Fluid Mechanics in SI Units

R. C. Hibbeler, Kai Beng Yap

Chapter 7

Differential Fluid Flow - all with Video Answers

Educators

Chapter Questions

Problem 1

As the top plate is pulled to the right with a constant velocity $\mathbf{U},$ the fluid between the plates has a linear velocity distribution as shown. Determine the rate of rotation of a fluid element and the shear-strain rate of the element located at $y$.

James Kiss

James Kiss

Numerade Educator

Problem 2

Determine the circulation at $r=30 \mathrm{~m}$ and at $r=50 \mathrm{~m}$ of a tornado, if the velocity at the center (eye) of a tornado is defined by $v_{r}=0, v_{\theta}=(0.3 r) \mathrm{m} / \mathrm{s}$, where $r$ is in meters.

James Kiss

Numerade Educator

Problem 3

Determine the circulation around the rectangular region when a uniform flow $\mathbf{V}$ is directed at an angle $\theta$ to the horizontal as shown.

James Kiss

Numerade Educator

Problem 4

The velocity components of a flow are given as $u=\left(3 x^{2}+3 y^{2}\right) \mathrm{m} / \mathrm{s}$ and $v=(-4 x y) \mathrm{m} / \mathrm{s},$ where $x$ and $y$ are in meters. If the flow is irrotational, determine the circulation around the rectangular region?

James Kiss

Numerade Educator

Problem 5

Consider the fluid element that has dimensions in polar coordinates as shown and whose boundaries are defined by the streamlines with velocities $v$ and $v+d v$. Show that the vorticity for the flow is given by $\zeta=-(v / r+d v / d r)$.

James Kiss

Numerade Educator

Problem 6

Determine the stream and potential functions for the two-dimensional flow field if $\mathbf{V}_{0}$ and $\theta$ are known.

James Kiss

Numerade Educator

Problem 7

A fluid is flowing with the stream function, $\Psi=(5 x+3 y),$ where $x$ and $y$ are in meters, determine the potential function and the magnitude of the velocity of a fluid particle at point $(2 \mathrm{~m}, 3 \mathrm{~m})$

James Kiss

Numerade Educator

Problem 8

A thick fluid is flowing along the channel of constant width. The velocity profile is given as $u=\left(5 \mathrm{y}^{2}\right) \mathrm{mm} / \mathrm{s}$, where $y$ is in millimeters. Determine the stream function for the flow and plot the streamlines for $\psi_{0}=0, \psi_{1}=2 \mathrm{~mm}^{2} / \mathrm{s},$ and $\psi_{2}=3 \mathrm{~mm}^{2} / \mathrm{s}$.

James Kiss

Numerade Educator

Problem 9

A thick fluid is flowing along the channel of constant width. The velocity profile is given as $u=\left(5 \mathrm{y}^{2}\right) \mathrm{mm} / \mathrm{s}$, where $y$ is in millimeters. Is it possible to determine the potential function for the flow? If so, what is it?

James Kiss

Numerade Educator

Problem 10

A two-dimensional flow is described by the stream function $\psi=\left(x y^{3}-x^{3} y\right) \mathrm{m}^{2} / \mathrm{s},$ where $x$ and $y$ are in meters. Show that the continuity condition is satisfied and determine if the flow is rotational or irrotational.

James Kiss

Numerade Educator

Problem 11

The distribution of linear velocity of the fluid flowing between two pates is shown in Fig. Determine the stream function. Does the potential function exist?

James Kiss

Numerade Educator

Problem 12

The distribution of linear velocity of the fluid flowing between two pates is shown in Fig. If the pressure at the top surface of the bottom plate is $500 \mathrm{~N} / \mathrm{m}^{2}$, determine the pressure at the bottom surface of the top plate. Take $\rho=1.2 \mathrm{Mg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 13

A two-dimensional flow field is defined by its components $u=(3 y) \mathrm{m} / \mathrm{s}$ and $v=(9 x) \mathrm{m} / \mathrm{s},$ where $x$ and $y$ are in meters. Determine if the flow is rotational or irrotational, and show that the continuity condition for the flow is satisfied. Also, find the stream function and the equation of the streamline that passes through point $(4 \mathrm{~m}, 3 \mathrm{~m}) .$ Plot this streamline.

