1) Consider an object of a parabolic shape with rotational symmetry z

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1 Umeå Univesitet, Fysik 1 Vitaly Bychkov Pov i teknisk fysik, Fluid Mechanics (Stömningsläa), , kl jälpmedel: Students may use any book including the tetbook Lectues on Fluid Dynamics. Students may not use thei lectue notes. 1) Conside an object of a paabolic shape with otational symmety / 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 hoiontal foce epeienced by ½ of the object cut symmetically by a vetical plane (e.g. the visible side of the object on Fig. 1)? ( p) R Fig. 1 ) A simple hydodynamic model of a heat is shown in Fig.. 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)

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

3 Umeå Univesitet, Fysik 3 Vitaly Bychkov Pov i teknisk fysik, Fluid Mechanics (Stömningsläa), , kl jälpmedel: Students may use any book including the tetbook Lectues on Fluid Dynamics. Students may not use thei lectue notes. 1) A gate shown in Fig. 1 holds a statified fluid of density (1 3 / ) in the 0 esevoi (with length L in y-diection). Find toque with espect to the ais A. (4 p) A Fig. 1. A gate holds fluid in the esevoi ) 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) pacel. 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 fied position in the Euleian efeence fame. (4 p) 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 comple unity, E is an eigenvalue of paticle enegy. Using the dimensional analysis, find how pessue of the degeneate gas depends on concentation. (4 p)

4 Umeå Univesitet, Fysik 4 Vitaly Bychkov 4) A viscous stationay flow in a bent cylindical pipe shown in Fig. 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 Fig.. 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 eo. The final esult may be pesented in a comple fom without educing it to a eal value. (4 p) u U cos( t) Fig. 3. Velocity distibution in a viscous laye poduced by bottom oscillations.

5 Umeå Univesitet, Fysik 5 Vitaly Bychkov Pov i teknisk fysik, stömningsläa (fluid mechanics), , kl jälpmedel: Students may use any book including the tetbook Lectues on Fluid Dynamics. Students may not use thei lectue notes. 1) Conside an inteface of two fluids of densities (uppe fluid) and 1 (lowe fluid). A body of mass M and otational paabolic shape / R is floating steadily at the inteface, see Fig. 1. Find depth h of the body immesed in the second fluid. (4 p) 1 h R Fig. 1. A body floating at the inteface between two fluids ) Fluid flows out of the tank though a spheical oifice at the bottom and foms a stationay jet (adius of the oifice is R ), see Fig.. The fluid level in the tank is kept constant. Find how adius of the jet vaies with height () assuming that the vetical velocity component in the jet is much lage than the tansvese one. (4p) Fig.. Configuation of a tank with an oifice at the bottom

6 Umeå Univesitet, Fysik 6 Vitaly Bychkov 3) Conside the Rayleigh-Taylo instability at an inteface of a heavy fluid of density suppoted by a light gas in a gavitational field g. The instability is suppessed by suface tension fo petubations with wave numbes lage than a cetain citical value k c. 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 the citical wave numbe k c depends on fluid density, gavity acceleation g and the coefficient of suface tension. (4 p) 4) Fluid in a cylindical tube consists of two layes of viscosities fo 0 a and 1 fo a R (see, Fig. 3). Pessue diffeence P between the ends of the tube causes plane-paallel fluid motion. Neglecting the effect of enty length find velocity distibution in the tube. int: Thee ae two bounday conditions at the inteface between the fluid layes. One condition is kinematical of velocity balance, while the othe one is a dynamical condition of stess balance at the inteface. (4p) Fig. 3. A viscous flow in a two adjacent fluid layes. 5) A unifom flow of velocity U comes on a plana plate of total length L as shown in Fig. 4. A bounday laye at the plate may be descibed by the Blasius velocity pofile with u Uf ( ), whee y / ( ) is a self-simila vaiable, () is width of the bounday laye and df / d at 0. Find the dag foce acting on the plate pe unit length in -diection. Fluid density is and kinematical viscosity is. (4p)

7 Umeå Univesitet, Fysik 7 Vitaly Bychkov y U Fig. 4. A bounday laye at a plana plate. L