FLOW ANALYSIS USING DIFFERENTIAL METHOD

FM 5.1 The velocity field of a flow is given by V = x ti+ y t− + x t j+xz k s , where x y, and z are in meters and t is in seconds. For fluid particles on the x -axis, the speed and direction of flow respectively, are

(A) 2 2x t² m s/ , 45c (B) 2 2x t² m s/ , 30c (C) 2x t² m s/ , 45c (D) 2 2xt²m s/ , 45c

FM 5.2 Consider the following steady, three-dimensional velocity field in Cartesian coordinates:

V =( , , )u v w =(axy²−1.5)i+(cy³−3.0)j+(dxy−4.5)k where a c, andd are constants. The flow field is incompressible at

(A) a=− c (B) d =− a

FM 5.3 The velocity field of a flow is given by

( ) ( ) m s

x y y

x y

V= i x j

+ −

+ ,

where x and y are in meters. The fluid speed along the y-axis and the angle between velocity vector and x-axis at points ( , )x y =( , )5 5 respectively, are

(A) 10m s, / 45c (B) 10m s, / −45c (C) 20m s, / −45c (D) 15m s, / 45c

FM 5.4 Consider the following statements regarding the creeping flow : (a) Unsteady term of Navier stokes equation is ignored.

(b) Inertial term of Navier stokes equation is ignored.

(d) Viscous term of Navier stokes equation is ignored.

Which of the statements given above is correct ?

(A) 1, 2 and 3 (B) 1, 2 and 4

FM 5.5 The velocity field of a flow is given by V=(x−y)i+(xy³−1 ) m sk , where x y, are in meters. What will be the location of any stagnation point in the flow field ?

(A) ( , )2 1 (B) ( , )1 2

FM 5.6 The velocity component of a steady, two dimensional, incompressible flow field is u=2x²−4xy. What will be the expression for v as a component of x and y ? (A) v=2xy−4y²+f x( ) (B) v=4xy−2y²+f x( )

FM 5.7 For incompressible fluids the volumetric dilatation rate must be zero. For what combination of constants a, b, c and e can the velocity components

u =ax+by

v =cx+ey w =0

be used to describe an incompressible flow field ?

(A) a+ =e (B) a+ + =b e

FM 5.8 A fluid field is given by V=x yi+y zj−yz km s, where x y, and z are in meters. What will be the acceleration at the point ( , , )2 1 3 ?

(A) 38m s/ ² (B) 40m s/ ²

FM 5.9 The viscosity of a body lotion as a function of temperature is listed in table below.

The specific gravity of the body lotion is about 1.5 and is not a strong function of temperature. The body lotion is squeezed through a small hole of diameter

D= in the lid of an inverted jar. The room and the lotion are at T= cC . Assume that Re must be less than 0.1 for the creeping flow approximation, the maximum speed of the lotion through the hole such that the flow can be approximated as creeping flow is

Viscosity of Body lotion at 16 percent moisture content

, C

T c μ, Poise (g/cm-s)

14 600

20 190

30 65

40 20

50 10

70 3

(A) 1.09m s (B) / 0.109m s/

FM 5.10 Two velocity components of a steady incompressible flow field are known as:

u =ax+bxy v =axz−byz²

where a and b are constants, the velocity component w as a function of x, y and z is

(A) w az byz bz3 f x y( , )

=− − + 3+ (B) w az byz bz3 f x y( , )

=− + + 3 +

= − + 3+ (D) w az byz bz3 f x y( , )

=− − − 3 +

FM 5.11 Consider a certain region of steady flow with the velocity field

V =( , )u v =(ax+b)i+ −( ay+cx)j The flow field is

(A) incompressible only (B) inviscid only

(D) neither incompressible nor inviscid

FM 5.12 The velocity in a certain two-dimensional flow field is given by the equation V =2xti−2ytj

where the velocity is in m s when x/ , y and t are in meter and seconds, respectively.

