Question 8.7: In the laminar flow of a fluid in a circular pipe, the veloc...

Introduction to Fluid Mechanics and Fluid Machines

In the laminar flow of a fluid in a circular pipe, the velocity profile is exactly a parabola. The rate of discharge is then represented by volume of a paraboloid. Prove that for this case the ratio of the maximum velocity to mean velocity is 2.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

See Fig. 8.15. For a paraboloid,
$v_{z}=v_{z_{\max }}\left[1-\left(\frac{r}{R}\right)^{2}\right]$

Q=\int v_{z} \mathrm{~d} A=\int_{0}^{R} v_{z_{\max }}\left[1-\left(\frac{r}{R}\right)^{2}\right](2 \pi r \mathrm{~d} r)

=2 \pi v_{z_{\max }}\left[\frac{r^{2}}{2}-\frac{r^{4}}{4 R^{2}}\right]_{0}^{R}=2 \pi v_{z_{\max }}\left[\frac{R^{2}}{2}-\frac{R^{2}}{4}\right]

=v_{z_{\max }}\left(\frac{\pi R^{2}}{2}\right)

v_{z_{\text {mean }}}=\frac{Q}{A}=\frac{v_{z_{\max }}\left(\pi R^{2} / 2\right)}{\left(\pi R^{2}\right)}=\frac{v_{z_{\max }}}{2}

Thus, $\frac{v_{z_{\max }}}{v_{z_{\text {mean }}}}=2$

8 6

Related Answered Questions

Question: 8.2

Two viscous, incompressible, immiscible fluids of same density (=ρ) but different viscosities (viscosity of the lower fluid layer = μ1 and that of the upper fluid layer = μ2 < μ1) flow in separate layers between parallel boundaries located at y = ± H, as shown in the Fig. 8.6. Thickness of each ...

Verified Answer:

Large lateral width of the plate renders the basic...

Question: 8.12

For the following thrust bearing (Fig. 8.22), show that the force on the straight slider in the x direction is the same as that on the guide. ...

Verified Answer:

It is given that the velocity profile is

\...

Question: 8.11

A cylindrical journal bearing supports a load directed vertically upwards, with the shaft rotating clockwise. Sketch the position of the shaft centre with respect to that of the bearing (hole), if no cavitation is present. Give explanation. No equations are required. ...

Verified Answer:

In this case, the pressure distribution is symmetr...

Question: 8.10

A slipper (slider) and plate (guide), both 0.5 m wide, constitutes a bearing as shown in Fig. 8.20. Density of the fluid, ρ = 9.00 kg/m^3 and viscosity, μ= 0.1 Ns/m^2. Find out the (i) load carrying capacity of the bearing, (ii) drag, and (iii) power lost in the bearing. ...

Verified Answer:

(i) Considering the width as b and using Eq. (8.82...

Question: 8.9

Consider steady flow of an incompressible Newtonian fluid (density = ρ, viscosity = μ) between two infinitely long concentric circular cylinders of inner radius a and outer radius b (Fig. 8.17). Both the cylinders rotate with the same angular velocity w. In addition, the inner cylinder moves ...

Verified Answer:

Infinitely long cylinder implies

\frac{\pa...

Question: 8.8

The velocity distribution for a fully developed laminar flow in a pipe is given by vz = – R^2 / 4 μ . ∂p / ∂z [ 1 – ( r / R )^2 ] Determine the radial distance from the pipe axis at which the velocity equals the average velocity ...

Verified Answer:

For a fully developed laminar flow in a pipe, we c...

Question: 8.6

The analysis of a fully developed laminar flow through a pipe can alternatively be derived from control volume approach. Derive the expression vz = R^2 / 4 μ ( – dp / dz ) ( 1 – r^2 / R^2 ) accordingly. Fig. 8.15 Fully developed laminar flow through a pipe ...

Verified Answer:

Let us have a look at Fig. 8.15. The fluid moves d...

Question: 8.5

A continuous belt (Fig. 8.13) passing upward through a chemical bath at velocity U0, picks up a liquid film of thickness h, density ρ, and viscosity μ. Gravity tends to make the liquid drain down, but the movement of the belt keep the fluid from running off completely. Assume that the flow is ...

Verified Answer:

The governing equation is

\mu \frac{\mathr...

Question: 8.4

Water at 60°C flows between two large flat plates. The lower plate moves to the left at a speed of 0.3 m/s. The plate spacing is 3 mm and the flow is laminar. Determine the pressure gradient required to produce zero net flow at a cross section. (μ = 4.7 × 10^–4 Ns/ m^2 at 60 °C) ...

Verified Answer:

Governing equation:

\mu \frac{\mathrm{d}^{...

Question: 8.3

Two infinite plates are h distance apart as in Fig. 8.10. There is a fluid of viscosity μ between the plates and the pressure is constant. The upper plate is moving at speed U = 4 m/s. The height of the channel h = 1.8 cm. Calculate the shear stress at the upper and lower walls if μ = 0.44 kg/m.s ...

Verified Answer:

Re

=\rho h U / \mu=(888)(1.8 / 100)(4) / 0...