Concept explainers
The expression for stream function.
The plot some streamlines of the flow.
Answer to Problem 63EP
The expression for the stream function is
The following figure represents the streamlines of the flow.
Explanation of Solution
Given information:
The incompressible flow filed for which the velocity
Write the expression for the velocity along
Here, the stream function along
Write the expression for the velocity along
Here, the stream function along
Write the expression for quadric stream function.
Here, the stream function is
Calculation:
Substitute
Integrate Equation (IV) with respect to
Here, the constant is
Substitute
Differentiate Equation (V) with respect to
Substitute
Substitute
Here, the constant is
Substitute
Substitute
Substitute
Substitute
Solve Equation (XII) by taking positive sign.
Solve Equation (XII) by taking negative sign.
The following table shows that the value of stream function with respect to value of
| | | |
| | | |
| | | |
| | | |
Plot the streamlines on y x plane by using Equation (XII) and the range of
Figure-
The Figure
Conclusion:
The expression for the stream function is
The following figure represents the streamlines of the flow.
Want to see more full solutions like this?
Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
- The velocity components of a flow field are given by: = 2x² – xy + z², v = x² – 4xy + y², w = 2xy – yz + y² (i) Prove that it is a case of possible steady incompressible fluid flow (ii) Calculate the velocity and acceleration at the point (2,1,3)arrow_forward4 = 3x2 – y represents a stream function in a two – dimensional flow. The velocity component in 'x' direction at the point (1, 3) is:arrow_forwardConverging duct flow is modeled by the steady, two- dimensional velocity field V = (u, v) = (U₁ + bx) i-by. For the case in which Ug = 3.56 ft/s and b = 7.66 s¯¹, plot several streamlines from x = 0 ft to 5 ft and y=-2 ft to 2 ft. Be sure to show the direction of the streamlines. (Please upload you response/solution using the controls provided below.)arrow_forward
- Two velocity components of a steady, incompressible flow field are known: u = 2ax + bxy + cy2 and ? = axz − byz2, where a, b, and c are constants. Velocity component w is missing. Generate an expression for w as a function of x, y, and z.arrow_forward6)arrow_forward1. Stagnation Points A steady incompressible three dimensional velocity field is given by: V = (2 – 3x + x²) î + (y² – 8y + 5)j + (5z² + 20z + 32)k Where the x-, y- and z- coordinates are in [m] and the magnitude of velocity is in [m/s]. a) Determine coordinates of possible stagnation points in the flow. b) Specify a region in the velocity flied containing at least one stagnation point. c) Find the magnitude and direction of the local velocity field at 4- different points that located at equal- distance from your specified stagnation point.arrow_forward
- The stream function relation is given as: Y = xy Find the equations for the components of velocity. Check if we satisfy continuity. Also, plot streamlines for a constant y=4 and y=1.arrow_forwardConsider steady flow of water through an axisymmetric garden hose nozzle. The axial component of velocity increases linearly from uz, entrance to uz, exit as sketched. Between z = 0 and z = L, the axial velocity component is given by uz = uz,entrance + [(uz,exit − uz,entrance)/L]z. Generate an expression for the radial velocity component ur between z = 0 and z = L. You may ignore frictional effects on the walls.arrow_forwardI am working on a practice exam and I wanted to be certain if I was going about the problem correctly.arrow_forward
- Need help on both parts pleasearrow_forwardFlow through a converging nozzle can be approximated by a one-dimensional velocity distribution u=vo (1+2). For the nozzle shown below, assume that the velocity varies linearly from u = vo at the entrance to u = 3v, at the exit. Compute the acceleration at the entrance and exit if vo=10m/s and L = 1m. x=0 X u= :326 x=Larrow_forwardConsider steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D or radius R = D/2 inclined at angle a. There is no applied pressure gradient (@P/x = 0). Instead, the fluid flows down the pipe due to gravity alone. We adopt the coordinate system shown, with x down the axis of the pipe. Derive an expression for the x- component of velocity u as a function of radius r and the other parameters of the problem. Calculate the volume flow rate and average axial velocity through the pipe. 10₂ α Pipe wall Fluid: p. p Rarrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY