Q2. The velocity field in a fluid flow is given by

Transcription

1 Kinematics of Flid Q. Choose the correct anser (i) streamline is a line (a) hich is along the path of a particle (b) dran normal to the elocit ector at an point (c) sch that the streamlines diide the passage into eqal nmber of parts (d) on hich tangent dran at an point gies the direction of elocit [ns.(d)] (ii) Streamline, pathline and streakline are identical hen (a) the flo is niform (b) the flo is stead (c) the flo elocities do not change steadil ith time (d) the flo is neither stead nor niform. [ns.(b)] (iii) The material acceleration is ero for a (a) stead flo (b) niform flo (c) stead and niform flo (d) nstead and non-niform flo [ns.(c)] (i) In a to dimensional flo in - plane, if ( and are the elocit components in the and directions respectiel) then the flid element ill ndergo (a) translation onl (b) translation and rotation (c) translation and deformation (d) rotation and deformation [ns.(c)] Q. The elocit field in a flid flo is gien b V tiˆtj ˆtkˆ Ealate the acceleration of a flid particle at (,-, ) at t s. Soltion cceleration is gien b DV V a V. V Dt t Here, V tiˆtj ˆtkˆ V Hence, iˆj ˆkˆ t V. V V

2 tiˆ tj ˆ tkˆ tiˆtj ˆ tj ˆtkˆ tkˆ ˆ ˆ ˆ ˆ ˆ t ti tj t tj tk t tk tiˆ6t ˆj 4t 4 t k 3 ˆ Finall, the acceleration field can be epressed as a iˆj ˆkˆ 3 t iˆ6 t ˆj 4t 4 t kˆ ˆ 6 ˆ 4 4 t i t j t t k 3 ˆ The acceleration ector at the point (,-, ) and at time t s can be fond b sbstitting the ales of,, and t in the aboe epression as 3 a iˆ 6 ˆ j 4 4 iˆ 8 ˆj 4kˆ kˆ Q3. Flid flos steadil throgh a conerging nole of length. Flo can be approimated as one-dimensional sch that the aial elocit aries linearl from entrance to eit. The elocities at entrance and eit are V and 4V respectiel. Find ot an epression of the acceleration of a particle floing throgh the nole. Soltion The conerging nole is shon in the figre belo. V 4V Since the aial elocit () aries linearl, let s consider B here, and B are constants and their ales are to be determined from the bondar conditions as gien belo. The appropriate bondar conditions are (refer to the aboe figre) t =, V, and t =, 4V

3 ppling the bondar conditions, the constants are fond to be 3V and B V Therefore, the elocit field can be epressed as 3V 3 V V 3V Hence, For stead, one-dimensional flo, acceleration can be ritten as a For the gien elocit field, the acceleration can be epressed as 3 3V 3V 3 a V Q4. Flid flos at a constant rate of Q throgh a conergent pipe of length haing inlet and otlet radii of R and R respectiel. ssming that the elocit to be aial and niform at an cross section, find ot the acceleration at the eit. Soltion Consider a section XX, at a distance from the inlet as shon in the figre belo. r R R X R X Radis of the pipe at section XX is gien b R R R R Velocit at section XX can be ritten as Q Q R R R R 3

4 Q R R 3 R R R The acceleration field can be epressed as a Q Q R R 3 R R R R R R cceleration at the eit is then Q Q R R a 3 R R R R R R Q R R 5 R Q5. three-dimensional elocit field is gien b,, 5. Find the eqation of streamline throgh (,,). Soltion The eqation of a streamline in three-dimensional flo is d d d Here, 5 Streamline in the -plane is gien b d d d d or or ln ln ln C or C Eqation of streamline passing throgh point (,,) is or 8 Streamline in the -plane is gien b d d 4

5 or d d 5 ln ln 5 ln C or or 5 C Eqation of streamline passing throgh point (,,) is or Q6. three-dimensional elocit field is gien b,, B C,, D,, E here, B, C, D, E are constants. Find the components of (a) the strain rates for the aboe elocit field (b) the rotational elocit, and (c) the orticit Soltion (a) Rate of linear strain along direction is Rate of linear strain along direction is Rate of linear strain along direction is Rate of olmetric strain is ol The shear strain rates are fond to be B B 5

6 6 (b) The components of the rotational elocit are as follos (c) The components of the orticit are as follos Q7. The elocit field in a flid medim is gien b ˆ ˆ ˆ 3 3 V i j t k. Determine the rotational elocit ector at the point,, and at time 3 t. Soltion Gien 3, and 3 t For a to-dimensional flo, rotation is gien b For the gien elocit field 6,,,, and Ths, 6 3 t point,, and time 3 t, e hae 3 4

7 The rotational elocit ector at the point,, and at timet 3 is 4iˆ ˆj 7