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Homework answers / question archive / 1) An accelerating flow is described by vx = Ax, where A is a constant

1) An accelerating flow is described by vx = Ax, where A is a constant. Find the rate-of-strain tensor and interpret its physical meaning.Derive an expression for the divergence of the rate of strain tensor.What form does the divergence of the stress tensor take for a Newtonian incompressible fluid?

2) Consider a cylinder of radius R rotating about its axis with an angular velocity of β. Calculate the circulation and relate it to the vorticity The velocity profile in a fluid is given as vx = A(1 − y2) Find the streamfunction. Find the flow across the system if the domain for y is from zero to one. Find the ratio of the average velocity to the maximum velocity.

3) Find the vorticity for a free vortex. The velocity profile is vθ = A/r where A is a constant. This profile can, for example, be observed if there is a rotating stirrer at the center of a large pool of liquid. Upon applying the curl operator in polar coordinates we find that the vorticity is zero!

4) A slab of thickness 2L is maintained at a surface temperature of Ts at both ends (x = ±L). There is a constant internal heat generation at a rate of Q? v. Find the temperature profile, the maximum temperature, and the heat being transferred to the surroundings.

5) Develop a dynamic model for a heating of a fluid in tank. Heat is supplied from a jacketed vessel with steam heating. The transfer rate across the system may be assumed to be UA(Ts − T ), where U is the overall heat transfer coefficient. Here Ts is the steam temperature and T is the process fluid temperature in the tank. The tank is assumed to be well mixed in this case.

6) A normal shock wave occurs in the flow of helium. The conditions at the point of shock are pressure 1/14.7 atm; temperature 5 ?C, and velocity 1250 m/s. γ = 1.66 for helium. Find the conditions at the point after the shock wave. Determine the entropy and enthalpy gain in the system.

7) A converging–diverging nozzle produces a velocity of Mach three at the exit at a flow rate of 1 kg/s. The reservoir conditions are pressure 90 kPa and temperature 298 K. Find the exit conditions. Find the conditions at the throat.

8) A Venturi type of pipe as shown in Fig. 2.13 is used for flow measurement. In this problem we examine the pressure profile and how the pressure measurements can be used to find the flow rate in the system. The analysis is done by writing a Bernoulli equation between section 1 and section 2 assuming no friction losses: p1/ ρ + 1 /2 v2 1 = p2/ ρ + 1/ 2 v2 2

9) A viscous liquid leaves a pipe and emerges as a jet as shown in Fig. 2.12. The diameter of the jet is smaller than that of the pipe from which the jet is emanating. Calculate this diameter from the energy balance. Assume that the flow in the pipe is occurring under laminar conditions.

10) A centrifugal pump has a diameter of 20 cm and delivers 0.4 m3/s of water at a tangential velocity of 9 m/s. The pump operates at 1200 r.p.m. The fluid enters axially near the center and is discharged in a direction tangential to the impeller. Determine the torque on the pump shaft and the power needed.

11) Set up the model equations for a two-compartment model shown in Fig. 1.12 for analysis of the distribution of a drug in a body.A jet of fluid is impinging on a blade. The fluid comes in the x-direction and leaves in the y-direction. See Fig. 2.9. It is required to calculate the force on the blade. Problems of this type are of importance in design of turbines.