Concept explainers
A steady, two-dimensional velocity field in the ay-plane is given by
(a) What are the primary dimensions (m. L,t. T,.. .) of coefficients a, b. c, and ci?
(b) What relationship between the coefficients is necessary in order for this flow to be incompressible?
(c) What relationship between the coefficients is necessary in order for this flow to be irrotational?
(d) Write the strain rate tensor for this (e) For the simplified case of d = -b, derive an equation for the streamlines of this tiow, namely, = function(x, a, b, c).
(a)
The primary dimensions (
Answer to Problem 120P
The primary dimension for
Explanation of Solution
Given information:
The flow is steady and two-dimensional.
The velocity field in the
Here,
Write the expression of the comparison for
Here, the velocity vector is
Write the expression for the velocity
Here, the distance is
Substitute
Substitute
Here, the length is
Write the expression of the comparison for
Here, the coefficient is
Substitute
Substitute
Write the expression of the comparison for
Here, the coefficient is
Substitute
V).
Write the expression of the comparison for
Substitute
Substitute
Here, the coefficient is
Conclusion:
The primary dimension for
(b)
The relationship between the coefficients in order for the given flow to be incompressible.
Answer to Problem 120P
The relation between the coefficients is
Explanation of Solution
Write the expression for the flow to be incompressible.
Here, the del operator is
Write the expression for del operator.
Here, the vector along
Substitute
Conclusion:
The relation between the coefficients is
(c)
The relationship between the coefficients in order for the given flow to be irrotational.
Answer to Problem 120P
The relation between the coefficients for the flow being irrotational is 0.
Explanation of Solution
Write the expression for the flow to be irrotational.
Here, the del operator is
Substitute
Here, the
Substitute
Conclusion:
The relation between the coefficients for the flow being irrotational is 0.
(d)
The strain rate tensor for the given flow.
Answer to Problem 120P
The strain rate tensor for two dimensional flow is
Explanation of Solution
Write the expression for the strain tensor for two-dimensional flow.
Here, the derivative in
Substitute
Conclusion:
The strain rate tensor for two dimensional flow is
(e)
An equation for the streamlines of the given flow, namely,
Answer to Problem 120P
The equation for the streamlines for
Explanation of Solution
Write the expression for the streamline equation.
Here, the differential with respect to
Substitute
Substitute
Integrate
Here, the integration constant is
Conclusion:
The equation for the streamlines for
Want to see more full solutions like this?
Chapter 4 Solutions
Fluid Mechanics: Fundamentals and Applications
- Velocity Field Assignment 4 2 -5 -4 2 -1 N -4 W- E Consider the steady, two-dimensional velocity field of wind as: V= (u, v)= (8 – 0.5x)i + (0.5 - 5y)j where x- an y- coordinates are in m, time in s, and the magnitude of the velocity is in m/s. Determine: (a) A stagnation point, if existed. (b) Sketch the velocity vector for the given coordinate on the map. (c) Sketch the relevant streamlines on a different graph. (d) Verify if the flow is rotational or irrotational flow. (e) Looking at the velocity vector, which section of the country will receive the most rain if the wind brings rainy season from the south-china sea?arrow_forward(a) Given the following steady, two-dimensional velocity field. [Diberi medan halaju yang mantap dan dua dimensi.] V = (u, v) = (8x + 6)ï + (-8y – 4)j (i) Is this flow field an incompressible flow? Prove your answer. (ii) Is this flow field irrotational? Prove your answer. (iii) Generate an expression for the velocity potential function if applicable.arrow_forward1. If u- 3x'yr and v = -6x'y'r answer the following questions giving reasons, Is this flow or fluid: (a) Real (Satisfies Continuity Principle). (b) Steady or unsteady. (c) Uniform or non-uniform. (d) One, two, or three dimensional. (e) Compressible or incompressible. Also, Find the acceleration at point (1,1). %3Darrow_forward
- A Fluid Mechanics, Third Edition - Free PDF Reader E3 Thumbnails 138 FLUID KINEMATICS Fluid Mechanies Fundamenteis and Applicationu acceleration); this term can be nonzero even for steady flows. It accounts for the effect of the fluid particle moving (advecting or convecting) to a new location in the flow, where the velocity field is different. For example, nunan A Çengel | John M. Cinbala consider steady flow of water through a garden hose nozzle (Fig. 4-8). We define steady in the Eulerian frame of reference to be when properties at any point in the flow field do not change with respect to time. Since the velocity at the exit of the nozzle is larger than that at the nozzle entrance, fluid particles clearly accelerate, even though the flow is steady. The accel- eration is nonzero because of the advective acceleration terms in Eq. 4-9. FLUID MECHANICS FIGURE 4-8 Flow of water through the nozzle of a garden hose illustrates that fluid par- Note that while the flow is steady from the…arrow_forwardProblem (5.9): In a three-dimensional incompressible fluid flow, the velocity components are: u = x? + z2 + 5, v = y2 + z2 - 3 (i) Determine the third component of velocity. (ii) Is the fluid flow irrotational? [Ans. (i) w = -2(x + y)z + f (x, y, t)(ii) No. ]arrow_forward4. Consider a velocity field V = K(yi + ak) where K is a constant. The vorticity, z , is (A) -K (B) K (C) -K/2 (D) K/2arrow_forward
- 3. The two-dimensional velocity field in a fluid is given by V 2ri+ 3ytj. (i) Obtain a parametric = equation for the pathline of the particle that passed through (1.1) at t = 0. (ii) Without calculating any equation: if I were to draw the streak-line at t = 0 of all points that passed through (1, 1) would it be the same or different? Justify yourself.arrow_forwardGiven the following steady, two-dimensional velocity field. [Diberi medan halaju yang mantap dan dua dimensi.] V = (u, v) = (6x + 4y – 2)i + (-6y + 4x – 5)j %3D (i) Is this flow field an incompressible flow? Prove your answer. [Adakah medan halaju ini adalah aliran tidak boleh mampat? Buktikan jawapan anda.] (ii) Is this flow field irrotational? Prove your answer. [Adakah medan halaju ini adalah aliran tidak berputar? Buktikan jawapan anda.] (iii) Generate an expression for the velocity potential function if applicable. [Terbitkan satu ungkapan bagi fungsi keupayaan halaju jika boleh dilakukan.]arrow_forwardA proposed harmonic function F(x, y, z) is given byF = 2x2 + y3 - 4xz +f(y)(a) If possible, fi nd a function f (y) for which the laplacianof F is zero. If you do indeed solve part (a), can your fi nalfunction F serve as (b) a velocity potential or (c) a streamfunction?arrow_forward
- Verify whether or not the following difference representation for the continuity equation for a 2-D steady incompressible flow has the conservation property: (Ui+1,j + U₁+1, j-1 — Ui, j — Ui,j-1) (Vi+¹, j — Vi+1,j-1). Ay + 2Ax where u and v are the x and y components of velocity, respectively.arrow_forwardA velocity field of the two-dimensional, time-dependent fluid flow is given by where t is time. Find the material derivative Du/Dt and hence calculate the acceleration of the fluid at any time t > 0 and any pont x > 0, y > 0. a) Incompressibility a) Is this flow incompressible (i.e. it has zero divergence)? Yes No ди Ət b) Time derivative of flow field Calculate the time derivative of the velocity. Represent your answer in the form i+ || 3 3 u(t, x, y) =r? (x² + y² ) i− {etxtyj X уј 3 a = c) Material derivative and acceleration Calculate the material derivative of the velocity and hence the acceleration a. Represent your answer in the form Du Dt || j i+ jarrow_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
- 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