See: Shahzad, H., Wang, X., Ghaffari, A. et al. Fluid structure interaction study of non-Newtonian Casson fluid in a bifurcated channel having stenosis with elastic walls. Sci Rep 12, Available on:

https://www.nature.com/articles/s41598-022-16213-3#:~:text=There%20are%20many%20non%2DNewtonian,of%20medicine%2C%20and%20synthetic%20lubricants.

Cassonian fluids are non-newtonian as J.C says above: basically the mathematical fit of the fluids properties are similar to the more common power law form. This formula has been widely used in blood flowing systems, as it allows fairly well for deformation of the particles and the walls (red and white cells in the case of blood). The power law form mu= k.(du/dx)^n is probably the most widely used as it is a simple curve fit from the visco-metric data: horses for courses!

The rheological equation of state for an isotropic flow of a Casson fluid can be expressed by equation (See image). In this Equation, π = eijeij; where eij is the (i, j)th component of the deformation rate. This means that π is the product of the component of deformation rate with itself. Also, πc is a critical value of this product based on the non-Newtonian model, μB is the plastic dynamic viscosity of the non-Newtonian fluid and Py is the yield stress of the fluid. If u and v are the fluid x- and y-components of velocity respectively; and C being the concentration field.

See Also:

Eldabe, N.T.M. and Salwa, M.G.E. (1995) Heat Transfer of MHD Non-Newtonian Casson Fluid Flow between Two Rotating Cylinders. Journal of the Physical Society of Japan, 64, 41.

Jamel Chahed Your rigorous mathematical approach to the fluid property using tensors is highly commendable and is commonly used in CFD codes. However, the best selection depends on the problem being solved, as parameters in the Casson equation have temperature dependencies that are very difficult to model: temperature effects are much simpler in the power law form. For isothermal cases, the choice is according to which form best fits the measured "effective viscosity" over the shear stress range required.

You are right Dear Dr. Jack Broughton the Power law model (Oswald Model) is relatively simple in formulation and the value of the exponent can account for the "thickening" or "thinning" behavior of a non-Newtonian fluid. However, the Oswald Model it is not suitable for describing the rheology of fluids with Visco-Plastic behavior, because it does not introduce the "Threshold Effect" in the stress tensors formulation. The "Casson Model" is precisely adapted to this situation and the fluids whose behavior fits with this formulation are called "Casson Fluids". Regarding the effect of taking into account the temperature field in the Casson model, one can refer for example to the paper by S.Pramanik "Casson fluid flow and heat transfer past an exponentially porous stretching surface in presence of thermal radiation, Ain Shams Engineering Journal (2014)5, 205–212". Available on:

Article Casson fluid flow and heat transfer past an exponentially po...

Dear Jamel Chahed I think that we are moving a long way from the simple question asked, but it is very interesting to discuss such detail! The Casson form of equation was developed by Casson in the 1950s, he was examining the rheology of slurries. It is one of a number of attempts to fit the complexities that occur over the full shear range of complex fluids. It used to be incorporated as an option in the Phoenix (CHAM) codes and is still widely used in 3-D CFD solutions.

What I am counselling is that one needs to take care in using any of these equations outside of their range of measurement: particularly in variable temperature fields.

Thank you Dear Jack Broughton for your comment. I very much appreciated what you said; "What I am counselling is that one needs to take care in using any of these equations outside of their range of measurement" This is exactly, word by word, what I always repeat to my Students, particularly in my course "SMTF; Single Phase and Multiphase Turbulent Flows" (In French on my RG-Profile), the introduction of which is at: Chapter SMTF-Chapitre 1 : Ecoulements Turbulents Mono & Multiphasiques

Speaking of repetition, I believe that part of our duty as teachers is to repeat. Here are some Proverbs, citations, and quotes on repetition (Translated from French):

“Error repeats itself over and over again in action, which is why truth must be repeated over and over again in speech”. Quote from Johann Wolfgang von Goethe; Maxims and reflections (1749-1832)

“We must constantly repeat to young people the truths that we want to engrave in their hearts; seek to inspire them with the horror of vice, and try to make them know evil as late as possible, so that reason can balance the ardor of the passions". Quote from Sosthène de La Rochefoucauld-Doudeauville; The Book of Thoughts (1861)

"Repetition is the strongest figure of speech". Quote from Napoleon Bonaparte; Maxims and Thoughts (1769-1821)

"The truth is always good to say, and even to repeat. Young artists do not know it: it is the first thing that a wise master should teach them". Quote from Jean Dutourd; Content and Form (1965)

“What is repeated is truer than what is invented”. Quote from Alfred Auguste Pilavoine; The thoughts, mixtures, and poems (1845)

“A repeated pleasure is doubly a pleasure”. Quote from Pedro Calderón de la Barca; In secret outrage, secret revenge (1642)

"Convinced people like to repeat themselves". Quote from Victor Cherbuliez; Fragile Loves (1880)

"Fine-minded people run the risk of repeating themselves because they never really know what tracks they have walked on". Quote from Suzanne Curchod; The mixtures and pansies (1798)

"To repeat is to persuade in detail". Quote from François Gaston de Lévis; Maxims and Reflections on Different Subjects (1810)

And as I am of advanced age, I will end with this quote from this scholar of Tahar Ben Jelloun, a French eminent intellectual and literary, of Moroccan origin:

"An old man has the right to repeat himself, it is an achievement and a privilege of age". Quote from Tahar Ben Jelloun; The Inn of the Poor (1997)

I also love this Arabic Proverb:

"In repetition, Interest"