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Fluid Mechanics 4th

Frank M. White

Chapter 4

Differential Relations for a Fluid Particle

Educators


Problem 1

An idealized velocity field is given by the formula
$$\mathbf{V}=4 t x \mathbf{i}-2 t^{2} y \mathbf{j}+4 x z \mathbf{k}$$
Is this flow field steady or unsteady? Is it two- or three-dimensional? At the point $(x, y, z)=(-1,1,0),$ compute $(a)$
the acceleration vector and ( $b$ ) any unit vector normal to the acceleration.

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Problem 2

Flow through the converging nozzle in Fig. $P 4.2$ can be approximated by the one-dimensional velocity distribution
$$u \approx V_{0}\left(1+\frac{2 x}{L}\right) \quad v \approx 0 \quad w \approx 0$$
(a) Find a general expression for the fluid acceleration in the nozzle. ( $b$ ) For the specific case $V_{0}=10 \mathrm{ft} / \mathrm{s}$ and $L=$ 6 in, compute the acceleration, in $g$ 's, at the entrance and at the exit.

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Problem 3

A two-dimensional velocity field is given by
$$\mathbf{V}=\left(x^{2}-y^{2}+x\right) \mathbf{i}-(2 x y+y) \mathbf{j}$$
in arbitrary units. At $(x, y)=(1,2),$ compute $(a)$ the accelerations $a_{x}$ and $a_{y},(b)$ the velocity component in the direction $\theta=40^{\circ},(c)$ the direction of maximum velocity, and $(d)$ the direction of maximum acceleration.

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Problem 4

Suppose that the temperature field $T=4 x^{2}-3 y^{3},$ in arbitrary units, is associated with the velocity field of Prob. 4.3. Compute the rate of change $d T / d t$ at $(x, y)=(2,1)$.

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Problem 5

The velocity field near a stagnation point (see Example 1.10 ) may be written in the form
$$u=\frac{U_{0} x}{L} \quad v=-\frac{U_{0} y}{L} \quad U_{0} \text { and } L \text { are constants }$$
(a) Show that the acceleration vector is purely radial. $(b)$ For the particular case $L=1.5 \mathrm{m},$ if the acceleration at $(x,$ $y)=(1 \mathrm{m}, 1 \mathrm{m})$ is $25 \mathrm{m} / \mathrm{s}^{2},$ what is the value of $U_{0} ?$

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Problem 6

Assume that flow in the converging nozzle of Fig. P4.2 has the form $\mathbf{V}=V_{0}[1+(2 x) / L] \mathbf{i} .$ Compute $(a)$ the fluid acceleration at $x=L$ and $(b)$ the time required for a fluid particle to travel from $x=0$ to $x=L$.

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Problem 7

Consider a sphere of radius $R$ immersed in a uniform stream $U_{0},$ as shown in Fig. P4.7. According to the theory of Chap. $8,$ the fluid velocity along streamline $A B$ is given by
$$\mathbf{V}=u \mathbf{i}=U_{0}\left(1+\frac{R^{3}}{x^{3}}\right) \mathbf{i}$$
Find $(a)$ the position of maximum fluid acceleration along $A B$ and $(b)$ the time required for a fluid particle to travel from $A$ to $B$.

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Problem 8

When a valve is opened, fluid flows in the expansion duct of Fig. 4.8 according to the approximation
$$\mathbf{V}=\mathbf{i} U\left(1-\frac{x}{2 L}\right) \tanh \frac{U t}{L}$$
Find $(a)$ the fluid acceleration at $(x, t)=(L, L / U)$ and $(b)$
the time for which the fluid acceleration at $x=L$ is zero. Why does the fluid acceleration become negative after condition (b)?

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Problem 9

A velocity field is given by $\mathbf{V}=\left(3 y^{2}-3 x^{2}\right) \mathbf{i}+C x y \mathbf{j}+0 \mathbf{k}$.
Determine the value of the constant $C$ if the flow is to be
$(a)$ incompressible and $(b)$ irrotational.

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Problem 10

Write the special cases of the equation of continuity for $(a)$ steady compressible flow in the $y z$ plane, $(b)$ unsteady incompressible flow in the $x z$ plane, $(c)$ unsteady compressible flow in the $y$ direction only, $(d)$ steady compressible flow in plane polar coordinates.

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Problem 11

Derive Eq. $(4.12 b)$ for cylindrical coordinates by considering the flux of an incompressible fluid in and out of the elemental control volume in Fig. 4.2.

