Simple Shear Flow

The sketch to the right depicts the velocity profile in a simple shear flow. The fluid is moving to the right and the magnitude of the fluid velocity increases linearly with y. The velocity can be written

u = Cy
v = w = 0

where C is a constant determining the slope of the profile.

Velocity profile in shear flow




To make the analysis a little easier, we consider the flow as it would be observed by someone moving with the velocity of point P. This person sees velocity profile to the right.

u = C(y - yP)
v = w = 0

Shear velocity profile seen by observer moving with velocity at point P


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Decomposition of Velocity Field

Any fluid velocity field can be decomposed into the sum of a rotational field and a deformation field.
u = uR + uD
where u is the velocity field (a vector),
u = uex + vey
uR is the rotational field,
uR = uRex + vRey
and uD is the deformation field.
uD = uDex + vDey



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Rotational Velocity Field of the Simple Shear Flow

The rotational velocity field for this simple shear flow is

uR = (C/2)(y - yP)
vR = -(C/2)(x - xP)

The rotational field is depicted in the following sketch. On the left, the velocity vectors are drawn at selected points. On the right the streamlines of the flow are shown.

Velocity vectors and streamlines for rotational field

In this purely rotational flow all fluid elements experience solid body rotation. In other words the fluid elements rotate without undergoing any deformation. The motion is simulated with the following animation. Press the triangle in the lower left corner to begin the animation.



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Deformation Velocity Field of the Simple Shear Flow

The deformation velocity field for this simple shear flow is

uD = (C/2)(y - yP)
vD = (C/2)(x - xP)

The deformation field is depicted in the following sketch. On the left, the velocity vectors are drawn at selected points. On the right the streamlines of the flow are shown.

Velocity vectors and streamlines for deformation velocity field

The deformation velocity field initially appears to be quite complex. The flow approaches point P from the upper left and lower right corners. It moves away from point P toward the upper right and lower left corners. The green lines are asymptotes for the approaching and leaving flow directions. These lines are also the principle directions of the deformation flow. Fluid elements oriented along the principle directions experience pure extensional strain. This motion is depicted in the following animation. The fluid elements shown as yellow boxes are initially aligned with the principle directions. Press the triangle in the lower left corner to begin the animation.



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