# GATE Questions & Answers of Fluid Mechanics Mechanical Engineering

#### Fluid Mechanics 81 Question(s) | Weightage 09 (Marks)

For a two-dimensional incompressible flow field given by $\overset\rightharpoonup u=A\left(x\widehat i-y\widehat j\right)$, where $A>0$ , which one of the following statements is FALSE?

A tank open at the top with a water level of 1 m, as shown in the figure, has a hole at a height of 0.5 m. A free jet leaves horizontally from the smooth hole. The distance X (in m) where the jet strikes the floor is

In a Lagrangian system, the position of a fluid particle in a flow is described as $x=x_\circ e^{-kt}$ and $y=y_\circ e^{-kt}$ where t is the time while $x_\circ$, $y_\circ$, and k are constants. The flow is

A solid block of 2.0 kg mass slides steadily at a velocity V along a vertical wall as shown in the figure below. A thin oil film of thickness h = 0.15 mm provides lubrication between the block and the wall. The surface area of the face of the block in contact with the oil film is 0.04 $m^2$ . The velocity distribution within the oil film gap is linear as shown in the figure. Take dynamic viscosity of oil as $7\times10^{-3}$ Pa-s and acceleration due to gravity as 10 $m/s^2$ . Neglect weight of the oil. The terminal velocity V (in m/s) of the block is _________ (correct to one decimal place).

The viscous laminar flow of air over a flat plate results in the formation of a boundary layer. The boundary layer thickness at the end of the plate of length L is $\delta_L$. When the plate length is increased to twice its original length, the percentage change in laminar boundary layer thickness at the end of the plate (with respect to $\delta_L$) is ________ (correct to two decimal places).

Air flows at the rate of 1.5 m3/s through a horizontal pipe with a gradually reducing crosssection as shown in the figure. The two cross-sections of the pipe have diameters of 400 mm and 200 mm. Take the air density as 1.2 kg/m3 and assume inviscid incompressible flow. The change in pressure $\left(p_2-p_1\right)$ (in kPa) between sections 1 and 2 is

For steady flow of a viscos incompressible fluid through a circular pipe of constant diameter, the averavge velocity in the fully developed region is constant. Which one of the following statements about the average velocity in the developing region is TRUE?

Consider the two-dimensional velocity filed given by $\style{font-family:'Times New Roman'}{\overrightarrow V=(5+a_1x+b_1y)\widehat i+(4+a_2x+b_2y)\widehat j,}$ where a1, b1, a2 and b2 are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?

Consider steady flow of an incomressible fluid through two long and straight pipes of diameters $d_1$ and $d_2$ arranged in series. Both pipes are of equal length and the flow is turbulent in both pipes. The friction factor for turbulent flow though pipes is of the form, $f=K(Re)^{-n}$, where $K$ and $n$ are known positive constants and Re is the Reynolds number. Neglecting minir losses, the ratio of the frictional pressure drop in pipe 1 to that  in pipe 2, $\style{font-family:'Times New Roman'}{\left(\frac{\triangle P_1}{\triangle P_2}\right)}$, is given by

For a steady flow, the velocity field is $\style{font-family:'Times New Roman'}{\overrightarrow V=(-x^2+3y)\widehat i+(2xy)\widehat j}$. The magnitude of the acceleration of a particle at (1,-1) is

For the stability of a floating body the

Consider a laminar flow at zero incidence over a flat plate. The shear stress at the wall is denoted by ${\tau }_{\omega }$ . The axial position x1 and x2 on the plate are measured from the leading edge in the direction of flow. If x2>x1 then

The arrangement shown in the figure measures the velocity V of a gas of density 1 kg/m2 flowing through a pipe. The acceleration due to gravity is 9.81 m/s2. If the manometric fluid is water (density 1000 kg/m3) and the velocity V is 20 m/s, the differential head h (in mm) between the two arms of the manometer is__________

A 60 mm-diameter water jet strikes a plate containing a hole of 40 mm diameter as shown in the figure. Part of the jet passes through the hole horizontally, and the remaining is deflected vertically. The density of water is 1000 kg/m3. If velocities are as indicates in the figure, the magnitude of horizontal force (in N) required to hold the plate is_______

