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Chapter 3: Problem 8

An incompressible fluid flows steadily past a circular cylinder as shown in Fig. P3.8. The fluid velocity along the dividing streamline \((-\infty \leq x \leq-a)\) is found to be \(V=V_{0}\left(1-a^{2} / x^{2}\right),\) where \(a\) is the radius of the cylinder and \(V_{0}\) is the upstream velocity. (a) Determine the pressure gradient along this streamline. (b) If the upstream pressure is \(p_{0},\) integrate the pressure gradient to obtain the pressure \(p(x)\) for \(-\infty \leq x \leq-a,\) (c) Show from the result of part (b) that the pressure at the stagnation point \((x=-a)\) is \(p_{0}+\rho V_{0}^{2} / 2,\) as expected from the Bernoulli equation.

Short Answer

Expert verified
The pressure gradient along the streamline for the fluid can be calculated using Euler's equation, further integrating this equation to form the pressure as a function of \(x\). At the stagnation point \( x = -a \), the pressure \( p \) is \( p_{0} + ρV_{0}^{2} / 2 \) as anticipated according to the Bernoulli's equation.

Step by step solution

01

Identify Given Values

The given equation denotes fluid velocity, \(V=V_{0}(1-a^{2} / x^{2})\), where \(a\) is the radius of the cylinder and \(V_{0}\) is the upstream velocity. An upstream pressure \(p_{0}\) is also mentioned.
02

Equation for Pressure Gradient

The pressure gradient can be calculated using the Euler's equation, which is a simplified form of the Navier-Stokes equation. This equation is stated as: \( \frac{dp}{dx} = -ρ \frac{du}{dt} = -ρu \frac{du}{dx}\), where \(ρ\) is fluid's density, u is the velocity and x denotes the displacement. Substituting \(V\) for the fluid's velocity into the Euler's equation, obtain the pressure gradient along the streamline. Here, only the \(x\) component of the Euler's equation is considered as the fluid velocity is along the \(x\)-axis.
03

Integrate Pressure Gradient

Having established the equation for pressure gradient, integrate this equation from \(x\) to \(-a\) to obtain the pressure as a function of \(x\). Remember, integration will provide a function of \(x\), in which \(C\) is the constant of integration.
04

Determine the Pressure at the Stagnation Point

At the stagnation point, fluid velocity \(V\) is 0 and the pressure is maximum, which gives the condition to find the integration constant \(C\). So, \(C = p_{0} + ρV_{0}^{2} / 2\). Thus, substitute the value of \(C\) in the pressure function \( p(x) \). So, the pressure \( p \) at \( x = -a \) is \( p_{0} + ρV_{0}^{2} / 2 \) as expected from the Bernoulli equation.

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