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Fluid Mechanics: Fundamentals and Applications

Yunus Cengel

Chapter 9

DIFFERENTIAL ANALYSIS OF FLUID FLOW - all with Video Answers

Educators

Chapter Questions

Problem 1

Explain the fundamental differences between a flow domain and a control volume.

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Problem 2

What does it mean when we say that two or more differential equations are coupled?

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Problem 3

For a three-dimensional, unsteady, incompressible flow field in which temperature variations are insignificant, how many unknowns are there? List the equations required to solve for these unknowns.

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Problem 4

For a three-dimensional, unsteady, compressible flow field in which temperature and density variations are significant, how many unknowns are there? List the equations required to solve for these unknowns. (Hint: Assume other flow properties like viscosity and thermal conductivity can be treated as constants.)

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Problem 5

The divergence theorem is expressed as

$$
\int_V \vec{\nabla} \cdot \vec{G} d V=\oint_A \vec{G} \cdot \vec{n} d A
$$

where $\vec{G}$ is a vector, $V$ is a volume, and $A$ is the surface area that encloses and defines the volume. Express the divergence theorem in words.

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Problem 6

Transform the position $\vec{x}=(4,3,-4)$ from Cartesian $(x, y, z)$ coordinates to cylindrical ( $r, \theta, z$ ) coordinates, including units. The values of $\vec{x}$ are in units of meters.

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Problem 7

A Taylor series expansion of function $f(x)$ about some $x$-location $x_0$ is given as

$$
\begin{aligned}
f\left(x_0+d x\right)= & f\left(x_0\right)+\left(\frac{d f}{d x}\right)_{x=x_0} d x \\
& +\frac{1}{2!}\left(\frac{d^2 f}{d x^2}\right)_{x=x_0} d x^2+\frac{1}{3!}\left(\frac{d^3 f}{d x^3}\right)_{x=x_0} d x^3+\cdots
\end{aligned}
$$

Consider the function $f(x)=\exp (x)=e^x$. Suppose we know the value of $f(x)$ at $x=x_0$, i.e., we know the value of $f\left(x_0\right)$, and we want to estimate the value of this function at some $x$ location near $x_0$. Generate the first four terms of the Taylor series expansion for the given function (up to order $d x^3$ as in the above equation). For $x_0=0$ and $d x=-0.1$, use your truncated Taylor series expansion to estimate $f\left(x_0+d x\right)$. Compare your result with the exact value of $e^{-0.1}$. How many digits of accuracy do you achieve with your truncated Taylor series?

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Problem 8

Let vector $\vec{G}$ be given by $\vec{G}=2 x z \vec{i}-\frac{1}{2} x^2 \vec{j}+z^2 \vec{k}$. Calculate the divergence of $\vec{G}$, and simplify as much as possible. Is there anything special about your result?

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Problem 9

Let vector $\vec{G}$ be given by $\vec{G}=4 x z \vec{i}-y^2 \vec{j}+y z \vec{k}$ and let $V$ be the volume of a cube of unit length with its corner at the origin, bounded by $x=0$ to $1, y=0$ to 1 , and $z=0$ to 1 (Fig. P9-9). Area $A$ is the surface area of the cube. Perform both integrals of the divergence theorem and verify that they are equal. Show all your work.

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Problem 10

The product rule can be applied to the divergence of scalar $f$ times vector $\vec{G}$ as: $\vec{\nabla} \cdot(f \vec{G})=\vec{G} \cdot \vec{\nabla} f+f \vec{\nabla} \cdot \vec{G}$. Expand both sides of this equation in Cartesian coordinates and verify that it is correct.

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Problem 11

The outer product of two vectors is a second-order tensor with nine components. In Cartesian coordinates, it is

$$
\vec{F} \vec{G}=\left[\begin{array}{lll}
F_x G_x & F_x G_y & F_x G_z \\
F_y G_x & F_y G_y & F_y G_z \\
F_z G_x & F_z G_y & F_z G_z
\end{array}\right]
$$

The product rule applied to the divergence of the product of two vectors $\vec{F}$ and $\vec{G}$ can be written as $\vec{\nabla} \cdot(\vec{F} \vec{G})=\vec{G}(\vec{\nabla}$ $\cdot \vec{F})+(\vec{F} \cdot \vec{\nabla}) \vec{G}$. Expand both sides of this equation in Cartesian coordinates and verify that it is correct.

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Problem 12

Use the product rule of Prob. 9-11 to show that $\vec{\nabla} \cdot(\rho \vec{V} \vec{V})=\vec{V} \vec{\nabla} \cdot(\rho \vec{V})+\rho(\vec{V} \cdot \vec{\nabla}) \vec{V}$.

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Problem 13

On many occasions we need to transform a velocity from Cartesian $(x, y, z)$ coordinates to cylindrical $(r, \theta, z)$ coordinates (or vice versa). Using Fig. P9-13 as a guide, transform cylindrical velocity components ( $u_p, u_\theta, u_z$ ) into Cartesian velocity components ( $u, v, w$ ). (Hint: Since the $z$-component of velocity remains the same in such a transformation, we need only to consider the $x y$-plane, as in Fig. P9-13.)

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Problem 14

Using Fig. P9-13 as a guide, transform Cartesian velocity components ( $u, v, w$ ) into cylindrical velocity components ( $u_r, u_\theta, u_z$ ). (Hint: Since the $z$-component of velocity remains the same in such a transformation, we need only to consider the $x y$-plane.)

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Problem 15

Beth is studying a rotating flow in a wind tunnel. She measures the $u$ and $v$ components of velocity using a hot-wire anemometer. At $x=0.50 \mathrm{~m}$ and $y=0.20 \mathrm{~m}, u=10.3 \mathrm{~m} / \mathrm{s}$ and $v=-5.6 \mathrm{~m} / \mathrm{s}$. Unfortunately, the data analysis program requires input in cylindrical coordinates ( $r, \theta)$ and $\left(u_r, u_\theta\right)$. Help Beth transform her data into cylindrical coordinates. Specifically, calculate $r, \theta, u_r$ and $u_\theta$ at the given data point.

