ur[[Ljava.lang.String;2M 2Wxp@ur[Ljava.lang.String;V{Gxp tnotA fluid flow is steady if:t∂/∂ttD/Dttboth ∂/∂t and D/Dtteither ∂/∂t or D/Dtt1|t4t1t0q~ t1tOfor m_35: the force is in direction 3 due to rotation with respect to 2|750000|uq~ tnotfDye is injected into a flow at a given point. In steady flow, the locus of all dyed particles forms a:tpathlinet streamlinet streaklinetnone of the abovet1|2|3|t6t1q~ q~ t2tuq~ tnothDye is injected into a flow at a given point. In unsteady flow, the locus of all dyed particles forms a:tpathlinet streamlinet streaklinetnone of the abovet3|t6t2q~ q~ t1tuq~ q~tA small neutrally-buoyant particle is released into a flow at a given point. In steady flow, the line that the particle makes forms a:tpathlinet streamlinet streaklinetnone of the abovet1|2|3|t6t2q~ q~ t1q~'uq~ q~tA small neutrally-buoyant particle is released into a flow at a given point. In unsteady flow, the line that the particle makes forms a:tpathlinet streamlinet streaklinetnone of the abovet1|t6t2q~ q~ t1q~'uq~ q~tXA circular cylinder has radius a=1cm and length L=1m. Its added mass in water can be estimated as m_22=___; m_55=___; and m_44=___. If the cylinder is translating through water with velocity U[2]=0.5m/s, assuming potential flow and ignoring the mass of the cylinder itself, the total amount of work required to bring the cylinder to rest is ___tttttt1t2t1q~ t2q~'uq~ tnotA certain body with added mass coefficients m_ij has constant velocities U_1,U_2 ≠ 0, and all other U_i, Ủ_i and Ω_i = 0. In terms of the added mass coefficients m_ij: the forces on the body (in the body coordinates) are F_1=___; F_2=___; F_3=___.ttttt0;0;0t1t2q~ q~ t2tuq~ tnotA certain body with added mass coefficients m_ij has constant velocities U_1,U_2 ≠ 0, and all other U_i, Ủ_i and Ω_i = 0. In terms of the added mass coefficients m_ij: the moment M_3 on the body (in the body coordinates) is:q~ t(U_1)²*m_21+(U_2)²*m_12t3U_1*U_2*m_22+U_2*U_1*m_11-(U_1)²*m_21+(U_2)²*m_12t4-U_1*U_2*m_22+U_2*U_1*m_11-(U_1)²*m_21+(U_2)²*m_12t4|q~Et2q~ q~ q~Et*what symmetry does the object have?|25000|uq~ q~TtA certain body with added mass coefficients m_ij has constant velocities U_1,U_2 ≠ 0, and all other U_i, Ủ_i and Ω_i = 0. In terms of the added mass coefficients m_ij: the linear momentum of the surrounding fluid in the x_1 direction is ___.tU_1*m_11tU_1*m_11+U_2*m_22tU_1*m_11+U_2*m_12q~ t3|q~Eq~Zq~ q~ q~Eq~[uq~ q~TtA certain body with added mass coefficients m_ij has constant velocities U_1,U_2 ≠ 0, and all other U_i, Ủ_i and Ω_i = 0. In terms of the added mass coefficients m_ij: the total kinetic energy in the fluid is:t;(1/2)*(m_12*(U_1)²+m_21*(U_2)²+m_11*U_1*U_2+m_22*U_2*U_1)t;(1/2)*(m_11*(U_1)²+m_22*(U_2)²+m_12*U_1*U_2+m_21*U_2*U_1)t4(1/2)*(m_11*(U_1)²+m_22*(U_2)²+2*m_12*m21*U_1*U_2)q~ t2|q~Eq~Zq~Eq~ q~Eq~[uq~ q~TtA certain body has nonzero added mass coefficients only on the diagonal, i.e., m_ij=m_iб_ij. For a body motion given by U_1=t and U_2=-t, and all other U_i, Ω_i=0. The forces and moments on the body in terms of m_i are F_1=___, F_2=___, F_3=___.tq~jq~jq~jt m_1;-m_2;0q~Eq~Zq~ q~ q~Zq~juq~ q~TtA certain body has nonzero added mass coefficients only on the diagonal, i.e., m_ij=m_iб_ij. For a body motion given by U_1=t and U_2=-t, and all other U_i, Ω_i=0. The total kinetic energy in the fluid at time t=1 is:t(1/2)*(m_1*t²+m_2*t²)t7(1/2)*(m_1*(U_1)²+m_2*(U_2)²+m_1*U_1*U_2+m_2*U_2*U_1)t0(1/2)*(m_1*(U_1)²+m_2*(U_2)²+2*m_1*m2*U_1*U_2)q~ t1|t3q~Zq~Eq~ q~Eq~[uq~ q~tA 2D circular cylinder of radius 1m moves in an unbounded fluid at U=15 m/s. Assuming potential flow, the total amount of work done to bring this cylinder to rest is ___ per unit width:t225ρπt 112.5ρπt 66.25ρπt450ρπt1|t1t3q~Eq~ t1tuq~ q~TtA sphere of volume 1 m³ accelerates at Ủ_1= 2m/s² while at the same time the surrounding fluid (density ρ=1 kg/m³) is accelerated at a_1= 1 m/s². The horizontal force on the sphere is F_1=t.5Nt.75Nt1Nt2Nq~qq~Eq~Eq~Eq~ q~EtWhat is the effect of fluid acceleration plus body acceleration|15000|Fluid acceleration: buoyancy force + added mass force|15000|uq~ q~TtA sphere of volume 1 m³ accelerates at Ủ_1= 2m/s² while at the same time the surrounding fluid (density ρ=1 kg/m³) is accelerated at a_1= 1 m/s². If a_1 remains the same, F_1 will vanish if Ủ_1=.t1.5m/s²t3m/s²t4m/s²t6m/s²q~gq~Eq~Zq~ q~ q~EtWrite the expression for F_1 in terms of Ủ and a_1|10000|What is the effect of fluid acceleration plus body acceleration?|15000|Fluid acceleration: buoyancy force + added mass force|15000|uq~ tnot0A two-dimensional square box of width 2m accelerates to the right at an acceleration of 1m/s² while at the same time the surrounding fluid also accelerates to the right at 1m/s² (both with respect to a fixed coordinate system). The total horizontal force on the square is ___ N/m to the [right] [left].t4ρ N/m to the leftt4ρ N/m to the rightt4 N/m to the leftt4 N/m to the rightt1|t2t2q~ q~ t1tuq~ q~tA sphere of radius R=1m and density ρ=2.5ρ(water) is released in a current of velocity U(t)=At where A=10 m/s². In the absence of gravity, calculate the initial (t=0) horizontal acceleration Ủ of the sphere:t10 m/s²t5 m/s²t 3.33 m/s²t 2.5 m/s²t3t2t2q~ q~ t1q~uq~ q~tA sphere of radius R=1m and density ρ=2.5ρ(water) is released in a current of velocity U(t)=At where A=10 m/s². If the current is absent but there is gravity, calculate the initial vertical acceleration of the sphere.ttttt3.26t2t2q~ q~ t3q~uq~ q~tA sphere of radius R=1m and density ρ=2.5ρ(water) is released in a current of velocity U(t)=At where A=10 m/s². If both the current and gravity are present, calculate the angle θ (relative to the vertical) the sphere will tend to move initially.t30˚t45˚t60˚t75˚t2|t2t2q~ q~ t1q~uq~ q~TtA 2D square body of dimension 2m * 2m in water travels at velocity U_x = 1m/s, U_y = 2m/s, acceleration dU/dt=(2,3)m/s², and no rotation (Ω=0). Per unit (1m) depth of this body, calculate the forces F_1, F_2 and moment M_3 on this body.t*9.5*ρ*m² N/m; 14.25*ρ*m² N/m; 16.6 N/mt*9.5*ρ*m² N/m; -14.25*ρ*m² N/m; 8.2 N/mt%-9.5*ρ*m² N/m; -14.25*ρ*m² N/m; 0t(-9.5*ρ*m² N/m; 14.25*ρ*m² N/m; 0 N/mq~aq~Eq~Zq~ q~ q~Et~Use expression for force and moment in terms of added mass coefficients|35000|Use body symmetry to simplify expressions|15000|uq~ q~Tt A 2D square body of dimension 2m * 2m in water travels at velocity U_x = 1m/s, U_y = 2m/s, acceleration dU/dt=(2,3)m/s², and no rotation (Ω=0). Per unit (1m) depth of this body, obtain the linear momentum of the fluid around the body in the x_1 and x_2 directions.t4.75*ρ*m²î + 9.5*ρ*m²ĵt9.5*ρ*m²î + 4.75*ρ*m²ĵt9.5*ρ*m²î + 9.5*ρ*m²ĵt4.75*ρ*m²î + 4.75*ρ*m²ĵq~qq~Eq~Zq~ q~ q~EtAssume initially the body at rest and that U_x,y is its velocity at the present time|35000|the way the body is brought to the speed U_x,y does not matter|250000|uq~ q~TtA 2D square body of dimension 2m * 2m$ in water travels at velocity U_x = 1m/s, U_y = 2m/s, acceleration dU/dt=(2,3)m/s², and no rotation (Ω=0). Per unit (1m) depth of this body, find the kinetic energy of the fluid.q~jq~jq~jq~jt11.q~rq~Zq~ q~ q~rt_what does the kinetic energy expression look like, in terms of added mass coefficients?|250000|uq~ ta1.gif|300|181|t_A two-dimensional ellipse has the following added mass coefficients: m_11= πρb², m_22=πρa², m_66 = (1/8)πρ(a²-b²)². Find the hydrodynamic forces F_1 and F_2 and moment M on the ellipse, per unit span, when its translational and angular velocities through an infinite inviscid fluid are the following functions of time t: U_1=0, U_2=0, Ω=t.t0; 0; 0t0; 0; -1/8*πρ(a²-b²)²t-8πρb²t; 0; 0t 4πρa(a+b)²q~gq~Eq~Zq~ q~ q~Et^what do forces and moments' expressions look like in terms of added mass coefficients?|350000|uq~ q~tbA two-dimensional ellipse has the following added mass coefficients: m_11= πρb², m_22=πρa², m_66 = (1/8)πρ(a²-b²)². Find the hydrodynamic forces F_1 and F_2 and moment M on the ellipse, per unit span, when its translational and angular velocities through an infinite inviscid fluid are the following functions of time t: U_1=4t², U_2=0, Ω=0.q~q~q~t8πρb²t; 0; 0q~aq~Eq~Zq~ q~ q~Eq~uq~ q~t=A two-dimensional ellipse has the following added mass coefficients: m_11= πρb², m_22=πρa², m_66 = (1/8)πρ(a²-b²)². Find the moment M on the ellipse, per unit span, when its translational and angular velocities through an infinite inviscid fluid are the following functions of time t: U_1=3t, U_2=2t, Ω=t.t"1/8πρ(a²-b²)+8t²πρ(b²-a²)t8πρ((a²-b²)+t²(b²-a²))t"1/6πρ(a²-b²)+6t²πρ(b²-a²)t"1/8πρ(a²-b²)+6t²πρ(b²-a²)q~Yq~Eq~Zq~ q~ q~Eq~uq~ q~t5A two-dimensional ellipse has the following added mass coefficients: m_11= πρb², m_22=πρa², m_66 = (1/8)πρ(a²-b²)². If the ellipse is stationary, but the infinite fluid is moving past it in the +x_2 direction at a velocity U=4t, find the hydrodynamic forces F_1 and F_2 and moment M on the ellipse.t0; 16πρ(a+b)²; 0t0; 4πρa(a+b); 0t0; 4πρa(a+b); 6πρb(a+b)t0; 16πρa(a+b); 6πρb(a+b)q~gq~Eq~Zq~ q~ q~Et;Buoyancy force plus inertia force due to added mass|250000|uq~ ta2.gif|450|158|tA slender vehicle operating in an infinite fluid of density ρ can be modeled as a circular cylinder of length L and radius r, along x_1 axis. Using strip theory, estimate the added mass coefficient m_22t ρ*π*a²*Ltρ*π*a²*L²/6tρ*π*a²*L³/12t0t1|t3t2q~ q~ t1q~'uq~ ta2.gif|450|158|tA slender vehicle operating in an infinite fluid of density ρ can be modeled as a circular cylinder of length L and radius r, along x_1 axis. Using strip theory, estimate the added moment of inertia m_55t ρ*π*a²*Ltρ*π*a²*L²/12tρ*π*a²*L³/24t0t3|t3t2q~ q~ t1q~'uq~ ta2.gif|450|158|tA slender vehicle operating in an infinite fluid of density ρ can be modeled as a circular cylinder of length L and radius r, along x_1 axis. Using strip theory, estimate the added moment of inertia m_44t ρ*π*a²*Ltρ*π*a²*L²/6tρ*π*a²*L³/12t0t4|t3t2q~ q~ t1q~'uq~ ta3.gif|400|106|tjA flat plate of triangular shape lies in the x_1 - x_3 plane, as shown. Assuming that A/L << 1, find m_22.tπρAL/2t π²ρAL/2t0tπρALt2|t3t2q~ q~ t1q~}uq~ ta3.gif|400|106|tjA flat plate of triangular shape lies in the x_1 - x_3 plane, as shown. Assuming that A/L << 1, find m_66.tπρAL/4t π²ρAL²/8t π²ρAL³/16t πρAL²/8t3|t3t2q~ q~ t1q~}uq~ ta3.gif|400|106|tA flat plate of triangular shape lies in the x_1 - x_3 plane, as shown. Assuming that A/L << 1, find what elements of the added-mass matrix are non-zero.ttttt 44|62|26|t3t2q~ q~ t3q~uq~ ta4.gif|350|189|tA buoy consists of a large sphere under a circular cylinder, as shown. The volume and added mass of the cylinder are negligible compared to those of the sphere. What is the equation of motion for heave.t (a³+r²h)d²x_3/dt² +gr²x_3=0t#(a²+2r³h)d²x_3/dt² +g*r³*x_3=0t(a/3)³d²x_3/dt² +g*r*h*x_3=0t!(2a³+r²h)d²x_3/dt² +gr²x_3=0t4|t1t3q~ q~ t1q~uq~ ta4.gif|350|189|tA buoy consists of a large sphere under a circular cylinder, as shown. The volume and added mass of the cylinder are negligible compared to those of the sphere. Estimate the buoy's natural frequency in heave.t!ω²=gr²/(2a³+r²h) rad²/sec²t ω²=gr²/(a³+r²h) rad²/sec²tω²=gh²/(a³+r³) rad²/sec²tω²=gh/(a³+r³)t1|t1t3q~ q~ t1q~uq~ ta5.gif|232|300|tA cone of negligible density is pivoted about the apex in a fluid of density ρ. The length L is much larger than its largest radius R_0. The moment of inertia around the apex (m_66) is:t0tπρR_0²L³/5t πρR²L³/5t πρR²L²/4t2|t3t2q~ q~ t1q~}uq~ ta5.gif|232|300|tA cone of negligible density is pivoted about the apex in a fluid of density ρ. The length L is much larger than its largest radius R_0. The center of buoyancy is at:ttttt2/3|L|t3t2q~ q~ t3q~uq~ ta5.gif|232|300|tA cone of negligible density is pivoted about the apex in a fluid of density ρ. The length L is much larger than its largest radius R_0. The natural roll frequency is:tω²=g/Lt ω²=10g/9Lt ω²=10g/3Lt ω²=5g/9Lt2|t1t2q~ q~ t1q~}uq~ ta6.gif|424|159|tA body is composed of two cones of elliptical cross-section. The cones are aligned as shown along the x_1-axis. Each section has minor axis of length 2b and major axis of 2a. The cones are arranged so that the major axis of the elliptical section is parallel to the x_2-axis for x_1>0 and parallel to the x_3-axis for x_1<0. The ratio a/b is constant at all sections and a(x_1)=c|x_1|. Each cone is of length L so that the composite body is of length 2L. L>>a and a>b. Which m_ij can be obtained by means of the slender-body approximation?q~jq~jq~jq~jt-22|23|24|25|26|33|34|35|36|44|45|46|55|56|66|q~Eq~Zq~ q~ q~rtConsider each cone separately|100000|For a position -L < x < L, write the expression for the ellipse axes|150000|Use added mass coefficients of a 2D ellipse and integrate along x_1|350000|uq~ ta6.gif|424|159|tbA body is composed of two cones of elliptical cross-section. The cones are aligned as shown along the x_1-axis. Each section has minor axis of length 2b and major axis of 2a. The cones are arranged so that the major axis of the elliptical section is parallel to the x_2-axis for x_1>0 and parallel to the x_3-axis for x_1<0. The ratio a/b is constant at all sections and a(x_1)=c|x_1|. Each cone is of length L so that the composite body is of