Questions and Answers of Introduction To Fluid Mechanics

A velocity field in polar coordinates is given with the radial velocity as \(V_{r}=-A / r\) and the tangential velocity as \(V_{\theta}=A / r\), where \(r\) is in meters and \(A=10 \mathrm{~m}^{2}\).
The flow of air near the Earth's surface is affected both by the wind and thermal currents. In certain circumstances the velocity field can be represented by \(\vec{V}=a
A velocity field is given by \(\vec{V}=a y t \hat{i}-b x \hat{j}\), where \(a=1 \mathrm{~s}^{-2}\) and \(b=4 \mathrm{~s}^{-1}\). Find the equation of the streamlines at any time \(t\). Plot several
Air flows downward toward an infinitely wide horizontal flat plate. The velocity field is given by \(\vec{V}=(a x \hat{i}-a y \hat{j})(2+\cos \omega t)\), where \(a=5 \mathrm{~s}^{-1}, \omega=2 \pi
Consider the flow described by the velocity field \(\vec{V}=\) \(B x(1+A t) \hat{i}+C y \hat{j}\), with \(A=0.5 \mathrm{~s}^{-1}\) and \(B=C=1 \mathrm{~s}^{-1}\). Coordinates are measured in meters.
Consider the velocity field \(V=a x \hat{i}+b y(1+c t) \hat{j}\), where \(a=b=2 \mathrm{~s}^{-1}\) and \(c=0.4 \mathrm{~s}^{-1}\). Coordinates are measured in meters. For the particle that passes
Consider the flow field given in Eulerian description by the expression \(\vec{V}=a x \hat{i}+b y t \hat{j}\), where \(a=0.2 \mathrm{~s}^{-1}, b=0.04 \mathrm{~s}^{-2}\), and the coordinates are
A velocity field is given by \(\vec{V}=a x t \hat{i}-b y \hat{j}\), where \(A=0.1 \mathrm{~s}^{-2}\) and \(b=1 \mathrm{~s}^{-1}\). For the particle that passes through the point \((x, y)=(1,1)\) at
Consider the garden hose of Fig. 2.5. Suppose the velocity field is given by \(\vec{V}=u_{0} \hat{i}+v_{0} \sin \left[\omega\left(t-x / u_{0}\right)\right] \hat{j}\), where the \(x\) direction is
Consider the velocity field of Problem 2.18. Plot the streakline formed by particles that passed through the point \((1,1)\) during the interval from \(t=0\) to \(t=3 \mathrm{~s}\). Compare with the
Streaklines are traced out by neutrally buoyant marker fluid injected into a flow field from a fixed point in space. A particle of the marker fluid that is at point \((x, y)\) at time \(t\) must have
Consider the flow field \(\vec{V}=a x t \hat{i}+b \hat{j}\), where \(a=1 / 4 \mathrm{~s}^{-2}\) and \(b=1 / 3 \mathrm{~m} / \mathrm{s}\). Coordinates are measured in meters. For the particle that
A flow is described by velocity field \(\vec{V}=a y^{2} \hat{i}+b \hat{j}\), where \(a=1 \mathrm{~m}^{-1} \mathrm{~s}^{-1}\) and \(b=2 \mathrm{~m} / \mathrm{s}\). Coordinates are measured in meters.