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Problem 16

A steady, two-dimensional, incompressible velocity field has Cartesian velocity components $u=C y /\left(x^2+y^2\right)$ and $v=-C x /\left(x^2+y^2\right)$, where $C$ is a constant. Transform these Cartesian velocity components into cylindrical velocity components $u_r$ and $u_\theta$, simplifying as much as possible. You should recognize this flow. What kind of flow is this? Answer: $0,-\mathrm{Cr}$, line vortex

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Problem 17

Consider a spiraling line vortex/sink flow in the $x y$ - or $r \theta$-plane as sketched in Fig. P9-17. The two-dimensional cylindrical velocity components ( $u_r, u_g$ ) for this flow field are $u_r=C / 2 \pi r$ and $u_\theta=\Gamma / 2 \pi r$, where $C$ and $\Gamma$ are constants ( $m$ is negative and $\Gamma$ is positive). Transform these two-dimensional cylindrical velocity components into two-dimensional Cartesian velocity components ( $u, \nu$ ). Your final answer should contain no $r$ or $\theta$-only $x$ and $y$. As a check of your algebra, calculate $V^2$ using Cartesian coordinates, and compare to $V^2$ obtained from the given velocity components in cylindrical components.

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Problem 18

Alex is measuring the time-averaged velocity components in a pump using a laser Doppler velocimeter (LDV). Since the laser beams are aligned with the radial and tangential directions of the pump, he measures the $u_r$ and $u_\theta$ components of velocity. At $r=6.20$ in and $\theta=30.0^{\circ}, u_r=1.37$ $\mathrm{ft} / \mathrm{s}$ and $u_\theta=3.82 \mathrm{ft} / \mathrm{s}$. Unfortunately, the data analysis program requires input in Cartesian coordinates $(x, y)$ in feet and $(u, v)$ in $\mathrm{ft} / \mathrm{s}$. Help Alex transform his data into Cartesian coordinates. Specifically, calculate $x, y, u$, and $v$ at the given data point.

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Problem 19

If a flow field is compressible, what can we say about the material derivative of density? What about if the flow field is incompressible?

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Problem 20

In this chapter we derive the continuity equation in two ways: by using the divergence theorem and by summing mass flow rates through each face of an infinitesimal control volume. Explain why the former is so much less involved than the latter.

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Problem 21

A two-dimensional diverging duct is being designed to diffuse the high-speed air exiting a wind tunnel. The $x$-axis is the centerline of the duct (it is symmetric about the $x$-axis), and the top and bottom walls are to be curved in such a way that the axial wind speed $u$ decreases approximately linearly from $u_1=300 \mathrm{~m} / \mathrm{s}$ at section 1 to $u_2=100 \mathrm{~m} / \mathrm{s}$ at section 2 (Fig. P9-21). Meanwhile, the air density $\rho$ is to increase approximately linearly from $\rho_1$ $=0.85 \mathrm{~kg} / \mathrm{m}^3$ at section 1 to $\rho_2=1.2 \mathrm{~kg} / \mathrm{m}^3$ at section 2 . The diverging duct is 2.0 m long and is 1.60 m high at section 1 (only the upper half is sketched in Fig. P9-21; the half-height at section 1 is 0.80 m ). (a) Predict the $y$-component of velocity, $v(x, y)$, in the duct. (b) Plot the approximate shape of the duct, ignoring friction on the walls. (c) What should be the half-height of the duct at section 2 ?
(1)

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Problem 22

Repeat Example 9-1 (gas compressed in a cylinder by a piston), but without using the continuity equation. Instead, consider the fundamental definition of density as mass divided by volume. Verify that Eq. 5 of Example 9-1 is correct.

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Problem 23

The compressible form of the continuity equation is $(\partial \rho / \partial t)+\vec{V} \cdot(\rho \vec{V})=0$. Expand this equation as far as possible in Cartesian coordinates $(x, y, z)$ and $(u, v, w)$.

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Problem 24

In Example 9-6 we derive the equation for volumetric strain rate, $(1 / V)(D V I D t)=\vec{\nabla} \cdot \vec{V}$. Write this as a word equation and discuss what happens to the volume of a fluid element as it moves around in a compressible fluid flow field (Fig. P9-24).

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Problem 25

Verify that the spiraling line vortex/sink flow in the $r \theta$-plane of Prob. 9-17 satisfies the two-dimensional incompressible continuity equation. What happens to conservation of mass at the origin? Discuss.

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Problem 26

Verify that the steady, two-dimensional, incompressible velocity field of Prob. 9-16 satisfies the continuity equation. Stay in Cartesian coordinates and show all your work.

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Problem 27

Consider the steady, two-dimensional velocity field given by $\vec{V}=(u, v)=(1.3+2.8 x) \vec{i}+(1.5-2.8 y) \vec{j}$. Verify that this flow field is incompressible.

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Problem 28

Imagine a steady, two-dimensional, incompressible flow that is purely radial in the $x y$ - or $r \theta$-plane. In other words, velocity component $u_r$ is nonzero, but $u_\theta$ is zero everywhere (Fig. P9-28). What is the most general form of velocity component $u_r$ that does not violate conservation of mass?

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Problem 29

Consider the following steady, three-dimensional velocity field in Cartesian coordinates: $\vec{V}=(u, v, w)=\left(a x y^2-b\right) \vec{i}$ $+c y^3 \vec{j}+d x y \vec{k}$, where $a, b, c$, and $d$ are constants. Under what conditions is this flow field incompressible? Answer: a $=-3 c$

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Problem 30

The $u$ velocity component of a steady, two-dimensional, incompressible flow field is $u=a x+b$, where $a$ and $b$ are constants. Velocity component $v$ is unknown. Generate an expression for $v$ as a function of $x$ and $y$.

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Problem 31

Imagine a steady, two-dimensional, incompressible flow that is purely circular in the $x y$ - or $r \theta$-plane. In other words, velocity component $u_\theta$ is nonzero, but $u_r$ is zero everywhere (Fig. P9-31). What is the most general form of velocity component $u_\theta$ that does not violate conservation of mass?

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Problem 32

The $u$ velocity component of a steady, two-dimensional, incompressible flow field is $u=a x+b y$, where $a$ and $b$ are constants. Velocity component $v$ is unknown. Generate an expression for $v$ as a function of $x$ and $y$.

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Problem 33

The $u$ velocity component of a steady, two-dimensional, incompressible flow field is $u=a x^2-b x y$, where $a$ and $b$ are constants. Velocity component $v$ is unknown. Generate an expression for $v$ as a function of $x$ and $y$.

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Problem 34

Consider steady flow of water through an axisymmetric garden hose nozzle (Fig. P9-34). The axial component of velocity increases linearly from $u_{z, \text { entrance }}$ to $u_{z \text {, exit }}$ as sketched. Between $z=0$ and $z=L$, the axial velocity component is given by $u_z=u_{z \text {, entrance }}+\left[\left(u_{z, \text { exit }}-u_{z, \text { entrance }}\right) / L\right] z$. Generate an expression for the radial velocity component $u_r$ between $z=0$ and $z=L$. You may ignore frictional effects on the walls.