James Kiss

Numerade Educator

Problem 14

A fluid has the velocity components shown. Determine the stream and potential functions. Plot the streamlines for $\psi_{0}=0, \psi_{1}=1 \mathrm{~m}^{2} / \mathrm{s}$, and $\psi_{2}=2 \mathrm{~m}^{2} / \mathrm{s}$.

James Kiss

Numerade Educator

Problem 15

The stream function of a flow field is defined by $\psi=3\left(2 x^{2}-y^{2}\right) \mathrm{m}^{2} / \mathrm{s}$, where $x$ and $y$ are in meters. Determine the flow per unit depth in $\mathrm{m}^{2} / \mathrm{s}$ that occurs through $A B, C B$ and $A C$ as shown.

James Kiss

Numerade Educator

Problem 17

The stream function of a fluid flow is defined as $\psi=(6 x-3 y) \mathrm{m}^{2} / \mathrm{s}$, where $x$ and $y$ are in meters. Determine the potential function, and show that the continuity condition is satisfied and that the flow is irrotational.

James Kiss

Numerade Educator

Problem 18

The stream function for a flow field is defined by $\psi=2 r^{3} \sin 2 \theta .$ Determine the magnitude of the velocity of fluid particles at point $r=1 \mathrm{~m}, \theta=(\pi / 3) \mathrm{rad}$, and plot the streamlines for $\psi_{1}=1 \mathrm{~m}^{2} / \mathrm{s}$ and $\psi_{2}=2 \mathrm{~m}^{2} / \mathrm{s}$.

James Kiss

Numerade Educator

Problem 19

Water flow through the horizontal channel is defined by the stream function $\psi=2\left(x^{2}-y^{2}\right) \mathrm{m}^{2} / \mathrm{s}$. If the pressure at $B$ is atmospheric, determine the pressure at point $(0.5 \mathrm{~m}, 0)$ and the flow per unit depth in $\mathrm{m}^{2} / \mathrm{s}$.

James Kiss

Numerade Educator

Problem 20

An ideal fluid flows into the corner formed by the two walls. If the stream function for this flow is defined by $\psi=\left(5 r^{4} \sin 4 \theta\right) \mathrm{m}^{2} / \mathrm{s}$, show that continuity for the flow is
satisfied. Also, plot the streamline that passes through point $r=2 \mathrm{~m}, \theta=(\pi / 6) \mathrm{rad},$ and find the magnitude of the velocity at this point.

James Kiss

Numerade Educator

Problem 21

The flat plate is subjected to the flow defined by the stream function $\psi=\left[8 r^{1 / 2} \sin (\theta / 2)\right] \mathrm{m}^{2} / \mathrm{s}$. Sketch the streamline that passes through point $r=4 \mathrm{~m}, \theta=\pi \mathrm{rad}$ and determine the magnitude of the velocity at this point.

James Kiss

Numerade Educator

Problem 22

Determine the potential function for the twodimensional flow field if $\mathbf{V}_{0}$ and $\theta$ are known.

James Kiss

Numerade Educator

Problem 23

The stream function for a concentric flow is defined by $\psi=-4 r^{2}$. Determine the velocity components $v_{r}$ and $v_{\theta}$, and $v_{x}$ and $v_{y} .$ Can the potential function be established? If so, what is it?

James Kiss

Numerade Educator

Problem 24

If the potential function for a two-dimensional flow is $\phi=(x y) \mathrm{m}^{2} / \mathrm{s}$, where $x$ and $y$ are in meters, determine the stream function, and plot the streamline that passes through the point $(1 \mathrm{~m}, 2 \mathrm{~m})$. What are the $x$ and $y$ components of the velocity and acceleration of fluid particles that pass through this point?