What will be the magnitude of the velocity and the acceleration respectively at the point x= =y m at the time t= ?

(A) 8.2m s, 0 / (B) 0, 5 66. m s/

FM 5.13 Consider the following steady, two dimensional, incompressible velocity field V =( , )u v =(2x+3)i+ −( 2y+4 )x² j

The pressure as a function of x and y is

(A) cannot be found (B) p= −r( x− )

FM 5.14 Consider the following statements regarding a laminar boundary layer on a flat plate as shown in figure below

(a) At a given x-location, if the Reynolds number were to increase, the boundary layer thickness would also increase.

(b) As outer flow velocity increase, so does the boundary layer thickness.

(d) As the fluid density increase, so does the boundary layer thickness Which one of these is true ?

(A) only (a) (B) Only (c)

FM 5.15 Consider a steady, incompressible, two-dimensional flow of fluid into a converging duct with straight walls as shown in figure below. The volume flow rate is vo and the velocity is in the radial direction only with ur a function of r only. If b be the width into the paper and at the inlet r=R and ur =u Rr( ), for inviscid flow everywhere, an expression for ur as a function of r R and , u Rr( ) is

(A) u ( ) r

Ru R

r= r (B) u ( )

R r u R

r= r

C) u R( ) r Ru

r = r (D) u R_r( )=R ru: _r

FM 5.16 Air flows parallel to a speed limit sign along the high-way at a speed of V= m s . The temperature of the air is 25 Cc (μ=1.849#10^-⁵kg m s/ ^- ) and the width w of the sign parallel to the flow direction is 0.45 m. What will be the boundary layer on the sign ?

(A) laminar (B) Transitional (C) Turbulent

(D) Laminar for a while and then becomes transitional

FM 5.17 The water flows through the curved hose as shown in figure with an increasing speed of V = tm s, where t is in seconds. If t= sec, what will be the resultant acceleration and its direction, respectively ?

(A) 6.71m s/ ², 63 4. c (B) 6m s/ ², 52.4c (C) 7.61m s/ ², 45 4. c (D) 6.71m s/ ², 72 2. c

FM 5.18 Air flows into the region between two parallel circular disks from a pipe as shown in figure. The fluid velocity in the gap between the disks is given by V =V R r, where R is the radius of the disk, r is the radial coordinate and V is the fluid velocity at the edge of the disk. If V = . m s, the acceleration at r= . m will be

(A) 7.63m s/ ² (B) 8.62m s/ ²

FM 5.19 A hydraulic jump is a rather sudden change in depth of a liquid layer as it flows in an open channel as shown in figure. The liquid depth changes from z to z in a relatively short distance with a corresponding changes in velocity from V = m s to V = m s and l= . m. What will be the average acceleration of the liquid as it flows across the hydraulic jump ?

(A) −60m s/ ² (B) −120m s/ ²

FM 5.20 A nozzle is used to accelerate the fluid from V to V in a linear fashion. If the flow is constant with V = m s at x = and V = m s at x = m, what will be the local and the convective acceleration, respectively ?

(A) 128 (2x+1)i, 0 (B) 0, 96(2x−1)i (C) 96(2x−1)i, 0 (D) 0, 128 (2x+1)i

FM 5.21 Oil flows past a sphere with an upstream velocity of 24m s as shown in figure. / The speed of the fluid along the front part of the sphere is V = V0sinq. If the radius of the sphere is 0.20 m, the streamwise and normal components of acceleration at point A in m s/ ² respectively, are

(A) 8863, 7437 (B) 7350, 8950

FM 5.22 The fluid velocity changes from 7m s at point A/ to 19m s at point B/ along the x-axis as shown in figure. The velocity is a linear function of distance along the streamline. If the flow is steady, what will be the acceleration at point C ?

