Fundamentals of Fluid Mechanics

Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Chapter 4

Fluid Kinematics - all with Video Answers

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Chapter Questions

Problem 1

Obtain a photograph/image of a situation in which a fluid is flowing. Print this photo and draw in some lines to represent how you think some streamlines may look. Write a brief paragraph to describe the acceleration of a fluid particle as it flows along cne of these streamlines.

Manish Jain

Manish Jain

Numerade Educator

Problem 2

The surface velocity of a river is measured at several locations $x$ and can be reasonably represented by
\[
V=V_{0}+\Delta V\left(1-e^{-a x}\right)
\]
where $V_{0}, \Delta V,$ and $a$ are constants. Find the Lagrangian description of the velocity of a fluid particle flowing along the surface if $x=$
0 at time $t=0$

Manish Jain

Numerade Educator

Problem 3

The velocity field of a flow is given by $\mathbf{V}=2 x^{2} t \hat{\mathbf{i}}+[4 y(t-1)$ $+2 x^{2}+1 j$ is where $x$ and $y$ are in meters and $t$ is in seconds. For fluid particles on the $x$ axis, determine the speed and direction of flow.

Narayan Hari

Numerade Educator

Problem 4

A two-dimensional velocity field is given by $u=1+y$ and $v=1 .$ Determine the equation of the streamline that passes through the origin. On a graph, plot this streamline.

Manish Jain

Numerade Educator

Problem 5

Streamlines are given in Cartesian coordinates by the equation.
\[
\psi=U\left(y-\frac{y}{x^{2}+y^{2}}\right) . \quad x^{2}-y^{2} \geq 1
\]
Plot the streamlines for $\psi=0,\pm 60.0625,$ and describe the physical situation represented by this equation. The parameter $U$ is an upstream uniform velocity of $1.0 \mathrm{ft} / \mathrm{s}$

Manish Jain

Numerade Educator

Problem 6

A flow can be visualized by plotting the velocity field as velocity vectors at representative locations in the flow as shown in Video $\mathrm{V} 4.2$ and Fig. E4.1. Consider the velocity field given in polar coordinates by $v_{r}=-10 / r,$ and $v_{\theta}=10 / r .$ This flow approximates a fluid swirling into a sink as shown in Fig. P4.6. Plot the velocity field at locations given by $r=1,2,$ and 3 with $\theta=0,30,60,$ and $90^{\circ}$

Narayan Hari

Numerade Educator

Problem 7

A car accelerates from rest to a final constant velocity $V_{f}$ and a police officer records the following velocities at various locations
$x$ along the highway:
$$\begin{array}{ll}
x=0 & V=0 \mathrm{mph} \\
x=100 \mathrm{ft} & V=34.8 \mathrm{mph} \\
x=200 \mathrm{ft} & V=47.6 \mathrm{mph} \\
x=300 \mathrm{ft} & V=52.3 \mathrm{mph} \\
x=400 \mathrm{ft} & V=54.0 \mathrm{mph} \\
x=1000 \mathrm{ft} & V=55.0 \mathrm{mph}=V_{f}
\end{array}$$
Find a mathematical Eulerian expression for the velocity $V$ traveled by the car as a function of the final velocity $V_{f}$ of the car and $x$ if $t=0$ at $V=0 .[$ Hint: Try an exponential fit to the data.

Manish Jain

Numerade Educator

Problem 8

The components of a velocity ficld are given by $u=x+y$ $v=x y^{3}+16,$ and $w=0 .$ Determine the location of any stagnation points $(\mathbf{V}=0)$ in the flow field.

Narayan Hari

Numerade Educator

Problem 9

A two-dimensional, unsteady velocity field is given by
\[
u=5 x(1+t) \text { and } v=5 y(-1+t)
\]
where $u$ is the $x$ -velocity component and $v$ the $y$ -velocity component. Find $x(t)$ and $y(t)$ if $x=x_{0}$ and $y=y_{0}$ at $t=0 .$ Do the velocity components represent an Eulerian description or a Lagrangian description?

Manish Jain

Numerade Educator

Problem 10

The velocity field of a flow is given by $u=-V_{0} y /\left(x^{2}+\right.$ $\left.y^{2}\right)^{1 / 2}$ and $v=V_{0} x /\left(x^{2}+y^{2}\right)^{1 / 2},$ where $V_{0}$ is a constant. Where in the flow field is the speed equal to $V_{0}$ ? Determine the equation of the streamlines and discuss the various characteristics of this flow.