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Problem 12

Spherical polar coordinates $(r, \theta, \phi)$ are defined in Fig. P4.12. The cartesian transformations are
$$\begin{aligned}
&x=r \sin \theta \cos \phi\\
&y=r \sin \theta \sin \phi\\
&z=r \cos \theta
\end{aligned}$$
The cartesian incompressible continuity relation $(4.12 a) \mathrm{can}$ be transformed to the spherical polar form
$$\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} v_{r}\right)+\frac{1}{r \sin \theta} \frac{\partial}{\partial \theta}\left(v_{\theta} \sin \theta\right)+\frac{1}{r \sin \theta} \frac{\partial}{\partial \phi}\left(v_{\phi}\right)=0$$
What is the most general form of $v_{r}$ when the flow is purely radial, that is, $v_{\theta}$ and $v_{\phi}$ are zero?

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Problem 13

A two-dimensional velocity field is given by
$$u=-\frac{K y}{x^{2}+y^{2}} \quad v=\frac{K x}{x^{2}+y^{2}}$$
where $K$ is constant. Does this field satisfy incompressible
continuity? Transform these velocities to polar components $v_{r}$ and $v_{\theta} .$ What might the flow represent?

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Problem 14

For incompressible polar-coordinate flow, what is the most general form of a purely circulatory motion, $v_{\theta}=v_{\theta}(r, \theta, t)$ and $v_{r}=0,$ which satisfies continuity?

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Problem 15

What is the most general form of a purely radial polar-coordinate incompressible-flow pattern, $v_{r}=v_{r}(r, \theta, t)$ and $v_{\theta}=0,$ which satisfies continuity?

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Problem 16

An incompressible steady-flow pattern is given by $u=x^{3}+$ $2 z^{2}$ and $w=y^{3}-2 y z .$ What is the most general form of the third component, $v(x, y, z),$ which satisfies continuity?

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Problem 17

A reasonable approximation for the two-dimensional incompressible laminar boundary layer on the flat surface in Fig. $\mathrm{P} 4.17$ is
$$u=U\left(\frac{2 y}{\delta}-\frac{y^{2}}{\delta^{2}}\right) \quad \text { for } y \leq \delta \quad \text { where } \delta=C x^{1 / 2}, C=\mathrm{const}$$
(a) Assuming a no-slip condition at the wall, find an expression for the velocity component $v(x, y)$ for $y \leq \delta .(b)$ Then find the maximum value of $v$ at the station $x=1 \mathrm{m}$ for the particular case of airflow, when $U=3 \mathrm{m} / \mathrm{s}$ and $\delta=$
$1.1 \mathrm{cm}$.

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Problem 18

A piston compresses gas in a cylinder by moving at constant speed $Y,$ as in Fig. P4.18. Let the gas density and length at $t=0$ be $\rho_{0}$ and $L_{0},$ respectively. Let the gas velocity vary linearly from $u=V$ at the piston face to $u=0$ at $x=L .$ If the gas density varies only with time, find an expression for $\rho(t)$.

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Problem 19

An incompressible flow field has the cylindrical components $v_{\theta}=C r, v_{z}=K\left(R^{2}-r^{2}\right), v_{r}=0,$ where $C$ and $K$ are constants and $r \leq R, z \leq L .$ Does this flow satisfy continuity? What might it represent physically?

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Problem 20

A two-dimensional incompressible velocity field has $u=$ $K\left(1-e^{-a y}\right),$ for $x \leq L$ and $0 \leq y \leq \infty,$ What is the most general form of $v(x, y)$ for which continuity is satisfied and $v=v_{0}$ at $y=0 ?$ What are the proper dimensions for constants $K$ and $a ?$

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Problem 21

Air flows under steady, approximately one-dimensional conditions through the conical nozzle in Fig. P4.21. If the speed of sound is approximately $340 \mathrm{m} / \mathrm{s}$, what is the minimum nozzle-diameter ratio $D_{e} / D_{0}$ for which we can safely neglect compressibility effects if $V_{0}=(a) 10 \mathrm{m} / \mathrm{s}$ and $(b)$ $30 \mathrm{m} / \mathrm{s} ?$

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Problem 22

Air at a certain temperature and pressure flows through a contracting nozzle of length $L$ whose area decreases linearly, $A \approx A_{0}[1-x /(2 L)] .$ The air average velocity increases nearly linearly from $76 \mathrm{m} / \mathrm{s}$ at $x=0$ to $167 \mathrm{m} / \mathrm{s}$ at $x=L$. If the density at $x=0$ is $2.0 \mathrm{kg} / \mathrm{m}^{3},$ estimate the density at $x=L$.

