In a steady, two-dimensional flow field in the xy - plane, the x-component of velocity is $u=a x+b y+c x^{2}$ where a, b, and c are constants with appropriate dimensions. Generate a general expression for velocity component v such that the flow field is incompressible.

Consider a steady, two-dimensional flow field in the xy-plane whose x-component of velocity is given by $u=a+b(x-c)^{2}$ where a, b, and c are constants with appropriate dimensions. Of what form does the y-component of velocity need to be in order for the flow field to be incompressible? In other words, generate an expression for v as a function of x, y, and the constants of the given equation such that the flow is incompressible.

A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is $ec{V}=(u, v)=\left(U+a_{1} x+b_{1} y ight) ec{i}+\left(V+a_{2} x+b_{2} y ight) ec{j}$. Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

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 $e^{-b r}$.)

Consider a steady, two-dimensional, incompressible flow field in the xy-plane. The linear strain rate in the x-direction is 2.5 s -1. Calculate the linear strain rate in the y-direction.

The velocity field of a flow is given by $ec{V}=k\left(x^{2}-y^{2} ight) ec{i}-2 k x y ec{j}$ where k is a constant. If the radius of curvature of a streamline is $R=\left[1+y^{\prime 2} ight]^{{3} / 2}\left|y^{\prime \prime} ight|$ determine the normal acceleration of a particle (which is normal to the streamline) passing through the position x = 1, y = 2.

We approximate the flow of air into a vacuum cleaner attachment by the following velocity components in the centerplane (the $x y$-plane):

$$u=\frac{-\dot{V} x}{\pi L} \frac{x^2+y^2+b^2}{x^4+2 x^2 y^2+2 x^2 b^2+y^4-2 y^2 b^2+b^4}$$

and

$$v=\frac{-\dot{V} y}{\pi L} \frac{x^2+y^2-b^2}{x^4+2 x^2 y^2+2 x^2 b^2+y^4-2 y^2 b^2+b^4}$$

where $b$ is the distance of the attachment above the floor, $L$ is the length of the attachment, and $\dot{V}$ is the volume flow rate of air being sucked up into the hose. Find the location of any stagnation point(s) in this flow field.

A tiny neutrally buoyant electronic pressure probe is released into the inlet pipe of a water pump and transmits 2000 pressure readings per second as it passes through the pump. Is this a Lagrangian or an Eulerian measurement? Explain.

Converging duct flow is modeled by the steady, two-dimensional velocity field of the earlier problem.

For the case in which $U_0=5.0\ \mathrm{ft} / \mathrm{s}$ and $b=4.6 \mathrm{~s}^{-1}$, consider an initially square fluid particle of edge dimension $0.5\ \mathrm{ft}$, centered at $x=0.5\ \mathrm{ft}$ and $y=1.0\ \mathrm{ft}$ at $t=0$. Carefully find and plot where the fluid particle will be and what it will look like at time $t=0.2$ s later. Comment on the fluid particle's distortion.

Water is flowing in a 3-cm-diameter garden hose at a rate of 30 L/min. A 20-cm nozzle is attached to the hose which decreases the diameter to 1.2 cm. The magnitude of the acceleration of a fluid particle moving down the centerline of the nozzle is (a) $9.81 m/s^2$ (b) $14.5 m/s^2$ (c) $25.4 m/s^2$ (d) $39.1 m/s^2$ (e) $47.6 m/s^2$

Consider the steady, two-dimensional velocity field given by $\vec{v}=(u, v)=(1.6+2.8 x) \vec{i}+(1.5-2.8 y) \vec{j}$. Show that this flow field is incompressible.

A steady, incompressible, two-dimensional (in the xy-plane) velocity field is given by $ec{V}=(0.523-1.88 x+3.94 y) ec{i}+(-2.44+1.26 x+1.88 y) ec{j}$. Calculate the acceleration at the point (x, y) = (-1.55, 2.07).

Consider the steady, two-dimensional velocity field given by $ec{V}=(u, v)=(1.6+1.8 x) ec{i}+(1.5-1.8 y) ec{j}$. Verify that this flow field is incompressible.

A pressurized tank of water has a 10-cm-diameter orifice at the bottom, where water discharges to the atmosphere. The water level is 2.5 m above the outlet. The tank air pressure above the water level is 250 kPa (absolute) while the atmospheric pressure is 100 kPa. Neglecting frictional effects, determine the initial discharge rate of water from the tank.