Narayan Hari

Numerade Educator

Problem 11

A velocity field is given by $\mathbf{V}=x \hat{\mathbf{i}}+x(x-1)(y+1) \hat{\mathbf{j}}$
where $u$ and $v$ are in $f t / s$ and $x$ and $y$ are in feet. Plot the streamline that passes through $x=0$ and $y=0 .$ Compare this streamline with the streakline through the origin.

Manish Jain

Numerade Educator

Problem 12

From time $t=0$ to $t=5$ hr radioactive steam is released from a nuclear power plant accident located at $x=-1$ mile and $y=3$ miles. The following wind conditions are expected:
$\mathbf{V}=10 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}$ mph for $0<t<3 \mathrm{hr}, \mathbf{V}=15 \hat{\mathbf{i}}+8 \hat{\mathbf{j}}$ mph for $3<t<10 \mathrm{hr},$ and $\mathbf{V}=5 \hat{\mathbf{i}}$ mph for $t>10$ hr. Draw to scale the expected streakline of the steam for $t=3,10,$ and 15 hr.

Manish Jain

Numerade Educator

Problem 13

The $x$ and $y$ components of a velocity field are given by $u=x^{2} y$ and $v=-x y^{2} .$ Determine the equation for the streamlines of this flow and compare it with those in Example $4.2 .$ Is the flow in this problem the same as that in Example $4.2 ?$ Explain.

Narayan Hari

Numerade Educator

Problem 14

In addition to the customary horizontal velocity components of the air in the atmosphere (the "wind"), there often are vertical air currents (thermals) caused by buoyant effects due to uneven heating of the air as indicated in Fig. $\mathrm{P} 4.14$. Assume that the velocity field in a certain region is approximated by $u=u_{0}$ $v=v_{0}(1-y / h)$ for $0<y<h,$ and $u=u_{0}, v=0$ for $y>h$ Plot the shape of the streamline that passes through the origin for values of $u_{0} / v_{0}=0.5,1,$ and 2.

Manish Jain

Numerade Educator

Problem 15

A test car is traveling along a level road at $88 \mathrm{km} / \mathrm{hr}$. In order to study the acceleration characteristics of a newly installed engine, the car accelerates at its maximum possible rate. The test crew records the following velocities at various locations along the level road:
$$\begin{array}{lll}
x=0 & V=88.5 \mathrm{km} / \mathrm{hr} \\
x=0.1 \mathrm{km} & V=93.1 \mathrm{km} / \mathrm{hr} \\
x=0.2 \mathrm{km} & V=98.3 \mathrm{km} / \mathrm{hr} \\
x=0.3 \mathrm{km} & V=104.0 \mathrm{km} / \mathrm{hr} \\
x=0.4 \mathrm{km} & V=110.3 \mathrm{km} / \mathrm{hr} \\
x=0.5 \mathrm{km} & V=117.2 \mathrm{km} / \mathrm{hr} \\
x=1.0 \mathrm{km} & V=164.5 \mathrm{km} / \mathrm{hr}
\end{array}$$
A preliminary study shows that these data follow an equation of the form $V=A\left(1+e^{B x}\right),$ where $A$ and $B$ are positive constants. Find $A$ and $B$ and a Lagrangian expression $V=V\left(V_{0}, t\right),$ where $V_{0}$ is the car velocity at time $t=0$ when $x=0$

Manish Jain

Numerade Educator

Problem 16

For any steady flow the streamlines and streaklines are the same. For most unsteady flows this is not true. However, there are unsteady flows for which the streamlines and streaklines are the same. Describe a flow field for which this is true.

Manish Jain

Numerade Educator

Problem 17

A tornado has the following velocity components in polar coordinates:
\[
V_{r}=-\frac{C_{1}}{r} \quad \text { and } \quad V_{\theta}=-\frac{C_{2}}{r}
\]
Note that the air is spiraling inward. Find an equation for the streamlines. $r$ and $\theta$ are polar coordinates.

Narayan Hari

Numerade Educator

Problem 18

(See The Wide World of Fluids article titled "Follow those particles," Section $4.1 .$ ) Two photographs of four particles in a flow past a sphere are superposed as shown in Fig. P4.18. The time interval between the photos is $\Delta t=0.002 \mathrm{s}$. The locations of the particles, as determined from the photos, are shown in the table.
(a) Determine the fluid velocity for these particles.
(b) Plot a graph to compare the results of part (a) with the theoretical velocity, which is given by $V=V_{0}\left(1+a^{3} / x^{3}\right),$ where $a$ is the sphere radius and $V_{0}$ is the fluid speed far from the sphere.
$$\begin{array}{ccc}
\hline \text { Particle } & x \text { at } t=0 \mathrm{s}(\mathrm{ft}) & x \text { at } t=0.002 \mathrm{s}(\mathrm{ft}) \\
\hline 1 & -0.500 & -0.480 \\
2 & -0.250 & -0.232 \\
3 & -0.140 & -0.128 \\
4 & -0.120 & -0.112
\end{array}$$

Manish Jain

Numerade Educator

Problem 19

Air flows steadily through a circular, constant-diameter duct. The air is perfectly inviscid, so the velocity profile is flat across cach flow area. However, the air density decreases as the air flows down the duct. Is this a one-, two-, or three-dimensional flow?