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Problem 23

A tank volume $\%$ contains gas at conditions $\left(\rho_{0}, p_{0}, T_{0}\right) .$ At time $t=0$ it is punctured by a small hole of area $A$. According to the theory of Chap. 9 , the mass flow out of such a hole is approximately proportional to $A$ and to the tank pressure. If the tank temperature is assumed constant and the gas is ideal, find an expression for the variation of density within the tank.

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Problem 24

Reconsider Fig. $\mathrm{P} 4.17$ in the following general way. It is known that the boundary layer thickness $\delta(x)$ increases monotonically and that there is no slip at the wall $(y=0) .$ Further, $u(x, y)$ merges smoothly with the outer stream flow, where $u \approx U=$ constant outside the layer. Use these facts to prove that $(a)$ the component $v(x, y)$ is positive everywhere within the layer, ( $b$ ) $v$ increases parabolically with $y$ very near the wall, and $(c) v$ is a maximum at $y=\delta$.

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Problem 25

An incompressible flow in polar coordinates is given by
$$\begin{aligned}
&v_{r}=K \cos \theta\left(1-\frac{b}{r^{2}}\right)\\
&v_{\theta}=-K \sin \theta\left(1+\frac{b}{r^{2}}\right)
\end{aligned}$$
Does this field satisfy continuity? For consistency, what
should the dimensions of constants $K$ and $b$ be? Sketch the surface where $v_{r}=0$ and interpret.

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Problem 26

Curvilinear, or streamline, coordinates are defined in Fig. P4.26, where $n$ is normal to the streamline in the plane of the radius of curvature $R$. Show that Euler's frictionless momentum equation (4.36) in streamline coordinates becomes
$$\begin{array}{l}
\frac{\partial V}{\partial t}+V \frac{\partial V}{\partial s}=-\frac{1}{\rho} \frac{\partial p}{\partial s}+g_{s} \\
-V \frac{\partial \theta}{\partial t}-\frac{V^{2}}{R}=-\frac{1}{\rho} \frac{\partial p}{\partial n}+g_{n}
\end{array}$$
Further show that the integral of Eq. (1) with respect to s is none other than our old friend Bernoulli's equation (3.76).

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Problem 27

A frictionless, incompressible steady-flow field is given by
$$\mathbf{V}=2 x y \mathbf{i}-y^{2} \mathbf{j}$$
in arbitrary units. Let the density be $\rho_{0}=$ constant and neglect gravity. Find an expression for the pressure gradient in the $x$ direction.

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Problem 28

If $z$ is "up," what are the conditions on constants $a$ and $b$ for which the velocity field $u-a y, v-b x, w-0$ is an exact solution to the continuity and Navier-Stokes equations for incompressible flow?

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Problem 29

Consider a steady, two-dimensional, incompressible flow of a newtonian fluid in which the velocity field is known, i.e. $u=-2 x y, v=y^{2}-x^{2}, w=0 .(a)$ Does this flow satisfy conservation of mass? ( $b$ ) Find the pressure field, $p(x, y)$ if the pressure at the point $(x=0, y=0)$ is equal to $p_{a}$.

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Problem 30

Show that the two-dimensional flow field of Example 1.10 is an exact solution to the incompressible Navier-Stokes equations $(4.38) .$ Neglecting gravity, compute the pressure field $p(x, y)$ and relate it to the absolute velocity $V^{2}=u^{2}+$ $v^{2} .$ Interpret the result.

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Problem 31

According to potential theory (Chap. 8 ) for the flow approaching a rounded two-dimensional body, as in Fig. P4.31, the velocity approaching the stagnation point is given by $u=U\left(1-a^{2} / x^{2}\right),$ where $a$ is the nose radius and $U$ is the velocity far upstream. Compute the value and position of the maximum viscous normal stress along this streamline.
Is this also the position of maximum fluid deceleration? Evaluate the maximum viscous normal stress if the fluid is SAE 30 oil at $20^{\circ} \mathrm{C},$ with $U=2 \mathrm{m} / \mathrm{s}$ and $a=6 \mathrm{cm}$.

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Problem 32

The answer to Prob. 4.14 is $v_{\theta}=f(r)$ only. Do not reveal this to your friends if they are still working on Prob. 4.14. Show that this flow field is an exact solution to the NavierStokes equations (4.38) for only two special cases of the function $\mathrm{f}(r)$. Neglect gravity. Interpret these two cases physically.

