Question

A viscous liquid is sheared between two parallel disks; the upper disk rotates and the lower one is fixed. The velocity field between the disks is given by V=e^θ​rωz/h (The origin of coordinates is located at the center of the lower disk; the upper disk is located at z = h.) What are the dimensions of this velocity field? Does this velocity field satisfy appropriate physical boundary conditions? What are they?

Answer 1

Answer:

For  lower disk :   V = e^θ​rω(0)/h  = 0

At the upper disk:  V = e^θ​rω(h)/h  = e^θ​rω

Hence The physical boundary conditions are satisfied

Explanation:

Velocity field ( V ) = e^θ​rωz/h

Upper disk located at  z = h

Determine the dimensions of the velocity field

velocity field is two-dimensional ; V = V( r , z )

applying the no-slip condition

condition : The no-slip condition must be satisfied

For  lower disk Vo = 0 when disk is at rest z = 0

∴  V = e^θ​rω(0)/h  = 0

At the upper disk  V = e^θ​rω  given that a upper disk it rotates at z = h

∴ V = e^θ​rω(h)/h  = e^θ​rω

Hence we can conclude that the velocity field satisfies the appropriate physical boundary conditions.