Supratim Pal

Numerade Educator

Problem 25

The horizontal flow confined by the walls is defined by the stream function $\psi=\left[4 r^{4 / 3} \sin \left(\frac{4}{3} \theta\right)\right] \mathrm{m}^{2} / \mathrm{s}$, where $r$ is in meters. Determine the magnitude of the velocity at point $r=2 \mathrm{~m}, \quad \theta=45^{\circ} .$ Is the flow rotational or irrotational? Can the Bernoulli equation be used to determine the difference in pressure between the two points $A$ and $B ?$

James Kiss

Numerade Educator

Problem 26

The horizontal flow between the walls is defined by the stream function $\psi=\left[4 r^{4 / 3} \sin \left(\frac{4}{3} \theta\right)\right] \mathrm{m}^{2} / \mathrm{s},$ where $r$ is in meters. If the pressure at the origin $O$ is $20 \mathrm{kPa}$, determine the pressure at $r=2 \mathrm{~m}, \quad \theta=45^{\circ} .$ Take $\rho=950 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 27

A fluid flows through a bend in a horizontal channel. The flow can be described as a free vortex for which $v_{r}=0$, $v_{\theta}=(4 / r) \mathrm{m} / \mathrm{s}$, where $r$ is in meters. Show that the flow is irrotational. If the pressure at point $A$ is $5 \mathrm{kPa}$, determine the pressure at point $B$. Take $\rho=1200 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 28

A two-dimensional flow is described by the potential function $\phi=\left(8 x^{2}-8 y^{2}\right) \mathrm{m}^{2} / \mathrm{s},$ where $x$ and $y$ are in meters. Show that the continuity condition is satisfied, and determine if the flow is rotational or irrotational. Also, establish the stream function for this flow, and plot the streamline that passes through point $(1 \mathrm{~m}, 0.5 \mathrm{~m})$.

James Kiss

Numerade Educator

Problem 29

The stream function for a horizontal flow near the corner is $\psi=(8 x y) \mathrm{m}^{2} / \mathrm{s},$ where $x$ and $y$ are in meters. Determine the $x$ and $y$ components of the velocity and the acceleration of fluid particles passing through point $(1 \mathrm{~m},$ $2 \mathrm{~m}$ ). Show that it is possible to establish the potential function. Plot the streamlines and equipotential lines that pass through point $(1 \mathrm{~m}, 2 \mathrm{~m})$.

James Kiss

Numerade Educator

Problem 30

The stream function for horizontal flow near the corner is defined by $\psi=(8 x y) \mathrm{m}^{2} / \mathrm{s}$, where $x$ and $y$ are in meters. Show that the flow is irrotational. If the pressure at point $A(1 \mathrm{~m}, 2 \mathrm{~m})$ is $150 \mathrm{kPa}$, determine the pressure at point $B(2 \mathrm{~m}, 3 \mathrm{~m})$. Take $\rho=980 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 31

The stream function for the flow field around the $90^{\circ}$ corner is $\psi=8 r^{2} \sin 2 \theta .$ Show that the continuity of flow is satisfied. Determine the $r$ and $\theta$ velocity components of a fluid particle located at $r=0.5 \mathrm{~m}, \theta=30^{\circ},$ and plot the streamline that passes through this point. Also, determine the potential function for the flow.

James Kiss

Numerade Educator

Problem 32

A fluid has velocitycomponents $u=2\left(x^{2}-y^{2}\right) \mathrm{m} / \mathrm{s}$ and $v=(-4 x y) \mathrm{m} / \mathrm{s},$ where $x$ and $y$ are in meters. Determine the stream function. Also show that the potential function exists, and find this function. Plot the streamlines and equipotential lines that pass through point $(1 \mathrm{~m}, 2 \mathrm{~m})$.

