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Chapter 6: Problem 2

An incompressible frictionless flow field is given by \(\vec{V}=(A x+B y) \hat{i}+(B x-A y) \hat{j},\) where \(A=2 \mathrm{s}^{-1}\) and \(B=2 \mathrm{s}^{-1}\) and the coordinates are measured in meters. Find the magnitude and direction of the acceleration of a fluid particle at point \((x, y)=(2,2) .\) Find the pressure gradient at the same point, if \(\vec{g}=-g j\) and the fluid is water.

Short Answer

Expert verified
The acceleration of the fluid particle is \(0 \ ms^-2\). The pressure gradient at the point (2, 2) is -9800 Pa/m in the x-direction and 0 Pa/m in the y-direction.

Step by step solution

01

Find the velocity

First, plug the given coordinates into the velocity vector. With \( (x, y) = (2,2) \) and \( A = B = 2 s^{-1} \), the velocity vector, \( \vec{V} \), becomes \( \vec{V} = ((2 s^{-1} * 2 m) + (2 s^{-1} * 2 m)) \hat{i} + ((2 s^{-1} * 2 m) - (2 s^{-1} * 2 m))\hat{j} = 8 \hat{i} ms^{-1}. \)
02

Calculate the acceleration

The acceleration of the fluid particle is calculated as the derivative of its velocity with respect to time. In this case, since the flow is steady, the Lagrangian and Eulerian accelerations are the same, and hence, the time derivative of velocity is zero. So, acceleration \( A = 0 \, ms^{-2} \).
03

Compute the pressure gradient

To determine the pressure gradient, we apply the momentum equation, also known as Euler鈥檚 equation, for incompressible and frictionless fluid: \( \frac{dp}{dx} = -\rho ( ag + \frac{du^2}{dx}; \ \frac{dp}{dy} = -\rho ( \frac{dv^2}{dy})\). Here \( \frac{dp}{dx} = -\rho ( ag + \frac{du^2}{dx} ) \) and \( \frac{dp}{dy} = -\rho ( \frac{dv^2}{dy} )\), where \( \rho \) is the fluid density, \( a \) acceleration and \( g \) gravity's effect. With acceleration \( a = 0 m/s^{2} \), and velocity \( u = v = 8 m/s \), we find \( \frac{dp}{dx} = -\rho g \) and \( \frac{dp}{dy} = 0 \). For water as fluid, \( \rho = 1000 kg/m^3 \) and \( g = 9.8 m/s^2 \). Hence, the pressure gradient \( \frac{dp}{dx} = -9800\n, Pa/m \) and \( \frac{dp}{dy} = 0\n Pa/m \).

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Most popular questions from this chapter

Show that any differentiable function \(f(z)\) of the complex number \(z=x+i y\) leads to a valid potential (the real part of \(f\), and a corresponding stream function (the negative of the imaginary part of \(f\) ) of an incompressible, irrotational flow. To do so, prove using the chain rule that \(f(z)\) automatically satisfies the Laplace equation. Then show that \(d f / d z=-u+i v\)

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