Stream Tube. When density do not depend explicitly on time then from continuity equation, we have V 2 V 1. δa 2. δa 1 PH6L24 1

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1 Stream Tube A region of the moving fluid bounded on the all sides by streamlines is called a tube of flow or stream tube. As streamline does not intersect each other, no fluid enters or leaves across the sides. V V 1 δa 1 δa When density do not depend explicitly on time then from continuity equation, we have ( vρ) = 0 PH6L4 1

2 = V s ( vρ) 0 ( vρ ). ds= 0 Where V is the volume of the streamtube between faces δa 1 and δa From divergence theorem v ρ and vρ are the mean values of ρvover δa and δa 1 vρ δa= vρ δa For a homogeneous, incompressible, Non-viscous fluids v δa= v δa 1 1 Thus, whenever stream tube is constricted, i.e., wherever streamlines gets crowded, the speed of flow of liquid is larger PH6L4

3 For the entire collection of the stream tubes occupying the whole cross section of the passage through which the fluid flows ρvds A is constant along the passage In the case of constant density vds is constant or v A= v A 1 1 A PH6L4 3

4 Bernoulli s Principle and Physical Significance of different terms In the steady flow of homogeneous, incompressible non-viscous fluid, P 1 v gz Constant where gravitional potential ρ + + = Ω =gz Along a very narrow stream tube In the case of the gases last term is negligible w.r.t. first two terms as the Density is very low for the gases Third term is gravitational potential energy of the unit mass Second term is kinetic energy of the unit mass First term also has units of energy per unit mass What energy this term represents? PH6L4 4

5 δl 1 δl P 1 P A B 1 1 A B Consider the volume of fluid element between A1 and A which is moved after some time to volume between B1 and B δlδa = δl δa = δv 1 1 Work done during the motion by the pressure forces is given by δlpδa δlpδa = ( P P) δv Work done during the motion by the pressure forces in moving the volume element from A1B1 to AB per unit mass is given by P ( P1 P) δv P1 P dp = = ρδv ρ ρ PH6L4 5 P1

6 P 0 P1 dp dp P = = ρ ρ ρ P Therefore, P/ρ term represents the work that will be done on the unit mass of the fluid by the pressure forces, as the element flows from a point where pressure is P to a point where pressure is zero. At two different point in the stream tube P 1 P 1 + v + gz = + v + gz ρ ρ P P ρ gz ( 1 z) = v v1 In the steady flow of an incompressible, homogeneous, nonviscous fluid, moving from position 1 to, the increase in kinetic energy per unit mass is equal to the work done on unit mass of the fluid by pressure and gravity forces. PH6L4 6

7 The three terms is therefore called as pressure energy, kinetic energy and gravitational energy P v,, z are known as pressure head, velocity head ρg g and gravity head and each having the dimention of length PH6L4 7

8 Application 1: Velocity of efflux from a reservoir P 1 Flow from a sharp edged orifice in a tank P Vena Contracta Following assumptions are made in order to simplify the problem steady state flow of Incompressible, homogeneous, nonviscous fluid (frictional effects can be neglected) Cross sectional area of the reservoir is very large compared to that of orifice. Hence the velocity of the fluid at the surface is negligibly small. Practically hydrostatic equilibrium condition exist for points at the surface. PH6L4 8

9 From the Bernoulli s principle, we can write P P atm v1 = 0 1 atm 1 P 1 P 1 + v + z = + v + z ρg g ρg g z = 0 when centre of the orifice is chosen as the horizontal plane for the gravitaional head P P = ρg( h z ) from the hydrostatic condition Putting these all values v = gh PH6L4 9

10 Identical relation for velocity of efflux is obtained for a discharge from one reservoir to another reservoir where h now stand for difference between the gravitational head across the orifice The rate of discharge Q, i.e., the volume of the fluid flowing out of the reservoir per unit time is given by Q Av A gh actual = αα c v = αd Q= Av = A gh Actual Velocity where αv = ; αc = Cofficient of contraction Ideal Velocity and αα = α is called coefficient of discharge c v d PH6L4 10

11 Application : Flow meters From the Bernoulli s principle for the streamline OA 1 1 P0 + ρv0 + ρgz0 = PA + ρva + ρgza For steady state, and OA taken as horizontal and velocity at A is zero from the definition of stagnation point 1 1 PA = P0 + ρv0 or PA P0 = ρv0 Thus pressure at the stagnation point exceed the pressure at any other point on the streamline OA exceeds by 1 ρv Dynamic pressure PH6L4 11

12 Vortex Line In the steady flow of homogeneous, incompressible, non-viscous fluid the Eulers equation is given by v 1 1 v ( v) = p v +Ω t ρ Instead of taking a dot product with velocity vector as we have taken earlier, we take dot product with vorticity vector χ χ φ ρ P 1. v Ω = = Therefore, the potential φ is constant along the vortex line PH6L4 1

13 P v + ρ + Ω = C May be satisfied by more than one stream line and more than one vortex line Circular flow with straight streamline?? Non-zero curl or circulation does not necessary mean that the streamlines are circular or curved Ex. v v = v ( y) ˆ x e x dvx ( y) = eˆ dy z 0 Flow is rotational even though the stream lines are parallel straight lines PH6L4 13