Consider inviscid, incompressible flow over a closed semi-cylinder with radius 0.5m. Assume the flow field over the top of the semi-cylinder is identical to the flow over a cylinder for Osesn. Assume the pressure on the bottom of the semi-cylinder is stagnation pressure, po. The freestream velocity is 3 m/s and the density is 1.23 kg/m?. a) What is the net lift force acting on the semi-cylinder? U b) What is the net drag force acting on the semi-cylinder? P. Now consider that the flow is viscous, but still incompressible and that the dynamic viscosity is 2x10 kg/(m*s). c) Estimate the drag on the semi-cylinder assuming it is equivalent to % the drag of a cylinder. Recall that Co=f(Re) for a cylinder as shown below. d) Do you estimate the boundary layer to be laminar or turbulent in part (c) and why? tttt

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a) Contsioer THE velbeine Fieb: V- xy i+ xyj (ij UNIT VECTORS AbNG X-, AND Y DIRECTTONS) IF THE FIUID DENSITY is CONOTANT, is CONSERVATION OF MASS SATİSFİED! CONSIDER THE FolbwiNG STREAM FUNCTION is THE Flow FielD IRROTATIONAL ? WHAT is THE VelocitY POTENTIAl ? C) CONSIDER THE STREAM FUNCTION DESCRIBING A Flow Field iN THE UPPER plaNE xy yoo. FOR THERE is A plATE @ y=0. ) i) is No-slip SATİS FIED @ PIATE (y=o) DRAW THE STREAMLINES FIND THE PRESSURE AS A FUNCTION OF THE PRESSURE O ORIGIN Po. (ASSOME NO GRAVitr).

Q/ Derive the Void fracrion equation below for the two-phase fluid inside the pipes for slug flow model We know that, = C₁₂j + C₂u For horizontal flow, u = 0 Therefore, u=CI For large A, C, can be given as: G = 1+127 {1-exp(-38())} Void fraction: a => 1 น, & ++ 6:55 ✓ HW

Q4: The velocity components for a two dimensional incompressible flow are u = - 6xy, v=-3x2 + 3y2 1) Is the flow satisfied the continuity equation?. 2) Obtain an expression of stream function 3) Obtain an expression for the velocity potential if it is exsits.

WATER Flows THOvGH A SPHON As SHOWN BE ow, THE TUBE HAS A CONSTANT DIAMETER, D= kM Pa 2.5cm Q = lo w% WATER 8: lo00 kg/u3 THE HeIGHT OF THE HGHEST POINT IN THE TUBE is h, AN» THE PRESSURE INSIDE is PH CONSIDER INviscio, INCOMPRESsiole Flow. a caloulaTE THE Velocire y oF THE FlUID INSIDE THE TUBE. b) caloulaTE THE HeiGHT DIFFERENCE l To THE arlET of THE TUBE IF WATER EXTS AT ATMOSPHERIC PRESSURE c) calaolaTE PH As A FUNCTION OF h. d) WHAT is THE HAKİMUM HEİGHT BEFORE THE PRESSURE WoulD BE NEGATIVE PiK . e) WHAT WoulD HAPPEN FOR A HeiGHTrh LARGER THAN THE VAlUE DEERMINED N C)

Consider steady, incompressible, parallel, laminar flow of a film of oil falling slowly down an infinite vertical wall. The oil film thickness is h, and gravity acts in the negative z-direction. There is no applied (forced) pressure driving the flow—the oil falls by gravity alone., except for the case in which the wall is inclined at angle ?. Generate expressions for both the pressure and velocity fields. As a check, make sure that your result agrees with that of when ? = 90°. [Hint: It is most convenient to use the (s, y, n) coordinate system with velocity components (us, ?, un), where y is into the page in Fig. Plot the dimensionless velocity profile us* versus n* for the case in which ? = 60°.]

Consider steady, incompressible, parallel, laminar flow of a viscous fluid falling between two infinite vertical walls. The distance between the walls is h, and gravity acts in the negative z-direction (downward in the figure). There is no applied (forced) pressure driving the flow—the fluid falls by gravity alone. The pressure is constant everywhere in the flow field. For the fluid falling between two parallel vertical walls, generate an expression for the volume flow rate per unit width (V·/L) as a function of ?, ? , h, and g. Compare your result to that of the same fluid falling along one vertical wall with a free surface replacing the second wall, all else being equal. Discuss the differences and provide a physical explanation.

An incompressible Newtonian liquid is confined between two concentric circular cylinders of infinite length— a solid inner cylinder of radius Ri and a hollow, stationary outer cylinder of radius Ro. The inner cylinder rotates at angular velocity ?i. The flow is steady, laminar, and two-dimensional in the r?-plane. The flow is also rotationally symmetric, meaning that nothing is a function of coordinate ? (u? and P are functions of radius r only). The flow is also circular, meaning that velocity component ur = 0 everywhere. Generate an exact expression for velocity component u? as a function of radius r and the other parameters in the problem. You may ignore gravity.

Consider a 2-dimensional incompressible flow field. The vertical component of velocity forthe flow field is given by 2y. The pressure at (x, y) = 0,0 is given by 3 bar absolute. The densityof the fluid is 1.2 Kg/m3 . Find. a) x-component of velocity; b) acceleration at point (x, y) = 2,1;c) pressure gradient at the same point; d) pressure gradient along the x-axis; e) check whetherthe flow is irrotational; f) find the potential function; g) find the stream function; h) equationfor streamline and sketch few streamlines.