Narayan Hari

Numerade Educator

Problem 20

A constant-density fluid flows in the converging, twodimensional channel shown in Fig. P4.20. The width perpendicular to the paper is quite large compared to the channel height. The velocity in the $z$ direction is zero. The channel half-height, $Y$, and the fluid $x$ velocity, $u,$ are given by
\[
Y=\frac{Y_{0}}{1+x / \ell} \quad \text { and } \quad u=u_{0}\left(1+\frac{x}{\ell}\right)\left[1-\left(\frac{y}{Y}\right)^{2}\right]
\]
Where $x, y, Y,$ and $\ell$ are in meters, $u$ is in $\mathrm{m} / \mathrm{s}, u_{0}=1.0 \mathrm{m} / \mathrm{s},$ and $Y_{0}=1.0 \mathrm{m} .$ (a) Is this flow steady or unsteady? Is it one-dimensional, two-dimensional, or three-dimensional?
(b) Plot the velocity distribution $u(y)$ at $x / \ell=0,0.5,$ and $1.0 .$ Use $y / Y$ values of 0
\[
\pm 0.2,\pm 0.4,\pm 0.6,\pm 0.8, \text { and }\pm 1.0
\]

Ajay Singhal

Numerade Educator

Problem 21

Pathlines and streaklines provide ways to visualize flows. Another technique would be to instantly inject a line of dye across streamlines and observe how this line moves as time increases. For example, consider the initially straight dye line injected in front of the circular cylinder shown in Fig. P4.21. Discuss how this dye line would appear at later times. How would you calculate the location of this line as a function of time? This type of line is called a timeline.

Manish Jain

Numerade Educator

Problem 22

Classify the following flows as one- , two-, or three-dimensional. Sketch a few streamlines for each.
(a) Rainwater flow down a wide driveway
(b) Flow in a straight horizontal pipe
(c) Flow in a straight pipe inclined upward at a $45^{\circ}$ angle
(d) Flow in a long pipe that follows the ground in hilly country
(e) Flow over an airplane
(f) Wind blowing past a tall telephone
(g) Flow in the impeller of a centrifugal pump

Manish Jain

Numerade Educator

Problem 23

The velocity components of $u$ and $v$ of a two-dimensional flow are given by
\[
u=a x+\frac{b x}{x^{2} y^{2}} \quad \text { and } \quad v=a y+\frac{b y}{x^{2} y^{2}}
\]
where $a$ and $b$ are constants. Calculate the acceleration.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 24

Air is delivered through a constant-diameter duct by a fan. The air is inviscid, so the fluid velocity profile is "flat" across each cross section. During the fan start-up, the following velocities were measured at the time $t$ and axial positions $x.$
$$\begin{array}{cccc}
& x=0 & x=10 \mathrm{m} & x=20 \mathrm{m} \\
\hline t=0 \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} \\
t=1.0 \mathrm{s} & V=1.00 \mathrm{m} / \mathrm{s} & V=1.20 \mathrm{m} / \mathrm{s} & V=1.40 \mathrm{m} / \mathrm{s} \\
t=2.0 \mathrm{s} & V=1.70 \mathrm{m} / \mathrm{s} & V=1.80 \mathrm{m} / \mathrm{s} & V=1.90 \mathrm{m} / \mathrm{s} \\
t=3.0 \mathrm{s} & V=2.10 \mathrm{m} / \mathrm{s} & V=2.15 \mathrm{m} / \mathrm{s} & V=2.20 \mathrm{m} / \mathrm{s}
\end{array}$$
Calculate the local acceleration, the convective acceleration. and the total acceleration at $t=1.0 \mathrm{s}$ and $x=10 \mathrm{m} .$ What is the local acceleration after the fan has reached a steady air flow rate?

Manish Jain

Numerade Educator

Problem 25

Water flows through a constant diameter pipe with a uniform velocity given by $\mathbf{V}=(8 / t+5) \hat{\mathbf{j}} \mathrm{m} / \mathrm{s},$ where $t$ is in seconds. Determine the acceleration at time $t=1,2,$ and $10 \mathrm{s}$

Narayan Hari

Numerade Educator

Problem 26

The velocity of air in the diverging pipe shown in Fig. P4.26 is given by $V_{1}=4 t \mathrm{ft} / \mathrm{s}$ and $V_{2}=2 t \mathrm{ft} / \mathrm{s},$ where $t$ is in seconds.
(a) Determine the local acceleration at points (1) and $(2) .$ (b) Is the average convective acceleration between these two points negative, zero, or positive? Explain.

Narayan Hari

Numerade Educator

Problem 27

A certain flow field has the velocity vector $\mathbf{V}=\frac{-2 x y z}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{i}+\frac{\left(x^{2}-y^{2}\right) z}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{j}+\frac{y}{x^{2}+y^{2}} \mathbf{k} .$ Find the acceleration vector for this flow.

Manish Jain

Numerade Educator

Problem 28

Determine the $x$ -component of the acceleration, $a_{x}$ along the centerline $(y=0)$ for the flow of Problem 4.20 . Can you determine the acceleration vector at a location not on the centerline? Why or why not?

Manish Jain

Numerade Educator

Problem 29

The velocity of the water in the pipe shown in Fig. P4.29 is given by $V_{1}=0.50 t \mathrm{m} / \mathrm{s}$ and $V_{2}=1.0 t \mathrm{m} / \mathrm{s},$ where $t$ is in seconds. Determine the local acceleration at points (1) and $(2) .$ Is the average convective acceleration between these two points negative, zero, or positive? Explain.

Narayan Hari

Numerade Educator

Problem 30

A shock wave is a very thin layer (thickness $=\ell$ ) in a highspeed (supersonic) gas flow across which the flow properties (velocity, density, pressure, etc.) change from state ( 1 ) to state (2) as shown in Fig. P4.30. If $V_{1}=1800 \mathrm{fps}, V_{2}=700 \mathrm{fps},$ and $\ell=10^{-4} \mathrm{in.}$ estimate the average deceleration of the gas as it flows across the shock wave. How many $g$ 's deceleration does this represent?

Narayan Hari

Numerade Educator

Problem 31

Estimate the average acceleration of water as it travels through the nozzie on your garden hose. List all assumptions and show all calculations.

Narayan Hari

Numerade Educator

Problem 32

A stream of water from the faucet strikes the bottom of the sink. Estimate the maximum acceleration experienced by the water particles. List all assumptions and show calculations.

Manish Jain

Numerade Educator

Problem 33

As a valve is opened, water flows through the diffuser shown in Fig. $\mathrm{P} 4.33$ at an increasing flowrate so that the velocity along the centerline is given by $\mathbf{V}=u \hat{\mathbf{i}}=V_{0}\left(1-e^{-\alpha}\right)(1-x / \ell) \hat{\mathbf{i}},$ where
$u_{0}, c,$ and $\ell$ are constants. Determine the acceleration as a function of $x$ and $t .$ If $V_{0}=10 \mathrm{ft} / \mathrm{s}$ and $\ell=5 \mathrm{ft},$ what value of $c$ (other than $c=0$ ) is needed to make the acceleration zero for any $x$ at $t=1 \mathrm{s} ?$ Explain how the acceleration can be zero if the flowrate is increasing with time.

Manish Jain

Numerade Educator

Problem 34

The fluid velocity along the $x$ axis shown in Fig. $\mathrm{P} 4.34$ changes from $6 \mathrm{m} / \mathrm{s}$ at point $A$ to $18 \mathrm{m} / \mathrm{s}$ at point $B$. It is also known that the velocity is a linear function of distance along the streamline. Determine the acceleration at points $A, B,$ and $C$ Assume steady flow.

Prem Bijarniya

Numerade Educator

Problem 35

A fluid flows along the $x$ axis with a velscity given by $\mathbf{V}=(x / t) \hat{\mathbf{i}},$ where $x$ is in feet and $t$ in seconds. (a) Plot the speed for $0 \leq x \leq 10$ ft and $t=3$ s. (b) Plot the speed for $x=7$ ft and $2 \leq t \leq 4 \mathrm{s},$ (c) Determine the local and convective acceleration.
(d) Show that the acceleration of any fluid particle in the flow is zero.
(e) Explain physically how the velocity of a particle in this unsteady flow remains constant throughout its motion.