James Kiss

Numerade Educator

Problem 33

The potential function of two-dimensional flow is defined as $f=\left(x^{2}+y^{2}\right) \mathrm{m}^{2} / \mathrm{s}$, where $x$ and $y$ are in meters, determine the stream function and plot the streamline that passes through the point $(2 \mathrm{~m}, 3 \mathrm{~m})$. Also, determine the velocity and acceleration of fluid particles passing through this point?

James Kiss

Numerade Educator

Problem 34

The velocity components of a fluid flow are given as $u=\left(x^{2}+y^{2}\right) \mathrm{m} / \mathrm{s}$ and $v=(x y) \mathrm{m} / \mathrm{s}$, where $x$ and $y$ are in meters. If the pressure at point $A(4 \mathrm{~m}, 2 \mathrm{~m})$ is $800 \mathrm{kPa},$ determine the pressure at point $B(2 \mathrm{~m}, 3 \mathrm{~m})$. Also, what is the potential function for the flow? Take $\gamma=10 \mathrm{kN} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 35

A fluid has velocity components of $u=(10 x y) \mathrm{m} / \mathrm{s}$ and $v=5\left(x^{2}-y^{2}\right) \mathrm{m} / \mathrm{s},$ where $x$ and $y$ are in meters. Determine the stream function, and show that the continuity condition is satisfied and that the flow is irrotational. Plot the streamlines for $\psi_{0}=0, \psi_{1}=1 \mathrm{~m}^{2} / \mathrm{s},$ and $\psi_{2}=2 \mathrm{~m}^{2} / \mathrm{s}$.

James Kiss

Numerade Educator

Problem 36

The velocity components of a fluid flow are given as $u=(8 x y) \mathrm{m} / \mathrm{s}$ and $v=4\left(x^{2}-y^{2}\right) \mathrm{m} / \mathrm{s}$, where $x$ and $y$ are in meters. Determine the potential function, and show that the continuity condition is satisfied and that the flow is irrotational.

James Kiss

Numerade Educator

Problem 37

A fluid flow has the potential function $\varphi=\left(x^{3}-8 x y^{2}\right) \mathrm{m}^{2} / \mathrm{s}$, where $x$ and $y$ are in meters. Determine the magnitude of the velocity at point $A(3 \mathrm{~m}, 1 \mathrm{~m})$. What is the difference in pressure between this point and the origin? Take $\rho=1020 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 38

Show that the equation that defines a sink will satisfy continuity, which in polar coordinates is written as $\frac{\partial\left(v_{r} r\right)}{\partial r}+\frac{\partial\left(v_{\theta}\right)}{\partial \theta}=0$.

James Kiss

Numerade Educator

Problem 39

Combine a source of strength $q$ with a free counterclockwise vortex, and sketch the resultant streamline for $\psi=0$.

James Kiss

Numerade Educator

Problem 40

Pipe $A$ provides a source flow of $8 \mathrm{~m}^{2} / \mathrm{s}$, whereas the drain, or sink, at $B$ removes $8 \mathrm{~m}^{2}$ s. Determine the stream function between $A B,$ and show the streamline for $\psi=0$.

James Kiss

Numerade Educator

Problem 41

Pipe $A$ provides a source flow of $8 \mathrm{~m}^{2} / \mathrm{s}$, whereas the drain, or sink, at $B$ removes $8 \mathrm{~m}^{2 /}$ s. Determine the stream function between $A B,$ and show the equipotential line for $\varphi=0$.

James Kiss

Numerade Educator

Problem 42

Determine the location of the stagnation point for a combined uniform flow of $8 \mathrm{~m} / \mathrm{s}$ and a source having a strength of $3 \mathrm{~m}^{2} / \mathrm{s}$. Plot the streamline passing through the stagnation point.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 43

Two sources, each having a strength of $2 \mathrm{~m}^{2} / \mathrm{s}$, are located as shown. Determine the $x$ and $y$ components of the velocity of fluid particles that pass point $(x, y)$. What is the equation of the streamline that passes through point $(0,8 \mathrm{~m})$ in Cartesian coordinates? Is the flow irrotational?

