Exercice 7 Soit un écoulement dont le potentiel des vitesses est donné par : \[ \emptyset=x^{2}-2 y-y^{2} \] 1. Trouver le champ des vitesses 2. Est-ce que ce champ vérifié l'équation de continuité 3. Est-ce que l'écoulement est plan 4. Est-ce que l'écoulement est permanent 5. A partir du potentiel des vitesses, trouver la fonction de courant. Exercice 8 Le champ des vitesses d'un écoulement bidimensionnel modélisant la houle s'exprime en formalisme Eulérien par : \[ \mathrm{u}=\mathrm{V}_{0} \cos (\omega \mathrm{t}), \mathrm{v}=\mathrm{V}_{0} \sin (\omega \mathrm{t}) \] Déterminer les trajectoires et les lignes de courant.
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Step 1: To find the velocity field from the velocity potential \(\emptyset = x^2 - 2y - y^2\), we use the relations for potential flow: \[ u = \frac{\partial \emptyset}{\partial x} \] \[ v = \frac{\partial \emptyset}{\partial y} \] Show more…
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The velocity components in a steady, incompressible, 2-dimensional flow field are given by V = U i + 0 j. a. Determine the corresponding stream function, ψ, and velocity potential, φ. b. If U = 1 m/s, determine the value of ψ at y = 0.1 m and the value of φ at x = 0.1 m. c. If ψ2 is a horizontal line at y = 0.5 m and the value of ψ along the x-axis is zero, calculate the volume flow rate per unit width between these two streamlines when U = 5 m/s. 17. Define briefly the following terms. a. Streamline: A streamline is a line that is tangent to the velocity vector at every point in a flow field. It represents the path followed by a fluid particle in steady flow. b. Equipotential line: An equipotential line is a line in a flow field where the velocity potential is constant. It represents a line of equal potential in the flow. c. Irrotational flow: Irrotational flow is a flow field in which the fluid particles do not rotate as they move. This means that the vorticity, which represents the local rotation of fluid particles, is zero everywhere in the flow field.
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For the velocity fields given below, determine: a. whether the flow field is one-, two-, or three-dimensional, and why. b. whether the flow is steady or unsteady, and why. (The quantities $a$ and $b$ are constants.) (1) $\vec{V}=\left[(a x+t) e^{\log y}\right]$ (2) $\vec{V}=(a x-b y)$ (3) $\vec{V}=a x \vec{i}+\left[e^{\ln x}\right]$ (4) $\vec{V}=a x \hat{i}+b x^{2} j+a x k$ (5) $\vec{V}=a x \hat{i}+\left[e^{b r}\right] \dot{j}$ (6) $\vec{V}=a x \hat{i}+b x^{2} \hat{j}+a y k$ (7) $\vec{V}=a x \hat{i}+\left[e^{\ln }\right] \hat{j}+a y \hat{k}$ (8) $\vec{V}=a x i+\left[e^{b y}\right] j+a z \hat{k}$
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