# Question 11.8: The steady, 2D inviscid ﬂow of an incompressible ﬂuid shown ...

The steady, 2D inviscid ﬂow of an incompressible ﬂuid shown in Figure 11.7 is described by u = kx, v = −ky, w = 0,  and  p(x, y) = p0 − ρ(k²/2)(x² + y²), where k is a constant, p0 is the pressure at the origin, and body forces have been neglected. This inviscid ﬂow model for a constant density ﬂow approaching a plane wall is referred to as plane stagnation point ﬂow. Show that this ﬂow satisﬁes the continuity and Euler equations, and comment on whether the no-slip, no-penetration conditions are or are not satisﬁed.

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Question: 11.11

## Figure 11.13 shows a crude hurricane model in which the ﬂow is circular, and the wind speed increases linearly with radius from 0 at the center of the eye to 150 km/h at R = 50 km. Estimate the pressure difference from the eye of the hurricane to the indicated location R. ...

Circular ﬂow implies circular streamlines with r p...
Question: 11.10

## Consider a constant density ﬂuid in solid body rotation in the absence of body forces as shown in Figure 11.12. Use the Euler equations in streamline coordinates to analyze the direction of pressure change. Use the Euler equations in cylindrical coordinates to conﬁrm your ﬁndings. ...

The velocity ﬁeld in solid body rotation is descri...
Question: 11.12

## A 2D steady, constant density, inviscid ﬂow of air is described by the velocity ﬁeld u = Ax, v = −Ay, where A = 1.5 s^−1 and the coordinates are measured in feet. Find the pressure difference between a point at (1, 1, 0) and a point at (2, 2, 0). Are these two points located on the same streamline? ...

We are asked to ﬁnd the pressure difference betwee...
Question: 11.9

## The streamlines for the 2D, inviscid, constant density ﬂow over a cylinder are shown in Figure 11.8. The streamfunction for this ﬂow is given in cylindrical coordinates by ψ(r, θ) = U∞r(1− R²/r²)sinθ, where U∞ is the freestream velocity and R is the cylinder radius. If the body force is neglected, ...

The streamfunction for a 2D constant density or in...
Question: 11.7

## In the Poiseuille ﬂow of a constant density, constant viscosity ﬂuid in a round pipe, (Figure 11.6), the velocity ﬁeld is given in cylindrical coordinates by u = vrer + vθeθ + vzez with components vr = 0, vθ = 0, and vz(r) = {[R²P(p1 − p2)]/4µL}[1−( r/RP)²]. Find the pressure distribution in this ...

The velocity ﬁeld must satisfy the continuity equa...
Question: 11.5

## Consider the ﬂow of an incompressible Newtonian ﬂuid between parallel plates with the top plate moving as shown in Figure 11.4. The velocity ﬁeld is u = U(y/h)i, where U is the speed of the moving plate and h is the gap between the plates. Find the stresses in this ﬂow. What can the momentum ...

We are asked to ﬁnd the stresses in a given ﬂow an...
Question: 11.4

## Consider an inﬁnitesimal CV ﬁlled with ﬂuid as shown in Figure 11.3A. Apply a momentum balance in the x direction to this CV to derive the x component of the momentum equation in Cartesian coordinates. ...

We are asked to derive the x component of the mome...
Question: 11.6

## In the channel ﬂow of a constant density, constant viscosity ﬂuid shown in Figure 11.5, suppose the complete description of the ﬂow is given in Cartesian coordinates by the velocity ﬁeld u = {[h²(p1 − p2)]/2µL}[1−(y/h)²], v = 0, and w = 0, and the pressure ﬁeld p(x) = p1 + [(p2 − p1)/L](x − x1). ...

We will ﬁrst substitute the three velocity compone...
Question: 11.1