Question

vn incompressible viscous fluid is placed between two large parallel plates as shownin Fig. 3. The bottom plate is fixed and the upper plate moves with a constant velocity,U. Forthese conditions the velocity distribution between the plates is linear and can be expressed as yu = U ---- bDetermine:(a) the volumetric dilatation rate(b) the rotation vector(c) the vorticity(d) the rate of angular deformation.

Answer 1

Answer:

a. 0

b. -½(U/b)k

c. (-U/b)k

d. U/b

Explanation:

Given

u = U(y/b)

a.

The rate of volumetric dilation is calculated by using the general formula;

u = U(y/b)

v = 0 and w = 0

The volumetric dilation rate is then given as

∆V = ∂u/∂x + ∂v/∂y + ∂w/∂z

∆V = ∂/∂x(U(y/b)) + ∂/∂y(0) + ∂/∂z(0)

∆V = 0 + 0 + 0

∆V = 0;

The volumetric dilation rate is zero than flow is an incompressible fluid.

b.

Calculating Rotation Vector

The rotation Vector of an element about one axis is the average of the Angular Velocity and the two perpendicular lines.

This is calculated as follows;

W = ½((∂w/∂x - ∂v/∂z)i + (∂u/∂z - ∂w/∂x)j + (∂v/∂x - ∂u/∂y)k)

W = (∂v/∂x - ∂u/∂y)k

Since vector W is not a 0 and everywhere the flow field is not irrational

W ≠ -½(U/b)k

c. The vorticity

The vorticity is the Vector that is twice the rotation Vector.

V = 2W

V = 2( -½(U/b)k)

V = (-U/b)k

d. The rate of angular deformation.

This is defined as

Y = ∂v/∂x + ∂u/∂y where ∂v/∂x = 0

Y = ∂u/∂y

Y = ∂/∂y(U(y/b))

Y = U/b