James Kiss

Numerade Educator

Problem 44

A fluid flow is originated from a source $O$ that is described by the potential function $\varphi=(5 \ln r) \mathrm{m}^{2} / \mathrm{s},$ where $r$ is in meters. Determine the stream function, and specify the velocity at point $r=8 \mathrm{~m}, u=30^{\circ}$.

James Kiss

Numerade Educator

Problem 45

The stream function of a free vortex is defined by $\psi=(-180 \ln r) \mathrm{m}^{2} / \mathrm{s}$, where $r$ is in meters. Determine the velocity of a particle at $r=3 \mathrm{~m}$ and the pressure at points on this streamline. Take $\rho=1.20 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 46

The source and sink of equal strength $q$ are located a distance $d$ from the origin as indicated. Determine the stream function for the flow, and draw the streamline that passes through the origin.

James Kiss

Numerade Educator

Problem 47

Two sources, each having a strength $q,$ are located as shown. Determine the stream function, and show that this is the same as having a single source with a wall along the $y$ axis.

James Kiss

Numerade Educator

Problem 48

The Rankine body is defined by the source and sink, each having a strength of $0.2 \mathrm{~m}^{2} / \mathrm{s}$. If the velocity of the uniform flow is $4 \mathrm{~m} / \mathrm{s}$, determine the longest and shortest dimensions of the body.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 49

The Rankine body is defined by the source and sink, each having a strength of $0.2 \mathrm{~m}^{2} / \mathrm{s}$. If the velocity of the uniform flow is $4 \mathrm{~m} / \mathrm{s}$, determine the equation in Cartesian coordinates that defines the boundary of the body.

James Kiss

Numerade Educator

Problem 50

The half body is defined by a combined uniform flow having a velocity of $U$ and a point source of strength $q$. Determine the pressure distribution along the top boundary of the half body as a function of $\theta$, if the pressure within the uniform flow is $p_{0}$. Neglect the effect of gravity. The density of the fluid is $\rho$.

James Kiss

Numerade Educator

Problem 51

A fluid flows over a half body for which $U=0.4 \mathrm{~m} / \mathrm{s}$ and $q=1.0 \mathrm{~m}^{2} / \mathrm{s}$. Plot the half body, and determine the magnitudes of the velocity and pressure in the fluid at the point $r=0.8 \mathrm{~m}$ and $\theta=90^{\circ} .$ The pressure within the uniform flow is 300 Pa. Take $\rho=850 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 52

A source $q$ is emitted from the wall while a flow occurs toward the wall. If the stream function is described as $\psi=(4 x y+8 \theta) \mathrm{m}^{2} / \mathrm{s},$ where $x$ and $y$ are in meters, determine the distance $d$ from the wall where the stagnation point occurs along the $y$ axis. Plot the streamline that passes through this point.

James Kiss

Numerade Educator

Problem 53

As water drains from the large cylindrical tank, its surface forms a free vortex having a circulation of $\Gamma$. Assuming water to be an ideal fluid, determine the equation $z=f(r)$ that defines the free surface of the vortex. Hint: Use the Bernoulli equation applied to two points on the surface.

James Kiss

Numerade Educator

Problem 54

A fluid has a uniform velocity of $U=10 \mathrm{~m} / \mathrm{s}$. A source $q=15 \mathrm{~m}^{2} / \mathrm{s}$ is at $x=2 \mathrm{~m}$, and a sink $q=-15 \mathrm{~m}^{2} / \mathrm{s}$ is at $x=-2 \mathrm{~m}$. Graph the Rankine body that is formed, and determine the magnitudes of the velocity and the pressure at point $(0,2 \mathrm{~m})$. The pressure within the uniform flow is $40 \mathrm{kPa}$. Take $\rho=850 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 55

Integrate the pressure distribution, Eq. 7-67, over the surface of the cylinder in Fig. $7-33 b$, and show that the resultant force is equal to zero.

