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Chapter 6: Problem 109

Consider the flow field represented by the potential function $\phi=A x^{2}+B x y-A y^{2} .$ Verify that this is an incompressible flow and determine the corresponding stream function.

Short Answer

Expert verified
By calculating the velocity vectors from the given potential function and substituting those in the continuity equation, it can be verified that the given flow is incompressible as the equation equals to zero. The stream function can then be determined by integrating the relations between the velocity vectors and the stream function.

Step by step solution

01

Verify the Incompressibility

The continuity equation for an incompressible flow in two dimensions is obtained when the divergence of the velocity vector is zero i.e \( \nabla \cdot \vec{v} = 0 \). The velocity vectors can be derived from the potential function as \( u = \frac{\partial \phi}{\partial x} \) and \( v = -\frac{\partial \phi}{\partial y} \). Substitute these expressions in the continuity equation and check if it equals zero.
02

Calculate Velocity Vectors

Calculate \( u = \frac{\partial \phi}{\partial x} \) and \( v = -\frac{\partial \phi}{\partial y} \). For \( u = \frac{\partial \phi}{\partial x} \), take the derivative of \( \phi \) with respect to \( x \) and likewise for \( v = -\frac{\partial \phi}{\partial y} \), take the derivative of \( \phi \) with respect to \( y \). This will give the expressions for \( u \) and \( v \).
03

Apply Continuity Equation

Substitute the expressions derived for \( u \) and \( v \) in the continuity equation \( \nabla \cdot \vec{v} = 0 \) which gives \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \). If this equation equals to zero, this verifies the flow is incompressible.
04

Determine the Stream Function

Once the flow has been verified as incompressible, determine the stream function \( \psi \). The stream function \( \psi \) is related to the velocity vectors as \( u = \frac{\partial \psi}{\partial y} \) and \( v = -\frac{\partial \psi}{\partial x} \). The stream function \( \psi \) can be obtained by integrating these expressions.

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Most popular questions from this chapter

The stream function of a flow field is \(\psi=A x^{2} y-B y^{3},\) where \(A=1 \mathrm{m}^{-1} \cdot \mathrm{s}^{-1}, B=\frac{1}{3} \mathrm{m}^{-1} \cdot \mathrm{s}^{-1},\) and the coordinates are measured in meters. Find an expression for the velocity potential.

Water flows in a circular duct. At one section the diameter is \(0.3 \mathrm{m}\), the static pressure is \(260 \mathrm{kPa}\) (gage), the velocity is \(3 \mathrm{m} / \mathrm{s},\) and the elevation is \(10 \mathrm{m}\) above ground level. At a section downstream at ground level, the duct diameter is \(0.15 \mathrm{m}\) Find the gage pressure at the downstream section if frictional effects may be neglected.

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 \((x, y)=(2,2)\)

The \(x\) component of velocity in an incompressible flow field is given by \(u=A x,\) where \(A=2 \mathrm{s}^{-1}\) and the coordinates are measured in meters. The pressure at point \((x, y)=(0,0)\) is \(p_{0}=190 \mathrm{kPa}\) (gage). The density is \(\rho=1.50 \mathrm{kg} / \mathrm{m}^{3}\) and the \(z\) axis is vertical. Evaluate the simplest possible \(y\) component of velocity. Calculate the fluid acceleration and determine the pressure gradient at point \((x, y)=(2,1) .\) Find the pressure distribution along the positive \(x\) axis.

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.
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