14.7 Viscosity and turbulence  (Page 7/14)

The arterioles (small arteries) leading to an organ constrict in order to decrease flow to the organ. To shut down an organ, blood flow is reduced naturally to 1.00% of its original value. By what factor do the radii of the arterioles constrict?

A spherical particle falling at a terminal speed in a liquid must have the gravitational force balanced by the drag force and the buoyant force. The buoyant force is equal to the weight of the displaced fluid, while the drag force is assumed to be given by Stokes Law, $F_{\text{s}}=6\pi r\eta v.$ Show that the terminal speed is given by $v=\frac{2R^{2}g}{9\eta }\left({\rho }_{\text{s}}-{\rho }_1\right)$ , where R is the radius of the sphere, ${\rho }_{\text{s}}$ is its density, and ${\rho }_1$ is the density of the fluid, and $\eta$ the coefficient of viscosity.

Using the equation of the previous problem, find the viscosity of motor oil in which a steel ball of radius 0.8 mm falls with a terminal speed of 4.32 cm/s. The densities of the ball and the oil are 7.86 and 0.88 g/mL, respectively.

A skydiver will reach a terminal velocity when the air drag equals his or her weight. For a skydiver with a large body, turbulence is a factor at high speeds. The drag force then is approximately proportional to the square of the velocity. Taking the drag force to be $F_{\text{D}}=\frac{1}{2}\rho A{v}^2,$ and setting this equal to the skydiver’s weight, find the terminal speed for a person falling “spread eagle.”

(a) Verify that a 19.0% decrease in laminar flow through a tube is caused by a 5.00% decrease in radius, assuming that all other factors remain constant. (b) What increase in flow is obtained from a 5.00% increase in radius, again assuming all other factors remain constant?

When physicians diagnose arterial blockages, they quote the reduction in flow rate. If the flow rate in an artery has been reduced to 10.0% of its normal value by a blood clot and the average pressure difference has increased by 20.0%, by what factor has the clot reduced the radius of the artery?

An oil gusher shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. Neglecting air resistance but not the resistance of the pipe, and assuming laminar flow, calculate the pressure at the entrance of the 50.0-m-long vertical pipe. Take the density of the oil to be $900{\text{kg/m}}^3$ and its viscosity to be $1.00({\text{N/m}}^2)\cdot \text{s}$ (or $1.00\text{Pa}\cdot \text{s}$ ). Note that you must take into account the pressure due to the 50.0-m column of oil in the pipe.

Concrete is pumped from a cement mixer to the place it is being laid, instead of being carried in wheelbarrows. The flow rate is 200 L/min through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the pump is $8.00\times {10}^6{\text{N/m}}^2$ . (a) Calculate the resistance of the hose. (b) What is the viscosity of the concrete, assuming the flow is laminar? (c) How much power is being supplied, assuming the point of use is at the same level as the pump? You may neglect the power supplied to increase the concrete’s velocity.

Verify that the flow of oil is laminar for an oil gusher that shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. The vertical pipe is 50 m long. Take the density of the oil to be $900{\text{kg/m}}^3$ and its viscosity to be $1.00({\text{N/m}}^2)\cdot \text{s}$ (or $1.00\text{Pa}\cdot \text{s}$ ).