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Chapter 6: Problem 10

In a two-dimensional frictionless, incompressible \(\left(\rho=1500 \mathrm{kg} / \mathrm{m}^{3}\right)\) flow, the velocity field in meters per second is given by \(\vec{V}=(A x+B y) \hat{i}+(B x-A y) \hat{j} ;\) the coordinates are measured in meters, and \(A=4 \mathrm{s}^{-1}\) and \(B=2 \mathrm{s}^{-1}\). The pressure is \(p_{0}=200 \mathrm{kPa}\) at point \((x, y)=(0,0) .\) Obtain an expression for the pressure field, \(p(x, y)\) in terms of \(p_{0}, A,\) and \(B,\) and evaluate at point \((x, y)=(2,2)\)

Short Answer

Expert verified
The pressure field \(p(x, y)\) would be a function of the fluid density \(\rho\), velocity field constants \(A\) and \(B\), and pressure at origin \(p_0\). This is obtained by integrating the Euler's equations over \(x\) and \(y\) respectively. The pressure at point (2,2) is then evaluated by substituting \(x=2\) and \(y=2\) into the derived expression for \(p(x, y)\).

Step by step solution

01

Derive Euler's equation for incompressible flow in the x and y directions

Euler's equation in vector format expresses the change in pressure as an opposite change in the kinetic energy of the fluid. In component form, Euler's equation for an incompressible static fluid where the only force is pressure is \[\frac{dp}{dx} = - \rho \frac{du}{dt} = - \rho u \frac{du}{dx} \quad (1)\]\[\frac{dp}{dy} = - \rho v \frac{dv}{dy} \quad (2)\]Take \(u=Ax+By\) and \(v =Bx-Ay\) as velocity components in the \(x\) and \(y\) directions respectively.
02

Integrate the Euler's equations to obtain the pressure field

To get the pressure field \(p(x,y)\), integrate the derived equations (1) and (2) over \(x\) and \(y\) with respect to the pressure, which yields \(p(x, y)\).Integration of equation (1) over \(x\):\[\int dp = -\rho \int u dx\]Integration of equation (2) over \(y\):\[\int dp = -\rho \int v dy\]
03

Calculate pressure at point (2,2)

Substitute the values of u and v obtained in Step 1 into the equations obtained in Step 2, and evaluate the pressure at \((x, y) = (2,2)\). Applying the given initial condition, \(p(0,0) = p_{0}\), for both integrations, evaluates the constant of integration.

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