How to derive the Cauchy stress tensor term in the momentum equations?

jinggca · September 9, 2022, 02:49

I am learning about the source term in the momentum equation, and I got to the point where the momentum equation looks like this:

$\rho\frac{D\mathbf{u}}{Dt} = -\nabla p + \nabla\cdot\bar{\bar{\tau}} + \mathbf{f}$ .

However, from this point onwards every source I came about provides the expression for $\bar{\bar{\tau}} = \mu\left(\nabla\mathbf{u}+(\nabla\mathbf{u})^T\right)+\lambda(\nabla\cdot\mathbf{u})\mathbf{I}$ without providing a derivation. I understand that constitutive laws are involved here that states the shear stress is proportional to the spatial velocity gradient, but there's a lot more to it. Could someone please point me to a source where a detailed derivation, or at least an explanation of where each term comes from is given? Thanks a lot in advance!

LuckyTran · September 9, 2022, 04:37

It comes directly from the constitutive law including both first and second viscosity. See Lame's constant.

It often appears in a general derivation of Navier-Stokes because if dynamic viscosity is a thermodynamic parameter (i.e. you can look it up from a table of simply temperature and pressure) then it automatically implies a specific value for Lame's constant. However, experimental measurements yields Lame's constants that are very different than this assumed value and even with the opposite sign. It is there as a reminder that the conventional Navier-Stokes equations, as difficult as they are to solve, are still not the ultimate form governing all fluid phenomenon. But also, it is the original Stoke's hypothesis.

Frank White's book contains a detailed derivation from Hooke's law but not in vector nabla notation. Otherwise it's identical to the same one on Wikipedia which you've already seen.

The way to derive it yourself is to take your unit cube and apply taylor series to the forces on each face and write down the face normal and face parallel (i.e. the shear) forces, example is given here in Cauchy equation. Note that when you go to derive Boussinesq's eddy viscosity closure, you will need to do the exact same thing for the Reynolds stresses.

sbaffini · September 9, 2022, 08:46

Quote:

Originally Posted by jinggca

I am learning about the source term in the momentum equation, and I got to the point where the momentum equation looks like this:

$\rho\frac{D\mathbf{u}}{Dt} = -\nabla p + \nabla\cdot\bar{\bar{\tau}} + \mathbf{f}$ .

However, from this point onwards every source I came about provides the expression for $\bar{\bar{\tau}} = \mu\left(\nabla\mathbf{u}+(\nabla\mathbf{u})^T\right)+\lambda(\nabla\cdot\mathbf{u})\mathbf{I}$ without providing a derivation. I understand that constitutive laws are involved here that states the shear stress is proportional to the spatial velocity gradient, but there's a lot more to it. Could someone please point me to a source where a detailed derivation, or at least an explanation of where each term comes from is given? Thanks a lot in advance!

Aris: Vectors, tensors and the basic equations of fluid mechanics

gives you as much as possible for an introductory fluid mechanics textbook.

And will also probably fix most issues you might have with whatever source you are using now (considering also this post).

jinggca · September 9, 2022, 18:00

Thanks for the in depth response! Will look into each topics in detail.

jinggca · September 9, 2022, 18:01

Thanks for the recommendation!

September 9, 2022, 02:49	How to derive the Cauchy stress tensor term in the momentum equations?	#1
jinggca New Member William Join Date: Sep 2022 Posts: 8 Rep Power: 4	I am learning about the source term in the momentum equation, and I got to the point where the momentum equation looks like this: $\rho\frac{D\mathbf{u}}{Dt} = -\nabla p + \nabla\cdot\bar{\bar{\tau}} + \mathbf{f}$ . However, from this point onwards every source I came about provides the expression for $\bar{\bar{\tau}} = \mu\left(\nabla\mathbf{u}+(\nabla\mathbf{u})^T\right)+\lambda(\nabla\cdot\mathbf{u})\mathbf{I}$ without providing a derivation. I understand that constitutive laws are involved here that states the shear stress is proportional to the spatial velocity gradient, but there's a lot more to it. Could someone please point me to a source where a detailed derivation, or at least an explanation of where each term comes from is given? Thanks a lot in advance!

September 9, 2022, 04:37		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	It comes directly from the constitutive law including both first and second viscosity. See Lame's constant. It often appears in a general derivation of Navier-Stokes because if dynamic viscosity is a thermodynamic parameter (i.e. you can look it up from a table of simply temperature and pressure) then it automatically implies a specific value for Lame's constant. However, experimental measurements yields Lame's constants that are very different than this assumed value and even with the opposite sign. It is there as a reminder that the conventional Navier-Stokes equations, as difficult as they are to solve, are still not the ultimate form governing all fluid phenomenon. But also, it is the original Stoke's hypothesis. Frank White's book contains a detailed derivation from Hooke's law but not in vector nabla notation. Otherwise it's identical to the same one on Wikipedia which you've already seen. The way to derive it yourself is to take your unit cube and apply taylor series to the forces on each face and write down the face normal and face parallel (i.e. the shear) forces, example is given here in Cauchy equation. Note that when you go to derive Boussinesq's eddy viscosity closure, you will need to do the exact same thing for the Reynolds stresses. jinggca likes this. Last edited by LuckyTran; September 9, 2022 at 13:56.

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September 9, 2022, 18:00		#4
jinggca New Member William Join Date: Sep 2022 Posts: 8 Rep Power: 4	Thanks for the in depth response! Will look into each topics in detail.

September 9, 2022, 18:01		#5
jinggca New Member William Join Date: Sep 2022 Posts: 8 Rep Power: 4	Thanks for the recommendation!