Manish Jain

Numerade Educator

Problem 36

A constant-density fluid flows through a converging section having an area $A$ given by
\[
A=\frac{A_{0}}{1+(x / \ell)}
\]
where $A_{0}$ is the area at $x=0 .$ Determine the velocity and acceleration of the fluid in Eulerian form and then the velocity and acceleration of a fluid particle in Lagrangian form. The velocity is $V_{0}$ at $x=0$ when $t=0$

Ajay Singhal

Numerade Educator

Problem 38

A hydraulic jump is a rather sudden change in depth of a liquid layer as it flows in an open channel as shown in Fig. P4.37 and Video $V 10.12 .$ In a relatively short distance (thickness $=\ell$ ) the liquid depth changes from $z_{1}$ to $z_{2},$ with a corresponding change in velocity from $V_{1}$ to $V_{2}$. If $V_{1}=1.20 \mathrm{ft} / \mathrm{s}, V_{2}=0.30 \mathrm{ft} / \mathrm{s},$ and $\ell=0.02 \mathrm{ft}$, estimate the average deceleration of the liquid as it flows across the hydraulic jump. How many $g$ 's deceleration does this represent?

Manish Jain

Numerade Educator

Problem 38

A fluid particle flowing along a stagnation streamline, as shown in Video $V 4.9$ and Fig. $P 4.38,$ slows down as it approaches the stagnation point. Measurements of the dye flow in the video indicate that the location of a particle starting on the stagnation streamline a distance $s=0.6 \mathrm{ft}$ upstream of the stagnation point at $t=0$ is given approximately by $s=0.6 e^{-0.5 y},$ where $t$ is in seconds and $s$ is in feet. (a) Determine the speed of a fluid particle es a function of time, $V_{\text {particle }}(t),$ as it flows along the streamline.
(b) Determine the speed of the fluid as a function of position along the streamline, $V=V(s)$
(c) Determine the fluid acceleration along the streamline as a function of position, $a_{s}=a_{s}(s)$

Manish Jain

Numerade Educator

Problem 39

A nozzle is designed to accelerate the fluid from $V_{1}$ to $V_{2}$ in a linear fashion. That is, $V=a x+b$, where $a$ and $b$ are constants. If the flow is constant with $V_{1}=10 \mathrm{m} / \mathrm{s}$ at $x_{1}=0$ and $V_{2}=25 \mathrm{m} / \mathrm{s}$ at $x_{2}=1 \mathrm{m},$ determine the local acceleration, the convective acceleration, and the acceleration of the fluid at points (1) and (2)

Narayan Hari

Numerade Educator

Problem 40

An incompressible fluid flows through the converging duct shown in Fig. P4.40 $a$ with velocity $V_{0}$ at the entrance. Measurements indicate that the actual velocity of the fluid near the wall of the duct along streamline $A-F$ is as shown in Fig. $\mathrm{P} 4.40 b$ Sketch the component of acceleration along this streamline, $a$, as a function of $s .$ Discuss the important characteristics of your result.

Manish Jain

Numerade Educator

Problem 41

Air flows steadily through a variable area pipe with a velocity of $\mathbf{V}=u(x) \hat{\mathbf{i}}$ ft $/ \mathrm{s},$ where the approximate measured values of $u(x)$ are given in the table. Plot the acceleration as a function of $x$ for $0 \leq x \leq 12$ in. Plot the acceleration if the flowrate is increased by a factor of $N$ (i.e., the values of $u$ are increased by a factor of $N$ ) for $N=2,4,10$
$$\begin{array}{cccc}
\hline x \text { (in.) } & u(\mathrm{ft} / \mathrm{s}) & x \text { (in.) } & u(\mathrm{ft} / \mathrm{s}) \\
\hline 0 & 10.0 & 7 & 20.1 \\
1 & 10.2 & 8 & 17.4 \\
2 & 13.0 & 9 & 13.5 \\
3 & 20.1 & 10 & 11.9 \\
4 & 28.3 & 11 & 10.3 \\
5 & 28.4 & 12 & 10.0 \\
6 & 25.8 & 13 & 10.0 \\
\hline
\end{array}$$

Ajay Singhal

Numerade Educator

Problem 42

As is indicated in Fig. $P 4.42,$ the speed of exhaust in a car's cxhaust pipe varies in time and distance because of the periodic nature of the engine's operation and the damping effect
with distance from the engine. Assume that the speed is given
\[
\text { by } \quad V=V_{0}\left[1+a e^{-b x} \sin (\omega t)\right], \quad \text { where } \quad V_{0}=8 \text { fps, } a=0.05
\]
$b=0.2 \mathrm{ft}^{-1},$ and $\omega=50 \mathrm{rad} / \mathrm{s} .$ Calculate and plot the fluid acceleration at $x=0,1,2,3,4,$ and 5 ft for $0 \leq t \leq \pi / 25$ s.

Manish Jain

Numerade Educator

Problem 43

Water flows down the face of the dam shown in Fig. $\mathrm{P} 4.43$ The face of the dam consists of two circular arcs with radii of 10 and $20 \mathrm{ft}$ as shown. If the speed of the water along streamline $\mathrm{A}-\mathrm{B}$ is approximately $V=(2 g h)^{1 / 2},$ where the distance $h$ is as indicated, plot the normal acceleration as a function of distance along the streamline, $a_{n}=a_{n}(s)$

Manish Jain

Numerade Educator

Problem 44

Water flows over the crest of a dam with speed $V$ as shown in Fig. $\mathrm{P} 4.44 .$ Determine the speed if the magnitude of the normal acceleration at point (1) is to equal the acceleration of gravity, $g$

Rashmi Sinha

Numerade Educator

Problem 45

Water flows under the sluice gate shown in Fig. P4.45. If $V_{1}=3 \mathrm{m} / \mathrm{s},$ what is the normal acceleration at point (1)$?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

A fluid flows past a sphere with an upstream velocity of $V_{0}=40 \mathrm{m} / \mathrm{s}$ as shown in Fig. $\mathrm{P} 4.46 .$ From a more advanced theory it is found that the speed of the fluid along the front part of the sphere is $V=\frac{3}{2} V_{0} \sin \theta .$ Determine the streamwise and normal components of acceleration at point $A$ if the radius of the sphere is $a=0.20 \mathrm{m}.$

Narayan Hari

Numerade Educator

Problem 47

Assume that the streamlines for the wingtip vortices from an airplane (see Fig. P4.15 and Video V4.6) can be approximated by circles of radius $r$ and that the speed is $V=K / r$, where $K$ is a constant. Determine the streamline acceleration, $a_{s}$, and the normal acceleration, $a_{n}$ for this flow.

Narayan Hari

Numerade Educator

Problem 48

The velocity components for steady flow through the nozzle shown in Fig. P4.48 are $u=-V_{0} x / \ell$ and $v=V_{0}[1+(y / \ell)]$ where $V_{0}$ and $\ell$ are constants. Determine the ratio of the magnitude of the acceleration at point (1) to that at point (2)

Narayan Hari

Numerade Educator

Problem 49

Water flows through the curved hose shown in Fig. P4.49 with an increasing speed of $V=10 t$ ft $f$ s, where $t$ is in seconds. For $t=2$ s determine (a) the component of acceleration along the streamline, (b) the component of acceleration normal to the streamline, and (c) the net acceleration (magnitude and direction).

Narayan Hari

Numerade Educator

Problem 50

Water flows though the slit at the bottom of a two-dimensional water trough as shown in Fig. P4.50. Throughout most of the trough the flow is approximately radial (along rays from $O$ ) with a velocity of $V=c / r,$ where $r$ is the radial coordinate and $c$ is a constant. If the velocity is $0.04 \mathrm{m} / \mathrm{s}$ when $r=0.1 \mathrm{m},$ determine the acceleration at points $A$ and $B$

Narayan Hari

Numerade Educator

Problem 51

Air flows from a pipe into the region between two parallel circular disks as shown in Fig. P4.51. The fluid velocity in the gap between the disks is closely approximated by $V=V_{0} R / r,$ where $R$ is the radius of the disk, $r$ is the radial coordinate, and $V_{0}$ is the fluid velocity at the edge of the disk. Determine the acceleration for $r=1,2,$ or 3 ft if $V_{0}=5 \mathrm{ft} / \mathrm{s}$ and $R=3 \mathrm{ft}.$

Manish Jain

Numerade Educator

Problem 52

Air flows into a pipe from the region between a circular disk and a cone as shown in Fig. P4.52. The fluid velocity in the gap between the disk and the cone is closely approximated by $V=V_{0} R^{2} / r^{2},$ where $R$ is the radius of the disk, $r$ is the radial coordinate, and $V_{0}$ is the fluid velocity at the edge of the disk. Determine the acceleration for $r=0.5$ and $2 \mathrm{ft}$ if $V_{0}=5 \mathrm{ft} / \mathrm{s}$ and $R=2 \mathrm{ft}$

Narayan Hari

Numerade Educator

Problem 53

Fluid flows through a pipe with a velocity of $2.0 \mathrm{ft} / \mathrm{s}$ and is being heated, so the fluid temperature $T$ at axial position $x$ increases at a steady rate of $30.0^{\circ} \mathrm{F} / \mathrm{min}$. In addition, the fluid temperature is increasing in the axial direction at the rate of $2.0^{\circ} \mathrm{F} / \mathrm{ft}$ Find the value of the material derivative $D T / D t$ at position $x.$

Manish Jain

Numerade Educator

Problem 54

A gas flows along the $x$ axis with a speed of $V=5 x \mathrm{m} / \mathrm{s}$ and
eressure of $p=10 x^{2} \mathrm{N} / \mathrm{m}^{2},$ where $x$ is in meters.
(a) Determine the time rate of change of pressure at the fixed location $x=1$
(b) Determine the time rate of change of pressure for a fluid particle flowing past $x=1 .$ (c) Explain without using any equations why the answers to parts (a) and (b) are different.

