1. Dean Flow, Separation, Branching, and Entrance Length
Juan M. Lopez
BIEN 501
Wednesday, April 4, 2007
Louisiana Tech University
Ruston, LA 71272

2. Dean Flow (Flow in a curved pipe)
*This is an adaptation of Dr. Jones Spring 1997 Physiological Fluid Mechanics lecture on Dean Flow.
Although we call Dean Flow “Flow in a curved pipe”, technically it is flow in a torus, as shown below:
The center axis of the torus is a distance R from the center, C, of the torus. The coordinates within the torus are the radial distance (r) from the axis (A). The angle (ψ) between r and the vertical line through A, and the distance ξ along the axis.
Figure 1 Dean Flow System Sketch, showing coordinate system
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3. Dean Flow – Governing Equations
The equations for this are determined from the expansion of the steady Navier-Stokes equations:
In this coordinate system, the Navier-Stokes Equations become
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4. Dean Flow-Governing Equations
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5. Dean Flow-Governing Equations
In conjunction with these, we must look at the continuity equation, , which becomes:
Notice that if the radius of curvature is extremely large, so that 1/R -> 0, then the equations reduce to those for cylindrical coordinates.
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6. Dean Flow-Boundary Conditions
The boundary conditions for this problem are:
Velocities must be finite for r->0
Velocity must be zero at the walls (no slip condition) i.e.
Notice that these boundary conditions are separable. That is, they can be expressed in the form u evaluated along a surface = This will impact the method used to convert the partial differential equation to a set of ordinary differential equations. For example, if the boundary condition on u were , we would have to seek solutions of a different form. On the other hand: would be a separable boundary condition and could be handled in a manner similar to that below.
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7. Conversion to a Linear Set of Partial Differential Equations
As usual, the first question to ask is, “ can we get rid of the nonlinearities on the left hand side of the Navier-Stoked equation?”
The technique Dean used was to assume that the solution will be a small perturbation on Poiseuille flow.
Certainly, if a/R is large enough, the torus looks locally like a straight pipe, and the solution must converge to Poiseuille flow in the limit of a/R ->0.
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8. Conversion to a Linear Set of Partial Differential Equations
Dean, therefore assumed:
He then assumed that are of order a/R, is of order 1, and that a/R << 1. From a practical standpoint, this means that any terms which are combinations of velocities, such
as , etc, will be of order and will be negligible. However, some of the convective accelerations will be maintained in the equation as a result of the Poiseuille flow term.
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9. Conversion to a Linear Set of Partial Differential Equations
For example, in the ξ-momentum equation the first term (multiplied by a) is:
The second (nonlinear) term is of order and will be neglected, but the first term is only of order a/R (because Aar is of the same order as and is thus maintained in the equation. However, the first term is linear and thus not as bothersome as the second.
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10. Conversion to a Linear Set of Partial Differential Equations
Applying this analysis to the r-momentum equation, it will be clearer if the equation is first multiplied by a. With this, and with the substitution of equations 4 and 5, the left hand side of the equation becomes:
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11. Conversion to a Linear Set of Partial Differential Equations
The first three terms are of order For the third term, is of order a/R, and expands to three terms, which is of order 1, , which is of order a/R, and , which is of order The only term of order a/R or greater is:
Similar analyses can be performed on the right side of the r-momentum equation and on the other two momentum equations. The result is:
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12. Conversion to a Linear Set of Partial Differential Equations
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13. Conversion to a Set of Linear Differential Equations
The next step is to obtain a set of ordinary differential equations. To do this, we must break each of the functions into components which are individually functions of r, ψ, and ξ. For example, we would like to have , where the superscripts are used to distinguish between the parts of , which depend only on the indicated variable. That is, depends on r only, depends on ψ only, and depends on ξ only.
First consider the ξ dependence of the dependent variables. Because the torus has circular symmetry about C, there is nothing to distinguish a cross section at one value of ξ from a cross-section at any other. Thus, the velocities cannot depend on ξ.
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14. Conversion to a Set of Linear Differential Equations
It can be shown as follows that is a function of r and ψ only. Since
the velocities do not depend on ξ, Equation 12 shows that is a function of r and ξ only. Thus, from simple integration, has the form , which can only happen if α is a function of ψ only. This is extremely important in terms of the solution to the partial differential equation. It means we can look for solutions to which are independent of ξ.
Dean then uses an eigenfunction expansion technique (separation of variables) to convert the partial differential equations into a set of ordinary differential equations which can be solved. He assumes:
Where, are functions of r only. These forms are then substituted into the Navier-Stokes equations to obtain ordinary differential equations.
Since the ψ-dependence is explicity given in Equation 14, these ordinary differential equations will have derivatives of r only.
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15. Conversion to a Set of Linear Differential Equations
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16. Solution of the Ordinary Differential Equations for vr and vψ
The method by which these equations are solved is fairly interesting. First, the pressure can be eliminated from equations 15, and 16 to obtain the differential equation:
Instinct would be to eliminate , but a better option is available.
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17. Solution of the Ordinary Differential Equations for vr and vψ