length 2L. L>>a and a>b. Thinking of the limits of applicability of the slender-body approximation for this object, if L≈a but a>>b, which m_ij would be suspect?tttttt1t2q~Eq~ t2q~}uq~ q~ZtA body is composed of two cones of elliptical cross-section. The cones are aligned as shown along the x_1-axis. Each section has minor axis of length 2b and major axis of 2a. The cones are arranged so that the major axis of the elliptical section is parallel to the x_2-axis for x_1>0 and parallel to the x_3-axis for x_1<0. The ratio a/b is constant at all sections and a(x_1)=c|x_1|. Each cone is of length L so that the composite body is of length 2L. L>>a and a>b. What is m_53?tπρL²(a²-b²)/4tπρ(a²-b²)²/4tπρL(a²-b²)/4tπρL(a²-b²)/2q~qq~Eq~Zq~ q~ q~EtqConsider each cone separately|500000|For a position -L < x < L, write the expression for the ellipse axes|350000|uq~ ta7.gif|350|144|tA semi-submersible platform has the configuration shown. The diameter of the uprights is 5m, and that of the pontoons is 10m. The volume displaced by the uprights is negligible compared to that of the pontoons. Estimate the added mass in heave, neglecting the effects of the free surface, the uprights and the interactions between the pontoons. Use the result to estimate the natural frequency in heave of the platform.tm_33=5000ρπ; ω=0.025√gtm_33=500ρπ; ω=0.25√gtm_33=5000ρπ; ω=0.025gtm_33=5000ρπ; ω=0.025gt1|t1t2q~Eq~ t2q~}uq~ ta8.gif|450|154|thA new class of submarine can be modeled by a cylinder of length L and radius R, with a vertical sail and horizontal elliptical wings of major and minor axis radii a and b and length h, as shown. Assuming that these main members are slender so that their longitudinal added mass may be ignored, and also neglecting the interactions among the members, find m_55.t⅓πρ((a² -b² )R²L³+abR)tπρ ((a² -b² )R +R²L³)t¼ πρ(a² -b² )RLt ¼ πρ((a² -b² )R +⅓R²L³)t4|t1t3ppt1q~}uq~ ta8.gif|450|154|tqA new class of submarine can be modeled by a cylinder of length L and radius R, with a vertical sail and horizontal elliptical wings of major and minor axis radii a and b and length h, as shown. Assuming that these main members are slender so that their longitudinal added mass may be ignored, and also neglecting the interactions among the members, find m_33 and m_35.tm_33=ρπ(R²L+2a²h); m_35=0tm_33=0; m_35=ρπ(R²L+2a²h)t!m_33=ρπLh(R+2a)²; m_35=ρπR²tm_33=ρπLh(R+a)²; m_35=0t1|t1t2q~Eq~ t1q~}uq~ ta8.gif|450|154|t6A new class of submarine can be modeled by a cylinder of length L and radius R, with a vertical sail and horizontal elliptical wings of major and minor axis radii a and b and length h, as shown. Assuming that these main members are slender so that their longitudinal added mass may be ignored, and also neglecting the interactions among the members, find the instantaneous force moment M(x,y,z)on the submarine at an instant when its 6 degree-of-freedom motions are: velocity [1,2,3,1,2,3] and acceleration [3,2,1,3,2,1]. You may leave your answers in terms of m_ij.tttttt1t2q~Eq~ t3q~}uq~ ta9.gif|450|142|tYAn underwater vehicle is to have a manipulator arm mounted on it. The designers of the vehicle must know the forces and moments acting on the arm and the objects it manipulates so that they can select appropriate actuator motors. You may