Tiny hydrogen bubbles are being used as tracers to visualize a flow. All the bubbles are generated at the origin \((x=0, y=0)\). The velocity field is unsteady and obeys the
A flow is described by velocity field \(\vec{V}=a \hat{i}+b x \hat{j}\), where \(a=2 \mathrm{~m} / \mathrm{s}\) and \(b=1 \mathrm{~s}^{-1}\). Coordinates are measured in meters. Obtain the equation
A flow is described by velocity field \(\vec{V}=a y \hat{i}+b t \hat{j}\), where \(a=0.2 \mathrm{~s}^{-1}\) and \(b=0.4 \mathrm{~m} / \mathrm{s}^{2}\). At \(t=2 \mathrm{~s}\), what are the
A flow is described by velocity field \(\vec{V}=a t \hat{i}+b \hat{j}\), where \(a=0.4 \mathrm{~m} / \mathrm{s}^{2}\) and \(b=2 \mathrm{~m} / \mathrm{s}\). At \(t=2 \mathrm{~s}\), what are the
The variation with temperature of the viscosity of air is represented well by the empirical Sutherland correlation\[\mu=\frac{b T^{1 / 2}}{1+S / T}\]Best-fit values of \(b\) and \(S\) are given in
The variation with temperature of the viscosity of air is correlated well by the empirical Sutherland equation\[\mu=\frac{b T^{1 / 2}}{1+S / T}\]Best-fit values of \(b\) and \(S\) are given for use
Some experimental data for the viscosity of helium at 1 atm areUsing the approach described correlate these data to the empirical Sutherland equation\[\mu=\frac{b T^{1 / 2}}{1+S / T}\](where \(T\) is
The velocity distribution for laminar flow between parallel plates is given by\[\frac{u}{u_{\max }}=1-\left(\frac{2 y}{h}\right)^{2}\]where \(h\) is the distance separating the plates and the origin
Calculate velocity gradients and shear stress for \(y=0,0.2\), 0.4 , and \(0.6 \mathrm{~m}\), if the velocity profile is a quarter-circle having its center \(0.6 \mathrm{~m}\) from the boundary. The
A very large thin plate is centered in a gap of width \(0.06 \mathrm{~m}\) with different oils of unknown viscosities above and below; one viscosity is twice the other. When the plate is pulled at a
A female freestyle ice skater, weighing \(100 \mathrm{lbf}\), glides on one skate at speed \(V=20 \mathrm{ft} / \mathrm{s}\). Her weight is supported by a thin film of liquid water melted from the
A block of mass \(10 \mathrm{~kg}\) and measuring \(250 \mathrm{~mm}\) on each edge is pulled up an inclined surface on which there is a film of SAE \(10 \mathrm{~W}-30\) oil at \(30^{\circ}
A 73-mm-diameter aluminum ( \(\mathrm{SG}=2.64)\) piston of \(100-\mathrm{mm}\) length resides in a stationary 75 -mm-inner-diameter steel tube lined with SAE \(10 \mathrm{~W}-30\) oil at
A vertical gap \(25 \mathrm{~mm}\) wide of infinite extent contains oil of specific gravity 0.95 and viscosity \(2.4 \mathrm{~Pa} \cdot \mathrm{s}\). A metal plate \(1.5 \mathrm{~m} \times 1.5
A cylinder 8 in. in diameter and \(3 \mathrm{ft}\) long is concentric with a pipe of 8.25 in. i.d. Between cylinder and pipe there is an oil film. What force is required to move the cylinder along
Crude oil at \(20^{\circ} \mathrm{C}\) fills the space between two concentric cylinders \(250 \mathrm{~mm}\) high and with diameters of \(150 \mathrm{~mm}\) and \(156 \mathrm{~mm}\). What torque is
The piston in Problem 2.40 is traveling at terminal speed. The mass \(m\) now disconnects from the piston. Plot the piston speed vs. time. How long does it take the piston to come within 1 percent of
A block of mass \(M\) slides on a thin film of oil. The film thickness is \(h\) and the area of the block is \(A\). When released, mass \(m\) exerts tension on the cord, causing the block to
A block \(0.1 \mathrm{~m}\) square, with \(5 \mathrm{~kg}\) mass, slides down a smooth incline, \(30^{\circ}\) below the horizontal, on a film of SAE 30 oil at \(20^{\circ} \mathrm{C}\) that is
A torque of \(4 \mathrm{~N} \cdot \mathrm{m}\) is required to rotate the intermediate cylinder at \(30 \mathrm{r} / \mathrm{min}\). Calculate the viscosity of the oil. All cylinders are \(450
A circular disk of diameter \(d\) is slowly rotated in a liquid of large viscosity \(\mu\), at a small distance \(h\) from a fixed surface. Derive an expression for the torque \(T\) necessary to
The fluid drive shown transmits a torque \(T\) for steadystate conditions ( \(\omega_{1}\) and \(\omega_{2}\) constant). Derive an expression for the slip \(\left(\omega_{1}-\omega_{2}\right)\) in