James Kiss

Numerade Educator

Problem 56

The diameter of pier of a bridge is 0.8 -m-diameter and it is subjected to the uniform flow of water at $5 \mathrm{~m} / \mathrm{s}$. Determine the maximum and minimum pressures exerted on the pier at a depth of $3 \mathrm{~m}$.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 57

A gas flows around the cylinder such that the pressure, measured at $A$, is $p_{A}=-5 \mathrm{kPa}$. Determine the velocity $U$ of the flow if the density of the gas $\rho=1.20 \mathrm{~kg} / \mathrm{m}^{3} .$ Can this velocity be determined if instead the pressure at $B$ is measured?

James Kiss

Numerade Educator

Problem 58

A cylinder is rotated in anticlockwise direction with a constant angular velocity of 120 rev/min and by applying the torque $\mathbf{T}$. If the wind is blowing at a constant speed of $20 \mathrm{~m} / \mathrm{s}$, determine the location of the stagnation points on the surface of the cylinder, and find the maximum pressure. The pressure within the uniform flow is $500 \mathrm{~Pa}$. Take $\rho_{a}=1.20 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 59

A cylinder is rotated in anticlockwise direction with a constant angular velocity of 120 rev/min and by applying the torque $\mathbf{T}$. If the wind is blowing at a constant speed of $20 \mathrm{~m} / \mathrm{s},$ determine the lift per unit length on the cylinder and the minimum pressure on the cylinder. The pressure within the uniform flow is $500 \mathrm{~Pa}$. Take $\rho_{a}=1.20 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 60

Air is flowing at $U=30 \mathrm{~m} / \mathrm{s}$ past the Quonset hut of radius $R=3 \mathrm{~m}$. Find the velocity and absolute pressure distribution along the $y$ axis for $3 \mathrm{~m} \leq y \leq \infty$. The absolute pressure within the uniform flow is $p_{0}=100 \mathrm{kPa}$. Take $\rho_{a}=1.23 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 61

The Quonset hut of radius $R$ is subjected to a uniform wind having a velocity $U$. Determine the resultant vertical force caused by the pressure that acts on the hut if it has a length $L$. The density of air is $\rho$.

James Kiss

Numerade Educator

Problem 62

The Quonset hut of radius $R$ is subjected to a uniform wind having a velocity $U$. Determine the speed of the wind and the gage pressure at point $A$. The density of air is $\rho$.

James Kiss

Numerade Educator

Problem 63

The 200-mm-diameter cylinder is subjected to a uniform horizontal flow having a velocity of $6 \mathrm{~m} / \mathrm{s}$. At a distance far away from the cylinder, the pressure is $150 \mathrm{kPa}$. Plot the variation of the velocity and pressure along the radial line $r$, at $\theta=90^{\circ}$, and specify their values at $r=0.1 \mathrm{~m}$, $0.2 \mathrm{~m}, 0.3 \mathrm{~m}, 0.4 \mathrm{~m}$, and $0.5 \mathrm{~m}$. Take $\rho=1.5 \mathrm{Mg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 64

The 200-mm-diameter cylinder is subjected to a uniform horizontal flow having a velocity of $6 \mathrm{~m} / \mathrm{s}$. At a distance far away from the cylinder, the pressure is $150 \mathrm{kPa}$. Plot the variation of the velocity and pressure along the radial line $r$, at $\theta=90^{\circ}$, and specify their values at $r=0.1 \mathrm{~m}$, $0.2 \mathrm{~m}, 0.3 \mathrm{~m}, 0.4 \mathrm{~m}$, and $0.5 \mathrm{~m}$. Take $\rho=1.5 \mathrm{Mg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 65

Determine the equation of the boundary of the half body formed by placing a source of $0.5 \mathrm{~m}^{2} / \mathrm{s}$ in the uniform flow of $8 \mathrm{~m} / \mathrm{s}$. Express the result in Cartesian coordinates.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 66

The half body is defined by a combined uniform flow having a velocity of $U$ and a point source of strength $q$. Determine the location $\theta$ on the boundary of the half body where the pressure $p$ is equal to the pressure $p_{0}$ within the uniform flow. Neglect the effect of gravity.

