Fluid mechanics: defition of shear flow [rate of deformation tensor]

[kd215]

fluid mechanics: defition of "shear flow" [rate of deformation tensor]

I'm studying old undergraduate chemical engineering notes for an exam in grad school. Can't recall what this really means, can anyone explain to me what "off-diagonal elements" means and why the trig function velocities would be or not be "off-diagonal elements". And can you explain what the question is talking about in general.

Problem statement: Consider the velocity field u = ([/x],[/y],[/z]), where: [/x](x,y,z)=constant*y*z*sin(constant*x)...(similar functions for y and z velocities)

and question: "Recall that the definition of "shear flow" is one for which the rate of deformation tensor [Δ][/ij] has only off-diagonal elements. Is this shear flow?" (y or n)

 

[kd215]

And those are velocities like u (sub x,y,z) just in each direction. Not sure how to write the notation in the posts

 

[Chestermiller]

If you are a chemical engineer, your first step should be to go back to Bird, Stewart, and Lightfoot, and look up the definition of the rate of deformation tensor. The components of the rate of deformation tensor in cartesian coordinates can be arranged in a 3x3 matrix. The diagonal elements of this matrix are equal to the partial derivatives of the three velocity components with respect to distance in the coordinate direction of the velocity components. If these three components of the matrix are equal to zero, the flow is considered to be a pure shear flow. The rate of deformation tensor does not specifically relate to the trigonometric functions, although, for a particular flow in which the spatial variation of the velocity components are expressed in terms of the trigonometric functions, they will of course come into play.

 

[Studiot]

Welcome to Physics Forums, KD215.

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As regards the technical part of your question. There are several mechanical properties that have the principal or normal property as diagonal elements of their matrix or tensor and other properties (parallel or cross products) as off diagonal. Examples as Inertia, stress, strain, displacement.