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Problem 35

Two velocity components of a steady, incompressible flow field are known: $u=a x+b x y+c y^2$ and $v=a x z-$ $b y z^2$, where $a, b$, and $c$ are constants. Velocity component $w$ is missing. Generate an expression for $w$ as a function of $x$, $y$, and $z$.

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Problem 36

What is significant about curves of constant stream function? Explain why the stream function is useful in fluid mechanics.

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Problem 37

What restrictions or conditions are imposed on stream function $\psi$ so that it exactly satisfies the two-dimensional incompressible continuity equation by definition? Why are these restrictions necessary?

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Problem 38

Consider two-dimensional flow in the $x y$-plane. What is the significance of the difference in value of stream function $\psi$ from one streamline to another?

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Problem 39

There are numerous occasions in which a fairly uniform free-stream flow of speed $V$ in the $x$-direction encounters a long circular cylinder of radius $a$ aligned normal to the flow (Fig. P9-39). Examples include air flowing around a car antenna, wind blowing against a flag pole or telephone pole, wind hitting electric wires, and ocean currents impinging on the submerged round beams that support oil platforms. In all these cases, the flow at the rear of the cylinder is separated and unsteady and usually turbulent. However, the flow in the front half of the cylinder is much more steady and predictable. In fact, except for a very thin boundary layer near the cylinder surface, the flow field can be approximated by the following steady, two-dimensional stream function in the $x y$ - or $r \theta$-plane, with the cylinder centered at the origin: $\psi=$ $V \sin \theta\left(r-a^2 / r\right)$. Generate expressions for the radial and tangential velocity components.

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Problem 40

Consider fully developed Couette flow-flow between two infinite parallel plates separated by distance $h$, with the top plate moving and the bottom plate stationary as illustrated in Fig. P9-40. The flow is steady, incompressible, and twodimensional in the $x y$-plane. The velocity field is given by $\vec{V}=(u, v)=(V y / h) \vec{i}+0 \vec{j}$. Generate an expression for stream function $\psi$ along the vertical dashed line in Fig. P9-40. For convenience, let $\psi=0$ along the bottom wall of the channel. What is the value of $\psi$ along the top wall?

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Problem 41

As a follow-up to Prob. 9-40, calculate the volume flow rate per unit width into the page of Fig. P9-40 from first principles (integration of the velocity field). Compare your result to that obtained directly from the stream function. Discuss.

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Problem 42

Consider the Couette flow of Fig. P9-40. For the case in which $V=10.0 \mathrm{ft} / \mathrm{s}$ and $h=1.20 \mathrm{in}$, plot several streamlines using evenly spaced values of stream function. Are the streamlines themselves equally spaced? Discuss why or why not.

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Problem 43

Consider fully developed, two-dimensional channel flow-flow between two infinite parallel plates separated by distance $h$, with both the top plate and bottom plate stationary, and a forced pressure gradient $d P / d x$ driving the flow as illustrated in Fig. P9-43. ( $d P / d x$ is constant and negative.) The flow is steady, incompressible, and two-dimensional in the $x y$-plane. The velocity components are given by $u$ $=(1 / 2 \mu)(d P / d x)\left(y^2-h y\right)$ and $v=0$, where $\mu$ is the fluid's viscosity. Generate an expression for stream function $\psi$ along the vertical dashed line in Fig. P9-43. For convenience, let $\psi$ $=0$ along the bottom wall of the channel. What is the value of $\psi$ along the top wall?

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Problem 44

As a follow-up to Prob. 9-43, calculate the volume flow rate per unit width into the page of Fig. P9-43 from first principles (integration of the velocity field). Compare your result to that obtained directly from the stream function. Discuss.

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Problem 45

Consider the channel flow of Fig. P9-43. The fluid is water at $20^{\circ} \mathrm{C}$. For the case in which $d P / d x=-20,000 \mathrm{~N} / \mathrm{m}^3$ and $h=1.20 \mathrm{~mm}$, plot several streamlines using evenly spaced values of stream function. Are the streamlines themselves equally spaced? Discuss why or why not.

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Problem 46

In the field of air pollution control, one often needs to sample the quality of a moving airstream. In such measurements a sampling probe is aligned with the flow as sketched in Fig. P9-46. A suction pump draws air through the probe at volume flow rate $\dot{V}$ as sketched. For accurate sampling, the
air speed through the probe should be the same as that of the airstream (isokinetic sampling). However, if the applied suction is too large, as sketched in Fig. P9-46, the air speed through the probe is greater than that of the airstream (superisokinetic sampling). For simplicity consider a twodimensional case in which the sampling probe height is $h$ $=4.5 \mathrm{~mm}$ and its width (into the page of Fig. P9-46) is W $=52 \mathrm{~mm}$. The values of the stream function corresponding to the lower and upper dividing streamlines are $\psi_l$ $=0.105 \mathrm{~m}^2 / \mathrm{s}$ and $\psi_u=0.150 \mathrm{~m}^2 / \mathrm{s}$, respectively. Calculate the volume flow rate through the probe (in units of $\mathrm{m}^3 / \mathrm{s}$ ) and the average speed of the air sucked through the probe

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Problem 47

Suppose the suction applied to the sampling probe of Prob. 9-46 were too weak instead of too strong. Sketch what the streamlines would look like in that case. What would you call this kind of sampling? Label the lower and upper dividing streamlines.

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Problem 48

Consider the air sampling probe of Prob. 9-46. If the upper and lower streamlines are 5.8 mm apart in the airstream far upstream of the probe, estimate the free stream speed $V_{\text {free stream }}$.

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Problem 49

Consider a steady, two-dimensional, incompressible flow field called a uniform stream. The fluid speed is $V$ everywhere, and the flow is aligned with the $x$-axis (Fig. P9-49). The Cartesian velocity components are $u=V$ and $v=0$. Generate an expression for the stream function for this flow. Suppose $V=8.9 \mathrm{~m} / \mathrm{s}$. If $\psi_2$ is a horizontal line at $y$ $=0.5 \mathrm{~m}$ and the value of $\psi$ along the $x$-axis is zero, calculate the volume flow rate per unit width (into the page of Fig. P9-49) between these two streamlines.

