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FLUID MECHANICS – 1 Semester 1 2011 - 2012 Week – 5 Class – 2 Kinematics of Fluids Compiled and modified by Sharma, Adam
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Review• Description of Fluid flow• Steady and unsteady flow• Uniform and non uniform flow• Dimensions of flow• Material derivative and acceleration• Differentiate between streamlines, pathlines and streaklines 2
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Objectives• Types of Motion - Translation, Deformation• Rotation• Vorticity• Existence of flow• Continuity equation• Irrotational flow
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OTHER KINEMATIC DESCRIPTIONSTypes of Motion or Deformation ofFluid ElementsIn fluid mechanics, an element may undergo fourfundamental types of motion or deformation:(a) translation, (b) rotation,(c) linear strain (also called extensional strain), and(d) shear strain.All four types of motion or deformation usually occursimultaneously.It is preferable in fluid dynamics to describe the motionand deformation of fluid elements in terms of rates such as velocity (rate of translation), angular velocity (rate of rotation), linear strain rate (rate of linear strain), and shear strain rate (rate of shear strain). Fundamental types of fluid element motion orIn order for these deformation rates to be useful in the deformation: (a) translation,calculation of fluid flows, we must express them in terms of (b) rotation, (c) linear strain,velocity and derivatives of velocity. and (d) shear strain. 4
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Volumetric strain rate or bulk strain rate: The rate of increase of volume of afluid element per unit volume.This kinematic property is defined as positive when the volume increases.Another synonym of volumetric strain rate is also called rate of volumetric dilatation, (theiris of your eye dilates (enlarges) when exposed to dim light).The volumetric strain rate is the sum of the linear strain rates in three mutuallyorthogonal directions. The volumetric strain rate is zero in an incompressible flow. Air being compressed by a piston in a cylinder; the volume of a fluid element in the cylinder decreases, corresponding to a negative rate of volumetric dilatation. 5
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Figure shows a general (although two-dimensional) situation in a compressiblefluid flow in which all possible motionsand deformations are presentsimultaneously.In particular, there is translation,rotation, linear strain, and shear strain.Because of the compressible nature ofthe fluid flow, there is also volumetricstrain (dilatation). A fluid element illustrating translation, rotation, linear strain, shear strain, and volumetric strain. 6
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• Vorticity is closed related to fluid rate of rotation.• If the vorticity at a point in a flow field is nonzero, the fluid particle that happens to occupy that point in space is rotating; the flow in that region is called rotational. The difference between• Likewise, if the vorticity in a region of the flow is zero (or rotational and irrotational negligibly small), fluid particles there are not rotating; flow: fluid elements in a the flow in that region is called irrotational. rotational region of the flow rotate, but those in an• Physically, fluid particles in a rotational region of flow irrotational region of the rotate end over end as they move along in the flow. flow do not. 7
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Existence of flow• Velocity components in accordance with the mass conservation principle are said to constitute a possible fluid flow.• Therefore the existence of a physically possible flow field is verified from the principle of conservation of mass.
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Continuity Equation• A rectangular parallelepiped is considered as the control volume in a Cartesian frame.• Let the fluid enter across the left face with a velocity u , and a density ρ . The velocity and density with which the ∂u ∂ρ fluid leave the right face will be u+ ∂x dx and ρ + ∂x dx , neglecting the higher order terms in dx.• The rate of mass entering the control volume is: ρ u dy dz• The rate of mass leaving the control volume is: ∂ρ ∂u = ρ + dx u + dx dy dz ∂x ∂x ∂ = ρ u + ( ρ u ) dx dy dz neglecting higher order terms in dx ