A Shape-Factor Method for Modeling Parallel and Axially-Varying Flow in Tubes and Channels of Complex Cross-Section Shapes

In the study of some industrial, biological and natural fluidic systems it is often necessary to model fluid flow through tubes, channels or passages of complex geometries. The complexity may arise from the cross-sectional shape, or from longitudinal cross-section variation, or from both. Typical cases include flow of molten metals or plastics through dies and moulds, blood flow, microfluidic applications, and flow in porous media, among many others. Characteristics of these flows are laminar state, incompressibility, small rates of flow and varied time patterns. One field where pertinent applications are being developed at a fast rate is Microfluidics (Cetin and Li., 2008; Chen et al., 2008; Forte et al., 2008; Gebauer and Bocek, 2002 ; Mathies and Huang, 1992; Sommer et al., 2008; Srivastava et al., 2005; Woolley and Mathies, 1994; Yeger et al., 2006.) . In this specific field, present microchannel manufacturing techniques produce typically non-circular capillaries (Sommer et al., 2008). Also the introduction of electrical or magnetic field induce plastic behavior in the working fluid. In particular, it is well known that blood is a biological fluid that behaves as a Newtonian fluid in arteries, veins and large capillaries, but becomes non-Newtonian in the smaller vessels, where the size of suspended particles is big as compared to the vessel s diameter size (Pedley ,2008). A relevant problem in this field as to the method presented in the next sections is the analysis of diseased arteries and veins for quirurgical interventions. Specifically, stenosed arteries are blood conduits of irregular geometry in which crosssection geometry usually varies along the vessel length. The above context implies that it is desirable, particularly for modeling and design purposes, to count with analytical techniques that can integrate variables such as the noncircular cross-section of conduits, axial variation of conduit geometry, and plastic flow in some cases. In this chapter it is presented a method of analysis that allows to address in a general way the problem here outlined. The standard analytical technique for tube flow problems is usually the search of specific solutions to the momentum equations with associated boundary and initial conditions (Batchelor, 2000). Otherwise numerical solutions are developed for some purposes (Xue et al.,1995).


Introduction
In the study of some industrial, biological and natural fluidic systems it is often necessary to model fluid flow through tubes, channels or passages of complex geometries.The complexity may arise from the cross-sectional shape, or from longitudinal cross-section variation, or from both.Typical cases include flow of molten metals or plastics through dies and moulds, blood flow, microfluidic applications, and flow in porous media, among many others.Characteristics of these flows are laminar state, incompressibility, small rates of flow and varied time patterns.One field where pertinent applications are being developed at a fast rate is Microfluidics (Cetin and Li., 2008;Chen et al., 2008;Forte et al., 2008;Gebauer and Bocek, 2002 ;Mathies and Huang, 1992;Sommer et al., 2008;Srivastava et al., 2005;Woolley and Mathies, 1994;Yeger et al., 2006.) .In this specific field, present microchannel manufacturing techniques produce typically non-circular capillaries (Sommer et al., 2008).Also the introduction of electrical or magnetic field induce plastic behavior in the working fluid.
In particular, it is well known that blood is a biological fluid that behaves as a Newtonian fluid in arteries, veins and large capillaries, but becomes non-Newtonian in the smaller vessels, where the size of suspended particles is big as compared to the vessel´s diameter size (Pedley ,2008).A relevant problem in this field as to the method presented in the next sections is the analysis of diseased arteries and veins for quirurgical interventions.Specifically, stenosed arteries are blood conduits of irregular geometry in which crosssection geometry usually varies along the vessel length.The above context implies that it is desirable, particularly for modeling and design purposes, to count with analytical techniques that can integrate variables such as the noncircular cross-section of conduits, axial variation of conduit geometry, and plastic flow in some cases.In this chapter it is presented a method of analysis that allows to address in a general way the problem here outlined.The standard analytical technique for tube flow problems is usually the search of specific solutions to the momentum equations with associated boundary and initial conditions (Batchelor, 2000).Otherwise numerical solutions are developed for some purposes (Xue et al.,1995).
The main aim of this chapter is, thus, to introduce and explore the potential use of a general analytical approach to irregular conduit flow, which makes it possible to determine velocity field, rate of flow, shear stress, recirculation regions and plug zones, this last when fluid plasticity is operant.The method already referred to has been developed by the authors through specific applications mainly during the past decade.In this chapter some previous results are organized within a common analytical pattern, together with novel material.This chapter includes sections for the general model, considering one velocity component and more than one velocity component versions, applications related to flow in straight tubes and to axially-varying flows, and a closing conclusion section.

The general model
The concept of "shape factor" herein used is applied to a function of spatial coordinates, such as when , a series of closed curves are determined for a range of some parameters contained in .One typical example is In this , are polar coorfinates, is an integer number and is a parameter such as that for the curve described by ( 1) is a circle, and as increases, the shape evolves to some limiting shape, controlled by .In all cases here considered, the maximum allowable value of is less than unity, and beyond that value, the curve is no longer a closed one.If is the critical, or maximum, allowable value of , then for the shape factor described by (1), is found to be which leads to more complex shapes.Some instances of these shapes are shown in Fig. 1.
For the purposes of this presentation, a general shape factor in polar coordinates can be defined as in which , … are boundary perturbation functions.For the case of channel flow, the structure of (4) may be the same, in which polar coordinated may be substituted by Cartesian coordinates.The specific characteristics of functions are determined by the nature of the equations of motion and associated boundary conditions.Two relevant cases can be highlighted, namely, flow with one velocity component, and flow with more than one velocity component.