For a laminar flow of water over a sphere, the drag coefficient CF is defined as ${C}_{F}=F/\left(\rho {U}^{\mathit{2}}{D}^{\mathit{2}}\right)$ , where F is the drag force, $\rho$ is the fluid density, U is the fluid velocity and D is the diameter of the sphere. The density of water is 1000 kg/m3. The diameter of the sphere is 100 mm and the velocity is 2 m/s, the drag coefficient is 0.5. If water now flows over another sphere of diameter 200 mm under dynamically similar conditions, the drag force (in N) on the sphere is___________

The instantaneous stream-wise velocity of a turbulent flow is given as follows:
$u(x,y,z,t)$=$\overset\_u(x,y,z)$+$u'(x,y,z,t)$
The time-average of the fluctuating velocity $u'(x,y,z,t)$ is

For a floating body, buoyant force acts at the

Consider two hydraulic turbines having identical specific speed and effective head at the inlet. If the speed ratio $\left(N_1/N_2\right)$ of the two turbines is 2, then the respective power ratio $\left(P_1/P_2\right)$ is _____________

An inverted U-tube manometer is used to measure the pressure difference between two pipes A and B, as shown in the figure. Pipe A is carrying oil (specifc gravity=0.8) and pipe B is carrying water. The densities of air and water are 1.16 $\mathrm{kg}/\mathrm m^3$ and 1000 $\mathrm{kg}/\mathrm m^3$, respectivly. The pressure difference between pipes A and B is _________ kPa.

Oil (kinematic viscosity,$v_{oil}=1.0\times10^{-5}\;\mathrm m^2/\mathrm s$) flows through a pipe of 0.5 m diameter with a velocity of 10 m/s. Water (kinematic viscosity, $v_{oil}=0.89\times10^{-6}\;\mathrm m^2/\mathrm s$) is flowing through a model pipe of diameter 20 mm. For satisfying the dynamic similarity, the velocity of water (in m/s) is __________

A steady laminar boundary layer is formed over a flat plate as shown in the figure. The free stream velocity of the fluid is U0. The velocity profile at the inlet a-b is uniform, while that at a downstream location c-d is given by $\mathrm u=U_0\left[2\left(\frac y\delta\right)-\left(\frac y\delta\right)^2\right]$
The ratio of the mass flow rate,$\overset.{M_{bd}}$, leaving through the horizontal section $b-d$ to that entering through the vertical section $a-b$ is ___________

The volumetric flow rate (per unit depth) between two streamlines having stream functions $\psi_1$ and $\psi_2$ is

The large vessel shown in the figure contains oil and water. A body is submerged at the interface of oil and water such that 45 percent of its volume is in oil while the rest is in water. The density of the body is _________ $\mathrm{kg}/\mathrm m^3$.

The specific gravity of oil is 0.7 and density of water is 1000 $\mathrm{kg}/\mathrm m^3$.

Acceleration due to gravity $\mathbf g\boldsymbol=\mathbf{10}\boldsymbol\;\mathbf m\boldsymbol/\mathbf s^\mathbf2$.

Consider fluid flow between two infinite horizontal plates which are parallel (the gap between them being 50 mm). The top plate is sliding parallel to the stationary bottom plate at a speed of 3 m/s. The flow between the plates is solely due to the motion of the top plate. The force per unit area (magnitude) required to maintain the bottom plate stationary is _________ $\mathrm N/\mathrm m^2$.

Viscosity of the fluid $\mu=0.44\;\mathrm{kg}/\mathrm m-\mathrm s$ and density $\rho=888\;\mathrm{kg}/\mathrm m^3$.

Consider a frictionless, massless and leak-proof plug blocking a rectangular hole of dimensions $2R\times L$ at the bottom of an open tank as shown in the figure. The head of the plug has the shape of a semi-cylinder of radius $R$. The tank is filled with a liquid of density $\rho$ up to the tip of the plug. The gravitational acceleration is $g$. Neglect the effect of the atmospheric pressure.

The force F required to hold the plug in its position is

A channel of width 450 mm branches into two sub-channels having width 300 mm and 200 mm as shown in figure. If the volumetric flow rate (taking unit depth) of an incompressible flow through the main channel is $0.9\;\mathrm m^3/\mathrm s$ and the velocity in the sub-channel of width 200 mm is 3 m/s, the velocity in the sub-channel of width 300 mm is _____________ m/s.

Assume both inlet and outlet to be at the same elevation.

