Critical Velocity and Reynold's Number

Critical velocity

Whether a flow is streamline or turbulent is decided by the velocity of the flow. The velocity beyond which a streamline flow becomes turbulent is called the critical velocity.

According to Osborne Reynolds, the critical velocity is given by

$$v_{c} = \dfrac{R_{n}\eta}{\rho d}$$

where $v_{c}$ = critical velocity of the fluid, $R_{n}$ = Reynolds number, $\eta$ = coefficient of viscosity, $\rho$ = density of the fluid, $d$ = diameter of the tube.

Reynolds number

Rearranging the same equation,

$$R_{n} = \dfrac{\rho v_{c} d}{\eta}$$

• Reynolds number is a pure number: it has no unit and no dimensions.
• For $R_{n} < 1000$ the flow of a fluid is streamline.
• For $R_{n} > 2000$ the flow is turbulent.
• For $R_{n}$ between 1000 and 2000 the flow is unsteady, changing from streamline to turbulent.

2.6.1 Viscosity

When water is poured from a glass it flows freely and quickly, but syrup or honey flows slowly and sticks to the container. The difference is due to fluid friction, both within the fluid itself and between the fluid and its surroundings. This property is called viscosity.

Observing a river, the water near the banks flows slowly and the water at the centre flows fastest, which shows that there is an opposing force between adjacent layers affecting their relative motion.

Definition: viscosity is that property of a fluid by virtue of which the relative motion between different layers of the fluid experiences a dragging force. This force is called the viscous drag.

• In liquids the viscous drag is due to short range molecular cohesive forces; in gases it is due to collisions between fast moving molecules.
• In both liquids and gases, as long as the relative velocity between the layers is small, the viscous drag is proportional to the relative velocity. In a turbulent flow the viscous drag increases rapidly and becomes proportional to higher powers of the relative velocity.
• The only fluid that is almost non-viscous is liquid helium at about 2 K.

Diagram to draw: a viscous flow through a pipe, drawn as arrows of different lengths — the central layer the longest (fastest) and the outermost layers the shortest (slowest); beside it a non-viscous flow, drawn as arrows of equal length, with no dragging force between layers.

Velocity gradient: the rate of change of velocity $dv$ with distance $dx$ measured from a stationary layer is called the velocity gradient, $dv/dx$.

2.6.2 Coefficient of viscosity

According to Newton's law of viscosity, for a streamline flow the viscous force $f$ acting on any layer is directly proportional to the area $A$ of the layer and to the velocity gradient:

$$f \propto A\left(\dfrac{dv}{dx}\right) \qquad\Longrightarrow\qquad f = \eta A\left(\dfrac{dv}{dx}\right)$$

where $\eta$ is a constant called the coefficient of viscosity of the liquid. Hence

$$\eta = \dfrac{f}{A\left(\dfrac{dv}{dx}\right)}$$

Definition: the coefficient of viscosity is the viscous force per unit area per unit velocity gradient.

• SI unit: N s m$^{-2}$, also called decapoise. CGS unit: poise; $1$ decapoise $= 10$ poise, so $1$ poise $= 0.1$ N s m$^{-2}$.
• Note carefully: $A$ in this expression is not the cross sectional area — it is the area of the layer parallel to the direction of the flow.
• For a plate of area $A$ dragged with velocity $dv$ over an oil layer of thickness $dx$ by a force $F$, $\eta = \dfrac{F\,dx}{A\,dv}$.

Effect of temperature on viscosity

• For most liquids the coefficient of viscosity decreases as the temperature increases, because at higher temperatures the molecules are farther apart and the cohesive forces are less effective.
• For gases the coefficient of viscosity increases with temperature, because the molecules move faster and collide more often, increasing the internal friction.

Standard results

• $v_{c} = \dfrac{R_{n}\eta}{\rho d}$ and $R_{n} = \dfrac{\rho v_{c} d}{\eta}$; $R_{n}$ is dimensionless.
• $R_{n} < 1000$ streamline, $R_{n} > 2000$ turbulent, in between unsteady.
• $f = \eta A(dv/dx)$; $\eta = \dfrac{f}{A(dv/dx)}$, unit N s m$^{-2}$ (decapoise).