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Chapter 6: Problem 37

The velocity potential for a given two-dimensional flow field is $$\phi=\left(\frac{5}{3}\right) x^{3}-5 x y^{2}$$ Show that the continuity equation is satisfied and determine the corresponding stream function.

Short Answer

Expert verified
The continuity equation for the given potential \( \left(\frac{5}{3}\right) x^{3}-5 x y^{2} \) is satisfied and the corresponding stream function is \( -\left(\frac{5}{3}\right) x^{3} + 5 x y^{2} \).

Step by step solution

01

Calculating the gradient of the given potential

The given potential is \( \phi=\left(\frac{5}{3}\right) x^{3}-5 x y^{2} \). The velocity field \( \vec{V} \) is equal to the gradient of \( \phi \). Therefore, first differentiate \( \phi \) with respect to \( x \) and \( y \) to find the two components of \( \vec{V} \). The differentiation yields \( \vec{V} = (5x^2-5y^2)\hat{i} - 10xy\hat{j} \).
02

Verifying the continuity equation

Now calculate the divergence of \( \vec{V} \). The divergence \( \nabla \cdot \vec{V} \) is obtained by differentiating the \( i \) and \( j \) components of \( \vec{V} \) with respect to \( x \) and \( y \) respectively, and adding them. The calculation yields \( \nabla \cdot \vec{V} = 10x - 10x = 0 \), which confirms the continuity equation is satisfied.
03

Determining the stream function \(\psi\)

The stream function \( \psi \) for a 2D flow is given by \( \psi = -\phi \). So, substitute the given potential \( \phi \) into this equation to find \( \psi \). This gives \( \psi = -\left(\frac{5}{3}\right) x^{3} + 5 x y^{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity Potential
In fluid dynamics, the velocity potential is a scalar function often denoted by \( \phi \), which describes the flow field of a fluid. It is useful because, for irrotational flows (where the vorticity is zero), the fluid velocity \( \vec{V} \) can be described as the gradient of \( \phi \). This means that \( \vec{V} = abla \phi \).

  • When you differentiate \( \phi \) with respect to spatial variables, you obtain the components of the velocity field.
  • In the given exercise, this potential yields \( \vec{V} = (5x^2-5y^2)\hat{i} - 10xy\hat{j} \).
Velocity potential is particularly helpful in simplifying complex problems, especially when dealing with potential flows. Once you know the velocity potential, finding the velocity components becomes straightforward as seen in this example.
Continuity Equation
The continuity equation in fluid mechanics is a mathematical statement of the principle of conservation of mass. For a two-dimensional incompressible flow, it is expressed as \( abla \cdot \vec{V} = 0 \). This implies that the net flow of fluid into any volume must equal the net flow out to ensure mass conservation.

In simpler terms:
  • The total amount of fluid entering a region must be equal to the total amount leaving for the flow to be steady and incompressible.
For our problem:
  • The velocity field \( \vec{V} = (5x^2-5y^2)\hat{i} - 10xy\hat{j} \) has a divergence of \( 10x - 10x = 0 \).
Thus, it satisfies the continuity equation confirming no accumulation of mass at any point in the flow. This ensures that the flow is consistent with the principle of mass conservation.
Stream Function
A stream function \( \psi \) is a tool used in fluid mechanics to describe flow fields, especially two-dimensional incompressible flows. The functional form of the stream function is such that the partial derivatives of \( \psi \) with respect to coordinates give the respective velocity components.

A key property of the stream function is:
  • The lines of constant \( \psi \) represent streamlines of the flow, which are the paths followed by fluid elements.
From the exercise:
  • The stream function relative to the given velocity potential \( \phi \) is \( \psi = -\left(\frac{5}{3}\right) x^{3} + 5 x y^{2} \).
Stream functions provide a handy way to visualize and describe the flow, making them immensely beneficial for both analysis and interpretation of fluid behavior. They also help to ensure that the velocity field automatically satisfies the continuity equation.

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Most popular questions from this chapter

A Bingham plastic is a fluid in which the stress \(\tau\) is related to the rate of strain \(d u / d y\) by $$\tau=\tau_{0}+\mu\left(\frac{d u}{d y}\right)$$ where \(\tau_{0}\) and \(\mu\) are constants. Consider the flow of a Eingham plastic between two fixed, horizontal, infinitely wide, flat plates. For fully developed flow with \(d u / d x=0\) and \(d p / d x\) constant. find \(u(y)\).

The \(x\) -velocity profile in a certain laminar boundary layer is approximated as follows $$u=U_{0} \sin \left(\frac{\pi}{2} \frac{y}{0.1 \sqrt{x}}\right)$$ Determine the \(y\) -velocity, \(v(x, y)\).

The velocity in a certain two-dimensional flow field is given by the equation $$\mathbf{v}=2 x t \hat{\mathbf{i}}-2 y t \hat{\mathbf{j}}$$ where the velocity is in ft's when \(x, y,\) and \(t\) are in feet and seconds, respectively. Determine expressions for the local and convective components of acceleration in the \(x\) and \(y\) directions. What is the magnitude and direction of the velocity and the acceleration at the point \(x=y=2 \mathrm{ft}\) at the time \(t=0 ?\)

A viscous fluid is contained between two infinitely long, vertical, concentric cylinders. The outer cylinder has a radius \(r_{o}\) and rotates with an angular velocity \(\omega\). The inner cylinder is fixed and has a radius \(r_{i} .\) Make use of the Navier-Stokes equations to obtain an exact solution for the velocity distribution in the gap. Assume that the flow in the gap is axisymmetric (neither velocity nor pressure are functions of angular position \(\theta\) within the gap) and that there are no velocity components other than the tangential component. The only body force is the weight.

The two-dimensional velocity field for an incompressible Newtonian fluid is described by the relationship $$\mathbf{V}=\left(12 x y^{2}-6 x^{3}\right) \hat{\mathbf{i}}+\left(18 x^{2} y-4 y^{3} \hat{\mathbf{j}}\right.$$ where the velocity has units of \(\mathrm{m} / \mathrm{s}\) when \(x\) and \(y\) are in meters. Determine the stresses \(\sigma_{x x}, \sigma_{y y},\) and \(\tau_{x y}\) at the point \(x=0.5 \mathrm{m}\) \(y=1.0 \mathrm{m}\) if pressure at this point is \(6 \mathrm{kPa}\) and the fluid is glycerin at \(20^{\circ} \mathrm{C}\). Shew these stresses on a sketch.
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