(A) 1440i m s/ ² (B) 1360im s/ ² (C) 1560i m s/ ² (D) 1560j m s/ ²

FM 5.23 The temperature of the exhaust in an pipe is given by the relation :

( ) cos( )

T=T₀ +ae⁻^bx +c wt

where T0=80 Cc , a=5, b=0.03 m⁻ , c=0.05, ω =125rad s/ and the exhaust speed is constant at 1.5m s/ . What will be the time rate of change of temperature of the fluid particles at x=4 m when t=0 ?

(A) −15.8cC s/ (B) 16.8cC s/

FM 5.24 A fluid flows steadily along the stream line as shown in figure. What is the angle between the acceleration and the stream line at point A ?

(A) 18.5c (B) 0c

FM 5.25 The velocity field for a steady two-dimensional incompressible flow in the xy -plane is given by V =( , )u v =V i_h^y +0j. The flow is

(A) Clockwise, Rotational (B) Counter clockwise, Rotational (C) Irrotational (D) Not determined

FM 5.26 What is the expression for the vorticity of the flow field described by V =x yi−xy j

(A) −(x²+y k²) (B) −(x²+y i²) (C) (x²−y j²) (D) (x²+y k²)

FM 5.27 The three components of velocity in a flow field are given by u =3 0. +2 0. x−y

v =2 0. x−2 0. y w =0 5. xy

The vorticity vector as a function of space ( , , )x y z is

(A) (0.5 )y i−(0.5 )x j+(3.0)k (B) (0.5 )x i+(0.5 )y j−(3.0)k (C) (0.5 )x i−(0.5 )y j+(3.0)k (D) (0.5 )x i+(0.5 )y j+(3.0)k

FM 5.28 A velocity field of a flow is given by V=(y− )i+ −(y )j, where V is in m s / and x and y are in meters. What will be the equation of stream line passing through the point ( , )5 3 ?

(A) y= +x ln(y− )+ (B) y= +x ln(y+ )+ (C) x= +y ln(y− )− (D) x= +y ln(y− )+

FM 5.29 The velocity field of a flow is given by

( ) ( ) m s

y x y x x y

V=− + i+ + j

where x and y are in meters. The equation of streamline will be (A) x + =y constant (B) x + =y constant (C) x + =y constant (D) x − =y constant

FM 5.30 Consider a uniform flow in the positive x direction combined with a free vortex located at the origin of the coordinate system. If the streamline ψ =0 passes through the point x= and y= , what will be the equation of this streamline ? (A) sinθ= 2πΓU ln4 (B) sin rU ln(4 )r

θ= 2πΓ

θ= πΓ ^ hr (D) sin 2 U ln ⁴ θ = πΓ _ i1 FM 5.31 A two-dimensional flow field given by the relation

V =^2x y² +xhi+^2xy²+ +y 1hj

where the velocity is in m s when x/ and y are in meters. What is the angular rotation of a fluid element located at ( , )x y =(0.5m, 1.0m) ?

(A) 0.56j rad s/ (B) 0.75j rad s/ (C) 0.75k rad s/ (D) 0. 57 irad s/

FM 5.32 The three components of velocity in a flow field are given by u = + +x y z

v =xy+yz+z w =−3xz−z²/2+4

The volumetric dilatation rate and an expression for the rotation vector respectively, are

(A) 2

y z i zj yk 2

−a + k + −2 , 0 (B) 0, y z i zj yk

2 2

−a + k + +2 (C) 0, zi y z j yk

2 5

2 2

−a + k − (D) 0, y z i zj yk

2 2

−a + k + −2

FM 5.33 For a certain two-dimensional steady flow field it is suggested that the x-component of velocity is given by

u = +a b x( −c)

where a b, and c are constants with appropriate dimensions. What will be the expression for v as a function of x y, and the constants of the given equation such that the flow is incompressible ?

