For a certain two-dimensional incompressible flow, velocity field is given by 2xy î - y?j. The streamlines for this flow are given by the family of curves
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Q: For a certain two-dimensional incompressible flow, velocity field is given by 2xy î - yj. The…
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- An incompressible velocity field is given by u=a(x°y²-y), v unknown, w=bxyz where a and b are constants. (a)What is the form of the velocity component for that the flow conserves mass? (b) Write Navier- Stokes's equation in 2-dimensional space with x-y coordinate system.Consider steady, incompressible, two-dimensional flow due to a line source at the origin. Fluid is created at the origin and spreads out radially in all directions in the xy-plane. The net volume flow rate of created fluid per unit width is V·/L (into the page of Fig), where L is the width of the line source into the page in Fig Since mass must be conserved everywhere except at the origin (a singular point), the volume flow rate per unit width through a circle of any radius r must also be V·/L. If we (arbitrarily) specify stream function ? to be zero along the positive x-axis (? = 0), what is the value of ? along the positive y-axis (? = 90°)? What is the value of ? along the negative x-axis (? = 180°)?A fluid flow is described (in Cartesian coordinates) by u = x2, v = 4xz. (a) Is this flow two-dimensional or three-dimensional? (b) Is this flow field steady or unsteady? (c) Find the simplest form of the z-component of velocity if the flow is incompressible.
- A Fluid Mechanics, Third Edition - Free PDF Reader E3 Thumbnails 138 FLUID KINEMATICS Fluid Mechanies Fundamenteis and Applicationu acceleration); this term can be nonzero even for steady flows. It accounts for the effect of the fluid particle moving (advecting or convecting) to a new location in the flow, where the velocity field is different. For example, nunan A Çengel | John M. Cinbala consider steady flow of water through a garden hose nozzle (Fig. 4-8). We define steady in the Eulerian frame of reference to be when properties at any point in the flow field do not change with respect to time. Since the velocity at the exit of the nozzle is larger than that at the nozzle entrance, fluid particles clearly accelerate, even though the flow is steady. The accel- eration is nonzero because of the advective acceleration terms in Eq. 4-9. FLUID MECHANICS FIGURE 4-8 Flow of water through the nozzle of a garden hose illustrates that fluid par- Note that while the flow is steady from the…Consider a uniform stream V, which assumed to be steady, incompressible, inviscid and two-dimensional flows from left to the right of this paper (x-direction). a) Determine the velocity potential of this flow. b) Determine the stream function of this flow.Find the vorticity of the fluid motion for the given velocity com- ponents. KINEMATICS OF FLUIDS (a) u A(x + y), v = - A(x + y) (b) u = 2Axz, (c) u Ay²+ By + C, v = A(c² + x² - z²) 1)=0
- 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).A velocity field of the two-dimensional, time-dependent fluid flow is given by where t is time. Find the material derivative Du/Dt and hence calculate the acceleration of the fluid at any time t > 0 and any pont x > 0, y > 0. a) Incompressibility a) Is this flow incompressible (i.e. it has zero divergence)? Yes No ди Ət b) Time derivative of flow field Calculate the time derivative of the velocity. Represent your answer in the form i+ || 3 3 u(t, x, y) =r? (x² + y² ) i− {etxtyj X уј 3 a = c) Material derivative and acceleration Calculate the material derivative of the velocity and hence the acceleration a. Represent your answer in the form Du Dt || j i+ jA proposed harmonic function F(x, y, z) is given byF = 2x2 + y3 - 4xz +f(y)(a) If possible, fi nd a function f (y) for which the laplacianof F is zero. If you do indeed solve part (a), can your fi nalfunction F serve as (b) a velocity potential or (c) a streamfunction?
- The velocity potential for non-viscous two-dimensional and uncompressed water flow in Cartesian coordinates is given as D= -(3x²y - y³) a) Find the corresponding current function. b) Find the pressure difference between points (1,2) and (4,4). Omit the height difference4-17 Converging duct flow is modeled by the steady, two-dimensional velocity field of Prob. 4-16. The pressure field is given by P = Po 2U,bx + b°(x² + y°) where P, is the pressure at x = 0. Generate an expression for the rate of change of pressure following a fluid particle.An idealized incompressible fl ow has the proposed threedimensionalvelocity distributionV = 4xy2i + f (y)j - zy2k Find the appropriate form of the function f ( y ) that satisfi esthe continuity relation.