Subhfan Fontanills

assuming the vessel is like a circular pipe and assuming the flow is fully developed and laminar (that is not very real in an artery) you can deduce that the velocity law is a parabola. From the basic geometry we know that the max value of such parabola is 3/2 the value of the average. See https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation

Aditya Kumar Mishra

maybe you are confusing with the concept of laminar sublayer followed from the transitional and logarithmic law for the statistically averaged turbulent velocity profile. The topic is discussing a fully developed laminar flow in confined region.

Dear Nicolette,

The velocity profile will be an axially asymmetric surface and can not be defined in a simple format at least in commonly used coordinate systems.

The exact solution for the steady laminar velocity profile w=w(x,y) obeys the governing elliptic equations.

Lap w = Re*dp/dz

w=0 on the BC.

You can find a solution expressed by a series in some textbook or you can solve numerically the Poisson equation.

In addition to Prof Denaro's numerical solution, it is also possible to write w in terms of a double Fourier sine series. If the boundaries of the square channel are at x=0, x=L, y=0 and y=L, then let

w = sum_n sum_m  B_nm  sin(n*pi*x/L)  sin(m*pi*y/L)

and then the appropriate formula for the double Fourier sine series gives

B_nm = 16(-1)^{n+m} / (n*m*pi^2).

....my solution was for Lap w = -1.  Any other constant on teh right hand side will have the same solution as I gave but multiplied by Re*dp/dz.

Formally, the answer to the original question is that there is no simple analytical solution, apart from the infinite sum.

I believe that it is true that an analytical solution exists for the eqiuilateral triangular-shaped duct.

If I remember correctly, the texbook of White illustrates the series-based solution for several shapes.

The expressions are also reported in the last pages here http://user.engineering.uiowa.edu/~me_160/lecture_notes/Ch6Oct2009-part1-updated.pdf

Prof. Denaro is correct.

The great book by Prof. White summarized the solution for Poiseuille Flow for non-circular ducts on Chapter (1991 edition).

Viscous Fluid Flow (McGraw-Hill Mechanical Engineering) 3rd Edition

by Frank M. White (Author)

Currently on sale in Amazon.

Larry

Dear Nicolette, in hydraulic applications I have found that a useful and good simple formula is u(x, y) = 2 Uavg x(L-x) y(L-y) / (L / 2)4, where L is the edge square length and (L / 2)4 is a normalization coefficient. Note that at boundary u = 0, while u(L/2, L/2) = 2 Uavg as in circular section pipe.  Gianluca

@Filippo Maria Denaro I am a Cardiovascular Sonography student and have a multiple choice question I could not find in the textbooks or online articles on velocity profile of laminar flow: In parabolic flow, the maximum velocity being at the center, is_________ the average speed.

Possible answers: 3 times faster, 1/2 faster, 5 times faster, 2 times faster.

I would appreciate anyone's input. Thank you.

Subhfan Fontanills

I am not sure about the nature of your question, are you asking for the relation between the maximum in a parabola y(x)=a+b*x + c*x^2 and its averaged value? That seems more a homework...

In general, in laminar flow near the surface velocity is zero, velocity increases logarithmically with increase in distance from the surface having maximum at the free surface.

1. If one considers a square duct of infinite width the velocity profile will indicate maximum velocity at the center and minimum at the upper and lower edges ( boundaries).

2. For finite width square duct intutively, we could expect velocity variations from the four edges (boundaries) of the to follow the same pattern., maximum at the center and near zero at the edges.

3. At the corners however one could expect average flow velocity from the corresponding sides.

It would appear parabolic in case 1 and combination of two parabolic curves for the case 2.

Thus, the flow velocity conditions at the corners of the duct remains questionable?

Using appropriate mathematical model you could attempt to give it a try.

Not exactly parabolic actually rectangular parabolic

Aditya Kumar Mishra a What is your definition of rectangular parabolic? How does something increase logarithmically? There isn't a free surface in this idealised pipe flow. So I am not at all sure about your comments!

@Filippo Maria Denaro It is a question for my Vascular Final. The question assumes parabolic flow in a normal blood vessel, I know the velocity in the center (laminae) is higher of those lamina closer to the vessel wall, the question is asking how much faster is velocity in the center compared to the average velocity of the blood flow in the entire vessel? Is there a set number or constant to know this?

Subhfan Fontanills

assuming the vessel is like a circular pipe and assuming the flow is fully developed and laminar (that is not very real in an artery) you can deduce that the velocity law is a parabola. From the basic geometry we know that the max value of such parabola is 3/2 the value of the average. See https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation

Aditya Kumar Mishra

maybe you are confusing with the concept of laminar sublayer followed from the transitional and logarithmic law for the statistically averaged turbulent velocity profile. The topic is discussing a fully developed laminar flow in confined region.

Thank you! @Filippo Maria Denaro 🙏🏻😊

Dear Dr. Rees I could have mistaken on terminology, could you kindly see the image and correct it.

Well this kind of velocity profile I was expecting on one side of the duct. It was also clearly mentioned that the profile is intutive completely.

Aditya Kumar Mishra

the velocity profile you posted has nothing to do with a fully developed laminar flow in a duct/pipe. Again, check for "wall turbulence" in literature and you will see the difference from laminar flow.

Certainly Professor Fillipo Maria Denaro. Thank you very much for the reference.

The explicit analytical solution for the flow in a square pipe is presented by F. Delplace in the paper: “Laminar flow of Newtonian liquids in ducts of rectangular cross-section a model for both physics and mathematics”. The author uses a Saint-Venant method to obtain an explicit expression for the velocity field expressed in the form of a series (eq. 7). The paper contains some very interesting mathematical results. I have attached a copy of the paper.