# Iran University of Science & Technology School of Mechanical Engineering Advance Fluid Mechanics

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1 1. Consider a sphere of radius R immersed in a uniform stream U0, as shown in 3 R Fig.1. The fluid velocity along streamline AB is given by V ui U i x Find (a) the position of maximum fluid acceleration along AB and (b) the time required for a fluid particle to travel from A to B. Fig.1 2. A piston compresses gas in a cylinder by moving at constant speed, as in Fig.2. Let the gas density and length at t = 0 be 0 and L 0, respectively. Let the gas velocity vary linearly from u = V at the piston face to u = 0 at x = L. If the gas density varies only with time, find an expression for (t). Fig.2 3. A viscous liquid of constant and falls due to gravity between two plates a distance 2h apart, as in Fig.1. The flow is fully developed, with a single velocity component w = w(x). There are no applied pressure gradients, only gravity. Solve the Navier-Stokes equation for the velocity profile between the plates. Fig.3

2 4. The velocity profile for laminar flow between two plates, as in Fig.4 is: 4u max y h y u, v w 0 2 h If the wall temperature is T w at both walls, use the incompressible flow energy equation to solve for the temperature distribution T(y) between the walls for steady flow. Fig.4 5. A two-dimensional, incompressible, frictionless fluid is guided by wedge-shaped walls into a small slot at the origin, as in Fig.5. The width into the paper is b, and the volume flow rate is Q. At any given distance r from the slot, the flow is radial inward, with constant velocity. Find an expression for the polar-coordinate stream function of this flow. Fig.5 Ky Kx 6. A two-dimensional incompressible flow is defined by u v x y x y where K = cte. Is this flow irrotational? If so, find its velocity potential, sketch a few potential lines, and interpret the flow pattern. 7. A Consider a viscous film of liquid draining uniformly down the side of a vertical rod of radius a, as in Fig.7. At some distance down the rod the film will approach a terminal or fully developed draining flow of constant outer radius b, with v z = v z (r), v = v r = 0. Assume that the atmosphere offers no shear resistance to the film motion. Derive a differential equation for v z, state the proper boundary conditions, and solve for the film velocity distribution. How does the film radius b relate to the total film volume flow rate Q?

3 Fig.7 8. The flow pattern in bearing lubrication can be illustrated by Fig.8, where a viscous oil (, ) is forced into the gap h(x) between a fixed slipper block and a wall moving at velocity U. If the gap is thin, h << L, it can be shown that the pressure and velocity distributions are of the form p = p(x), u = u(y), v = w = 0. Neglecting gravity, reduce the Navier- Stokes equations to a single differential equation for u(y). What are the 1 dp proper boundary conditions? Integrate and show that: 2 y u y yh U 1, 2 dx h where h = h(x) may be an arbitrary slowly varying gap width. Fig The viscous oil in Fig.9 is set into steady motion by a concentric inner cylinder moving axially at velocity U inside a fixed outer cylinder. Assuming constant pressure and density and a purely axial fluid motion, solve Navier-Stokes equations for the fluid velocity distribution v z (r). What are the proper boundary conditions? Fig. 9

4 10. Find the resultant velocity vector induced at point A in Fig. 10 by the uniform stream, vortex, and line source. Fig A Rankine half-body is formed as shown in Fig. 11. For the stream velocity and body dimension shown, compute (a) the source strength m in m 2 /s, (b) the distance a, (c) the distance h, and (d) the total velocity at point A. Fig Wind at U and p flows past a Quonset hut which is a half-cylinder of radius a and length L (Fig. 12). The internal pressure is pi. Using inviscid theory, derive an expression for the upward force on the hut due to the difference between pi and ps. Fig It is desired to simulate flow past a two-dimensional ridge or bump by using a streamline which passes above the flow over a cylinder, as in Fig. 13. The bump is to

5 be a/2 high, where a is the cylinder radius. What is the elevation h of this streamline? What is U max on the bump compared with stream velocity U? Fig The steady plane flow in Fig. 14 has the polar velocity components v = Ωr and v r =0. Determine the circulation around the path shown. Fig Consider three equal sources m in a triangular configuration: one at (a/2, 0), one at (-a/2, 0), and one at (0, a). Plot the streamlines for this flow. Are there any stagnation points? Hint: Try the MATLAB contour command. 16. Air at 20 C and 1 atm flows at 20 m/s past the flat plate in Fig. 1. A pitot stagnation tube, placed 2 mm from the wall, develops a manometer head h = 16 mm of Meriam red oil, SG = Use this information to estimate the downstream position x of the pitot tube. Assume laminar flow. Fig.16

6 17. Suppose you buy a 4- by 8-ft sheet of plywood and put it on your roof rack. (See Fig. 2.) You drive home at 35 mi/h. (a) Assuming the board is perfectly aligned with the airflow, how thick is the boundary layer at the end of the board? (b) Estimate the drag on the sheet of plywood if the boundary layer remains laminar. (c) Estimate the drag on the sheet of plywood if the boundary- layer is turbulent (assume the wood is smooth), and compare the result to that of the laminar boundary-layer case. Fig For flow past a cylinder of radius R as in Fig. 3, the theoretical inviscid velocity distribution along the surface is U = 2U 0 sin (x/r), where U 0 is the oncoming stream velocity and x is the arc length measured from the nose. Compute the laminar separation point x sep and sep by Thwaites method, and compare with the digital computer solution x sep /R = ( sep = ) given by R. M. Terrill in Fig The cross section of a cylinder is shown in Fig. 4. Assume that on the front surface the velocity is given by potential theory, V =2U sin, from which the surface pressure is computed by Bernoulli s equation. In the separated flow on the rear, the pressure is assumed equal to its value at = 90. Compute the theoretical drag coefficient and compare with Table 7.2 (White). Fig.19

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