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Problem 33

From Prob. 4.15 the purely radial polar-coordinate flow which satisfies continuity is $v_{r}=f(\theta) / r,$ where $f$ is an arbitrary function. Determine what particular forms of $f(\theta)$ satisfy the full Navier-Stokes equations in polar-coordinate form from Eqs. (D.5) and (D.6).

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Problem 34

The fully developed laminar-pipe-flow solution of Prob. $3.53, v_{z}=u_{\max }\left(1-r^{2} / R^{2}\right), v_{\theta}=0, v_{r}=0,$ is an exact solution to the cylindrical Navier-Stokes equations (App. D). Neglecting gravity, compute the pressure distribution in the pipe $p(r, z)$ and the shear-stress distribution $\tau(r, z),$ using $R$ $u_{\max },$ and $\mu$ as parameters. Why does the maximum shear occur at the wall? Why does the density not appear as a parameter?

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Problem 35

From the Navier-Stokes equations for incompressible flow in polar coordinates (App. D for cylindrical coordinates), find the most general case of purely circulating motion $v_{\theta}(r)$, $v_{r}=v_{z}=0,$ for flow with no slip between two fixed concentric cylinders, as in Fig. P4.35.

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Problem 36

A constant-thickness film of viscous liquid flows in laminar motion down a plate inclined at angle $\theta$, as in Fig. P4.36 The velocity profile is
$$u=C y(2 h-y) \quad v=w=0$$
Find the constant $C$ in terms of the specific weight and viscosity and the angle $\theta$. Find the volume flux $Q$ per unit width in terms of these parameters.

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Problem 37

A viscous liquid of constant $\rho$ and $\mu$ falls due to gravity between two plates a distance $2 h$ apart, as in Fig. P4.37. The flow is fully developed, with a single velocity component $w=w(x) .$ There are no applied pressure gradients, only gravity. Solve the Navier-Stokes equation for the velocity profile between the plates.

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Problem 38

Reconsider the angular-momentum balance of Fig. 4.5 by adding a concentrated body couple $C_{z}$ about the $z$ axis [6]. Determine a relation between the body couple and shear stress for equilibrium. What are the proper dimensions for $C_{z} ?$ (Body couples are important in continuous media with microstructure, such as granular materials.)

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Problem 39

Problems involving viscous dissipation of energy are dependent on viscosity $\mu,$ thermal conductivity $k,$ stream velocity $U_{0}$, and stream temperature $T_{0} .$ Group these parameters into the di mensionless Brinkman number, which is proportional to $\mu$.

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Problem 40

As mentioned in Sec. 4.11 , the velocity profile for laminar flow between two plates, as in Fig. P4.40, is If the wall temperature is $T_{w}$ at both walls, use the incompressible-flow energy equation (4.75) to solve for the temperature distribution $T(y)$ between the walls for steady flow.

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Problem 41

The approximate velocity profile in Prob. 3.18 and Fig. $\mathrm{P} 3.18$ for steady laminar flow through a duct, was suggested as
$$u=u_{\max }\left(1-\frac{y^{2}}{b^{2}}\right)\left(1-\frac{z^{2}}{h^{2}}\right)$$
With $v=w=0,$ it satisfied the no-slip condition and gave a reasonable volume-flow estimate (which was the point of Prob. 3.18 ). Show, however, that it does not satsify the $x$ momentum Navier-Stokes equation for duct flow with constant pressure gradient $\partial p / \partial x<0 .$ For extra credit, explain briefly how the actual exact solution to this problem is obtained (see, for example, Ref. 5, pp. 120-121).

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Problem 42

In duct-flow problems with heat transfer, one often defines an average fluid temperature. Consider the duct flow of Fig. P4.40 of width $b$ into the paper. Using a control-volume integral analysis with constant density and specific heat, derive an expression for the temperature arising if the entire duct flow poured into a bucket and was stirred uniformly. Assume arbitrary $u(y)$ and $T(y) .$ This average is called the cup-mixing temperature of the flow.

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Problem 43

For the draining liquid film of Fig. $\mathrm{P} 4.36,$ what are the appropriate boundary conditions $(a)$ at the bottom $y=0$ and (b) at the surface $y=h ?$

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Problem 44

Suppose that we wish to analyze the sudden pipe-expansion flow of Fig. $\mathrm{P} 3.59,$ using the full continuity and NavierStokes equations. What are the proper boundary conditions to handle this problem?

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Problem 45

Suppose that we wish to analyze the sudden pipe-expansion flow of Fig. $\mathrm{P} 3.59,$ using the full continuity and NavierStokes equations. What are the proper boundary conditions to handle this problem?

