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 Chris December 16, 2003 11:14

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

Chris,

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?

Nicola

 chris December 16, 2003 16:05

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

Chris,

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,

Nicola

 Chris December 17, 2003 07:12

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

No, it doesn't

 m malik December 17, 2003 16:01

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