For a certain two-dimensional incompressible flow, velocity field is given by $2xy\;\widehat i-y^2\widehat j$ . The streamlines for this flow are given by the family of curves

Grashof number signifies the ratio of

Consider a fully developed steady laminar flow of an incompressible fluid with viscosity $\mu$ through a circular pipe of radius R. Given that the velocity at a radial location of $R/2$ from the centerline of the pipe is $U_1$ , the shear stress at the wall is  $K\mu U_1/R$, where K is __________

The water jet exiting from a stationary tank through a circular opening of diameter 300 mm impinges on a rigid wall as shown in the figure. Neglect all minor losses and assume the water level in the tank to remain constant. The net horizontal force experienced by the wall is ___________ kN.

Density of water is 1000 kg/m3.

Acceleration due to gravity g = 10 m/s2.

For a two-dimensional flow, the velocity field is are the basis vectors in the x-y Cartesian coordinate system. Identify the CORRECT statements from below

(1) The flow is incompressible.
(3) $y-$ component of acceleration, ${a}_{y}=\frac{-y}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$
(4) $x-$ component of acceleration ${a}_{x}=\frac{-\left(x+y\right)}{{\left({x}^{2}+{y}^{2}\right)}^{2}}$

Consider fully developed flow in a circular pipe with negligible entrance length effects. Assuming the mass flow rate , density and friction factor to be constant, if the length of the pipe is doubled and the diameter is halved, the head loss due to friction will increase by a factor of

The Blasius equation related to boundary layer theory is a

Match the following pairs:

 Equation Physical Interpretation P $\nabla\times\overrightarrow V=0$ I Incompressible continuity equation Q $\nabla\cdot\overrightarrow V=0$ II Steady flow R $\frac{D\overrightarrow V}{Dt}=0$ III Irrotational flow S $\frac{\partial\overrightarrow V}{\partial t}=0$ IV Zero acceleration of fluid particle

The velocity field on an incompressible flow is given by $V=(a_1x+a_2y+a_3z)\boldsymbol i+(b_1x+b_2y+b_3z)\boldsymbol j+(c_1x+c_2y+c_3z)\boldsymbol k$, where $a_1=2$ and $c_3=-4$ . The value of $b_2$ is________.

Air $(p=1.2\;kg/m^3$ and kinematic viscosity , $v=2\times10^{-5}m^2/s)$ with a velocity of 2 m/s flows over the top surface of a flat plate of length 2.5 m . If the average value of friction coefficient is ${C}_{f}=\frac{1.328}{\sqrt{R{e}_{x}}}$, the total drag force (in N) per unit width of the plate is _____.

Water ($\rho$ = 1000kg/m3) flows through a venturimeter with inlet diameter 80 mm and throat diameter 40 mm.The inlet and throat gauge pressures are measured to be 400 kPa and 130 kPa respectively . Assuming the venturimeter to be horizontal and neglecting friction , the inlet velocity (in m/s) is_______.

If the fluid velocity for a potential flow is given by $\mathbf V(x,y)=u(x,y)\mathbf i+v(x,y)\mathbf j$ with usual notations, then the slope of potential line at $(x,y)$ is

Within a boundary layer for a steady incompressible flow, the Bernoulli equation

The head loss for a laminar incompressible flow through a horizontal circular pipe is $h_1$ Pipe length and fluid remaining the same, if the average flow velocity doubles and the pipe diameter reduces to half its previous value, the head loss is $h_2$. The ratio $h_2/h_1$ is

For a fully developed laminar flow of water (dynamic viscosity 0.001 Pa-s) through a pipe of radius 5 cm, the axial pressure gradient is - 10 Pa/m The magnitude of axial velocity (in m/s) at a radial location of 0.2 cm is________.

Couette flow is characterized by

Three parallel pipes connected at the two ends have flow-rates Q1, Q2 and Q3 respectively, and the corresponding frictional head losses are hL1, hL2 and hL3 respectively. The correct expressions for total flow rate (Q) and frictional head loss across the two ends (hL) are

A Prandtl tube (Pitot-static tube with $C=1$) is used to measure the velocity of water. The differential manometer reading is 10 mm of liquid column with a relative density of 10. Assuming g = 9.8 m/s2, the velocity of water (in m/s) is _______.

