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engineering fluid mechanics

Questions and Answers of Engineering Fluid Mechanics

The cross-sectional area of a rectangular duct is divided into 16 equal rectangular areas, as shown in Fig. P5.131. The axial fluid velocity measured in feet per second in each smaller area is shown.
A small fan moves air at a mass flowrate of \(0.004 \mathrm{lbm} / \mathrm{s}\). Upstream of the fan, the pipe diameter is 2.5 in., the flow is laminar, the velocity distribution is parabolic, and
Air enters a radial blower with zero angular momentum. It leaves with an absolute tangential velocity, \(V_{\theta}\), of \(200 \mathrm{ft} / \mathrm{s}\). The rotor blade speed at rotor exit is
Water enters a pump impeller radially. It leaves the impeller with a tangential component of absolute velocity of \(10 \mathrm{~m} / \mathrm{s}\). The impeller exit diameter is \(60 \mathrm{~mm}\),
Water enters an axial-flow turbine rotor with an absolute velocity tangential component, \(V_{\theta}\), of \(15 \mathrm{ft} / \mathrm{s}\). The corresponding blade velocity, \(U\), is \(50
An inward flow radial turbine (see Fig. P5.136) involves a nozzle angle, \(\alpha_{1}\), of \(60^{\circ}\) and an inlet rotor tip speed, \(U_{1}\), of \(30 \mathrm{ft} / \mathrm{s}\). The ratio of
The velocity in a certain two-dimensional flow field is given by the equation\[ \mathbf{V}=2 x \hat{\mathbf{i}}-2 y t \hat{\mathbf{j}} \]where the velocity is in \(\mathrm{ft} / \mathrm{s}\) when
The velocity in a certain flow field is given by the equation\[ \mathbf{V}=x \hat{\mathbf{i}}+x^{2} z \hat{\mathbf{j}}+y z \hat{\mathbf{k}} \]Determine the expressions for the three rectangular
The flow in the plane two-dimensional channel shown in Figure P6.3 has \(x\) - and \(y\)-components of velocity given by\[ \begin{aligned} &
The three components of velocity in a flow field are given by\[ \begin{aligned} u & =x^{2}+y^{2}+z^{2} \\ v & =x y+y z+z^{2} \\ w & =-3 x z-z^{2} / 2+4 \end{aligned} \](a) Determine the
Determine an expression for the vorticity of the flow field described by\[ \mathbf{V}=-x y^{3} \hat{\mathbf{i}}+y^{4} \hat{\mathbf{j}} \]Is the flow irrotational?
According to Eq. 6.134, the \(x\)-velocity in fully developed laminar flow between parallel plates is given by\[ u=\frac{1}{2 \mu}\left(\frac{\partial p}{\partial x}\right)\left(y^{2}-h^{2}\right)
For a certain incompressible, two-dimensional flow field the velocity component in the \(y\) direction is given by the equation\[ v=3 x y+x^{2} y \]Determine the velocity component in the \(x\)
An incompressible viscous fluid is placed between two large parallel plates as shown in Fig. P6.8. The bottom plate is fixed and the upper plate moves with a constant velocity, \(U\). For these
A viscous fluid is contained in the space between concentric cylinders. The inner wall is fixed, and the outer wall rotates with an angular velocity \(\omega\). Fig. P6.9a Assume that the velocity
Air is delivered through a constant-diameter duct by a fan. The air is inviscid, so the fluid velocity profile is "flat" across each cross section. During the fan start-up, the following velocities
For a certain incompressible flow field it is suggested that the velocity components are given by the equations\[ u=2 x y \quad v=-x^{2} y \quad w=0 \]Is this a physically possible flow field?
