When the constitutive equation for a Newtonian fluid was derived in class, one of the postulates that was adopted was that a fluid element undergoing a solid body rotation does not exhibit any deformation. Because we have defined a Newtonian fluid to be a material for which stress is linearly related to rate of strain, then we must conclude that for a solid body rotation, no additional stresses are present in the fluid beyond those which are present when the fluid is at rest. Therefore, the terms describing a pure rotation type of flow field were not included in the derivation of the constitutive equation. Consider a flow undergoing a solid body rotation. In polar coordinates, such a flow field is given by: u, =0 and u, = W,r , %3D where w, is a constant. (10 points) а. Compute the components of the rate of strain tensor for this flow field. b. Compute the vorticity and the angular velocity at a point in the flow field.

When the constitutive equation for a Newtonian fluid was derived in class, one of the postulates that was adopted was that a fluid element undergoing a solid body rotation does not exhibit any deformation. Because we have defined a Newtonian fluid to be a material for which stress is linearly related to rate of strain, then we must conclude that for a solid body rotation, no additional stresses are present in the fluid beyond those which are present when the fluid is at rest. Therefore, the terms describing a pure rotation type of flow field were not included in the derivation of the constitutive equation. Consider a flow undergoing a solid body rotation. In polar coordinates, such a flow field is given by: u, =0 and u, = W,r , %3D where w, is a constant. (10 points) а. Compute the components of the rate of strain tensor for this flow field. b. Compute the vorticity and the angular velocity at a point in the flow field.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

When the constitutive equation for a Newtonian fluid was derived in class, one of the postulates that was adopted was that a fluid element undergoing a solid body rotation does not exhibit any deformation. Because we have defined a Newtonian fluid to be a material for which stress is linearly related to rate of strain, then we must conclude that for a solid body rotation, no additional stresses are present in the fluid beyond those which are present when the fluid is at rest. Therefore, the terms describing a pure rotation type of flow field were not included in the derivation of the constitutive equation.

Consider a flow undergoing a solid body rotation. In polar coordinates, such a flow field is given by:

\[ u_r = 0 \quad \text{and} \quad u_\theta = \omega_0 r, \]

where \( \omega_0 \) is a constant. (10 points)

a. Compute the components of the rate of strain tensor for this flow field.

b. Compute the vorticity and the angular velocity at a point in the flow field.

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