Flow with one velocity component
These are flows in straight tubes of constant cross-section.In these cases the axial velocity can be modeled as where, for the sake of simplicity, only one boundary perturbation function has been considered.Functions are to be determined from the equation of motion in terms of a standard regular perturbation scheme around the small parameter .

Flows with more than one velocity component
These are mainly flows with axial variation of tube or channel geometry.In these cases the solution procedure will usually involve the use of a stream function .In such problems both and the velocity components should be zero at the boundary, a condition that can be met by defining , , , , , ⋯ where again functions have to be determined from the equations of motion.The definition of is given for every specific application in the corresponding section.In the following some specific applications of this method of analysis are presented.
In this is the so-called unsteadiness number of the flow, which measures the relative importance of a temporal inertia force against a steady viscous force, and where density, reference tube radius, dynamic viscosity, and reference time.A convenient solution of (7) (Letelier et al., 1995) for round tubes (ie for / can be worked out by postulating Where for , , … ∞.Equation ( 10) meets the no-slip boundary condition , .After substituting (10) in ( 7) it is found that all functions can be expressed in terms of so that the axial velocity takes the form where is related to the forcing function as follows In these expressions can have any finite positive value.
According to (5), it is found The constants in these equations are obtained by putting the coefficients of all powers of , for any / , , … ,equal to zero in (13).The result is ( 17) and so on.Higher order terms in can be obtained in like fashion.An example of velocity profiles is shown in figure 1 for , .at two semi-axes (cf Fig. 1).In this case the tube contour is an approximate hexagon and , ie a purely oscillatory flow is described.The structure of (4) makes it possible to apply a regular perturbation method of solution around the dimensionless parameter .Since is bounded for a given value of , and is always less than unity, the solution becomes actually an exact one when enough terms are obtained.

Steady plastic flow
In this application it is considered steady flow of a Bingham plastic.Here ( 5) is also applicable, and the equation of motion, in terms of shear stress, as defined below, is and the following equations are found Equations (28-29) are the result of substituting ( 25) in ( 23) and of ordering terms in powers of through a linearization procedure.From (30-31) it is found Functions , , and following can be found equating terms in orders of in (5), ie (36) In this both functions are continuous for , and so can be built higher order functions.Isovel plots and plug zones for selected instances of flow are shown in figure 3. A plug zone is such that inside its limiting boundary the shear stress is less than the yield stress. which was transformed in a Fourier series in the range in terms of sine and cosine functions that allow a modeling similar, but more complex, to that already described.Examples of typical streamline and isovelocity patterns are shown in figure 7 and 8.

Conclusion
The method here described can lead to very accurate solutions for the velocity field and related variables such as shear stress, rate of flow and pressure in a great variety of flows in tubes and channels.Symbolic software presently available, such Maple and MathCAD make it possible to obtain and compute higher order solutions that, in some cases, may have complex algebraic structures.The fact that for all cases here considered, ie cases where , is much less than unity (cf table 2), leads to a regular perturbation scheme that in most cases requires terms up to second order to achieve enough accuracy.The cases when and deserve special mention.For the shape factor (1) describes an excentric circle, and for an ellipse.In this last instance is not bounded and can take any finite value, which implies that the perturbation scheme would break down if .So that, in this particular case, the method is limited to elliptical cross-sections of axes ratio close to unity.The method can be expanded to many more complex flow geometries.This possibility is implicit in the more general shape factor (3), which makes it necessary to develop a compound perturbation scheme, in terms of more than one perturbation parameter.The structure of the shape factor (1) determines that the analysis, especially for , is more sensible to the perturbation parameter for , ie close to the wall conduit.This requires a careful analysis of series convergency which should define the order of the higher order term considered.On the other hand, in the case of flow in straight tubes, in all cases studied, in a considerable region around the conduit axis, say for ., the flow variables are independent of the boundary geometry and take the values of the corresponding flow in round tubes.

Acknowledgment
The authors acknowledge the financial support provided, at different stages of the work here presented, by FONDECYT-CONICYT and DICYT at the University of Santiago of Chile.
unsteady flow For incompressible, developed and isothermal flow, the equation of motion are the standard Navier-Stokes and continuity equations.In dimensionless variables they are Ω (7) www.intechopen.comAShape-Factor Method for Modeling Parallel and Axially-Varying Flow in Tubes and Channels of Complex Cross-Section Shapes 473
invariant of the rate of deformation tensor.The dimensionless yield stress is (24)The momentum equation (20) is the standard one for parallel steady flow.Its structure has been made consistent with (21-22) and with the standard mathematical ordering of terms.The constitutive expressions (21-22) come from the applicable form of the Bingham fluid model
.intechopen.comA Shape-Factor Method for Modeling Parallel and Axially-Varying Flow in Tubes and Channels of Complex Cross-Section Shapes