The velocity components of an incompressible, two-dimensional velocity field are given by the equations\[ \begin{aligned} u & =y^{2}-x(1+x) \\ v & =y(2 x+1) \end{aligned} \]Show that the flow is
Consider the two-dimensional channel flow of Problem 6.3. Show that the given velocities satisfy conservation of mass in both differential and control volume forms.Problem 6.3The flow in the plane
The \(x\)-velocity profile in a certain laminar boundary layer is approximated as follows\[ u=U_{0} \sin \left(\frac{\pi}{2} \frac{y}{0.1 \sqrt{x}}\right) \]Determine the \(y\)-velocity, \(v(x,
For each of the following stream functions, with units of \(\mathrm{m}^{2} / \mathrm{s}\), determine the magnitude and the angle the velocity vector makes with the \(x\) axis at \(x=1 \mathrm{~m},
The stream function for an incompressible, two-dimensional flow field is\[ \psi=a y-b y^{3} \]where \(a\) and \(b\) are constants. Is this an irrotational flow? Explain.
For a certain two-dimensional flow field\[ \begin{aligned} & u=0 \\ & v=V \end{aligned} \](a) What are the corresponding radial and tangential velocity components? (b) Determine the
In a certain steady, two-dimensional flow field the fluid density varies linearly with respect to the coordinate \(x\) : that is, \(ho=A x\) where \(A\) is a constant. If the \(x\) component of
In a two-dimensional, incompressible flow field, the \(x\) component of velocity is given by the equation \(u=2 x\). (a) Determine the corresponding equation for the \(y\) component of velocity if
The stream function for an incompressible flow field is given by the equation\[ \psi=3 x^{2} y-y^{3} \]where the stream function has the units of \(\mathrm{m}^{2} / \mathrm{s}\) with \(x\) and
Consider the incompressible, two-dimensional flow of a nonviscous fluid between the boundaries shown in Fig. P6.22. The velocity potential for this flow field is\[ \phi=x^{2}-y^{2} \](a) Determine
A fluid with a density of \(2000 \mathrm{~kg} / \mathrm{m}^{3}\) flows steadily between two flat plates as shown in Fig. P6.23. The bottom plate is fixed and the top one moves at a constant speed in
We derived the differential equation(s) of linear momentum by considering the motion of a fluid element. Derive the linear momentum equation(s) by considering a small control volume, like we did for
By considering the rotational equilibrium of a fluid mass element, show that \(\tau_{x y}=\tau_{y x}\).
The stream function for a given two-dimensional flow field is\[ \psi=5 x^{2} y-(5 / 3) y^{3} \]Determine the corresponding velocity potential.
A certain flow field is described by the stream function\[ \psi=A \theta+B r \sin \theta \]where \(A\) and \(B\) are positive constants. Determine the corresponding velocity potential and locate
Integrate Bernoulli's equation for compressible flow, Eq. (6.56), for an ideal gas undergoing an isothermal (constant temperature) process along a streamline.Eq. (6.56) 3/2 + P + gz = constant
Integrate Bernoulli's equation for compressible flow, Eq. (6.56), for a fluid whose pressure \(p\) and density \(ho\) obey the expression \(p=C ho^{n}\) along a streamline, where \(C\) and \(n\) are
The velocity potential for a certain inviscid flow field is\[ \phi=-\left(3 x^{2} y-y^{3}\right) \]where \(\phi\) has the units of \(\mathrm{ft}^{2} / \mathrm{s}\) when \(x\) and \(y\) are in feet.
The stream function for a two-dimensional, nonviscous, incompressible flow field is given by the expression\[ \psi=-2(x-y) \]where the stream function has the units of \(\mathrm{ft}^{2} /
The velocity potential for a certain inviscid, incompressible flow field is given by the equation\[ \phi=2 x^{2} y-\left(\frac{2}{3}\right) y^{3} \]where \(\phi\) has the units of \(\mathrm{m}^{2}
Consider the two-dimensional air flow around the corner, as shown in Fig. P6.33. The \(x\) - and \(y\)-direction velocities are\[ u=\frac{v_{0}}{L} \sin \mathrm{h}\left(\frac{x}{L}\right) \cosh
The streamlines for an incompressible, inviscid, two-dimensional flow field are all concentric circles, and the velocity varies directly with the distance from the common center of the streamlines;
The velocity potential\[ \phi=-k\left(x^{2}-y^{2}\right) \quad(k=\text { constant }) \]may be used to represent the flow against an infinite plane boundary, as illustrated in Fig. P6.35. For flow
Water is flowing between wedge-shaped walls into a small opening as shown in Fig. P6.36. The velocity potential with units \(\mathrm{m}^{2} / \mathrm{s}\) for this flow is \(\phi=-2 \ln r\) with
As illustrated in Fig. P6.38, a tornado can be approximated by a free vortex of strength \(\Gamma\) for \(r>R_{c}\), where \(R_{c}\) is the radius of the core. Velocity measurements at points
The motion of a liquid in an open tank is that of a combined vortex consisting of a forced vortex for \(0 \leq r \leq 2 \mathrm{ft}\) and a free vortex for \(r>2 \mathrm{ft}\). The velocity
Water flows through a two-dimensional diffuser having a \(20^{\circ}\) expansion angle as shown in Fig. P6.40. Assume that the flow in the diffuser can be treated as a radial flow emanating from a
When water discharges from a tank through an opening in its bottom, a vortex may form with a curved surface profile, as shown in Fig. P6.41. Assume that the velocity distribution in the vortex is the
Water flows over a flat surface at \(4 \mathrm{ft} / \mathrm{s}\), as shown in Fig. P6.42. A pump draws off water through a narrow slit at a volume rate of \(0.1 \mathrm{ft}^{3} / \mathrm{s}\) per
Two sources, one of strength \(m\) and the other with strength \(3 m\), are located on the \(x\) axis as shown in Fig. P6.43. Determine the location of the stagnation point in the flow produced by
The velocity potential for a spiral vortex flow is given by \(\phi=\) \((\Gamma / 2 \pi) \theta-(m / 2 \pi) \ln r\), where \(\Gamma\) and \(m\) are constants. Show that the angle, \(\alpha\), between
For a free vortex determine an expression for the pressure gradient(a) along a streamline, (b) normal to a streamline. Assume that the streamline is in a horizontal plane, and express your answer in
The Wide World of Fluids article titled "Some Hurricane Facts,". Consider a category five hurricane that has a maximum wind speed of \(160 \mathrm{mph}\) at the eye wall, \(10 \mathrm{mi}\) from the
Consider the flow of a liquid of viscosity \(\mu\) and density \(ho\) down an inclined plate making an angle \(\theta\) with the horizontal. The film thickness is \(t\) and is constant. The fluid
A horizontal oil-hearing stratum is \(3 \mathrm{~m}\) high and, after hydraulic fracturing ("fracking"), provides a volume flow rate of \(Q=1000 \mathrm{~m}^{3} / \mathrm{hr}\). The flow moves
Show that the circulation of a free vortex for any closed path that does not enclose the origin is zero.
Show that the circulation of a free vortex for any closed path that encloses the origin is \(\Gamma\).
Potential flow against a flat plate (Fig. P6.51a) can be described with the stream function\[ \psi=A x y \]where \(A\) is a constant. This type of flow is commonly called a stagnation point flow
The combination of a uniform flow and a source can be used to describe flow around a streamlined body called a halfbody. Assume that a certain body has the shape of a half-body with a thickness of
Show that if \(\Phi_{1}\) and \(\Phi_{2}\) are both solutions of Laplace's equation, the sum \(\left(\Phi_{1}+\Phi_{2}\right)\) is also a solution.