A block that is \(a \mathrm{~mm}\) square slides across a flat plate on a thin film of oil. The oil has viscosity \(\mu\) and the film is \(h \mathrm{~mm}\) thick. The block of mass \(M\) moves at
In a food-processing plant, honey is pumped through an annular tube. The tube is \(L=2 \mathrm{~m}\) long, with inner and outer radii of \(R_{i}=5 \mathrm{~mm}\) and \(R_{o}=25 \mathrm{~mm}\),
SAE \(10 \mathrm{~W}-30\) oil at \(100^{\circ} \mathrm{C}\) is pumped through a tube \(L=10 \mathrm{~m}\) long, diameter \(D=20 \mathrm{~mm}\). The applied pressure difference is \(\Delta p=5
The lubricant has a kinematic viscosity of \(2.8 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) and specific gravity of 0.92 . If the mean velocity of the piston is \(6 \mathrm{~m} / \mathrm{s}\),
Calculate the approximate viscosity of the oil. V = 0.6 ft/s 2 ft x 2 ft square plate W= 25 lb P2.54 13 5 12 0.05 in. oil flim
Calculate the approximate power lost in friction in this ship propeller shaft bearing. -1m- 0.36 m d shaft 200 rpm Oil- 0.23 mm = 0.72 Pa.s P2.55
Fluids of viscosities \(\mu_{1}=0.1 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\) and \(\mu_{2}=0.15 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\) are contained between two plates (each plate is
A concentric cylinder viscometer may be formed by rotating the inner member of a pair of closely fitting cylinders. The annular gap is small so that a linear velocity profile will exist in the liquid
A concentric cylinder viscometer is driven by a falling mass \(M\) connected by a cord and pulley to the inner cylinder, as shown. The liquid to be tested fills the annular gap of width \(a\) and
A shock-free coupling for a low-power mechanical drive is to be made from a pair of concentric cylinders. The annular space between the cylinders is to be filled with oil. The drive must transmit
A shaft with outside diameter of \(18 \mathrm{~mm}\) turns at 20 revolutions per second inside a stationary journal bearing \(60 \mathrm{~mm}\) long. A thin film of oil \(0.2 \mathrm{~mm}\) thick
A proposal has been made to use a pair of parallel disks to measure the viscosity of a liquid sample. The upper disk rotates at height \(h\) above the lower disk. The viscosity of the liquid in the
The cone and plate viscometer shown is an instrument used frequently to characterize non-Newtonian fluids. It consists of a flat plate and a rotating cone with a very obtuse angle (typically
A viscometer is used to measure the viscosity of a patient's blood. The deformation rate (shear rate)-shear stress data is shown below. Plot the apparent viscosity versus deformation rate. Find the
A concentric-cylinder viscometer is shown. Viscous torque is produced by the annular gap around the inner cylinder. Additional viscous torque is produced by the flat bottom of the inner cylinder as
Design a concentric-cylinder viscometer to measure the viscosity of a liquid similar to water. The goal is to achieve a measurement accuracy of \(\pm 1\) percent. Specify the configuration and
A cross section of a rotating bearing is shown. The spherical member rotates with angular speed \(\omega\), a small distance, \(a\), above the plane surface. The narrow gap is filled with viscous
Small gas bubbles form in soda when a bottle or can is opened. The average bubble diameter is about \(0.1 \mathrm{~mm}\). Estimate the pressure difference between the inside and outside of such a
You intend to gently place several steel needles on the free surface of the water in a large tank. The needles come in two lengths: Some are \(5 \mathrm{~cm}\) long, and some are \(10 \mathrm{~cm}\)
According to Folsom [6], the capillary rise \(\Delta h\) (in.) of a waterair interface in a tube is correlated by the following empirical expression:\[\Delta h=A e^{-b \cdot D}\]where \(D\) (in.) is
Calculate and plot the maximum capillary rise of water \(\left(20^{\circ} \mathrm{C}\right)\) to be expected in a vertical glass tube as a function of tube diameter for diameters from 0.5 to \(2.5
Calculate the maximum capillary rise of water \(\left(20^{\circ} \mathrm{C}\right)\) to be expected between two vertical, clean glass plates spaced \(1 \mathrm{~mm}\) apart.
Calculate the maximum capillary depression of mercury to be expected in a vertical glass tube \(1 \mathrm{~mm}\) in diameter at \(15.5^{\circ} \mathrm{C}\).
