Chapter 6, Problem 6.1P

Interpretation Introduction

Interpretation: The expression for the fluid velocity in terms of the distance from the centre of the shaft is to be derived.

Concept introduction:

A shaft is a concentric or rotating element in a machine which is useful in transmitting power having circular cross-section.

For a linear velocity profile, the equation of the velocity around the shaft can be given as,

  $\text{V}\text{=}\text{A}\text{+}\text{Br}$  ..... (1)

V = Fluid velocity, a function of distance from the shaft

A and B are constants

r = Distance from the centre of the shaft

Answer to Problem 6.1P

The expression for the fluid velocity is $\text{V}\text{=}\frac{{\text{(r}}_{\text{0}}\text{-}\text{r)}}{{\text{(r}}_{\text{0}}\text{-}{\text{r}}_{\text{i}}\text{)}}\frac{\text{ωd}}{\text{2}}$ .

Explanation of Solution

Let r₀ be the distance from the centre to the outer boundary of the shaft and r_i be the distance from the centre to the inner boundary of the shaft.

The equation for the outer boundary can be written as,

  ${\text{V(r}}_{\text{0}}\text{)}\text{=}\text{A}\text{+}{\text{Br}}_{\text{0}}$  ..... (2)

The equation for the inner boundary can be written as,

  ${\text{V(r}}_{\text{i}}\text{)}\text{=}\text{A}\text{+}{\text{Br}}_{\text{i}}$  ..... (3)

Now as per the question, the conditions given are,

  $\begin{array}{l}{\text{V(r}}_{\text{0}}\text{)}\text{=}\text{0}\\ {\text{V(r}}_{\text{i}}\text{)}\text{=}\frac{\text{ωd}}{\text{2}}\end{array}$

d = Shaft diameter

  $\text{ω}$ = Angular velocity

Substitute the above conditions in equation (2) and equation (3),

  $\begin{array}{l}\text{0}\text{=}\text{A}\text{+}{\text{Br}}_{\text{0}}\\ \text{A}\text{=}{\text{-Br}}_{\text{0}}\mathrm{.......}\text{(4)}\end{array}$

Substitute equation (4) in equation (3) along with inner boundary condition,

  $\begin{array}{l}\frac{\text{ωd}}{\text{2}}\text{=}{\text{-Br}}_{\text{0}}\text{+}{\text{Br}}_{\text{i}}\\ {\text{B(r}}_{\text{i}}\text{-}{\text{r}}_{\text{0}}\text{)}\text{=}\frac{\text{ωd}}{\text{2}}\\ \text{B}\text{=}\frac{\text{ωd}}{{\text{2(r}}_{\text{i}}\text{-}{\text{r}}_{\text{0}}\text{)}}\\ \text{B}\text{=}\text{-}\frac{\text{ωd}}{{\text{2(r}}_{\text{0}}\text{-}{\text{r}}_{\text{i}}\text{)}}\mathrm{.......}\text{(5)}\end{array}$

Substitute equation (5) in equation (4),

  $\text{A}\text{=}\frac{{\text{r}}_{\text{0}}\text{ωd}}{{\text{2(r}}_{\text{0}}\text{-}{\text{r}}_{\text{i}}\text{)}}\mathrm{.......}\text{(6)}$

Substitute equation (6) and equation (5) in equation (1),

  $\begin{array}{l}\text{V}\text{=}\frac{{\text{r}}_{\text{0}}\text{ωd}}{{\text{2(r}}_{\text{0}}\text{-}{\text{r}}_{\text{i}}\text{)}}\text{-}\frac{\text{rωd}}{{\text{2(r}}_{\text{0}}\text{-}{\text{r}}_{\text{i}}\text{)}}\\ \text{V}\text{=}\frac{{\text{(r}}_{\text{0}}\text{-}\text{r)}}{{\text{(r}}_{\text{0}}\text{-}{\text{r}}_{\text{i}}\text{)}}\frac{\text{ωd}}{\text{2}}\mathrm{.......}\text{(7)}\end{array}$

Conclusion

The expression for the fluid velocity is $\text{V}\text{=}\frac{{\text{(r}}_{\text{0}}\text{-}\text{r)}}{{\text{(r}}_{\text{0}}\text{-}{\text{r}}_{\text{i}}\text{)}}\frac{\text{ωd}}{\text{2}}$ .