About the Hershel-Bulkley model of Non-Newtonian fluids

tiger0004 · October 24, 2013, 23:47

Hi,everyone!
Lately I have been using the Herschel-Bulkley model to model the behaviors of a Non-Newtonian fluid. The model states that if the stress of the fluid is less than the yield stress tau0,then the fluid would act like rigid(pseudoplastic).
The mathematical description of the HB model is tau=tau0+k*gamma^n,in which tau is the stress, tau0 is the yield stress,gamma is the shear rate and k and n are fluid constants.
Here is the question. In some fluid field, if I don't previously know in which area the shear stress would be smaller than tau0 (plug area), how can I find it out? I mean, according to the HB equation, because gamma and tau, tau0 point to the same direction, so no matter what value set gamma to be, the absolute value of tau is always no less than that of tau0. (tau,tau0 and gamma always have the same sign.) For example, if k=1,n=1,gamma>0, then tau(positive)=tau0(positive)+gamma(positive), tau>tau0;if gamma<0, then tau(negative)=tau0(negative)+gamma(negative),abs(t au)>abs(tau0). In both case, there would be no plug area in the fluid.
Can anyone help me? Thanks!

triple_r · October 25, 2013, 16:43

As you said, when shear stress is higher than the yield stress, then and only then the fluid will experience shear deformation. In other words, if tau > tau0, then $\dot{\gamma}$ > 0. Now, the equation shows exactly the same behavior, if you have a $\dot{\gamma}$ that is non zero, then you must have had a shear stress that was higher than the yield stress.

In order to see the plug flow region, you must write the equations in terms of shear stress, and not strain. For an example, you can take a look at some of the solutions for a Bingham plastic fluid in most non-Newtonian fluid flow text books.

By the way, in the constitutive equation that you have, neither tau, nor $\dot{\gamma}$ can be negative. They are "magnitudes" of tensors. Usually they are defined as:

$\dot{\gamma}=\sqrt{\frac{1}{2}\dot{\gamma}:\dot{\gamma}}$

and

$\dot{\tau}=\sqrt{\frac{1}{2}\dot{\tau}:\dot{\tau}}$

in these definitions, the left hand side symbols are scalars (and these are the ones that appear your constitutive equation), and the right hand side symbols are tensors. also,

means double-inner product of tensors (multiply corresponding components with each other, and then sum all of the products).

Hope this helps.

tiger0004 · October 25, 2013, 22:43

Quote:

Originally Posted by triple_r

As you said, when shear stress is higher than the yield stress, then and only then the fluid will experience shear deformation. In other words, if tau > tau0, then $\dot{\gamma}$ > 0. Now, the equation shows exactly the same behavior, if you have a $\dot{\gamma}$ that is non zero, then you must have had a shear stress that was higher than the yield stress.

In order to see the plug flow region, you must write the equations in terms of shear stress, and not strain. For an example, you can take a look at some of the solutions for a Bingham plastic fluid in most non-Newtonian fluid flow text books.

By the way, in the constitutive equation that you have, neither tau, nor $\dot{\gamma}$ can be negative. They are "magnitudes" of tensors. Usually they are defined as:

$\dot{\gamma}=\sqrt{\frac{1}{2}\dot{\gamma}:\dot{\gamma}}$

and

$\dot{\tau}=\sqrt{\frac{1}{2}\dot{\tau}:\dot{\tau}}$

in these definitions, the left hand side symbols are scalars (and these are the ones that appear your constitutive equation), and the right hand side symbols are tensors. also,

means double-inner product of tensors (multiply corresponding components with each other, and then sum all of the products).

Hope this helps.

Thank you triple_r! I know sometimes my fundamental concepts are ambiguous. I referred to some text books and scientific papers, and found out that usually the shear rate is calculated first and then then shear stress is calculated with the constitution equation. In some simple cases, like in a tube or between two plates, the shear stress can be obtained when knowing the pressure distribution. Is this always the way to find out the shear stress without the shear rate?

October 24, 2013, 23:47	About the Hershel-Bulkley model of Non-Newtonian fluids	#1
tiger0004 New Member Join Date: Oct 2013 Posts: 2 Rep Power: 0	Hi,everyone! Lately I have been using the Herschel-Bulkley model to model the behaviors of a Non-Newtonian fluid. The model states that if the stress of the fluid is less than the yield stress tau0,then the fluid would act like rigid(pseudoplastic). The mathematical description of the HB model is tau=tau0+k*gamma^n,in which tau is the stress, tau0 is the yield stress,gamma is the shear rate and k and n are fluid constants. Here is the question. In some fluid field, if I don't previously know in which area the shear stress would be smaller than tau0 (plug area), how can I find it out? I mean, according to the HB equation, because gamma and tau, tau0 point to the same direction, so no matter what value set gamma to be, the absolute value of tau is always no less than that of tau0. (tau,tau0 and gamma always have the same sign.) For example, if k=1,n=1,gamma>0, then tau(positive)=tau0(positive)+gamma(positive), tau>tau0;if gamma<0, then tau(negative)=tau0(negative)+gamma(negative),abs(t au)>abs(tau0). In both case, there would be no plug area in the fluid. Can anyone help me? Thanks!

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October 25, 2013, 16:43		#2
triple_r Senior Member Reza Join Date: Mar 2009 Location: Appleton, WI Posts: 116 Rep Power: 17	As you said, when shear stress is higher than the yield stress, then and only then the fluid will experience shear deformation. In other words, if tau > tau0, then $\dot{\gamma}$ > 0. Now, the equation shows exactly the same behavior, if you have a $\dot{\gamma}$ that is non zero, then you must have had a shear stress that was higher than the yield stress. In order to see the plug flow region, you must write the equations in terms of shear stress, and not strain. For an example, you can take a look at some of the solutions for a Bingham plastic fluid in most non-Newtonian fluid flow text books. By the way, in the constitutive equation that you have, neither tau, nor $\dot{\gamma}$ can be negative. They are "magnitudes" of tensors. Usually they are defined as: $\dot{\gamma}=\sqrt{\frac{1}{2}\dot{\gamma}:\dot{\gamma}}$ and $\dot{\tau}=\sqrt{\frac{1}{2}\dot{\tau}:\dot{\tau}}$ in these definitions, the left hand side symbols are scalars (and these are the ones that appear your constitutive equation), and the right hand side symbols are tensors. also, means double-inner product of tensors (multiply corresponding components with each other, and then sum all of the products). Hope this helps.