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Chapter 4: Problem 8
The components of a velocity ficld are given by \(u=x+y\) \(v=x y^{3}+16,\) and \(w=0 .\) Determine the location of any stagnation points \((\mathbf{V}=0)\) in the flow field.
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A fluid particle flowing along a stagnation streamline, as shown in Video \(V 4.9\) and Fig. \(P 4.38,\) slows down as it approaches the stagnation point. Measurements of the dye flow in the video indicate that the location of a particle starting on the stagnation streamline a distance \(s=0.6 \mathrm{ft}\) upstream of the stagnation point at \(t=0\) is given approximately by \(s=0.6 e^{-0.5 y},\) where \(t\) is in seconds and \(s\) is in feet. (a) Determine the speed of a fluid particle es a function of time, \(V_{\text {particle }}(t),\) as it flows along the streamline. (b) Determine the speed of the fluid as a function of position along the streamline, \(V=V(s)\) (c) Determine the fluid acceleration along the streamline as a function of position, \(a_{s}=a_{s}(s)\)
A test car is traveling along a level road at \(88 \mathrm{km} / \mathrm{hr}\). In order to study the acceleration characteristics of a newly installed engine, the car accelerates at its maximum possible rate. The test crew records the following velocities at various locations along the level road: $$\begin{array}{lll} x=0 & V=88.5 \mathrm{km} / \mathrm{hr} \\ x=0.1 \mathrm{km} & V=93.1 \mathrm{km} / \mathrm{hr} \\ x=0.2 \mathrm{km} & V=98.3 \mathrm{km} / \mathrm{hr} \\ x=0.3 \mathrm{km} & V=104.0 \mathrm{km} / \mathrm{hr} \\ x=0.4 \mathrm{km} & V=110.3 \mathrm{km} / \mathrm{hr} \\ x=0.5 \mathrm{km} & V=117.2 \mathrm{km} / \mathrm{hr} \\ x=1.0 \mathrm{km} & V=164.5 \mathrm{km} / \mathrm{hr} \end{array}$$ A preliminary study shows that these data follow an equation of the form \(V=A\left(1+e^{B x}\right),\) where \(A\) and \(B\) are positive constants. Find \(A\) and \(B\) and a Lagrangian expression \(V=V\left(V_{0}, t\right),\) where \(V_{0}\) is the car velocity at time \(t=0\) when \(x=0\)
The velocity field of a flow is given by \(\mathbf{V}=2 x^{2} t \hat{\mathbf{i}}+[4 y(t-1)\) \(+2 x^{2}+1 j\) is where \(x\) and \(y\) are in meters and \(t\) is in seconds. For fluid particles on the \(x\) axis, determine the speed and direction of flow.
For any steady flow the streamlines and streaklines are the same. For most unsteady flows this is not true. However, there are unsteady flows for which the streamlines and streaklines are the same. Describe a flow field for which this is true.
As is indicated in Fig. \(P 4.42,\) the speed of exhaust in a car's cxhaust pipe varies in time and distance because of the periodic nature of the engine's operation and the damping effect with distance from the engine. Assume that the speed is given \\[ \text { by } \quad V=V_{0}\left[1+a e^{-b x} \sin (\omega t)\right], \quad \text { where } \quad V_{0}=8 \text { fps, } a=0.05 \\] \(b=0.2 \mathrm{ft}^{-1},\) and \(\omega=50 \mathrm{rad} / \mathrm{s} .\) Calculate and plot the fluid acceleration at \(x=0,1,2,3,4,\) and 5 ft for \(0 \leq t \leq \pi / 25\) s.
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