Water usually is assumed to be incompressible when evaluating static pressure variations. Actually it is 100 times more compressible than steel. Assuming the bulk modulus of water is constant,
The viscous boundary layer velocity profile shown in Fig. 2.15 can be approximated by a cubic equation,\[u(y)=a+b\left(\frac{y}{\delta}\right)+c\left(\frac{y}{\delta}\right)^{3}\]The boundary
In a food industry process, carbon tetrachloride at \(20^{\circ} \mathrm{C}\) flows through a tapered nozzle from an inlet diameter \(D_{\text {in }}=50 \mathrm{~mm}\) to an outlet diameter of
What is the Reynolds number of water at \(20^{\circ} \mathrm{C}\) flowing at \(0.25 \mathrm{~m} / \mathrm{s}\) through a \(5-\mathrm{mm}\)-diameter tube? If the pipe is now heated, at what mean water
A supersonic aircraft travels at \(2700 \mathrm{~km} / \mathrm{hr}\) at an altitude of \(27 \mathrm{~km}\). What is the Mach number of the aircraft? At what approximate distance measured from the
SAE 30 oil at \(100^{\circ} \mathrm{C}\) flows through a 12 -mm-diameter stainless-steel tube. What is the specific gravity and specific weight of the oil? If the oil discharged from the tube fills a
A seaplane is flying at \(100 \mathrm{mph}\) through air at \(45^{\circ} \mathrm{F}\). At what distance from the leading edge of the underside of the fuselage does the boundary layer transition to
An airliner is cruising at an altitude of \(5.5 \mathrm{~km}\) with a speed of \(700 \mathrm{~km} / \mathrm{hr}\). As the airliner increases its altitude, it adjusts its speed so that the Mach number
Because the pressure falls, water boils at a lower temperature with increasing altitude. Consequently, cake mixes and boiled eggs, among other foods, must be cooked different lengths of time.
Ear "popping" is an unpleasant phenomenon sometimes experienced when a change in pressure occurs, for example in a fastmoving elevator or in an airplane. If you are in a two-seater airplane at \(3000
When you are on a mountain face and boil water, you notice that the water temperature is \(195^{\circ} \mathrm{F}\). What is your approximate altitude? The next day, you are at a location where it
Your pressure gauge indicates that the pressure in your cold tires is \(0.25 \mathrm{MPa}\) gage on a mountain at an elevation of \(3500 \mathrm{~m}\). What is the absolute pressure? After you drive
A \(125-\mathrm{mL}\) cube of solid oak is held submerged by a tether as shown. Calculate the actual force of the water on the bottom surface of the cube and the tension in the tether. Patm Oil 0.5
The tube shown is filled with mercury at \(20^{\circ} \mathrm{C}\). Calculate the force applied to the piston. Diameter, D 50 mm = h = 25 mm P3.6 d = 10 mm - F H = 200 mm
Calculate the absolute and gage pressure in an open tank of crude oil \(2.4 \mathrm{~m}\) below the liquid surface. If the tank is closed and pressurized to \(130 \mathrm{kPa}\), what are the
An open vessel contains carbon tetrachloride to a depth of \(6 \mathrm{ft}\) and water on the carbon tetrachloride to a depth of \(5 \mathrm{ft}\). What is the pressure at the bottom of the vessel?
A hollow metal cube with sides \(100 \mathrm{~mm}\) floats at the interface between a layer of water and a layer of SAE \(10 \mathrm{~W}\) oil such that \(10 \%\) of the cube is exposed to the oil.
