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Sample Examples ˗ Modified Format ˗Fluid Mechanics Problems

Example 1: A shaft of diameter, d, is pulled through a cylindrical bearing as shown below.

A lubricant with dynamic viscosity, μ, fills the gap of width, b, between the shaft and the bearing.

The force P pulls the shaft at a constant speed, V_o. Assume the velocity distribution in the gap

is given by v(y) = V_o( 1 ˗ y/b) where y is the coordinate directed from the shaft to the bearing.

Find an expression for the shearing stress, τ, in the fluid in terms of V_o, μ, and b.

a. τ = μV_o/b b. τ = 2μV_o/b c. τ = ˗ μV_o/b d. . τ = ˗ 2μV_o/b e. None of these

Example 2: In a steady, two-dimensional flow field the fluid density varies linearly with respect

to the coordinate, x. i.e. ρ = Ax where A is a constant. If the x component of velocity, u, is

given by u = y, find an expression for the y component, v.

a. v = - y² / 2x + f(x) b. v = y² / 2x + f(x) c. v = - y² / x + f(x)

d. v = y² / 2x + f(x) e. None of these.

Example 3: The velocity components in a two-dimensional flow field are u = ˗ 4 m/sec and

v = ˗ 2 m/sec. Find an expression for the stream function, ψ.

a. ψ = 2x + 4y + C b. ψ = 2x ˗ 4y + C c. ψ = 4x + 4y + C d. 4x ˗ 4y + C

Example 4: The following form for Bernoulli's equation,

P / ρ + ½ V² + g z = Constant along streamline

Holds for the following conditions:

a. Inviscid, steady flow, compressible or incompressible along a streamline

b. Inviscid, steady flow, incmpressible flow along a streamline

c. Inviscid, steady flow, irrotational flow, compressible or incompressible

d. Inviscid, steady, incompressible, irrotational flow

e. All of these