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Problem 46

Fluid from a large reservoir at temperature $T_{0}$ flows into a circular pipe of radius $R$. The pipe walls are wound with an electric-resistance coil which delivers heat to the fluid at a rate $q_{\mathrm{w}}$ (energy per unit wall area). If we wish to analyze this problem by using the full continuity, Navier-Stokes, and energy equations, what are the proper boundary conditions for the analysis?

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Problem 47

A two-dimensional incompressible flow is given by the velocity field $\mathbf{V}=3 y \mathbf{i}+2 x \mathbf{j},$ in arbitrary units. Does this flow satisfy continuity? If so, find the stream function $\psi(x, y)$ and plot a few streamlines, with arrows.

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Problem 48

Determine the incompressible two-dimensional stream function $\psi(x, y)$ which represents the flow field given in Example 1.10.

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Problem 49

Investigate the stream function $\psi=K\left(x^{2}-y^{2}\right), K=\mathrm{con}-$ stant. Plot the streamlines in the full $x y$ plane, find any stagnation points, and interpret what the flow could represent.

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Problem 50

Investigate the polar-coordinate stream function $\psi=$ $K r^{1 / 2} \sin \frac{1}{2} \theta, K=$ constant. Plot the streamlines in the full $x y$ plane, find any stagnation points, and interpret.

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Problem 51

Investigate the polar-coordinate stream function $\psi=$ $K r^{2 / 3} \sin (2 \theta / 3), K=$ constant. Plot the streamlines in all except the bottom right quadrant, and interpret.

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Problem 52

A two-dimensional, incompressible, frictionless fluid is guided by wedge-shaped walls into a small slot at the origin, as in Fig. P4.52. The width into the paper is $b,$ and the volume flow rate is $Q$. At any given distance $r$ from the slot, the flow is radial inward, with constant velocity. Find an expression for the polar-coordinate stream function of this flow.

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Problem 53

For the fully developed laminar-pipe-flow solution of Prob. $4.34,$ find the axisymmetric stream function $\psi(r, z) .$ Use this result to determine the average velocity $V=Q / A$ in the pipe as a ratio of $u_{\max }$.

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Problem 54

An incompressible stream function is defined by
$$\psi(x, y)=\frac{U}{L^{2}}\left(3 x^{2} y-y^{3}\right)$$
where $U$ and $L$ are (positive) constants. Where in this chapter are the streamlines of this flow plotted? Use this stream function to find the volume flow $Q$ passing through the rectangular surface whose corners are defined by $(x, y, z)=$
$(2 L, 0,0),(2 L, 0, b),(0, L, b),$ and $(0, L, 0) .$ Show the direction of $Q$.

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Problem 55

In spherical polar coordinates, as in Fig. P4.12, the flow is called axisymmetric if $v_{\phi}=0$ and $\partial / \partial \phi=0,$ so that $v_{r}=$ $v_{r}(r, \theta)$ and $v_{\theta}=v_{\theta}(r, \theta) .$ Show that a stream function $\psi(r$,
$\theta$ ) exists for this case and is given by
$$v_{r}=\frac{1}{r^{2} \sin \theta} \frac{\partial \psi}{\partial \theta} \quad v_{\theta}=-\frac{1}{r \sin \theta} \frac{\partial \psi}{\partial r}$$
This is called the Stokes stream function $[5, \mathrm{p} .204]$.

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Problem 56

Investigate the velocity potential $\phi=K x y, K=\mathrm{constant.}$ Sketch the potential lines in the full $x y$ plane, find any stagnation points, and sketch in by eye the orthogonal streamlines. What could the flow represent?

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Problem 57

Determine the incompressible two-dimensional velocity potential $\phi(x, y)$ which represents the flow field given in Example $1.10 .$ Sketch a few potential lines and streamlines.

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Problem 58

Show that the incompressible velocity potential in plane polar coordinates $\phi(r, \theta)$ is such that
$$v_{r}=\frac{\partial \phi}{\partial r} \quad v_{\theta}=\frac{1}{r} \frac{\partial \phi}{\partial \theta}$$
Further show that the angular velocity about the $z$ -axis in such a flow would be given by
$$2 \omega_{z}=\frac{1}{r} \frac{\partial}{\partial r}\left(r v_{\theta}\right)-\frac{1}{r} \frac{\partial}{\partial \theta}\left(v_{r}\right)$$
Finally show that $\phi$ as defined above satisfies Laplace's equation in polar coordinates for incompressible flow.