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Problem 50

Consider the steady, two-dimensional, incompressible flow field of Prob. 9-33, for which the $u$ velocity component is $u=a x^2-b x y$, where $a=0.45(\mathrm{ft} \cdot \mathrm{s})^{-1}$, and $b$ $=0.75(\mathrm{ft} \cdot \mathrm{s})^{-1}$. Let $v=0$ for all values of $x$ when $y=0$ (that is, $v=0$ along the $x$-axis). Generate an expression for the stream function and plot some streamlines of the flow. For consistency, set $\psi=0$ along the $x$-axis, and plot in the range $0<x<3 \mathrm{ft}$ and $0<y<4 \mathrm{ft}$.

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Problem 51

A uniform stream of speed $V$ is inclined at angle $\alpha$ from the $x$-axis (Fig. P9-51). The flow is steady, two-dimensional, and incompressible. The Cartesian velocity components are $u=V \cos \alpha$ and $v=V \sin \alpha$. Generate an expression for the stream function for this flow.

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Problem 52

A steady, two-dimensional, incompressible flow field in the $x y$-plane has the following stream function: $\psi=a x^2$ $+b x y+c y^2$, where $a, b$, and $c$ are constants. (a) Obtain expressions for velocity components $u$ and $v$. (b) Verify that the flow field satisfies the incompressible continuity equation.

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Problem 53

For the velocity field of Prob. 9-52, plot streamlines $\psi=0,1,2,3,4,5$, and $6 \mathrm{~m}^2 / \mathrm{s}$. Let constants $a, b$, and $c$ have the following values: $a=0.50 \mathrm{~s}^{-1}, b$ $=-1.3 \mathrm{~s}^{-1}$, and $c=0.50 \mathrm{~s}^{-1}$. For consistency, plot streamlines between $x=-2$ and 2 m , and $y=-4$ and 4 m . Indicate the direction of flow with arrows.

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Problem 54

A steady, two-dimensional, incompressible flow field in the $x y$-plane has a stream function given by $\psi=a x^2-b y^2$ $+c x+d x y$, where $a, b, c$, and $d$ are constants. (a) Obtain expressions for velocity components $u$ and $v$. (b) Verify that the flow field satisfies the incompressible continuity equation.

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Problem 55

Repeat Prob. 9-54, except make up your own stream function. You may create any function $\psi(x, y)$ that you desire, as long as it contains at least three terms and is not the same as an example or problem in this text. Discuss.

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Problem 56

A steady, incompressible, two-dimensional CFD calculation of flow through an asymmetric two-dimensional branching duct reveals the streamline pattern sketched in Fig. P9-56, where the values of $\psi$ are in units of $\mathrm{m}^2 / \mathrm{s}$, and $W$ is the width of the duct into the page. The values of stream function $\psi$ on the duct walls are shown. What percentage of the flow goes through the upper branch of the duct?

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Problem 57

If the average velocity in the main branch of the duct of Prob. 9-56 is $11.4 \mathrm{~m} / \mathrm{s}$, calculate duct height $h$ in units of cm . Obtain your result in two ways, showing all your work. You may use the results of Prob. 9-56 in only one of the methods.

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Problem 58

Consider steady, incompressible, axisymmetric flow $(r, z)$ and $\left(u_r, u_z\right)$ for which the stream function is defined as $u_r=-(1 / r)(\partial \psi / \partial z)$ and $u_z=(1 / r)(\partial \psi / \partial r)$. Verify that $\psi$ so defined satisfies the continuity equation. What conditions or restrictions are required on $\psi$ ?

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Problem 59

Consider steady, incompressible, two-dimensional flow due to a line source at the origin (Fig. P9-59). Fluid is created at the origin and spreads out radially in all directions in the $x y$-plane. The net volume flow rate of created fluid per unit width is $\dot{V} / L$ (into the page of Fig. P9-59), where $L$ is the width of the line source into the page in Fig. P9-59. Since mass must be conserved everywhere except at the origin (a singularity point), the volume flow rate per unit width through a circle of any radius $r$ must also be $\dot{V} / L$. If we (arbitrarily) specify stream function $\psi$ to be zero along the positive $x$-axis $(\theta=0)$, what is the value of $\psi$ along the positive $y$-axis $\left(\theta=90^{\circ}\right)$ ? What is the value of $\psi$ along the negative $x$ axis $\left(\theta=180^{\circ}\right)$ ?

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Problem 60

Repeat Prob. 9-59 for the case of a line sink instead of a line source. Let $\dot{V} / L$ be a positive value, but the flow is everywhere in the opposite direction.

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Problem 61

Consider the garden hose nozzle of Prob. 9-34. Generate an expression for the stream function corresponding to this flow field.

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Problem 62

Consider the garden hose nozzle of Probs. (9) 9-34 and 9-61. Let the entrance and exit nozzle diameters be 0.50 and 0.14 in , respectively, and let the nozzle length be 2.0 in . The volume flow rate through the nozzle is $2.0 \mathrm{gal} / \mathrm{min}$. (a) Calculate the axial speeds ( $\mathrm{ft} / \mathrm{s}$ ) at the nozzle entrance and at the nozzle exit. (b) Plot several streamlines in the $r z$-plane inside the nozzle, and design the appropriate nozzle shape.

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Problem 63

Flow separates at a sharp corner along a wall and forms a recirculating separation bubble as sketched in Fig. P9-63 (streamlines are shown). The value of the stream function at the wall is zero, and that of the uppermost streamline shown is some positive value $\psi_{\text {upper }}$. Discuss the value of the stream function inside the separation bubble. In particular, is it positive or negative? Why? Where in the flow is $\psi$ a minimum?

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Problem 64

A graduate student is running a CFD code for his MS research project and generates a plot of flow streamlines (contours of constant stream function). The contours are of equally spaced values of stream function. Professor I. C. Flows looks at the plot and immediately points to a region of the flow and says, "Look how fast the flow is moving here!" What did Professor Flows notice about the streamlines in that region and how did she know that the flow was fast in that region?

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Problem 65

Streaklines are shown in Fig. P9-65 for flow of water over the front portion of a blunt, axisymmetric cylinder aligned with the flow. Streaklines are generated by introducing air bubbles at evenly spaced points upstream of the field of view. Only the top half is shown since the flow is symmetric about the horizontal axis. Since the flow is steady, the streaklines are coincident with streamlines. Discuss how you can tell from the streamline pattern whether the flow speed in a particular region of the flow field is (relatively) large or small.