Think of this as an ordinary differential equation that describes the quantity defined by:
The equation falls into the class known as equidimensional ordinary differential equations and has the solution:
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18. Solution of the Ordinary Differential Equations for vr and vψ
Now, solve the continuity equation for , and substitute this into Equation 17. The result is:
The solution for this is:
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19. Solution of the Ordinary Differential Equations for vr and vψ
Substitution for this into the continuity equation gives :
Now, we can apply our boundary conditions and solve for the individual velocity profiles.
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20. Application of Boundary Conditions
From our first boundary condition, B = E = 0. Otherwise, the velocity along the axis would be infinite. It remains to apply the non-slip boundary conditions. This is straightforward, and it leads to the following solution for :
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21. Solution for
To obtain the differential equation for , combine Equation 25 with Equation 17. When this is done, the resulting differential equation is solved, and the result is as follows:
An excellent example of Dean Flow solutions and an application that can be extremely useful from such flows is found in Gelfgat et al The manuscript has been added to the course documents in case you are interested in looking at the work. The following figures stem from this work.
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22. Streamlines Louisiana Tech University Ruston, LA 71272
Figure 2 Streamlines for a variety of Dean Numbers [Gelfgat et al. 2003]
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23. Streamlines Louisiana Tech University Ruston, LA 71272
Figure 3 Description from manuscript of Dean Flow streamlines figure [Gelfgat et al. 2003]
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24. Nondimensional form of the Equations, NonD Parameters
It is instructive to nondimensionalized the equations for velocity. First, a good reference for velocity is the Poiseuille velocity at the center of the tube. In the notation used here (See Equation 5), this is:
The Reynolds number (based on tube radius) is then:
The nondimensional radius is:
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25. Nondimensional form of the Equations, NonD Parameters
The solutions for the three velocity components are then:
Examination of Equation 34 shows that there are two nondimensional parameters which are important. One is a/R, which has already been talked about, and which determines the importance of the second term in square brackets.
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26. Nondimensional form of the Equations, NonD Parameters
The other is , which determines the importance of the terms in curly brackets. The square root of this number,
is called the Dean number. This can be used to determine, approximately, when the Dean’s flow solution is valid.
By assumption (Equation 5), must be small, which means that the term in curly brackets in Equation 34 must be small. Thus, not only must a/R be small enough to make the terms it modifies be much less than 1, but also must be small enough to make the terms it modifies much less than 1. It will generally be more difficult to satisfy the condition on the Dean number than to satisfy the condition on a/R. For a relatively low Reynolds number, approximately 1000, the term that magnifies the curly brackets is about the same magnitude as a/R, so the terms in the curly brackets determine the validity of the solution unless the Reynolds number is very low.
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27. Relation to Textbook (5.4)
The solution in your textbook is much more simplified explanation, and does not cover any of the derivation. The main items in the text that are of note are as follows:
There is also a term, of much smaller magnitude, that applies due to the circulating flow within the torus. As opposed to Poiseuille flow where we have no ψ-circulation, we now have this circulation, and so must add a shear term:
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28. Flow Separation (4.6) From Section 4.6 of your text:
Comes from Boundary Layer Theory
When flow reversal or recirculation occurs, the boundary layer is said to be “Separated”.
The location at which the flow first reverses is known as the “Separation Point”
The location where the fluid again moves in the same direction is known as the “Reattachment Point”.
In biological systems, flow separation arises in lungs and blood flow.
Implicated in artherosclerosis
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29. Flow Separation (4.6)
When a change in cross sectional area occurs, the velocity must change.
To compensate for the momentum balance, the pressure must change.
A positive pressure field rather than a decreasing pressure due to flow is a necessary, but not sufficient, condition for flow separation.
Flow reversal occurs when the adverse pressure field is large enough to overcome the viscous forces at the wall and the inertial components of the fluid elements.
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Ruston, LA 71272

30. Branching Flows (5.5)
Branching flows exist everywhere in blood flow analysis.
Flow separation is a major part of the study of branching flows.
As was mentioned before, flow separation is implicated in the formation of atherosclerosis. This is illustrated in several of the branching flow diagrams in your textbook.
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Ruston, LA 71272

31. Entrance Length (5.3)
We often talk quite a bit about “fully developed flow”.
This occurs after the boundary layers have converged or become asymptotic.
However, we can estimate the entrance length required to achieve fully developed flow in a simple circular pipe by:
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32. Entrance Length (5.3) Equation 38 is for steady flow only.
For unsteady flows, including pulsatile flows, this length can vary and exhibit oscillations.
Flow in the major arteries is not fully developed.
Thus, these flows are inherently two or three-dimensional and sensitive to the inlet flow conditions.
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33. References
Gelfgat, A.Y., Yarin, A.L., Bar-Yoseph, P.Z., (2003), Dean vortices-induced enhancement of mass transfer trough an interface separating two immiscible liquids, Physics of Fluids, Volume 15, No. 2, pp
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34. Questions?
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Ruston, LA 71272