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Problem 59

Consider the simple flow defined by $\mathbf{V}=x \mathbf{i}-y \mathbf{j},$ in arbitrary units. At $t=0,$ consider the rectangular fluid element defined by the lines $x=2, x=3$ and $y=2, y=3 .$ Determine, and draw to scale, the location of this fluid element at $t=0.5$ unit. Relate this new element shape to whether the flow is irrotational or incompressible.

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Problem 60

Liquid drains from a small hole in a tank, as shown in Fig. P4.60, such that the velocity field set up is given by $v_{r} \approx 0$ $v_{z} \approx 0, v_{\theta}=\omega R^{2} / r,$ where $z=H$ is the depth of the water far from the hole. Is this flow pattern rotational or irrotational? Find the depth $z_{C}$ of the water at the radius $r=R$.

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Problem 61

Investigate the polar-coordinate velocity potential $\phi=$ $K r^{1 / 2} \cos \frac{1}{2} \theta, K=$ constant. Plot the potential lines in the full $x y$ plane, sketch in by eye the orthogonal streamlines, and interpret.

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Problem 62

Show that the linear Couette flow between plates in Fig. 1.6 has a stream function but no velocity potential. Why is this so?

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Problem 63

Find the two-dimensional velocity potential $\phi(r, \theta)$ for the polar-coordinate flow pattern $v_{r}=Q / r, v_{\theta}=K / r,$ where $Q$ and $K$ are constants.

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Problem 64

Show that the velocity potential $\phi(r, z)$ in axisymmetric cylindrical coordinates (see Fig. 4.2 ) is defined such that
$$v_{r}=\frac{\partial \phi}{\partial r} \quad v_{z}=\frac{\partial \phi}{\partial z}$$
Further show that for incompressible flow this potential satisfies Laplace's equation in $(r, z)$ coordinates.

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Problem 65

A two-dimensional incompressible flow is defined by
$$u=-\frac{K y}{x^{2}+y^{2}} \quad v=\frac{K x}{x^{2}+y^{2}}$$
where $K=$ constant. Is this flow irrotational? If so, find its velocity potential, sketch a few potential lines, and interpret the flow pattern.

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Problem 66

A plane polar-coordinate velocity potential is defined by
$$\phi=\frac{K \cos \theta}{r} \quad K=\mathrm{const}$$
Find the stream function for this flow, sketch some streamlines and potential lines, and interpret the flow pattern.

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Problem 67

A stream function for a plane, irrotational, polar-coordinate flow is
$$\psi=C \theta-K \ln r \quad C \text { and } K=\mathrm{const}$$
Find the velocity potential for this flow. Sketch some streamlines and potential lines, and interpret the flow pattern.

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Problem 68

Find the stream function and plot some streamlines for the combination of a line source $m$ at $(x, y)=(0,+a)$ and an equal line source placed at $(0,-a)$.

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Problem 69

Find the stream function and plot some streamlines for the combination of a counterclockwise line vortex $K$ at $(x, y)$ $=(+a, 0)$ and an equal line vortex placed at $(-a, 0)$.

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Problem 70

Superposition of a source of strength $m$ at $(-a, 0)$ and a $\operatorname{sink}(\text { source of strength }-m)$ at $(a, 0)$ was discussed briefly in this chapter, where it was shown that the velocity potential function is
$$\phi=\frac{1}{2} m \cdot \ln \frac{(x+a)^{2}+y^{2}}{(x-a)^{2}+y^{2}}$$
A doublet is formed in the limit as $a$ goes to zero (the source and sink come together) while at the same time their strengths $m$ and $-m$ go to infinity and minus infinity, respectively, with the product $a \cdot m$ remaining constant. $(a)$ Find the limiting value of velocity potential for the doublet. Hint: Expand the natural logarithm as an infinite series of the form
$$\ln \frac{1+\epsilon}{1-\epsilon}=2\left(\epsilon+\frac{\epsilon^{3}}{3}+\cdots\right)$$
as $\epsilon$ goes to zero. ( $b$ ) Rewrite your result for $\phi_{\text {doublet }}$ in cylindrical coordinates.

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Problem 71

Find the stream function and plot some streamlines for the combination of a counterclockwise line vortex $K$ at $(x, y)=$ $(+a, 0)$ and an opposite (clockwise) line vortex placed at $(-a, 0)$.

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Problem 72

A coastal power plant takes in cooling water through a vertical perforated manifold, as in Fig. P4.72. The total volume flow intake is $110 \mathrm{m}^{3} / \mathrm{s}$. Currents of $25 \mathrm{cm} / \mathrm{s}$ flow past the manifold, as shown. Estimate $(a)$ how far downstream and
(b) how far normal to the paper the effects of the intake are felt in the ambient 8 -m-deep waters.

