, and the curve BC is symmetrical. Find also the horizontal force in x-direction on one side of the body. h C

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1 Umeå Univesitet, Fysik 1 Vitaly Bychkov Pov i teknisk fysik, Fluid Dynamics (Stömningsläa), , kl jälpmedel: Students may use any book including the textbook Lectues on Fluid Dynamics. Mino notes in the books ae allowed. 1) A body of total mass M, total length L in z-diection and the coss-section shown in Fig. 1 is floating steadily in wate (density ). Find how much of the body is unde the wate (find h). Assume that the keel-line AC is definitely unde the wate. The shape of the keel fom B to A is given by x ( D / 2)sin y / D, and the cuve BC is symmetical. Find also the hoizontal foce in x-diection on one side of the body. D A h C D x y B Fig. 1. A floating body. 2) Conside an ideal flow in the geomety simila to that of blacksmith bellows (Fig. 2). The gas flow between two plates u ; u ) is ceated by otational motion of the top plate ( in cylindical coodinates ( ; ) with angula velocity d / dt and with the bottom plate fixed. ee is the total angle between the plates; do not confuse it with the angula vaiable. The back side of the system (at adius R ) is closed by a igid wall, so that the plate motion ceates a adial flow in the diection of the coodinate oigin. Find velocity distibution between the plates (both components). Fo simplicity assume that u does not depend on.

2 Umeå Univesitet, Fysik 2 Vitaly Bychkov u z () R Fig. 2. The geomety of blacksmith bellows. Fig. 3. Ai flow in a stack. 3) Conside suface waves at the inteface of two immiscible quantum gases. The dispesion elation fo these waves involves atomic mass M, scatteing length a, concentation n (i.e. numbe of atoms pe unit volume) and the Plank constant. Dimension of the Plank constant may be found fom the Schödinge equation i / t E, whee is a the wave function, i is a complex unity, E is enegy. Using the dimensional analysis, find how fequency of the suface waves depends on 4 the wave numbe k, if (na) 1/? (4 p) 4) Ai comes to the buning chambe of a stove as shown in Fig. 3, gets heated and leaves though a long cylindical stack of height and adius R, with R. Assume that density of the heated ai is educed by a facto of in compaison with the usual ai density outside, in /. Find the dischage out of the stack taking into account viscous foces (the viscosity coefficient is ). Assume that the flow is lamina and incompessible befoe and afte the chambe, and the pessue is almost unifom in the chambe (but not in the stack!). Neglect the effect of enty length in the stack. 5) A sphee of adius R and density 0 is sinking in a viscous fluid of viscosity and density f 0 with 1. Initial ball velocity is zeo. Find the ball velocity vesus time assuming that the flow is quasi-stationay and Re<<1.

3 Umeå Univesitet, Fysik 3 Vitaly Bychkov Pov i teknisk fysik, Fluid Mechanics (Stömningsläa), , kl jälpmedel: Students may use any book including the textbook Lectues on Fluid Dynamics. Students may not use thei lectue notes. 2 1) Conside an object of a paabolic shape with otational symmety z / R as shown in Fig. 1. Total mass of the object is M. The object is placed tightly at the bottom of a esevoi of depth (fluid density is ). ow lage is the foce needed to detach the object fom the bottom? (1 p) ow lage is the foce needed to lift it afte the detachment? (1 p) ow lage is the hoizontal foce expeienced by ½ of the object cut symmetically by a vetical plane (e.g. the visible side of the object on Fig. 1)? (2 p) z R Fig. 1 2) A simple hydodynamic model of a heat is shown in Fig. 2. We imitate a heat by a cylindical vessel of height with adius vaying in time as R( t) R0 (1 sin t). Assuming an ideal flow, find both velocity components in the model. (4 p) 3) Capillay waves ae waves of vey shot wavelengths at the inteface sepaating two substances. Dynamics of these waves is contolled by fluid density and suface tension, instead of gavitational acceleation. By definition, the coefficient of suface tension couples pessue incease because of cuved inteface P / R and the local adius of cuvatue, R. Using the dimensional analysis, find how fequency and phase velocity of capillay waves depend on the wave numbe k, density and suface tension? (4 p)

4 Umeå Univesitet, Fysik 4 Vitaly Bychkov z R(t) Fig. 2. 4) An infinitely long cylinde of adius R otates with fequency as shown in fig. 3. The cylinde is suounded by a fluid of viscosity 1, which fills the space between R 2R. The est is filled by anothe fluid of viscosity 2, 2 R. Velocity and stess ae continuous at the inteface between the fluids. Find stess at the cylinde u u suface. int: In the pesent geomety stess is calculated as. (4 p) 2R R Fig. 3 5) A flame font acceleates exponentially in a channel of width 2 R with non-slip bounday conditions at the walls. Flame acceleation ceates pessue gadient P / e ˆ x exp( t) diected paallel to the channel walls. Assuming a plane-paallel flow, find velocity distibution in the channel. (4 p)

5 Umeå Univesitet, Fysik 5 Vitaly Bychkov Pov i teknisk fysik, Fluid Mechanics (Stömningsläa), , kl jälpmedel: Students may use any book including the textbook Lectues on Fluid Dynamics. Students may not use thei lectue notes ) A gate shown in Fig. 1 holds a statified fluid of density (1 3z / ) in the 0 esevoi (with length L in y-diection). Find toque with espect to the axis A. (4 p) z x 2 A Fig. 1. A gate holds fluid in the esevoi 2) Conside pulsations of a spheically symmetic sta close to the cente, so that density of evey pacel of the stella matte (gas) vaies in time as ( t) 0(1 sin t). The gas velocity has only one adial component u aˆ u( t, ). Find the gas velocity. int: Density vaiations of a gas pacel ae diffeent fom density vaiations at a fixed position in the Euleian efeence fame. (4 p) pacel 3) Pessue of a non-elativistic degeneate electon gas depends only on electon concentation n (numbe of electons pe unit volume). The fomula fo pessue involves also electon mass m and the Plank constant. Dimension of the Plank constant may be detemined fom the Schödinge equation i / t E, whee is a paticle wave function, i is a complex unity, E is an eigenvalue of paticle enegy. Using the dimensional analysis, find how pessue of the degeneate gas depends on concentation. (4 p)

6 Umeå Univesitet, Fysik 6 Vitaly Bychkov 4) A viscous stationay flow in a bent cylindical pipe shown in Fig. 2 may be descibed locally as a Poiseuille flow fo any coss-section. Find the foce acting on the pipe, if the flow dischage is Q, pessue at the entance is P 1, fluid density is, fluid viscosity is, total tube length is L and the coss-sectional aea of the pipe is S. (4 p) y x Fig. 2. A bent cylindical pipe with a flow. 5) A laye of viscous fluid of thickness is suppoted fom below by a plate, while the uppe fluid suface is fee, see Fig. 3. Detemine velocity distibution in the fluid, if the plate oscillates as U cos( t). Viscous stess at a fee suface is zeo. The final esult may be pesented in a complex fom without educing it to a eal value. (4 p) u x U cos( t) Fig. 3. Velocity distibution in a viscous laye poduced by bottom oscillations.