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Chris December 16, 2003 11:14

Calculation of velocity gradient
I am trying to calculate absolute velocity gradient for my simulation, i was told that for a three dimensional case this would be:

SQRT ((DU/DY)^2+(DU/DZ)^2+(DV/DX)^2+(DV/DZ)^2+(DW/DX)^2+(DW/DY)^2) is this correct?

Nicola December 16, 2003 12:58

Re: Calculation of velocity gradient

the velocity gradient is a matrix. May be you are asked to calculate the norm of the gradient of the absolute velocity in a specified direction. Is your problem a general one, or do it pertain a particular situation?


chris December 16, 2003 16:05

Re: Calculation of velocity gradient
My flow is non-newtonian shear thinning so i need the shear rate (velocity gradient), the problem is general in as much as the flow is reasonably simple but 3 dimensional.

Nicola December 17, 2003 06:44

Re: Calculation of velocity gradient

in some cases, the constitutive equation is:

Tij = visc * STij

where Tij is the part of the stress tensor depending on the shear rate tensor Sij, and visc (the dynamic viscosity) is written as:

visc = f(e)

where e = [1/2 (Sij Sij)]^0.5, so visc depends on the effective strain rate e.

In these cases, you first need to find the gradients of the X,Y,Z velocity components, then you have to evaluate the strain tensor components and the effective strain rate e. So, Tij and Sij are tensors, while only e is a scalar. Which is the constitutive equation of your fluid? Is it similar to the previously described one?

Best regards,


Chris December 17, 2003 07:12

Re: Calculation of velocity gradient
All i am looking for is an input into my viscosity equation, the viscosity follows the power law equation i.e. visc=m*gammadot^n-1 where gammadot is the shear rate. I am looking at 3D flow in the cartesian co-ordinate system, so escentially i need the resultant absolute velocity gradient is that, does that correspond to the formula in my first post?

Nicola December 17, 2003 10:18

Re: Calculation of velocity gradient
No, it doesn't

m malik December 17, 2003 16:01

Re: Calculation of velocity gradient
If you are looking for gamma_dot: gamma_dot = sqrt(0.5*second_invariant) second_invariant = second invariant of the rate of

deformation tensor

= 4((du/dx)^2 + (dv/dy)^2 + + (dw/dz)^2)

+ 2((du/dy)(dv/dx) + (dv/dz)(dw/dy)

+ (dw/dx)(du/dz))

Chris December 18, 2003 07:24

Re: Calculation of velocity gradient
thank you for that, do you possibly have a reference where i could find a full derivation, also on another point what would be the shear rate in terms of DU/DX etc etc

m malik December 18, 2003 09:35

Re: Calculation of velocity gradient
Some on my desk are: 1.JN Reddy and DK Gartling, The FEM in Heat Transfer and Fluid Mechanics, CRC Press, 1994. 2. RB Bird, WE Stewart, and EN Lightfoot, Transport Phenomena, John Wiley, 2002. 3. RG Owens and TN Phillips, Computational Rheology, Imperial College Press, 2002.

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