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Chapter 6: Problem 98

Consider a two-dimensional fluid flow: \(u=a x+b y\) and \(v=c x+d y,\) where \(a, b, c\) and \(d\) are constant. If the flow is incompressible and irrotational, find the relationships among \(a, b, c,\) and \(d .\) Find the stream function and velocity potential function of this flow.

Short Answer

Expert verified
The relationships among the constants are given by \(a + d = 0\) and \(b - c = 0\), and the stream function and potential function of the flow are \(\Psi = a x y + 0.5 b y^{2}\) and \(\Phi = 0.5 a x^{2} + b x y\) respectively.

Step by step solution

01

Applying Incompressible Flow Condition

Incompressible flow is one whose density is constant. Mathematically, an incompressible fluid satisfies the continuity equation: \n\[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \]\nSubstitute the given u and v into the equation to find one of the relationships: \n\[a + d = 0\]
02

Applying Irrotational Flow Condition

Irrotational flow is one in which the fluid particles have no rotational motion. Mathematically, it is described by the condition: \n\[ \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = 0 \]\nSubstitute v and u into the equation to find the other relationship: \n\[c - b = 0\]
03

Derive Stream Function

The stream function, often denoted by the symbol Psi (蠄), is defined such that the component velocities are given by the derivatives of 蠄 in the y and x directions respectively. So: \n\[u = \frac{\partial \Psi}{\partial y}, v = -\frac{\partial \Psi}{\partial x}\]\nSo let us integrate the \(u = a x + b y\) w.r.t y and \(v = c x + d y\) w.r.t x (considering the constants of integration as zero): \n\[\Psi = a x y + 0.5 b y^{2}\]
04

Derive the velocity Potential Function

The velocity potential, often denoted by the symbol Phi (蠁), is defined such that the component velocities are given by the derivatives of 蠁 in the x and y directions respectively. So: \n\[u = \frac{\partial \Phi}{\partial x}, v = \frac{\partial \Phi}{\partial y}\]\nSo let us integrate the \(u = a x + b y\) w.r.t x and \(v = c x + d y\) w.r.t y (considering the constants of integration as zero): \n\[\Phi = 0.5 a x^{2} + b x y\]

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Most popular questions from this chapter

In a laboratory experiment, water flows radially outward at moderate speed through the space between circular plane parallel disks. The perimeter of the disks is open to the atmosphere. The disks have diameter \(D=150 \mathrm{mm}\) and the spacing between the disks is \(h=0.8 \mathrm{mm} .\) The measured mass flow rate of water is \(\dot{m}=305 \mathrm{g} / \mathrm{s}\). Assuming frictionless flow in the space between the disks, estimate the theoretical static pressure between the disks at radius \(r=50 \mathrm{mm} .\) In the laboratory situation, where some friction is present, would the pressure measured at the same location be above or below the theoretical value? Why?

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The stream function of a flow field is \(\psi=A x^{3}+\) \(B\left(x y^{2}+x^{2}-y^{2}\right),\) where \(\psi, x, y, A,\) and \(B\) are all dimensionless. Find the relation between \(A\) and \(B\) for this to be an irrotational flow. Find the velocity potential.

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