The flow in the impeller of a centrifugal pump is modeled by the superposition of a source and a free vortex. The impeller has an outer diameter of \(0.5 \mathrm{~m}\) and an inner diameter of \(0.3
One end of a pond has a shoreline that resembles a halfbody as shown in Fig. P6.55. A vertical porous pipe is located near the end of the pond so that water can be pumped out. When water is pumped at
A Rankine oval is formed by combining a source-sink pair, each having a strength of \(36 \mathrm{ft}^{2} / \mathrm{s}\) and separated by a distance of \(12 \mathrm{ft}\) along the \(x\) axis, with a
A 15-mph wind flows over a Quonset hut having a radius of \(12 \mathrm{ft}\) and a length of \(60 \mathrm{ft}\), as shown in Fig. P6.57. The upstream pressure and temperature are equal to those
An ideal fluid flows past an infinitely long, semicircular "hump" located along a plane boundary, as shown in Fig. P6.58. Far from the hump the velocity field is uniform, and the pressure is
Assume that the flow around the long circular cylinder of Fig. P6.59 is nonviscous and incompressible. Two pressures, \(p_{1}\) and \(p_{2}\), are measured on the surface of the cylinder, as
Consider the steady potential flow around the circular cylinder shown in Fig. 6.26. On a plot show the variation of the magnitude of the dimensionless fluid velocity, \(V / U\), along the positive
A highway sign has the shape of a half-cylinder with radius \(1.0 \mathrm{~m}\) and length \(25.0 \mathrm{~m}\). A rearward \(10-\mathrm{km} / \mathrm{hr}\) wind is blowing over the sign as shown in
Air at \(25^{\circ} \mathrm{C}\) flows normal to the axis of an infinitely long cylinder of \(1.0-\mathrm{m}\) radius. The cylinder is rotating at \(10 \mathrm{rad} / \mathrm{s}\), and the approach
The velocity potential for a cylinder (Fig. P6.63) rotating in a uniform stream of fluid is\[ \phi=U r\left(1+\frac{a^{2}}{r^{2}}\right) \cos \theta+\frac{\Gamma}{2 \pi} \theta \]where \(\Gamma\)
The Wide World of Fluids article titled "A Sailing Ship without Sails,". Determine the magnitude of the total force developed by the two rotating cylinders on the Flettner "rotor-ship" due to the
Consider the possibility of using two rotating cylinders to replace the conventional wings on an airplane for lift. Consider an airplane flying at \(700 \mathrm{~km} / \mathrm{hr}\) through the
Air at \(25^{\circ} \mathrm{C}\) flows normal to the axis of an infinitely long cylinder of \(1.0-\mathrm{m}\) radius. The approach velocity is \(100 \mathrm{~km} / \mathrm{hr}\). Find the rotational
Determine the shearing stress for an incompressible Newtonian fluid with a velocity distribution of \(\mathbf{V}=\left(3 x y^{2}-4 x^{3}\right) \mathbf{i}+\) \(\left(12 x^{2} y-y^{3}\right)
The two-dimensional velocity field for an incompressible Newtonian fluid is described by the relationship\[ \mathbf{V}=\left(12 x y^{2}-6 x^{3}\right) \hat{\mathbf{i}}+\left(18 x^{2} y-4
The velocity of a fluid particle moving along a horizontal streamline that coincides with the \(x\) axis in a plane, two-dimensional, incompressible flow field was experimentally found to be
"Stokes's first problem" involves the instantaneous acceleration at time \(t=0\) of a flat plate to a constant velocity \(U_{0}\) while in contact with a "semi-infinite," static fluid as shown in
Oil (SAE 30) at \(15.6{ }^{\circ} \mathrm{C}\) flows steadily between fixed, horizontal, parallel plates. The pressure drop per unit length along the channel is \(30 \mathrm{kPa} / \mathrm{m}\), and
Two fixed, horizontal, parallel plates are spaced 0.4 in. apart. A viscous liquid \(\left(\mu=8 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}, S G=0.9\right)\) flows between the
A liquid of constant density \(ho\) and constant viscosity \(\mu\) flows down a wide, long inclined flat plate. The plate makes an angle \(\theta\) with the horizontal. The velocity components do not
We will see in Chapter 8 that the pressure drop in fully developed pipe flow is sometimes computed with the aid of a friction factor, defined by\[ f=\frac{\Delta p}{\frac{1}{2} ho V^{2}}