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Problem 66

A sketch of flow streamlines (contours of constant stream function) is shown in Fig. P9-66E for steady, incompressible, two-dimensional flow of air in a curved duct. (a) Draw arrows on the streamlines to indicate the direction of flow. (b) If $h=2.0 \mathrm{in}$, what is the approximate speed of the air at point $P$ ? (c) Repeat part (b) if the fluid were water instead of air. Discuss. Answers: (b) $0.78 \mathrm{ft} / \mathrm{s}$, (c) $0.78 \mathrm{ft} / \mathrm{s}$

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Problem 67

We briefly mention the compressible stream function $\psi_\rho$ in this chapter, defined in Cartesian coordinates as $\rho u$ $=\left(\partial \psi_\rho / \partial y\right)$ and $\rho \nu=-\left(\partial \psi_\rho / \partial x\right)$. What are the primary dimensions of $\psi_\rho$ ? Write the units of $\psi_\rho$ in primary SI units and in primary English units.

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Problem 68

In Example 9-2 we provide expressions for $u, v$, and $\rho$ for flow through a compressible converging duct. Generate an expression for the compressible stream function $\psi_\rho$ that describes this flow field. For consistency, set $\psi_\rho=0$ along the $x$-axis.

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Problem 69

In Prob. 9-21 we developed expressions for $u, v$, and $\rho$ for flow through the compressible, twodimensional, diverging duct of a high-speed wind tunnel. Generate an expression for the compressible stream function $\psi_\rho$ that describes this flow field. For consistency, set $\psi_\rho=0$ along the $x$-axis. Plot several streamlines and verify that they agree with those you plotted in Prob. 9-21. What is the value of $\psi_\rho$ at the top wall of the diverging duct?

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Problem 70

Steady, incompressible, two-dimensional flow over a newly designed small hydrofoil of chord length $c=9.0 \mathrm{~mm}$ is modeled with a commercial computational fluid dynamics (CFD) code. A close-up view of flow streamlines (contours of constant stream function) is shown in Fig. P9-70. Values of the stream function are in units of $\mathrm{m}^2 / \mathrm{s}$. The fluid is water at room temperature. (a) Draw an arrow on the plot to indicate the direction and relative magnitude of the velocity at point $A$. Repeat for point $B$. Discuss how your results can be used to explain how such a body creates lift. (b) What is the approximate speed of the air at point $A$ ? (Point $A$ is between streamlines 1.65 and 1.66 in Fig. P9-70.)

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Problem 71

Time-averaged, turbulent, incompressible, two-dimensional flow over a square block of dimension $h=1 \mathrm{~m}$ is modeled with a commercial computational fluid dynamics (CFD) code. A close-up view of flow streamlines (contours of constant stream function) is shown in Fig. P9-71. The
fluid is air at room temperature. Note that contours of constant compressible stream function are plotted in Fig. P9-71, even though the flow itself is approximated as incompressible. Values of $\psi_\rho$ are in units of $\mathrm{kg} / \mathrm{m} \cdot \mathrm{s}$. (a) Draw an arrow on the plot to indicate the direction and relative magnitude of the velocity at point $A$. Repeat for point $B$. (b) What is the approximate speed of the air at point $B$ ? (Point $B$ is between streamlines 5 and 6 in Fig. P9-71.)

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Problem 72

The general control volume equation for conservation of linear momentum is

$$
\begin{aligned}
& \int_{\mathrm{CV}} \rho \vec{g} d V+\int_{\mathrm{CS}} \sigma_{\mathrm{II}} \cdot \vec{n} d A \\
&=\int_{\mathrm{CV}} \frac{\partial}{\partial t}(\rho \vec{V}) d V+\int_{\mathrm{CI}}(\rho \vec{V}) \vec{V} \cdot \vec{n} d A
\end{aligned}
$$

Discuss the meaning of each term in this equation. The terms are labeled for convenience. Write the equation as a word equation.

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Problem 73

An airplane flies through the sky at constant velocity $\vec{V}_{\text {aiplane }}$ (Fig. P9-73C). Discuss the velocity boundary conditions on the air adjacent to the surface of the airplane from two frames of reference: (a) standing on the ground, and (b) moving with the airplane. Likewise, what are the far-field velocity boundary conditions of the air (far away from the airplane) in both frames of reference?

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Problem 74

What are constitutive equations, and to which fluid mechanics equation are they applied?

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Problem 75

What is mechanical pressure $P_m$, and how is it used in an incompressible flow solution?

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Problem 76

What is the main distinction between a Newtonian fluid and a non-Newtonian fluid? Name at least three Newtonian fluids and three non-Newtonian fluids.

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Problem 77

Define or describe each type of fluid: (a) viscoelastic fluid, (b) pseudoplastic fluid, (c) dilatant fluid, (d) Bingham plastic fluid.

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Problem 78

A stirrer mixes liquid chemicals in a large tank (Fig. P9-78). The free surface of the liquid is exposed to room air. Surface tension effects are negligible. Discuss the boundary conditions required to solve this problem. Specifically, what are the velocity boundary conditions in terms of cylindrical coordinates $(r, \theta, z)$ and velocity components $\left(u_r, u_\theta, u_z\right)$ at all surfaces, including the blades and the free surface? What pressure boundary conditions are appropriate for this flow field? Write mathematical equations for each boundary condition and discuss.

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Problem 79

Repeat Prob. 9-78, but from a frame of reference rotating with the stirrer blades at angular velocity $\omega$.

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Problem 80

Consider liquid in a cylindrical tank. Both the tank and the liquid rotate as a rigid body (Fig. P9-80). The free surface of the liquid is exposed to room air. Surface tension effects are negligible. Discuss the boundary conditions required to solve this problem. Specifically, what are the velocity boundary conditions in terms of cylindrical coordinates $(r, \theta, z)$ and velocity components ( $u_r, u_{\theta,}, u_z$ ) at all surfaces, including the tank walls and the free surface? What pressure boundary conditions are appropriate for this flow field? Write mathematical equations for each boundary condition and discuss.

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Problem 81

The $r \theta$-component of the viscous stress tensor in cylindrical coordinates is given by

$$
\tau_{r \theta}=\tau_{\theta r}=\mu\left[r \frac{\partial}{\partial r}\left(\frac{u_\theta}{r}\right)+\frac{1}{r} \frac{\partial u_r}{\partial \theta}\right]
$$

Some authors write this component instead as

$$
\tau_{r \theta}=\tau_{\theta r}=\mu\left[\frac{1}{r}\left(\frac{\partial u_r}{\partial \theta}-u_\theta\right)+\frac{\partial u_\theta}{\partial r}\right]
$$

Are these the same? In other words is Eq. 2 equivalent to Eq. 1 , or do these other authors define their viscous stress tensor differently? Show all your work.