Which of the following statements are TRUE, when the cavitation parameter σ = 0?
(i) the local pressure is reduced to vapor pressure
(ii) cavitation starts
(iii) boiling of liquid starts
(iv) cavitation stops

For a completely submerged body with centre of gravity ‘G’ and centre of buoyancy ‘B’, the condition of stability will be

For a fully developed flow of water in a pipe having diameter 10 cm, velocity 0.1 m/s and kinematic viscosity 10−5 m2/s, the value of Darcy friction factor is _______

In a simple concentric shaft-bearing arrangement, the lubricant flows in the 2 mm gap between the shaft and the bearing. The flow may be assumed to be a plane Couette flow with zero pressure gradient. The diameter of the shaft is 100 mm and its tangential speed is 10 m/s. The dynamic viscosity of the lubricant is 0.1 kg/m.s. The frictional resisting force (in newton) per 100 mm length of the bearing is _______

The difference in pressure (in N/m2) across an air bubble of diameter 0.001 m immersed in water (surface tension = 0.072 N/m) is _______

A spherical balloon with a diameter of 10 m, shown in the figure below is used for advertisements. The balloon is filled with helium (RHe = 2.08 kJ/kg.K) at ambient conditions of 15°C and 100 kPa. Assuming no disturbances due to wind, the maximum allowable weight (in newton) of balloon material and rope required to avoid the fall of the balloon (Rair = 0.289 kJ/kg.K) is _______

Consider laminar flow of water over a flat plate of length 1 m. If the boundary layer thickness at a distance of 0.25 m from the leading edge of the plate is 8 mm, the boundary layer thickness (in mm), at a distance of 0.75 m, is _______

Consider the turbulent flow of a fluid through a circular pipe of diameter, D. Identify the correct pair of statements.

 I. The fluid is well-mixed II. The fluid is unmixed III. ReD < 2300 IV. ReD > 2300

A siphon is used to drain water from a large tank as shown in the figure below. Assume that the level of water is maintained constant. Ignore frictional effect due to viscosity and losses at entry and exit. At the exit of the siphon, the velocity of water is

A fluid of dynamic viscosity 2 × 10−5 kg/m.s and density 1 kg/m3 flows with an average velocity of 1 m/s through a long duct of rectangular (25 mm × 15 mm) cross-section. Assuming laminar flow, the pressure drop (in Pa) in the fully developed region per meter length of the duct is _______

A flow field which has only convective acceleration is

Consider the following statements regarding streamline(s):

 (i) It is a continuous line such that the tangent at any point on it shows the velocity vector at that point (ii) There is no flow across streamlines (iii) $\frac{\mathrm{dx}}{u}=\frac{\mathrm{dy}}{v}=\frac{\mathrm{dz}}{w}$ is the differential equation of a streamline, where u, v and w are velocities in directions x, y and z, respectively (iv) In an unsteady flow, the path of a particle is a streamline

Which one of the following combinations of the statements is true?

Consider a velocity field $\overrightarrow V=K\left(y\widehat i+x\widehat K\right)$ where K is a constant. The vorticity, ΩZ , is

For steady, fully developed flow inside a straight pipe of diameter D, neglecting gravity effects, the pressure drop over a length L and the wall shear stress ${\tau }_{\omega }$ are related by

A hinged gate of length 5 m, inclined at 30° with the horizontal and with water mass on its left, is shown in the figure below. Density of water is 1000 kg/m3. The minimum mass of the gate in kg per unit width (perpendicular to the plane of paper), required to keep it closed is

Oil flows through a 200 mm diameter horizontal cast iron pipe (friction factor, $f=0.0225$) of length 500 m. The volumetric flow rate is 0.2 m3/s. The head loss (in m) due to friction is (assume g = 9.81 m/s2)

An incompressible fluid flows over a flat plate with zero pressure gradient. The boundary layer thickness is 1 mm at a location where the Reynolds number is 1000. If the velocity of the fluid alone is increased by a factor of 4, then the boundary layer thickness at the same location, in mm will be

A large tank with a nozzle attached contains three immiscible, inviscid fluids as shown. Assuming that the changes in h1, h2 and h3 are negligible, the instantaneous discharge velocity is

A streamline and an equipotential line in a flow field

Figure shows the schematic for the measurement of velocity of air (density = 1.2kg /m3) through a constant-area duct using a pitot tube and a water-tube manometer. The differential head of water (density = 1000 kg /m3) in the two columns of the manometer is 10mm. Take acceleration due to gravity as  9.8m/ s2. The velocity of air in m/s is

For the stability of a floating body, under the influence of gravity alone, which of the following is TRUE?