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Problem 73

A two-dimensional Rankine half-body, $8 \mathrm{cm}$ thick, is placed in a water tunnel at $20^{\circ} \mathrm{C}$. The water pressure far upstream along the body centerline is 120 kPa. What is the nose radius of the half-body? At what tunnel flow velocity will cavitation bubbles begin to form on the surface of the body?

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Problem 74

Find the stream function and plot some streamlines for the combination of a uniform stream i $U$ and a clockwise line vortex $-K$ at the origin. Are there any stagnation points in the flow field?

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Problem 75

Find the stream function and plot some streamlines for the combination of a line source $2 m$ at $(x, y)=(+a, 0)$ and a line source $m$ at $(-a, 0) .$ Are there any stagnation points in the flow field?

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Problem 76

Air flows at $1.2 \mathrm{m} / \mathrm{s}$ along a flat surface when it encounters a jet of air issuing from the horizontal wall at point $A,$ as in Fig. $4.76 .$ The jet volume flow is $0.4 \mathrm{m}^{3} / \mathrm{s}$ per unit depth into the paper. If the jet is approximated as an inviscid line source, ( $a$ ) locate the stagnation point $S$ on the wall. ( $b$ ) How far vertically will the jet flow extend into the stream?

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Problem 77

A tornado is simulated by a line $\operatorname{sink} m=-1000 \mathrm{m}^{2} / \mathrm{s}$ plus a line vortex $K=+1600 \mathrm{m}^{2} / \mathrm{s}$. Find the angle between any streamline and a radial line, and show that it is independent of both $r$ and $\theta .$ If this tornado forms in sea-level standard air, at what radius will the local pressure be equivalent to 29 inHg?

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Problem 78

The solution to Prob. 4.68 (do not reveal!) can represent a line source $m$ at $(0,+a)$ near a horizontal wall $(y=0) .$ [The other source at $(0,-a)$ represents an "image" to create the wall.] Find ( $a$ ) the magnitude of the maximum flow velocity along the wall and ( $b$ ) the point of minimum pressure along the wall. Hint: Use Bernoulli's equation.

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Problem 79

Study the combined effect of the two viscous flows in Fig. 4.16. That is, find $u(y)$ when the upper plate moves at speed $V$ and there is also a constant pressure gradient $(d p / d x)$. Is superposition possible? If so, explain why. Plot representative velocity profiles for $(a)$ zero, $(b)$ positive, and $(c)$ negative pressure gradients for the same upper-wall speed $V$.

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Problem 80

Oil, of density $\rho$ and viscosity $\mu,$ drains steadily down the side of a vertical plate, as in Fig. P4.80. After a development region near the top of the plate, the oil film will become independent of $z$ and of constant thickness $\delta$. Assume that $w=w(x)$ only and that the atmosphere offers no shear resistance to the surface of the film. ( $a$ ) Solve the NavierStokes equation for $w(x),$ and sketch its approximate shape.
(b) Suppose that film thickness $\delta$ and the slope of the velocity profile at the wall $[\partial w / \partial x]_{\text {wall }}$ are measured with a laser-Doppler anemometer (Chap. 6). Find an expression for oil viscosity $\mu$ as a function of $\left(\rho, \delta, g,[\partial w / \partial x]_{\text {wall }}\right)$.

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Problem 81

Modify the analysis of Fig. 4.17 to find the velocity $u_{\theta}$ when the inner cylinder is fixed and the outer cylinder rotates at angular velocity $\Omega_{0} .$ May this solution be added to Eq. (4.146) to represent the flow caused when both inner and outer cylinders rotate? Explain your conclusion.

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Problem 82

A solid circular cylinder of radius $R$ rotates at angular velocity $\Omega$ in a viscous incompressible fluid which is at rest far from the cylinder, as in Fig. P4.82. Make simplifying assumptions and derive the governing differential equation and boundary conditions for the velocity field $v_{\theta}$ in the fluid. Do not solve unless you are obsessed with this problem. What is the steady-state flow field for this problem?

