ _-l V i _ — dh dh к _ ä s b _ a
dh _ 2äb p _ Pdh ä + b n Wo т _ т dh П Wo T - T T _ m q "djk
Re _Wh Fr _v/a n Fe _ ReFr We _XWo dh
D _ v v 2 DD _ D |-dt) Wo vWo ) ,Wo ndh V 2Wo V 2 _ , ndh
Here, the superscript parameters.
introduces the dimensional
v-V = o pV ■v V = -v P + v ■ 5 V ■VT = aV2T
(2a) (2b) (2c)
In fully developed condition, except the pressure, derivatives of all of the parameters with respect to the main flow direction (0) are zero. Here, dp 50 is a negative constant. Generally, pressure drop of flow inside a curved duct is defined based on the pressure gradient in pitch curved direction [11, 18, 19]:
1 d-P = -G, R 50
Wo - Gd
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