What are the three major assumptions used in the derivation of the Bernoulli equation?

What is stagnation pressure? Explain how it can be measured.

Does the amount of mass entering a control volume have to be equal to the amount of mass leaving during an unsteady-flow process?

Define pressure head, velocity head, and elevation head for a fluid stream and express them for a fluid stream whose pressure is P. velocity is V. and elevation is z.

The velocity of a fluid flowing in a pipe is to be measured by two different Pitot-type mercury manometers shown in the figure. Would you expect both manometers to predict the same velocity for flowing water? If not, which would be more accurate? Explain. What would your response be if air were flowing in the pipe instead of water?

Consider a device with one inlet and one outlet. If the volume flow rates at the inlet and at the outlet are the same, is the flow through this device necessarily steady? Why?

While traveling on a dirt road, the bottom of a car hits a sharp rock and a small hole develops at the bottom of its gas tank. If the height of the gasoline in the tank is 30 cm, determine the initial velocity of the gasoline at the hole. Discuss how the velocity will change with time and how the flow will be affected if the lid of the tank is closed tightly.

Using the results of the above problem and the fundamental definition of linear strain rate (the rate of increase in length per unit length), develop an expression for the linear strain rate in the $y$-direction $\left(\varepsilon_{y y}\right)$ of fluid particles moving down the channel. Compare your result to the general expression for $\varepsilon_{y y}$ in terms of the velocity field, i.e., $\varepsilon_{y y}=\partial \nu / \partial y$. (Hint: Take the limit as time $t \rightarrow 0$. You may need to apply a truncated series expansion for $e^{-b r}$.)

A weather balloon is launched into the atmosphere by meteorologists. When the balloon reaches an altitude where it is neutrally buoyant, it transmits information about weather conditions to monitoring stations on the ground. Is this a Lagrangian or an Eulerian measurement? Explain.

A Pitot-static probe is used to measure the speed of an aircraft flying at 3000 m. If the differential pressure reading is 3 kPa, determine the speed of the aircraft.

Explain the importance of the Reynolds transport theorem in fluid mechanics, and describe how the linear momentum equation is obtained from it.

Indicate whether the following assertions are true or incorrect for each one: The Euler-Bernoulli beam theory is more accurate than the Timoshenko theory.

Solve each inequality and graph the solution set on a number line. -7x>21

In the derivation of the Bernoulli equation for regions of inviscid flow, we rewrite the steady, incompressible Euler equation into a form showing that the gradient of three scalar terms is equal to the velocity vector crossed with the vorticity vector, noting that z is vertically upward, $ec{ abla}\left(rac{P}{ ho}+rac{V^{2}}{2}+g z ight)=ec{V} imes ec{\zeta}$. We then employ some arguments about the direction of the gradient vector and the direction of the cross product of two vectors to show that the sum of the three scalar terms must be constant along a streamline. In this problem you will use a different approach to achieve the same result. Namely, take the dot product of both sides of the Euler equation with velocity vector V and apply some fundamental rules about the dot product of two vectors. Sketches may be helpful.