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Problem 82

Engine oil at $T=60^{\circ} \mathrm{C}$ is forced between two very large, stationary, parallel flat plates separated by a thin gap height $h=2.5 \mathrm{~mm}$ (Fig. P9-82). The plate dimensions are $L$ $=1.5 \mathrm{~m}$ and $W=0.75 \mathrm{~m}$. The outlet pressure is atmospheric, and the inlet pressure is 1 atm gage pressure. Estimate the volume flow rate of oil. Also calculate the Reynolds number of the oil flow, based on gap height $h$ and average velocity $V$. Is the flow laminar or turbulent? Answers: 9.10 $\times 10^{-4} \mathrm{~m}^3 / \mathrm{s}, 14.5$, laminar

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Problem 83

Consider the following steady, two-dimensional, incompressible velocity field: $\vec{V}=(u, v)=(a x+b) \vec{i}+(-a y+$ $\left.c x^2\right) \vec{j}$, where $a, b$, and $c$ are constants. Calculate the pressure as a function of $x$ and $y$.

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Problem 84

Consider the following steady, two-dimensional, incompressible velocity field: $\vec{V}=(u, v)=\left(-a x^2\right) \vec{i}+(2 a x y) \vec{j}$, where $a$ is a constant. Calculate the pressure as a function of $x$ and $y$.

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Problem 85

Consider steady, two-dimensional, incompressible flow due to a spiraling line vortex/sink flow centered on the $z$-axis. Streamlines and velocity vectors are shown in Fig. P9-85. The velocity field is $u_r=C / r$ and $u_9=K / r$, where $C$ and $K$ are constants. Calculate the pressure as a function of $r$ and $\theta$.

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Problem 86

Consider the steady, two-dimensional, incompressible velocity field, $\vec{V}=(u, v)=(a x+b) \vec{i}+(-a y+c) \vec{j}$, where $a, b$, and $c$ are constants. Calculate the pressure as a function of $x$ and $y$.

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Problem 87

Consider steady, incompressible, parallel, laminar flow of a viscous fluid falling between two infinite vertical walls (Fig. P9-87). The distance between the walls is $h$, and gravity acts in the negative $z$-direction (downward in the figure). There is no applied (forced) pressure driving the flowthe fluid falls by gravity alone. The pressure is constant everywhere in the flow field. Calculate the velocity field and sketch the velocity profile using appropriate nondimensionalized variables.

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Problem 88

For the fluid falling between two parallel vertical walls (Prob. 9-87), generate an expression for the volume flow rate per unit width ( $\dot{V} / L)$ as a function of $\rho, \mu, h$, and $g$. Compare your result to that of the same fluid falling along one vertical wall with a free surface replacing the second wall (Example 9-17), all else being equal. Discuss the differences and provide a physical explanation.

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Problem 89

Repeat Example 9-17, except for the case in which the wall is inclined at angle $\alpha$ (Fig. P9-89). Generate expressions for both the pressure and velocity fields. As a check, make sure that your result agrees with that of Example 9-17 when $\alpha=90^{\circ}$. [Hint: It is most convenient to use the ( $s, y, n$ ) coordinate system with velocity components ( $u_s, v, u_n$ ), where $y$ is into the page in Fig. P9-89. Plot the dimensionless velocity profile $u_s^*$ versus $n^*$ for the case in which $\alpha=60^{\circ}$.]

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Problem 90

For the falling oil film of Prob. 9-89, generate an expression for the volume flow rate per unit width of oil falling down the wall $(\dot{V} / L)$ as a function of $\rho, \mu, h$, and $g$. Calculate $(\dot{V} / L)$ for an oil film of thickness 5.0 mm with $\rho$ $=888 \mathrm{~kg} / \mathrm{m}^3$ and $\mu=0.80 \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}$.

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Problem 91

The first two viscous terms in the $\theta$-component of the Navier-Stokes equation (Eq. 9-62c) are $\mu\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u_\theta}{\partial r}\right)-\frac{u_\theta}{r^2}\right]$.
Expand this expression as far as possible using the product rule, yielding three terms. Now combine all three terms into one term. (Hint: Use the product rule in reverse-some trial and error may be required.)

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Problem 92

An incompressible Newtonian liquid is confined between two concentric circular cylinders of infinite length a solid inner cylinder of radius $R_i$ and a hollow, stationary outer cylinder of radius $R_o$ (Fig. P9-92; the $z$-axis is out of the page). The inner cylinder rotates at angular velocity $\omega_i$. The flow is steady, laminar, and two-dimensional in the $r \theta$ plane. The flow is also rotationally symmetric, meaning that nothing is a function of coordinate $\theta$ ( $u_\theta$ and $P$ are functions of radius $r$ only). The flow is also circular, meaning that velocity component $u_r=0$ everywhere. Generate an exact expression for velocity component $u_\theta$ as a function of radius $r$ and the other parameters in the problem. You may ignore gravity. (Hint: The result of Prob. 9-91 is useful.)

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Problem 93

Analyze and discuss two limiting cases of Prob. 9-92: (a) The gap is very small. Show that the velocity profile approaches linear from the outer cylinder wall to the inner cylinder wall. In other words, for a very tiny gap the velocity profile reduces to that of simple two-dimensional Couette flow. (Hint: Define $y=R_o-r, h=$ gap thickness $=R_o-$ $R_i$, and $V=$ speed of the "upper plate" $\left.=R_i \omega_i\right)(b)$ The outer cylinder radius becomes infinite, while the inner cylinder radius becomes very small. What kind of flow does this approach?

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Problem 94

Repeat Prob. 9-92 for the more general case. Namely, let the inner cylinder rotate at angular velocity $\omega_i$ and let the outer cylinder rotate at angular velocity $\omega_o$. All else is the same as Prob. 9-92. Generate an exact expression for velocity component $u_\theta$ as a function of radius $r$ and the other parameters in the problem. Verify that when $\omega_o=0$ your result simplifies to that of Prob. 9-92.

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Problem 95

Analyze and discuss a limiting case of Prob. 9-94 in which there is no inner cylinder ( $R_i=\omega_i=0$ ). Generate an expression for $u_9$ as a function of $r$. What kind of flow is this? Describe how this flow could be set up experimentally. Answer: $\omega_\sigma r$

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Problem 96

Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius $R_i$ and outer radius $R_o$ (Fig. P9-96). Ignore the effects of gravity. A constant negative pressure gradient $\partial P / \partial x$ is applied in the $x$-direction, $(\partial P / d x)=\left(P_2-P_1\right) /\left(x_2-x_1\right)$, where $x_1$ and $x_2$ are two arbitrary locations along the $x$-axis,
and $P_1$ and $P_2$ are the pressures at those two locations. The pressure gradient may be caused by a pump and/or gravity. Note that we adopt a modified cylindrical coordinate system here with $x$ instead of $z$ for the axial component, namely, ( $r$, $\theta, x)$ and $\left(u_r, u_\theta, u\right)$. Derive an expression for the velocity field in the annular space in the pipe.

