The Bernoulli equation is valid for steady, inviscid, incompressible flows with constant acceleration of gravity. Consider flow on a planet where the acceleration of gravity varies with height so that $g=g_{0}-c z$, where $g_{0}$ and c are constants. Integrate “F = ma” along a streamline to obtain the equivalent of the Bernoulli equation for this flow.

The table lists the relative abundance of mosquitoes (as measured by the mosquito-positive rate) versus the flow rate (measured as a percentage of maximum flow) of canal networks in Saga City, Japan.

(a) Make a scatter plot of the data.

(b) Find and graph the regression line.

(c) Use the linear model in part (b) to estimate the mosquito-positive rate if the canal flow is 70% of the maximum.

$$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Flow rate } \\ \text { (percent) } \end{array} & \begin{array}{c} \text { Mosquito positive } \\ \text { rate (percent) } \end{array} \\ \hline 0 & 22 \\ 10 & 16 \\ 40 & 12 \\ 60 & 11 \\ 90 & 6 \\ 100 & 2 \\ \hline \end{array}$$

A $5$-mm-diameter capillary tube is used as a viscometer for nils. When the flow rate is $0.071 \mathrm{~m}^3 / \mathrm{h}$, the measured pressure drop per unit length is $375\ \mathrm{kPa} / \mathrm{m}$. Compute the viscosity of the fluid. Is the flow laminar? Can you also find the density of the fluid?

Water enters a 10-cm-diameter pipe steadily with a uniform velocity of 3 m/s and exits with the turbulent flow velocity distribution given by $u=u_{\max }(1-r / R)^{1 / 7}.$ If the pressure drop along the pipe is 10 kPa, determine the drag force exerted on the pipe by water flow.

The given figure provides steady-state operating data for a well-insulated device with air entering at one location and exiting at another with a mass flow rate of $10 \mathrm{~kg} / \mathrm{s}$. Considering ideal gas behavior and negligible potential energy effects, determine (a) the direction of flow and (b) the power, in $\mathrm{kW}$.

Design an experiment to measure the viscosity of liquids using a vertical funnel with a cylindrical reservoir of height h and a narrow flow section of diameter D and length L. Making appropriate assumptions, obtain a relation for vis­cosity in terms of easily measurable quantities such as den­sity and volume flow rate. Is there a need for the use of a cor­rection factor?

Make a system similar to that shown in the given figure, in which air pressure at $400\ \mathrm{kPa}$ above the kerosene at $25^{\circ} \mathrm{C}$ is used to cause the flow. The horizontal distance between the two tanks is $32 \mathrm{~m}$. The desired minimum flow rate is $500 \mathrm{~L} / \mathrm{min}$.

What does the data pair (886,68) represent?

Length (mi) and Water Flow (1,000 ft$^3$/s) of Rivers Length: 2540, 1980, 1460, 1420, 1290, 1040, 886, 774, 724, 659 Flow: 76, 225, 41, 58, 56, 57, 68, 67, 67, 41

The air flow in a compressed air line is divided into two equal streams by a $T$-fitting in the line. The compressed air enters this $2.5-\mathrm{cm}$ diameter fitting at $1.6 \mathrm{~MPa}$ and $40^{\circ} \mathrm{C}$ with a velocity of $50 \mathrm{~m} / \mathrm{s}$. Each outlet has the same diameter as the inlet, and the air at these outlets has a pressure of $1.4 \mathrm{~MPa}$ and a temperature of $36^{\circ} \mathrm{C}$. Determine the velocity of the air at the outlets and the rate of change of flow energy (flow power) across the $T$-fitting.

Choose a long-throated flume from the given table that will carry a range of flow from $30\ \mathrm{gal} / \mathrm{min}$ to $500\ \mathrm{gal} / \mathrm{min}$. Calculate the head for each of these flows and then compute the flow that would result from four additional heads spaced approximately equally between them.

A flying airplane produces swirling flow near the end of its wings. In certain circumstances this flow can be approximated by the velocity field

$$u=-K y /\left(x^{2}+y^{2}\right) \text{and} v=K x /\left(x^{2}+y^{2}\right),$$

where K is a constant depending on various parameters associated with the airplane (i.e., its weight, speed) and x and y are measured from the center of the swirl. (a) Show that for this flow the velocity is inversely proportional to the distance from the origin. That is, $V=K /\left(x^{2}+y^{2}\right)^{1 / 2}$ (b) Show that the streamlines are circles.

The $x$ component of velocity in a steady, incompressible flow field in the $x y$ plane is $u=A\left(x^5-10 x^3 y^2+5 x y^4\right)$, where $A=$ $2 \mathrm{~m}^{-4} \cdot \mathrm{s}^{-1}$ and $x$ is measured in meters. Find the simplest $y$ component of velocity for this flow field. Evaluate the acceleration of a fluid particle at point $(x, y)=(1,3)$.

A flow is represented by the velocity field $\vec{V}=$ $\left(x^7-21 x^5 y^2+35 x^3 y^4-7 x y^6\right) \hat{i}+\left(7 x^6 y-35 x^4 y^3+21 x^2 y^5-y^7\right) \hat{j}$. Determine if the field is (a) a possible incompressible flow and (b) irrotational.

A highly viscous Newtonian liquid ($\rho=1300 \mathrm{kg} / \mathrm{m}^{3};$ $\mu= 6.0 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$) is contained in a long, vertical, 150-mm-diameter tube. Initially, the liquid is at rest, but when a valve at the bottom of the tube is opened flow commences. Although the flow is slowly changing with time, at any instant the velocity distribution is essentially parabolic; that is, the flow is quasi-steady. Some measurements show that the average velocity, V, is changing in accordance with the equation V = 0.1 t, with V in m/s when t is in seconds. (a) Show on a plot the velocity distribution ($v_{z}$ vs. r) at t = 2 s, where $v_{z}$ is the velocity and r is the radius from the center of the tube. (b)Verify that the flow is laminar at this instant.