Manish Jain

Numerade Educator

Problem 55

Assume the temperature of the exhaust in an exhaust pipe can be approximated by $T=T_{0}\left(1+a e^{-b x}\right)[1+c \cos (\omega t)],$ where
\[
T_{0}=100^{\circ} \mathrm{C}, a=3, b=0.03 \mathrm{m}^{-1}, c=0.05, \text { and } \omega=100 \mathrm{rad} / \mathrm{s}
\]
If the exhaust speed is a constant $3 \mathrm{m} / \mathrm{s}$, determine the time rate of change of temperature of the fluid particles at $x=0$ and $x=4 \mathrm{m}$ when $t=0$

Manish Jain

Numerade Educator

Problem 56

A bicyclist leaves from her home at 9 A.M. and rides to a beach $40 \mathrm{mi}$ away. Because of a breeze off the ocean, the temperature at the beach remains $60^{\circ} \mathrm{F}$ throughout the day. At the cyclist's home the temperature increases linearly with time, going from $60^{\circ} \mathrm{F}$ at 9 A.M. to $80^{\circ} \mathrm{F}$ by 1 P.M. The temperature is assumed to vary linearly as a function of position between the cyclist's home and the beach. Determine the rate of change of temperature observed by the cyclist for the following conditions: (a) as she pedals 10 mph through a town 10 mi from her home at 10 A.M.; (b) as she eats lunch at a rest stop $30 \mathrm{mi}$ from her home at noon; (c) as she arrives enthusiastically at the beach at 1 P.M., pedaling 20 mph.

Ajay Singhal

Numerade Educator

Problem 57

The following pressures for the air flow in Problem 4.24 were measured:
$$\begin{array}{cccc}
& x=0 & x=10 \mathrm{m} & x=20 \mathrm{m} \\
\hline t=0 \mathrm{s} & p=101 \mathrm{kPa} & p=101 \mathrm{kPa} & p=101 \mathrm{kPa} \\
t=1.0 \mathrm{s} & p=121 \mathrm{kPa} & p=116 \mathrm{kPa} & p=111 \mathrm{kPa} \\
t=2.0 \mathrm{s} & p=141 \mathrm{kPa} & p=131 \mathrm{kPa} & p=121 \mathrm{kPa} \\
t=3.0 \mathrm{s} & p=171 \mathrm{kPa} & p=151 \mathrm{kPa} & p=131 \mathrm{kPa} .
\end{array}$$
Find the local rate of change of pressure $\partial p / \partial t$ and the convective rate of change of pressure $V$ on $/ \partial x$ at $t=2.0 \mathrm{s}$ and $x=10 \mathrm{m}.$

Manish Jain

Numerade Educator

Problem 58

Obtain a photograph/image of a situation in which a fluid is flowing. Print this photo and draw a control volume through which the fluid flows. Write a brief paragraph that describes how the fluid flows into and out of this control volume.

Manish Jain

Numerade Educator

Problem 59

In the region just downstream of a sluice gate, the water may develop a reverse flow region as is indicated in Fig. P4.59 and Video $V 10.9 .$ The velocity profile is assumed to consist of two uniform regions, one with velocity $V_{a}=10$ fps and the other with $V_{b}=3$ fps. Determine the net flowrate of water zeross the portion of the control surface at section (2) if the channel is $20 \mathrm{ft}$ wide.

Manish Jain

Numerade Educator

Problem 60

At time $t=0$ the valve on an initially empty (perfect vacuum, $\rho=0$ ) tank is opened and air rushes in. If the tank has a volume of $V_{0}$ and the density of air within the tank increases as $\rho=\rho_{\infty}\left(1-e^{-b t}\right),$ where $b$ is a constant, determine the time rate of change of mass within the tank.

Manish Jain

Numerade Educator

Problem 61

From calculus, one obtains the following formula (Leibnitz rule) for the time derivative of an integral that contains time in both the integrand and the limits of the integration:
\[
\frac{d}{d t} \int_{x_{1}(t)}^{x_{2}(t)} f(x, t) d x=\int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial t} d x+f\left(x_{2}, t\right) \frac{d x_{2}}{d t}-f\left(x_{1}, t\right) \frac{d x_{1}}{d t}
\]
Discuss how this formula is related to the time derivative of the total amount of a property in a system and to the Reynolds transport theorem.