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Problem 97

Consider again the pipe annulus sketched in Fig. P9-96. Assume that the pressure is constant everywhere (there is no forced pressure gradient driving the flow). However, let the inner wall be moving at steady velocity $V$ to the right. The outer wall is still stationary. (This is a kind of axisymmetric Couette flow.) Generate an expression for the $x$-component of velocity $u$ as a function of $r$ and the other parameters in the problem.

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Problem 98

Repeat Prob. 9-97 except swap the stationary and moving walls. In particular, let the inner wall be stationary, and let the outer pipe wall be moving at steady velocity $V$ to the right, all else being equal. Generate an expression for the $x$-component of velocity $u$ as a function of $r$ and the other parameters in the problem.

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Problem 99

Consider a modified form of Couette flow in which there are two immiscible fluids sandwiched between two infinitely long and wide, parallel flat plates (Fig. P9-99). The flow is steady, incompressible, parallel, and laminar. The top plate moves at velocity $V$ to the right, and the bottom plate is stationary. Gravity acts in the $-z$-direction (downward in the figure). There is no forced pressure gradient pushing the fluids through the channel-the flow is set up solely by viscous effects created by the moving upper plate. You may ignore surface tension effects and assume that the interface is horizontal. The pressure at the bottom of the flow $(z=0)$ is equal to $P_0$. (a) List all the appropriate boundary conditions on both velocity and pressure. (Hint: There are six required boundary conditions.) (b) Solve for the velocity field. (Hint: Split up the solution into two portions, one for each fluid. Generate expressions for $u_1$ as a function of $z$ and $u_2$ as a function of z.) (c) Solve for the pressure field. (Hint: Again split up the solution. Solve for $P_1$ and $P_2$.) (d) Let fluid 1 be water and let fluid 2 be unused engine oil, both at $80^{\circ} \mathrm{C}$. Also let $h_1=5.0 \mathrm{~mm}, h_2=8.0 \mathrm{~mm}$, and $V=10.0 \mathrm{~m} / \mathrm{s}$. Plot $u$ as a function of $z$ across the entire channel. Discuss the results.

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Problem 100

Consider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter $D$ or radius $R=D / 2$ inclined at angle $\alpha$ (Fig. P9-100). There is no applied pressure gradient $(\partial P / \partial x=0)$. Instead, the fluid flows down the pipe due to gravity alone. We adopt the coordinate system shown, with $x$ down the axis of the pipe. Derive an expression for the $x$-component of velocity $u$ as a function of radius $r$ and the other parameters of the problem. Calculate the volume flow and average axial velocity rate through the pipe.

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Problem 101

Explain why the incompressible flow approximation and the constant temperature approximation usually go hand in hand.

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Problem 102

For each part, write the official name for the differential equation, discuss its restrictions, and describe what the equation represents physically.
(a) $\frac{\partial \rho}{\partial t}+\vec{\nabla} \cdot(\rho \vec{V})=0$
(b) $\frac{\partial}{\partial t}(\rho \vec{V})+\vec{\nabla} \cdot(\rho \vec{V} \vec{V})=\rho \vec{g}+\vec{\nabla} \cdot \sigma_{i j}$
(c) $\rho \frac{D \vec{V}}{D t}=-\vec{\nabla} P+\rho \vec{g}+\mu \nabla^2 \vec{V}$

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Problem 103

List the six steps used to solve the Navier-Stokes and continuity equations for incompressible flow with constant fluid properties. (You should be able to do this without peeking at the chapter.)

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Problem 104

For each statement, choose whether the statement is true or false and discuss your answer briefly. For each statement it is assumed that the proper boundary conditions and fluid properties are known.
(a) A general incompressible flow problem with constant fluid properties has four unknowns.
(b) A general compressible flow problem has five unknowns.
(c) For an incompressible fluid mechanics problem, the continuity equation and Cauchy's equation provide enough equations to match the number of unknowns.
(d) For an incompressible fluid mechanics problem involving a Newtonian fluid with constant properties, the continuity equation and the Navier-Stokes equation provide enough equations to match the number of unknowns.

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Problem 105

Discuss the relationship between volumetric strain rate and the continuity equation. Base your discussion on fundamental definitions.

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Problem 106

Repeat Example 9-17, except for the case in which the wall is moving upward at speed $V$. As a check, make sure that your result agrees with that of Example 9-17 when $V$ $=0$. Nondimensionalize your velocity profile equation using the same normalization as in Example 9-17, and show that a Froude number and a Reynolds number emerge. Plot the profile $w^*$ versus $x^*$ for cases in which $\mathrm{Fr}=0.5$ and $\mathrm{Re}=0.5$, 1.0 , and 5.0. Discuss.

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Problem 107

For the falling oil film of Prob. 9-106, calculate the volume flow rate per unit width of oil falling down the wall $(\dot{V} / L)$ as a function of wall speed $V$ and the other parameters in the problem. Calculate the wall speed required such that there is no net volume flow of oil either up or down. Give your answer for $V$ in terms of the other parameters in the problem, namely, $\rho, \mu, h$, and $g$. Calculate $V$ for zero volume flow rate for an oil film of thickness 5.0 mm with $\rho$ $=888 \mathrm{~kg} / \mathrm{m}^3$ and $\mu=0.80 \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}$.

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Problem 108

Consider steady, two-dimensional, incompressible flow in the $x z$-plane rather than in the $x y$-plane. Curves of constant stream function are shown in Fig. P9-108. The nonzero velocity components are ( $u, w$ ). Define a stream function such that flow is from right to left in the $x z$-plane when $\psi$ increases in the $z$-direction.

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Problem 109

Consider the following steady, three-dimensional velocity field in Cartesian coordinates: $\vec{V}=(u, v, w)=\left(a x z^2\right.$ $-b y) \vec{i}+c x y z \vec{j}+\left(d z^3+e x z^2\right) \vec{k}$, where $a, b, c, d$, and $e$ are constants. Under what conditions is this flow field incompressible? What are the primary dimensions of constants $a, b$, $c, d$, and $e$ ?