Compressed nitrogen \((140 \mathrm{lbm}\) ) is stored in a spherical tank of diameter \(D=2.5 \mathrm{ft}\) at a temperature of \(77^{\circ} \mathrm{F}\). What is the pressure inside the tank? If the
If at the surface of a liquid the specific weight is \(\gamma_{o}\), with \(z\) and \(p\) both zero, show that, if \(E=\) constant, the specific weight and pressure are given
In the deep ocean the compressibility of seawater is significant in its effect on \(ho\) and \(p\). If \(E=2.07 \times 10^{9} \mathrm{~Pa}\), find the percentage change in the density and pressure at
Assuming the bulk modulus is constant for seawater, derive an expression for the density variation with depth, \(h\), below the surface. Show that the result may be written\[ho \approx ho_{0}+b
An inverted cylindrical container is lowered slowly beneath the surface of a pool of water. Air trapped in the container is compressed isothermally as the hydrostatic pressure increases. Develop an
A water tank filled with water to a depth of \(16 \mathrm{ft}\) has an inspection cover (1 in. \(\times 1\) in.) at its base, held in place by a plastic bracket. The bracket can hold a load of \(9
A partitioned tank as shown contains water and mercury. What is the gage pressure in the air trapped in the left chamber? What pressure would the air on the left need to be pumped to in order to
Consider the two-fluid manometer shown. Calculate the applied pressure difference. P1 -Water- | = 10.2 mm P2 P3.17 Carbon tetrachloride
The manometer shown contains water and kerosene. With both tubes open to the atmosphere, the free-surface elevations differ by \(H_{0}=20.0 \mathrm{~mm}\). Determine the elevation difference when a
Determine the gage pressure in \(\mathrm{kPa}\) at point \(a\), if liquid \(A\) has \(\mathrm{SG}=1.20\) and liquid \(B\) has \(\mathrm{SG}=0.75\). The liquid surrounding point \(a\) is water, and
With the manometer reading as shown, calculate \(p_{x}\). Oil (SG 0.85) |30 in. 60 in. Mercury Px P3.20
Calculate \(p_{x}-p_{y}\) for this inverted U-tube manometer. 60 in. Oil(SG 0.90) 10 in. Water 20 in. Px P3.21
An inclined gauge having a tube of 3-mm bore, laid on a slope of 1:20, and a reservoir of 25-mm-diameter contains silicon oil (SG \(0.84)\). What distance will the oil move along the tube when a
Water flows downward along a pipe that is inclined at \(30^{\circ}\) below the horizontal, as shown. Pressure difference \(p_{A}-p_{B}\) is due partly to gravity and partly to friction. Derive an
A reservoir manometer has vertical tubes of diameter \(D=18 \mathrm{~mm}\) and \(d=6 \mathrm{~mm}\). The manometer liquid is Meriam red oil. Develop an algebraic expression for liquid deflection
A rectangular tank, open to the atmosphere, is filled with water to a depth of \(2.5 \mathrm{~m}\) as shown. A U-tube manometer is connected to the tank at a location \(0.7 \mathrm{~m}\) above the
The sketch shows a sectional view through a submarine. Calculate the depth of submergence, \(y\). Assume the specific weight of seawater is \(10.0 \mathrm{kN} / \mathrm{m}^{3}\). 60" Atmos. pressure
The manometer reading is 6 in. when the funnel is empty (water surface at \(A\) ). Calculate the manometer reading when the funnel is filled with water. 10 ft -5 ft d- A -Water -Mercury P3.27
A reservoir manometer is calibrated for use with a liquid of specific gravity 0.827 . The reservoir diameter is 5 / 8 in. and the vertical tube diameter is 3/16 in. Calculate the required distance
The inclined-tube manometer shown has \(D=96 \mathrm{~mm}\) and \(d=8 \mathrm{~mm}\). Determine the angle, \(\theta\), required to provide a 5:1 increase in liquid deflection, \(L\), compared with
The inclined-tube manometer shown has \(D=76 \mathrm{~mm}\) and \(d=8 \mathrm{~mm}\), and is filled with Meriam red oil. Compute the angle, \(\theta\), that will give a \(15-\mathrm{cm}\) oil
A barometer accidentally contains 6.5 inches of water on top of the mercury column (so there is also water vapor instead of a vacuum at the top of the barometer). On a day when the temperature is
A water column stands \(50 \mathrm{~mm}\) high in a \(2.5-\mathrm{mm}\) diameter glass tube. What would be the column height if the surface tension were zero? What would be the column height in a
Consider a small-diameter open-ended tube inserted at the interface between two immiscible fluids of different densities. Derive an expression for the height difference \(\Delta h\) between the
Based on the atmospheric temperature data of the U.S. Standard Atmosphere of Fig. 3.3, compute and plot the pressure variation with altitude, and compare with the pressure data of Table A.3.Data From

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