(A) v=− b x( −c y) +f x( ) (B) v= b x( −c y) +f x( ) (C) v=− b x( −c y) −f x( ) (D) v= b x( −c y) −f x( )

FM 5.34 The radial velocity component in an incompressible, two-dimensional flow field is vr =2r+3 sinr² q, vz =0

What will be the corresponding tangential velocity component v_θ, required to satisfy conservation of mass ?

(A) v_θ= rθ- r cosθ+f r( ) (B) v_θ=- rθ- r cosθ+f r( ) (C) v_θ= r θ- rcosθ+f r( ) (D) v_θ=- rθ+ r cosθ+f r( )

FM 5.35 The streamlines in a two dimensional flow field are all concentric circles as shown in figure below. The velocity is given by the equation v_θ=ωr where ω is the angular velocity of the rotating mass of fluid. What will be the circulation around the path ABCD ?

(A) ω(a²-b²)θ1 (B) ω θΔ (b²-a²) (C) ω(b²-a²)θ2 (D) ω θΔ (a²-b²)

FM 5.36 A steady, two-dimensional flow field in the xy-plane has the following stream function

ψ =2x²+5xy+3y² The flow field is

(A) continuous (B) compressible (C) incompressible (D) not determined

FM 5.37 The stream function corresponding to the velocity potential φ = -x xy which passes through the origin is

(A) ψ =3x²-y³ (B) ψ =3x y² -y³ (C) ψ =3x y² +y³ (D) ψ =3y-x y² ³

Common Data For Linked Answer Q. 38 and 39

Consider a steady two dimensional, incompressible flow field called a uniform stream as shown in figure below. The fluid speed is V everywhere. The Cartesian velocity components are

u=V and v =

FM 5.38 The expression for the stream function for this flow is

(A) ψ =Vy+C (B) ψ =Vx+C

FM 5.39 The ψ2 is a horizontal line at y= . m and the value of ψ along the x-axis is zero. If V = m s, the volume flow rate per unit width between these two streamlines is

(A) 5m s³/ (B) 10m s³/

FM 5.40 A uniform stream of speed V = . m s is inclined at angle α from the x-axis as shown in figure below. The cartesian velocity components are

u =2 5. cosa v =2 5. sina

If the flow is steady, two dimensional and incompressible, the stream function for this flow is

(A) ψ=2 5. (ysinα+xcosα)+C (B) ψ=2.5 (ycosα+xsinα)+C (C) ψ=2.5 (ycosα-xsinα)+C (D) ψ=2.5 (ysinα-xcosα)+C

FM 5.41 The velocity distribution for two dimensional flow of a viscous fluid between wide parallel plates is parabolic and it is given by

u U

h y

= c: −a kD

The corresponding stream function and velocity potential respectively, are (A) U y

y C

ψ = c : - _{a k}D+ , Not possible (B) U y x

y C

ψ = c : - a kD+ , φ = -y xy (C) Not possible, U y

y C

φ = c : - a kD+ (D) ψ = -y xy , U y x

y C

φ = c : - a kD+

FM 5.42 Consider fully developed Couette flow as shown in figure, with top plate moving and bottom plate stationary. The velocity field for above steady, incompressible and two dimensional flow in the xy-plane is given by

V =( , )u v =aV hyki+0j

If stream function ψ =0 along the bottom wall of the channel, the value of stream function along the top wall is

(A) zero (B) Vh

ψ =- 2 (C) Vh

ψ = 2 (D) Vh

ψ =2

FM 5.43 The stream function for an incompressible flow field is described by ψ =3x y² −y³

where the stream function has the units of m s²/ with x and y in meters.