The Wide World of Fluids article titled "10 Tons on 8 psi,". A massive, precisely machined, 6-ft-diameter granite sphere rests on a 4-ft-diameter cylindrical pedestal as shown in Fig. P6.75. When the
The bearing shown in Fig. P6.76 consists of two parallel discs of radius \(R\) separated from each other by a small distance \(h(h)\), and angular symmetry exists \((\partial / \partial
A Bingham plastic is a fluid in which the stress \(\tau\) is related to the rate of strain \(d u / d y\) by\[ \tau=\tau_{0}+\mu\left(\frac{d u}{d y}\right) \]where \(\tau_{0}\) and \(\mu\) are
Two horizontal, infinite, parallel plates are spaced a distance \(b\) apart. A viscous liquid is contained between the plates. The bottom plate is fixed, and the upper plate moves parallel to the
An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as is shown in Fig. P6.79. The two plates move in opposite directions with constant velocities, \(U_{1}\) and
Two immiscible, incompressible, viscous fluids having the same densities but different viscosities are contained between two infinite, horizontal, parallel plates (Fig. P6.80). The bottom plate is
The disc shown in Fig. P6.81 has a diameter \(D\) of \(1 \mathrm{~m}\) and a rotational speed of \(1800 \mathrm{rpm}\). It is positioned \(4 \mathrm{~mm}\) from a solid boundary. The gap is filled
A viscous fluid (specific weight \(=80 \mathrm{lb} / \mathrm{ft}^{3}\); viscosity \(=\) \(0.03 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}\) ) is contained between two infinite, horizontal
A flat block is pulled along a horizontal flat surface by a horizontal rope perpendicular to one of the sides. The block measures \(1.0 \mathrm{~m} \times 1.0 \mathrm{~m}\), has a mass of \(100
A viscosity motor/pump is shown in Fig. P6.84. The rotor is concentric within a stationary housing. The clearance \(h\) between the housing and the rotor is small compared to the width \(w\) and
A vertical shaft passes through a bearing and is lubricated with an oil having a viscosity of \(0.2 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\) as shown in Fig. P6.85. Assume that the flow
A viscous fluid is contained between two long concentric cylinders. The geometry of the system is such that the flow between the cylinders is approximately the same as the laminar flow between two
Verify that the momentum correction factor \(\beta\) for fully developed, laminar flow in a circular tube is \(4 / 3\).
Verify that the kinetic energy correction factor \(\alpha\) for fully developed, laminar flow in a circular tube is 2.0.
A simple flow system to be used for steady-flow tests consists of a constant head tank connected to a length of 4-mm-diameter tubing as shown in Fig. P6.89. The liquid has a viscosity of \(0.015
(a) Show that for Poiseuille flow in a tube of radius \(R\) the magnitude of the wall shearing stress, \(\tau_{r z}\), can be obtained from the relationship\[ \left|\left(\tau_{r z}\right)_{\text
An infinitely long, solid, vertical cylinder of radius \(R\) is located in an infinite mass of an incompressible fluid. Start with the Navier-Stokes equation in the \(\theta\) direction and derive an
We will see in Chapter 8 that the pressure drop in fully developed pipe flow is sometimes computed with the aid of a friction factor, defined by\[ f=\frac{\Delta p}{\frac{1}{2} ho V^{2}}
A liquid (viscosity \(=0.002 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\); density \(=1000 \mathrm{~kg} / \mathrm{m}^{3}\) ) is forced through the circular tube shown in Fig. P6.93. A differential
Fluid with kinematic viscosity \(u\) flows down an inclined circular pipe of length \(\ell\) and diameter \(D\) with flow rate \(Q\). Find the vertical drop per unit length of the pipe so that the
Blood flows at volume rate \(Q\) in a circular tube of radius \(R\). The blood cells concentrate and flow near the center of the tube, while the cell-free fluid (plasma) flows in the outer region.
An incompressible Newtonian fluid flows steadily between two infinitely long, concentric cylinders as shown in Fig. P6.96. The outer cylinder is fixed, but the inner cylinder moves with a

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