Oil (SAE 30) at 15.6 oC flows steadily between fixed, horizontal, parallel plates. The pressure drop per unit length along the channel is 35 kPa/m, and the distance between the plates is 9 mm. The flow is laminar. Determine: (a) the volume rate of flow (per meter of width), (b) the magnitude of the shearing stress acting on the bottom plate, and (c) the velocity along the centerline of the channel.

(a) The volume rate of flow per meter width = 5.6*10⁻³ m²/s

(b) The shear stress acting on the bottom plate = 157.5 N/m²

(c) The velocity along the centerline of the channel = 0.93 m/s

Explanation:

(a)

Calculating the distance of plate from centre line using the formula;

h = d/2

where h = distance of plate

d = diameter of flow = 9 mm

Substituting, we have;

h = 9/2

= 4.5 mm = 4.5*10^-3 m

Calculating the volume flow rate using the formula;

Q = (2h³/3μ) * (Δp/L)

Where;

Q = volume flow rate

h = distance of plate = 4.5*10^-3 m

μ = dynamic viscosity = 0.38 N. s/m²

(Δp/L) = Pressure drop per unit length = 35 kPa/m = 35000 Pa

Substituting into the equation, we have;

Q = (2*0.0045³/3*0.38) * (35000)

= (1.8225*10⁻⁷/1.14) * (35000)

= 1.60*10⁻⁷ * 35000

= 5.6*10⁻³ m²/s

Therefore, the volume flow rate = 5.6*10⁻³ m³/s

(b) Calculating the shear stress acting at the bottom plate using the formula;

τ = h * (Δp/L)

= 0.0045 * 35000

= 157.5 N/m²

(c) Calculating the velocity along the centre of the channel using the formula;

u (max) = h²/2μ) * (Δp/L)

= (0.0045²/2*0.38) * 35000

=2.664*10⁻⁵ * 35000

= 0.93 m/s