The rate of flow across the straight path AB as shown in figure below is

(A) 1.5m s³/ (B) 2m s³/

FM 5.44 The velocity components for an incompressible, plane flow are vr =Ar⁻ +Br⁻ cosq

v_θ =Br⁻ sinq

where A and B are constants. What is the corresponding stream function ? (A) ψ=Ar^- +Bθsinθ+C (B) ψ=Aθ+Br^- sinθ+C

Common Data For Linked Answer Q. 45 and 46

Consider a steady, two-dimensional, incompressible, irrotational velocity field specified by its velocity potential function as :

φ =4 5. (x²−y²)+2x−4y

FM 5.45 The velocity components u and v respectively, are

(A) 9x+2, 9y−4 (B) − −9y 4, 9x+2 (C) 9y−4, 9x+2 (D) 9x+2, −9y−4

FM 5.46 The stream function in this region is

(A) ψ =-9xy-2y-4x+C (B) ψ =9xy+2y+4x+C (C) ψ =-9xy+2y+4x+C (D) ψ =9xy-2y-4x+C

FM 5.47 Consider the following steady, two-dimensional, incompressible velocity field V =( , )u v =(2x+1)i+ −( 2y+3)j

What will be the velocity potential function and the flow type ?

(A) Rotational, φ =- + +x x f y( ) (B) Rotational, φ = + +x x f y( ) (C) Irrotational, φ = + +x x f y( ) (D) Irrotational, φ =- + +x x f y( )

FM 5.48 A reverse flow region may develop by water in the region just downstream of a sluice gate as shown in figure. The velocity profile is to be consist of two uniform regions, one with velocity 3m s and the other with velocity 0.9/ m s. If the / channel is 6.1 m wide, what will be the net flow rate of water across the portion of the control surface at section (2) ?

(A) 4.50m s³/ (B) 7.50m s³/ (C) 3 75. m s³/ (D) 2 95. m s³/

FM 5.49 A fluid having a viscosity of 0.003N s m⁻ / ² flowing with an average velocity of 100mm s in a 2 mm diameter tube. What will be the magnitude of the wall / shearing stress (τrz) ?

(A) 1.5 Pa (B) 1.20Pa

Common Data For Q. 50 and 51

A layer of SAE 30 oil flows down a vertical plate as shown in figure with a velocity profile of

V h

V⁰ hx x j

= b − l

where V0 and h are constants. The width of the plate is b.

FM 5.50 The shear stress at the edge of the layer (x =h) is

(A) 2V h₀μ (B)

h V0μ

FM 5.51 The flow rate across the surface AB is (A) 2V bh

3 0 (B) 1 5. V h0

2 0 (D) 3h V b

2 0

Common Data For Q. 52 and 53

SAE 30 oil at 15.6 Cc (μ=0.38N s m^- / ²) flows steadily between fixed, horizontal, parallel plates. The pressure drop per unit length along the channel is 30kPa m / and the distance between the plates is 4 mm. Consider the flow is laminar.

FM 5.52 The volume rate (per meter of width) of the flow is

(A) 42.1#10⁻⁴m s²/ (B) 4.21#10⁻⁴m s²/ (C) 4.21#10⁻²m s³/ (D) 4.21#10⁻⁴m s³/

FM 5.53 The magnitude of the shearing stress acting on the bottom plate is

(A) 50N m/ ² (B) 75N m/ ²

FM 5.54 Consider the following statements regarding the D’Alembert’s paradox with the irrotational flow approximation:

(a) The pressure drag force on any non-lifting body of any shape immersed in a uniform stream is zero.

(b) The aerodynamic drag force on any lifting body on any shape immersed in a uniform stream is zero.

(d) The aerodynamic drag force on any non-lifting body of any shape immersed in a non-uniform stream is zero.

Which of the statement given above is correct.