The maximum velocity of a one-dimensional incompressible fully developed viscous flow, between two fixed parallel plates, is 6ms-1. The mean velocity (in ms-1) of the flow is

A phenomenon is modeled using n dimensional variables with k primary dimensions. The number of non-dimensional variables is

Velocity vector of a flow field is given as$\stackrel{\to }{v}=2xy\stackrel{^}{i}-{x}^{2}z\stackrel{^}{j}$.the velocity vector at (1,1,1)is

A lightly loaded full journal bearing has a journal of 50mm, bush bore of 50.05mm and bush length of 20mm. if rotational speed of journal is 1200rpm and average viscosity of liquid lubricant is 0.03 Pa s, the power loss (in W) will be

Consider steady, incompressible and irrotational flow through a reducer in a horizontal pipe where the diameter is reduced from 20 cm to 10 cm. The pressure in the 20 cm pipe just upstream of the reducer is 150 kPa. The fluid has a vapour pressure of 50 kPa and a specific weight of 5 kN/m3. Neglecting frictional effects, the maximum discharge (in m3/s) that can pass through the reducer without causing cavitation is

You are asked to evaluate assorted fluid flows for their suitability in a given laboratory application. The following three flow choices, expressed in terms of the two-dimensional velocity fields in the xy-plane, are made available.

P.   u = 2y, v = -3x
Qu = 3xy, v = 0
R.  u = -2x, v = 2y

Which flow(s) should be recommended when the application requires the flow to be incompressible and irrotational?

Water at 25 °C is flowing through a 1.0 km long G.I pipe of 200 mm diameter at the rate of 0.07 m3/s. If value of Darcy friction factor for this pipe is 0.02 and density of water is 1000 kg/m3, the pumping power (in kW) required to maintain the flow is

The velocity profile of a fully developed laminar flow in a straight circular pipe, as shown in the figure, is given by the expression $u\left(r\right)=-\frac{{R}^{2}}{4\mu }\left(\frac{dp}{dx}\right)\left(1-\frac{{r}^{2}}{{R}^{2}}\right)$ where $\frac{dp}{dx}$ is a constant.

The average velocity of fluid in the pipe is

For the continuity equation given by $\overrightarrow\nabla\bullet\overrightarrow V=0$ to be valid, where $\overrightarrow V$ is the velocity vector, which one of the following is a necessary condition?

A journal bearing has shaft diameter of 40mm and a length of 40mm. The shaft is rotating at 20 rad/s and the viscosity of the lubricant is 20 mPa.s. The clearance is 0.020mm .The loss of torque due to the viscosity of the lubricant is approximately

The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed V towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and inviscid.

The radial velocity vr at any radium r, when the gap width is h, is

The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed V towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and inviscid.

The radial component of the fluid acceleration at r = R is

Consider an incompressible laminar boundary layer flow over a flat plate of length L, aligned with the direction of an oncoming uniform free stream. If F is the ratio of the drag force on the front half of the plate to the drag force on the rear half, then

In a steady flow through a nozzle, the flow velocity on the nozzle axis is given by v = uo(1+3x/L)i, where x is the distance along the axis of the nozzle from its inlet plane and L is the length of the nozzle. The time required for a fluid particle on the axis to travel from the inlet to the exit lane of the nozzle is

Consider a steady incompressible flow through a channel as shown below.

The velocity profile is uniform with a value of uo at the inlet section A. The velocity profile at section B downstream is

$\mathrm{u}=\left\{\begin{array}{lc}{\mathrm{V}}_{\mathrm{m}}\frac{\mathrm{y}}{\delta },& 0\le \mathrm{y}\le \delta \\ {\mathrm{V}}_{\mathrm{m}},& \delta \le \mathrm{y}\le H-\delta \\ {\mathrm{V}}_{\mathrm{m}}\frac{H-\mathrm{y}}{\delta },& H-\delta \le \mathrm{y}\le H\end{array}\right\$

The ratio Vm/uo is

$\mathrm{u}=\left\{\begin{array}{lc}{\mathrm{V}}_{\mathrm{m}}\frac{\mathrm{y}}{\delta },& 0\le \mathrm{y}\le \delta \\ {\mathrm{V}}_{\mathrm{m}},& \delta \le \mathrm{y}\le H-\delta \\ {\mathrm{V}}_{\mathrm{m}}\frac{H-\mathrm{y}}{\delta },& H-\delta \le \mathrm{y}\le H\end{array}\right\$
The ratio $\frac{{\mathrm{p}}_{\mathrm{A}}-{\mathrm{p}}_{\mathrm{B}}}{\frac{1}{2}{{\mathrm{\rho u}}_{\mathrm{o}}}^{2}}$(where pA and pB are the pressures at section A and B, respectively, and ρ is the density of the fluid) is