Concept explainers
Streaklines are traced out by neutrally buoyant marker fluid injected into a flow field from a fixed point in space. A particle of the marker fluid that is at point (x, y) at time t must have passed through the injection point (x0, y0) at some earlier instant t = τ. The time history of a marker particle may be found by solving the pathline equations for the initial conditions that x = x0, y = y0 when t = τ. The present locations of particles on the streakline are obtained by setting τ equal to values in the range 0 ≤ τ ≤ t. Consider the flow field
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Fox and McDonald's Introduction to Fluid Mechanics
- For a given hypothetical flow the velocity at time t = 0 to t = 10 s was u = 0, v = 4m/s. Then from time t = 10 s to t = 15 s, the velocity was u = - 2 m/s, v = 2 m/s. A dye streak was started at a point in the flow field at time t = 0, and the path of a particle in the fluid was also traced. Draw to scale the streakline, pathline of the particle and streamlines at t = 15 s.arrow_forwardIn an experiment to determine the force acting on an oval-shaped body, a model is mounted in the test section of a wind tunnel. The test section itself has rectangular cross section with height 2h and width l (into the page). The model spans the entire width of the test section. The model is subject to a steady, uniform, incompressible flow of velocity V₁. The pressure gauge indicates a pressure difference (Ap) between the test section exit (location 2) and the test section entrance (location 1). The velocity profile downstream of the model is shown in the figure. Use the control volume shown. Find the force acting on the model in terms of given quantities V₁, h, l, Ap, and density, p. Ap V ALarrow_forwardA Newtonian fluid flows in the annular space created by a concentric pipe and rod moving to the right at a constant velocity V (this could be the configuration of a wire coating process). The flow is the result of the shear stress created by the moving rod. The flow is steady and incompressible. Assume u, is only a function of r, both u, and ue (as well as their derivatives) are zero, the pipe is horizontal, and the pressure gradient in the z direction is constant. Derive an expression for the velocity profile uz as a function of r. Note: R, is the inside radius of the outer pipe and R; is the radius of the moving rod. R. R; Varrow_forward
- 1. 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_forwardTwo-dimensional irrotational fluid flow is conveniently described by a complex poten- tial f(z) = u(x, v) + iv(x, y). We label the real part, u(x, y), the velocity potential, and the imaginary part, v(x, y), the stream function. The fluid velocity V is given by V = Vu. If f(z) is analytic: 11.2.11 (a) Show that df/dz= Vx – i Vy. (b) Show that V · V = 0 (no sources or sinks). (c) Show that V x V=0 (irrotational, nonturbulent flow).arrow_forward1. Answer the following questions: (a) What is the physical meaning of the following: D a +V.v at Dt where V is the velocity vector of the flow field. (b) Let the viscous stress tensor be denoted by 7. How is the surface (vector) force f, acting by the fluid on a surface element ds (with unit normal în ) computed? Give your answer in vector notation and also in index notation. What is the physical meaning of Ty ? (c) Write down the work done on a material volume of fluid by the viscous surface force in vector notation and also in index notation. (d) Write down the amount of conduction heat flux 'q' (a scalar) on a surface element ds (with unit normal în ) in vector notation and also in index notation.arrow_forward
- In a fluid flow, the density of the fluid is constant for incompressible flow Select one: True Falsearrow_forward4. The velocity vectors of three flow fileds are given as V, = axĩ + bx(1+1)}+ tk , V, = axyi + bx(1+t)j , and V3 = axyi – bzy(1+t)k where coefficients a and b have constant values. Is it correct to say that flow field 1 is one-, flow filed 2 is two-, and flow filed 3 is three-dimensional? Are these flow fields steady or unsteady?arrow_forwardThe velocity component in the y-direction is given as v = 3x - 4y for the steady, inviscid and two- dimensional flow of an incompressible fluid. The only body force is the gravity, g, and it acts in the negative y-direction. The density of the fluid is p. For an irrotational flow, determine a) The velocity component in the x-direction, if it is zero at the origin b) The acceleration vector: ) and c) The pressure field, if the pressure is Po at the origin d) The stream function(arrow_forward
- Given a velocity field as V(x, y, z) = axî – ayĵ %3D With units of velocity in m/sec; x and y in meters; and the constant coefficient a = 0.1 sec. a) Determine the equation for the streamline passing through the point (x, y, 0) = (2, 8, 0). b) Determine the velocity of a particle at the point (2, 8, 0). c) If we mark the particle passing through the point (xo, Yo, 0) at to = 0, determine the location of the particle at time t = 20 sec. d) Show that the equation of the particle path (the pathline) matches the equation of the streamline.arrow_forwardFind the total distance traveled by a particle given its velocity function Question Find the total distance traveled by a particle according to the velocity function v(t) =-4t + 6 m/sec over the time interval 0, 5|. Enter your answer as an exact fraction if necessary and do not include units in your answer. Provide your answer below: • Previous Nexarrow_forwardFluid Mechanics Assis.Professor Dr. Hasan Ali Jurmut HOMEWORK Q/A two-dimensional flow field has velocities along the x and y directions given by u = x2t and v = 2xyt, respectively, wheret is time. Determine the acceleration .If the time is(5 sec.) at point (3,5). Q/Consider steady, incompressible and flow through a reducer in a horizontal pipe where the diameter is reduced from 20 cm to 10 cm. The pressure in the 20 cm and 10cm is 150 and 50 kPa respectively. Compute the flow rate in m²/s. 2 150 kPa 50 kPa V, V, 10 cm 20 cm 20 cm IIarrow_forward
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