(A) only c (B) b and c

FM 5.55 The two dimensional velocity field for an incompressible, Newtonian fluid is described by the relation V =^12xy²−6x³hi+^18x y² −4y³hj where the velocity has units of m s when x/ and y are in meters. If pressure at point x=0. m , y= .0 m is 6 kPa and the fluid is glycerin at 20 Cc (μ=1.50N s m^- / ²), the stresses σxx, σyy and τxy at this point respectively, are

(A) −6.02 kPa, −5.98 kPa, 45.0 kPa (B) 45 kPa, −6.02 kPa, 5.98 kPa (C) 5.98 kPa, −6.02 kPa, −45 kPa (D) −5.98 kPa, −6.02 kPa, 45 Pa

FM 5.56 The stream function for the flow of a non-viscous, incompressible fluid in the vicinity of a corner as shown in figure is given by the equation

ψ 2r sin 3 4

4 3 q

What will be the expression for the pressure gradient along the boundary θ =3π/4

? (A) r

p r

2 = r (B)

r p 2 r

2 = r

p r

2 = r (D)

r p 2 r

2 =− r

FM 5.57 A wire of diameter d is stretched along the centerline of a pipe of diameter D. For a given pressure drop per unit length of pipe and .

Dd = , by how much the pressure of the wire does reduce the flow rate ?

(A) 31.95% (B) 42.6%

FM 5.58 An object having the general shape of a half-body is placed in a stream of fluid and at a great distance upstream the velocity is U as shown in figure below. If the body forces are neglected and the fluid is nonviscous and incompressible, What will be the pressure difference between the stagnation point and point A in terms of U and fluid density ?

(A) 0 879. ρU² (B) 0.703ρU²

FM 5.59 A viscous fluid is contained between two long concentric cylinders. Assuming that the flow between the cylinders is approximately the same as the laminar flow between two infinite parallel plates. The inner cylinder is fixed. What is the expression for the torque required to rotate the outer cylinder with an angular velocity ω in terms of the geometry of the system, viscosity of the fluid, and angular velocity ?

(A) T r l 1 2

rr o 4

i o

= p w− (B) T

r r r l 2

o i

= p mw−

r r r l 2p wo3

= − (D) T r l

1 2

p wo

= −

FM 5.60 A bearing is lubricated with an oil having a viscosity of 0.2N s m⁻ / ² through which a vertical shaft passes as shown in figure below. If the flow characteristics in the gap between the shaft and bearing are the same as those for laminar flow between infinite parallel plates with zero pressure gradient in the direction of flow. What will be the torque required to overcome viscous resistance when the shaft is turning at 80rev/ min ?

(A) 0.355 N m⁻ (B) 0.266N m⁻

FM 5.61 A drop of water in a rain cloud has a diameter of 45 mμ . The air temperature is 25 Cc and its pressure is standard atmospheric pressure. How fast does the air have to move vertically so that the drop will remain suspended in air ?

(μair = . # ^- g m s^- )

(A) 0.00596m s (B) / 0.0596m s/

FM 5.62 A viscous, incompressible fluid flows between the two infinite, vertical, parallel plates as shown in figure. If the flow is laminar, steady and uniform, what will be the expression for the pressure gradient in the direction of flow in terms of mean velocity by using the Navier Stokes equations ?

(A) y p

V g

2 = m −r (B)

y p

V g

2 = m +r

V g

2 =− m +r (D)

y p

V g

2 =− m −r

FM 5.63 The velocity distribution for free vortex flow in a horizontal, two-dimensional bend through which an ideal fluid flows can be approximated is shown in figure.

What will be the discharge (per unit width normal to plane of paper) through the channel ?

(A) q C p Dr

= (B) q =C rDp

= (D) q C

p Dr

***********

SOLUTIONS

FM 5.1 Option (A) is correct.

We have V =2x t² i+[4 (y t−1)+2x t² ]j+xz²km s/ For particle on the x-axis, y= =z

So V =2x t² i+2x t² j Thus on the x-axis the velocity is

V = u + +v w = ( x t) +( x t) + =2 2x t² m s/

From continuity equation, condition for incompressibility x

Thus equation (i) becomes

ay + cy =0 ay =−3cy²

a =−3c

Hence at a=− c, the given velocity field will be incompressible.