James Kiss

Numerade Educator

Problem 67

The pipe is built from four quarter segments that are glued together. If it is exposed to a uniform airflow having a velocity of $8 \mathrm{~m} / \mathrm{s}$, determine the resultant force the pressure exerts on the quarter segment $A B$ per unit length of the pipe. Take $\rho=1.22 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 68

The laminar flow of a fluid has velocity components $u=6 x$ and $v=-6 y,$ where $y$ is vertical. Use the NavierStokes equations to determine the pressure in the fluid, $p=p(x, y)$, if at point $(0,0), p=0$. The density of the fluid is $\rho$.

James Kiss

Numerade Educator

Problem 69

The steady laminar flow of an ideal fluid toward the fixed surface has a velocity of $u=\left[10\left(1+1 /\left(8 x^{3}\right)\right)\right] \mathrm{m} / \mathrm{s}$ along the horizontal streamline $A B$. Use the Navier-Stokes equations and determine the variation of the pressure along this streamline, and plot it for $-2.5 \mathrm{~m} \leq x \leq-0.5 \mathrm{~m}$. The pressure at $A$ is $5 \mathrm{kPa}$, and the density of the fluid is $\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}$.

James Kiss

Numerade Educator

Problem 70

Liquid is confined between a top plate having an area $A$ and a fixed surface. A force $\mathbf{F}$ is applied to the plate and gives the plate a velocity $\mathbf{U}$. If this causes laminar flow, and the pressure does not vary, show that the Navier-Stokes and continuity equations indicate that the velocity distribution for this flow is defined by $u=U(y / h)$, and that the shear stress within the liquid is $\tau_{x y}=F / A$.

James Kiss

Numerade Educator

Problem 71

Fluid having a density $\rho$ and viscosity $\mu$ fills the space between the two cylinders. If the outer cylinder is fixed, and the inner one is rotating at $\omega,$ apply the Navier-Stokes equations to determine the velocity profile assuming laminar flow.

James Kiss

Numerade Educator

Problem 72

The channel for a liquid is formed by two fixed plates. If laminar flow occurs between the plates, show that the Navier-Stokes and continuity equations reduce to $\partial^{2} u / \partial y^{2}=(1 / \mu) \partial p / \partial x$ and $\partial p / \partial y=0 .$ Integrate these equations to show that the velocity profile for the flow is $u=(1 /(2 \mu))(d p / d x)\left[y^{2}-(d / 2)^{2}\right] .$ Neglect the effect of gravity.

James Kiss

Numerade Educator

Problem 73

The sloped open channel has steady laminar flow at a depth $h$. Show that the Navier-Stokes equations reduce to $\partial^{2} u / \partial y^{2}=-(\rho g \sin \theta) / \mu$ and $\partial p / \partial y=-\rho g \cos \theta .$ Integrate these equations to show that the velocity profile is $u=[(\rho g \sin \theta) / 2 \mu]\left(2 h y-y^{2}\right) \quad$ and the shear-stress distribution is $\tau_{x y}=\rho g \sin \theta(h-y)$.

James Kiss

Numerade Educator

Problem 74

A horizontal velocity field is defined by $u=2\left(x^{2}-y^{2}\right) \mathrm{m} / \mathrm{s}$ and $v=(-4 x y) \mathrm{m} / \mathrm{s}$. Show that these expressions satisfy the continuity equation. Using the Navier-Stokes equations, show that the pressure distribution is defined by $p=C-\rho V^{2} / 2-\rho g z$.

James Kiss

Numerade Educator