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.

 

[PJC01]

Fluid mechanics is a branch of physics that studies the behavior of fluids (liquids and gases) and their interactions with forces. Shear flow is a type of flow where the fluid particles move parallel to each other in layers, with one layer sliding over the adjacent layer. This results in a shearing force between the layers, causing the fluid to deform. The rate of deformation tensor [Δ][/ij] is a mathematical representation of the deformation of a fluid, and it describes how the fluid changes its shape over time. In this context, "off-diagonal elements" refer to the elements of the rate of deformation tensor that are not on the main diagonal, meaning they do not correspond to the fluid's normal deformation in the x, y, and z directions.

In the given velocity field, all three components (x, y, and z) have a constant and a trigonometric function. This means that the velocity is not constant in any direction, and therefore, the rate of deformation tensor will have non-zero off-diagonal elements. This indicates that the fluid is undergoing shear flow, as the fluid particles are moving in different directions with respect to each other, resulting in shear deformation.

In general, the question is asking if the given velocity field represents shear flow, based on the definition of shear flow and the characteristics of the velocity field. By analyzing the components of the velocity field and the resulting rate of deformation tensor, we can determine that this is indeed shear flow.