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Problem 110

Simplify the Navier-Stokes equation as much as possible for the case of an incompressible liquid being accelerated as a rigid body in an arbitrary direction (Fig. P9-110). Gravity acts in the $-z$-direction. Begin with the incompressible vector form of the Navier-Stokes equation, explain how and why some terms can be simplified, and give your final result as a vector equation.

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Problem 111

Simplify the Navier-Stokes equation as much as possible for the case of incompressible hydrostatics, with gravity acting in the negative $z$-direction. Begin with the incompressible vector form of the Navier-Stokes equation, explain how and why some terms can be simplified, and give your final result as a vector equation.

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Problem 112

Bob will use a computational fluid dynamics code to model steady flow of an incompressible fluid through a twodimensional sudden contraction as sketched in Fig. P9-112. Channel height changes from $H_1=12.0 \mathrm{~cm}$ to $H_2=4.6 \mathrm{~cm}$. Uniform velocity $\vec{V}_1=18.5 \vec{i} \mathrm{~m} / \mathrm{s}$ is to be specified on the left boundary of the computational domain. The CFD code uses a numerical scheme in which the stream function must be specified along all boundaries of the computational domain. As shown in Fig. P9-112, $\psi$ is specified as zero along the entire bottom wall of the channel. (a) What value of $\psi$ should Bob specify on the top wall of the channel? (b) How should Bob specify $\psi$ on the left side of the computational domain? (c) Discuss how Bob might specify $\psi$ on the right side of the computational domain.

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Problem 113

For each of the listed equations, write down the equation in vector form and decide if it is linear or nonlinear. If it is nonlinear, which term(s) make it so? (a) Incompressible continuity equation, (b) compressible continuity equation, and ( $c$ ) incompressible Navier-Stokes equation.

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Problem 114

A boundary layer is a thin region near a wall in which viscous (frictional) forces are very important due to the no-slip boundary condition. The steady, incompressible, twodimensional, boundary layer developing along a flat plate aligned with the free-stream flow is sketched in Fig. P9-114. The flow upstream of the plate is uniform, but boundary layer thickness $\delta$ grows with $x$ along the plate due to viscous effects. Sketch some streamlines, both within the boundary layer and above the boundary layer. Is $\delta(x)$ a streamline? (Hint: Pay particular attention to the fact that for steady, incompressible, two-dimensional flow the volume flow rate per unit width between any two streamlines is constant.)

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Problem 115

A group of students is designing a small, round (axisymmetric), low-speed wind tunnel for their senior design project (Fig. P9-115E). Their design calls for the axial component of velocity to increase linearly in the contraction section from $u_{z, 0}$ to $u_{z, L}$. The air speed through the test section is to be $u_{z L}=120 \mathrm{ft} / \mathrm{s}$. The length of the contraction is $L=3.0 \mathrm{ft}$, and the entrance and exit diameters of the contraction are $D_0=5.0 \mathrm{ft}$ and $D_L=1.5 \mathrm{ft}$, respectively. The air is at standard temperature and pressure. (a) Verify that the flow can be approximated as incompressible. (b) Generate an expression for the radial velocity component $u_{\mathrm{r}}$ between $z=0$ and $z=L$, staying in variable form. You may ignore frictional effects (boundary layers) on the walls. (c) Generate an expression for the stream function $\psi=$ function of $r$ and $z$. (d) Plot some streamlines and design the shape of the contraction, assuming that frictional effects along the walls of the wind tunnel contraction are negligible.

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Problem 116

We approximate the flow of air into a vacuum cleaner's floor attachment by the stream function $\psi$ $=\frac{-\dot{V}}{2 \pi L} \arctan \frac{\sin 2 \theta}{\cos 2 \theta+b^2 / r^2}$ in the center plane (the $x y$ plane) in cylindrical coordinates, where $L$ is the length of the attachment, $b$ is the height of the attachment above the floor, and $\dot{V}$ is the volume flow rate of air being sucked into the hose. Shown in Fig. P9-116 is a three-dimensional view with the floor in the $x z$-plane; we model a two-dimensional slice of the flow in the $x y$-plane through the centerline of the attachment. Note that we have (arbitrarily) set $\psi=0$ along the positive $x$-axis $(\theta=0)$. (a) What are the primary dimensions of the given stream function? (b) Nondimensionalize the stream function by defining $\psi^*=(2 \pi L / \dot{V}) \psi$ and $r^*=r / b$. (c) Solve your nondimensionalized equation for $r^*$ as a function of $\psi^*$ and $\theta$. Use this equation to plot several nondimensional streamlines of the flow. For consistency, plot in the range -2 $<x^*<2$ and $0<y^*<4$, where $x^*=x / b$ and $y^*=y / b$. (Hint: $\psi^*$ must be negative to yield the proper flow direction.)

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Problem 117

Look up the definition of Poisson's equation in one of your math textbooks or on the Internet. Write Poisson's equation in standard form. How is Poisson's equation similar to Laplace's equation? How do these two equations differ?

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Problem 118

Water flows down a long, straight, inclined pipe of diameter $D$ and length $L$ (Fig. P9-118). There is no forced pressure gradient between points 1 and 2 ; in other words, the water flows through the pipe by gravity alone, and $P_1=P_2$ $=P_{\text {atm }}$.The flow is steady, fully developed, and laminar. We
adopt a coordinate system in which $x$ follows the axis of the pipe. (a) Use the control volume technique of Chap. 8 to generate an expression for average velocity $V$ as a function of the given parameters $\rho, g, D, \Delta z, \mu$, and $L$. (b) Use differential analysis to generate an expression for $V$ as a function of the given parameters. Compare with your result of part (a) and discuss. (c) Use dimensional analysis to generate a dimensionless expression for $V$ as a function of the given parameters. Construct a relationship between your $\Pi$ 's that matches the exact analytical expression.

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Problem 119

A block slides down a long, straight, inclined wall at speed $V$, riding on a thin film of oil of thickness $h$ (Fig. P9-119). The weight of the block is $W$, and its surface area in contact with the oil film is $A$. Suppose $V$ is measured, and $W$, $A$, angle $\alpha$, and viscosity $\mu$ are also known. Oil film thickness $h$ is not known. (a) Generate an exact analytical expression for $h$ as a function of the known parameters $V, A, W, \alpha$, and $\mu$. (b) Use dimensional analysis to generate a dimensionless expression for $h$ as a function of the given parameters. Construct a relationship between your $\Pi$ 's that matches the exact analytical expression of part (a).

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