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Problem 83

The flow pattern in bearing lubrication can be illustrated by Fig. $\mathrm{P} 4.83,$ where a viscous oil $(\rho, \mu)$ is forced into the gap $h(x)$ between a fixed slipper block and a wall moving at velocity $U .$ If the gap is thin, $h \ll L,$ it can be shown that the pressure and velocity distributions are of the form $p=p(x),$ $u=u(y), v=w=0 .$ Neglecting gravity, reduce the NavierStokes equations (4.38) to a single differential equation for $u(y) .$ What are the proper boundary conditions? Integrate and show that
$$u=\frac{1}{2 \mu} \frac{d p}{d x}\left(y^{2}-y h\right)+U\left(1-\frac{y}{h}\right)$$
where $h=h(x)$ may be an arbitrary slowly varying gap width. (For further information on lubrication theory, see Ref. $16 .)$

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Problem 84

Consider a viscous film of liquid draining uniformly down the side of a vertical rod of radius $a,$ as in Fig. $\mathrm{P} 4.84 .$ At some distance down the rod the film will approach a terminal or fully developed draining flow of constant outer ra$\operatorname{dius} b,$ with $v_{z}=v_{z}(r), v_{\theta}=v_{r}=0 .$ Assume that the at-
mosphere offers no shear resistance to the film motion. Derive a differential equation for $v_{2},$ state the proper boundary conditions, and solve for the film velocity distribution. How does the film radius $b$ relate to the total film volume
flow rate $Q ?$

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Problem 85

A flat plate of essentially infinite width and breadth oscillates sinusoidally in its own plane beneath a viscous fluid, as in Fig. $\mathrm{P} 4.85 .$ The fluid is at rest far above the plate. Making as many simplifying assumptions as you can, set up the governing differential equation and boundary conditions for finding the velocity field $u$ in the fluid. Do not solve (if you can solve it immediately, you might be able to get exempted from the balance of this course with credit).

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Problem 86

SAE 10 oil at $20^{\circ} \mathrm{C}$ flows between parallel plates $8 \mathrm{cm}$ apart, as in Fig. P4.86. A mercury manometer, with wall pressure taps $1 \mathrm{m}$ apart, registers a 6 -cm height, as shown. Estimate the flow rate of oil for this condition.

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Problem 87

Suppose in Fig. 4.17 that neither cylinder is rotating. The fluid has constant $\left(\rho, \mu, k, c_{p}\right) .$ What, then, is the steadyflow solution for $v_{\theta}(r) ?$ For this condition, suppose that the inner and outer cylinder surface temperatures are $T_{i}$ and $T_{o}$, respectively. Simplify the differential energy equation appropriately for this problem, state the boundary conditions, and find the temperature distribution in the fluid. Neglect gravity.

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Problem 88

The viscous oil in Fig. $\mathrm{P} 4.88$ is set into steady motion by a concentric inner cylinder moving axially at velocity $U$ inside a fixed outer cylinder. Assuming constant pressure and density and a purely axial fluid motion, solve Eqs. (4.38) for the fluid velocity distribution $v_{z}(r) .$ What are the proper boundary conditions?

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Problem 89

Modify Prob. 4.88 so that the outer cylinder also moves to the left at constant speed $V$. Find the velocity distribution $v_{z}(r) .$ For what ratio $V / U$ will the wall shear stress be the same at both cylinder surfaces?

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Problem 90

A $5-$ cm-diameter rod is pulled steadily at $2 \mathrm{m} / \mathrm{s}$ through a fixed cylinder whose clearance is filled with SAE 10 oil at $20^{\circ} \mathrm{C},$ as in Fig. $\mathrm{P} 4.90 .$ Estimate the (steady) force required to pull the inner rod.

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Problem 91

Consider two-dimensional, incompressible, steady Couette flow (flow between two infinite parallel plates with the upper plate moving at constant speed and the lower plate stationary, as in Fig. $4.16 a$ ). Let the fluid be nonnewtonian, with its viscous stresses given by
$$\begin{aligned}
\tau_{x x}=a\left(\frac{\partial u}{\partial x}\right)^{c} & \tau_{y y}=a\left(\frac{\partial v}{\partial y}\right)^{c} \quad \tau_{z z}=a\left(\frac{\partial w}{\partial z}\right)^{c} \\
\tau_{x y}=\tau_{y x}=\frac{1}{2} a\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)^{c} & \tau_{x z}=\tau_{z x}=\frac{1}{2} a\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right)^{c} \\
\tau_{y z}=& \tau_{z y}=\frac{1}{2} a\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)^{c}
\end{aligned}$$
where $a$ and $c$ are constants of the fluid. Make all the same assumptions as in the derivation of Eq. $(4.140) .(a)$ Find the velocity profile $u(y) .(b)$ How does the velocity profile for this case compare to that of a newtonian fluid?

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