FM 5.3 Option (B) is correct. Comparing with velocity field, we get

u (x y )

The resultant velocity does not depends on the coordinates.

So, velocity remains constant in all directions.

Now tanθ u

FM 5.4 Option (C) is correct.

When the creeping flow approximation is made, only pressure and viscous terms remain. The other three terms (i.e. unsteady term, Inertial term and gravity term) are very small compared to these two and can be ignored.

FM 5.5 Option (C) is correct.

We have V =(x−y)i+(xy³−16)k On comparing with velocity field, we get

u = −x y, v=xy − ...(i)

From continuity equation for two-dimensional incompressible flow x

Substituting in equation (i), y

The volumetric dilation rate for incompressible fluid is V

With the velocity distribution given x Thus, from equation (i), we get

a+e =0

FM 5.8 Option (B) is correct.

For the given fluid flow field

u=x y, 2 =u xy, 2 =u x , 2 =u , 2 =u

v=y z, So, acceleration vector becomes

a =axi+ayj+azk =28i+ +9j 27k

Density of body lotion

lotion

ρ =S G. .lotion#rwater =1.5#1000=1500kg m/ ³ Viscosity of lotion at 30 Cc

lotion

μ =65 /g cm s⁻ =6.5kg m s/ ⁻ Now from Reynolds number definition

Remax

Continuity equation for three-dimensional, incompressible flow field is x

Now equation (i) becomes z w 2

2 =− −a by+bz Integrate the above equation, we get

w =−az−byz+bz +f x y( , )

FM 5.11 Option (C) is correct.

First check for Incompressibility : From continuity equation Therefore flow is incompressible

Second check for Inviscid Flow :

For flow to be inviscid, the viscous term of Navier stokes equation be zero.

Therefore Navier stokes equation, for x-momentum viscous terms x

Therefore, x-momentum viscous term is zero.

Now for y-momentum viscous terms x ,

Since the viscous terms are identically zero in both components of Navier-stokes equation, this region of flow can indeed be considered inviscid.

FM 5.12 Option (B) is correct.

From expression for velocity, the components are u= xt, v=− yt and w=

Also u =2#2#0=0

From Navier-stokes equation, x-component of steady-two dimensional flow udx Hence, the x-momentum is

[4x 6]

Now y-component of Navier stokes equation for this flow udx Hence, y-momentum is

[16x² 24x 4y 8x²] Now we calculate cross-differentiation

x y

Hence these are not equal, the given velocity distribution is not an exact solution of Navier-stokes equation. Thus we are unable to calculate a steady, incompressible, two-dimensional pressure field with the given velocity field.

FM 5.14 Option (B) is correct.

(a) False : If the Reynolds number at a given x location were to increase, all else being equal, viscous forces would decrease in magnitude relative to inertial forces,

rendering the boundary layer thinner.

(b) False : Actually as V increases, so does Re and the boundary layer thickness decrease with increasing Reynolds number.

(c) True : Since μ appears in the denominator of the Reynolds Number, Re decreases as μ increases, causing the boundary layer thickness to increase.

(d) False : Since ρ appears in the numerator of the Reynolds number (Re) increases as ρ increases, causing the boundary layer thickness to decrease.

FM 5.15 Option (A) is correct.

If the flow is inviscid, we could not enforce the no-slip condition at the walls.

Thus at any r location, the volume flow rate must be the same.

vo =Area#velocity=rbD #q ur

Substituting equation (ii) into equation (i) u_r ( )

r Ru R_r

FM 5.16 Option (D) is correct.

Since the air flow is parallel to the sign, this flow is that of a flat plate boundary layer and Reynolds number for this flow

Rex VL the boundary layer is laminar for a while and then becomes transition by the trailing edge of fin.

FM 5.17 Option (A) is correct.

We have V= t, t= sec, R= m

The component of the acceleration along the stream line is

In document ME Vol 2 FM (Page 176-200)