Manish Jain

Numerade Educator

Problem 62

Air enters an elbow with a uniform spead of $10 \mathrm{m} / \mathrm{s}$ as shown in Fig. $\mathrm{P} 4.62 .$ At the exit of the elbow, the velocity profile is not uniform. In fact, there is a region of separation or reverse flow. The fixed control volume $A B C D$ coincides with the system at time $t=0 .$ Make a sketch to indicate (a) the system at time $t=0.01 \mathrm{s}$ and (b) the fluid that has entered and exited the control volume in that time period.

Manish Jain

Numerade Educator

Problem 63

A layer of oil flows down a vertical plate as shown in Fig. P4.63 with a velocity of $\mathbf{V}=\left(V_{0} / h^{2}\right)\left(2 h x-x^{2}\right) \hat{\mathbf{j}}$ where $V_{0}$ and $h$ are constants.
(a) Show that the fluid sticks to the plate and that the shear stress at the edge of the layer $(x=h)$ is zero.
(b) Determine the flowrate across surface $A B .$ Assume the width of the plate is $b$. (Note: The velocity profile for laminar flow in a pipe has a similar shape. See Video $\mathbf{V} 6.13 .$

Manish Jain

Numerade Educator

Problem 64

Figure $\mathrm{P} 4.64$ shows a fixed control volume. It has a volume $V_{0}=1.0 \mathrm{ft}^{3},$ a flow area $A=1.0 \mathrm{ft}^{2},$ and a length $\ell_{0}=1.0 \mathrm{ft}$
Position $x$ represents the center of the control volume where the fluid velocity $V_{0}=1.0 \mathrm{ft} / \mathrm{s}$ and the density $\rho_{0}=1.800 \mathrm{slug} / \mathrm{ft}^{3}$
Also, at position $x$ the fluid density does not change locally with time but decreases in the axial direction at the linear rate of 0.25 slug/ft $^{4}$. Use the system or Lagrangian approach to evaluate dp/dt. Compare this result with that of the material derivative and flux terms.

Ajay Singhal

Numerade Educator

Problem 65

Find $D V / D t$ for the system in Problem 4.64.

Manish Jain

Numerade Educator

Problem 66

Water enters a 5 -ft-wide, 1 -ft-deep channel as shown in Fig. P4.66. Across the inlet the water velocity is $6 \mathrm{ft} / \mathrm{s}$ in the center portion of the channel and $1 \mathrm{ft}$ s in the remainder of it. Farther downstream the water flows at a uniform 2 ft/s velocity across the cntire channel. The fixed control volume $A B C D$ coincides with the system at time $t=0 .$ Make a sketch to indicate (a) the system at time $t=0.5 \mathrm{s}$ and (b) the fluid that has entered and exited the control volume in that time period.

Manish Jain

Numerade Educator

Problem 67

Figure $P 4.67$ illustrates a system and fixed control volume at time $t$ and the system at a short time $\delta t$ later. The system temperature is $T=100^{\circ} \mathrm{F}$ at time $t$ and $T=103^{\circ} \mathrm{F}$ at time $t+\delta t,$ where $\delta t=0.1 \mathrm{s} .$ The system mass, $m,$ is 2.0 slugs, and 10 percent of it moves out of the control volume in $\delta t=0.1$ s. The energy per unit mass $\tilde{u}$ is $c_{v} T,$ where $c_{v}=32.0$ Btu/slug $\cdot^{\circ} \mathrm{F}$. The energy $U$ of the system at any time $t$ is $m \tilde{u}$. Use the system or Lagrangian approach to evaluate $D U / D t$ of the system. Compare this result with the $D U / D t$ evaluated with that of the material time derivative and flux terms.

Manish Jain

Numerade Educator

Problem 68

The wind blows across a field with an approximate velocity profile as shown in Fig. P4.68. Use Eq. 4.16 with the parameter $b$ equal to the velocity to determine the momentum flowrate across the vertical surface $A-B,$ which is of unit depth into the paper.

Manish Jain

Numerade Educator

Problem 69

Water flows from a nozzle with a speed of $V=10 \mathrm{m} / \mathrm{s}$ and is collected in a container that moves toward the nozzle with a speed of $V_{c v}=2 \mathrm{m} / \mathrm{s}$ as shown in Fig. $\mathrm{P} 4.69 .$ The moving control surface consists of the inner surface of the container. The system consists of the water in the container at time $t=0$ and the water between the nozzle and the tank in the constant diameter stream at $t=0 .$ At time $t=0.1$ s what volume of the system remains outside of the control volume? How much water has entered the control volume during this time period? Repeat the problem for $t=0.3 \mathrm{s}$

Manish Jain

Numerade Educator