assume the following simplifications for this problem: Idealize the arm as two circular cylinders each of length l having radii r_1 and r_2 respectively, with r_1=2r_2. Idealize the arm's load as a spherical package of radius r_0. The radius r_1 is small compared to r_0, and r_0 is small compared to l. In the coordinate system shown, estimate m_11 and m_44.q~jq~jq~jq~jq~jq~Eq~Zq~Eq~ q~Zq~juq~ ta10.gif|400|308|tHousing for certain underwater sensor equipment has a geometry shown below. The sphere has radius a, and the cylinders have radius 0.5a and length 4a. The density of the device can be assumed to be uniform and have a value of twice that of water. To get it to the sea floor, the device is lowered into the water and then released, find its initial acceleration for a vertical orientation and for a horizontal orientation.q~jq~jq~jq~jq~jq~Eq~Zq~Eq~ q~Zq~juq~ ta10.gif|400|308|tHousing for certain underwater sensor equipment has a geometry shown below. The sphere has radius a, and the cylinders have radius 0.5a and length 4a. The density of the device can be assumed to be uniform and have a value of twice that of water. To get it to the sea floor, the device is lowered into the water and then released. Assuming deep water, calculate the terminal drop velocity V_3 for the device falling in a horizontal orientation for very small a; and then for large a.tttttt1t2q~Eq~ t2q~uq~ q~tAn ROV measuring water salinity is moving with velocity V(x,y,z,t)=(2xt,+4y²,-3t). The salinity S of the water changes with the tidal currents, and is given by S(x,y,z,t)=2xcos(at). Find the rate of change of salinity of the water (DS/Dt) as measured by the ROV.t-2axsin(at)+4xtcos(at)t4axsin(at)-2xtcos(at)t 4x²cos(at)t 2sin(2at)t1|t5t3q~ q~ t1q~uq~ q~tTo determine the temperature θ in a certain part of the ocean, a fixed probe measures a rate of change of temperature given by bt, while a heavy probe dropped from the surface and reaching a constant downward velocity of -W records a rate of change of temperature in that area given by -aW+bt. If the temperature in that region is known to be independent of the horizontal coordinate, i.e. θ=θ(z,t), then the temperature there is given by θ(z,t)= constant + ___.t bt²/2+azt bz²/2+att az²/2+btt at²/2+bzt1|t5t3q~ q~ t1q~uq~ q~tThe velocity field in a certain part of the ocean is given by u=3*cos(3x+4y)e^5z, v=4*cos(3x+4y)e^5z and w=c*sin(3x+4y)e^5z. The constant c must be c=___.ttttt5t6t2q~ q~ t2q~uq~ q~tThe conservation of mass equation depends on the assumption(s) of: [constant density] [irrotationality] [inviscid fluid] [incompressibility] [Newtonian fluid] [matter cannot be created].tttttmatter|created|t7t2q~ q~ t3q~uq~ q~tTA tanker grounds on a reef and begins to leak a neutrally buoyant chemical into the ocean. The trace of the chemical as shown in a picture taken from the sky forms a ___. At some time the captain of the vessel abandons ship and jumps into the ocean. Assuming that he drifts with the current, the trajectory he moves through describes a ___.tttttstreakline;pathlinet6t2q~ q~ t2q~uq~ q~tIn a certain rescue operation, a marker is dropped onto the water from a helicopter. The trajectory traced by the drifting marking forms a:t streamlinetpathlinet streaklinetnone of the abovet2|t6t1q~ q~ t1q~uq~ q~tIn a certain rescue operation, a marker is dropped onto the water from a helicopter. Eventually, the marker is observed to again pass the point where it was dropped. The flow must be:t rotationalt irrotationalt can't telltt3|t6t1q~ q~ t1q~'uq~ q~tIn a certain rescue operation, a marker is dropped onto the water from a helicopter. To learn more about the flow, the helicopter pilot releases a large number of similar markers at the same point in quick succession. The line connecting these markers at a later instant is a:t streamlinetpathlinet streaklinetnone of the abovet3|t6t1q~ q~ t1q~'uq~ q~tIn a certain rescue operation, a marker is dropped onto the water from a helicopter. To learn more about the flow, the helicopter pilot releases a large number of similar markers at the same point in quick succession. It is observed that all of the markers have identical trajectories and all eventually pass through the drop-off point after equal time intervals. The flow is most likely:t rotationalt irrotationalt can't telltt1|t1t1q~ q~ t1q~'uq~ q~tOf the following: (a) velocity; (b) pressure; (c) shear stress; (d) density; (e) vorticity; (f) velocity potential; (g) mass flux; (h) momentum flux. The scalar quantities are (write only the letters):tttttb|d|f|g|t8t1q~ q~ t3q~uq~ q~tOf the following: (a) velocity; (b) pressure; (c) shear stress; (d) density; (e) vorticity; (f) velocity potential; (g) mass flux; (h) momentum flux. The vector quantities are (write only the letters):ttttta|e|h|t8t1q~ q~ t3q~uq~ q~tOf the following: (a) velocity; (b) pressure; (c) shear stress; (d) density; (e) vorticity; (f) velocity potential; (g) mass flux; (h) momentum flux. The tensor quantities are (write only the letters):tttttc|t8t1q~ q~ t3q~'uq~ q~tA small thin flat disk of surface area A is oriented in a fluid with directional cosines given by (n_1,n_2,n_3). If the stress tensor there is tau_ij, i,j=1,2,3, the net force on the disk is given by F_i=__, i=1,2,3.t+F_i = A(tau_i1+tau_i2+tau_i3)*(n_1+n_2+n_3)t0F_i = 1/A * (tau_i1+tau_i2+tau_i3)*(n_1+n_2+n_3)t)F_i = A(tau_i1*n_1+tau_i2*n_2+tau_i3*n_3)t.F_i = 1/A * (tau_i1*n_1+tau_i2*n_2+tau_i3*n_3)t3t7t2q~ q~ t1q~}uq~ q~tIn marine hydrodynamics, incompressibility is often a valid assumption because the [velocity, pressure, shear stresses, temperature, gravity] is [much greater, comparable, much smaller] than that of sound waves.tttttvelocity|smaller|t5t1t0q~8t3q~}uq~ q~tThe stress tensor in a flow is given by tau_ij=i+j; i,j=1,2,3. The force acting on a small surface, area ∂A, with unit normal (into the surface) given by n=(1,-1,-1)/√3 is ∂F=(___,___,___)t(1,-1,-1) ∂A/√3t(2,-3,-4) ∂A/√3t(-4,-5,-6) ∂A /√3t(-5,-6,-7) ∂A /√3t4|t7t2t0q~Ct1q~}uq~ tnotAlthough fluids, such as water, are really made up of discreet molecules, we are able to describe their behavior by differential equations by virtue of a ___ hypothesis. Fluids differ from solids in that a fluid at rest cannot sustain ___.tttttcontinum|shear|t7t2t0q~Ot3tuq~ q~FtA spherical gas bubble of radius R and internal pressure p_0 is formed in water. If the surface tension coefficient between the gas and water is Sigma, the pressure outside the bubble must be p=___.t p=p_0 +R/Σt p=p_0 +2R/Σt p=p_0 +Σ/Rt p=p_0 +2Σ/Rt4|t7t2q~Oq~Ot1q~Q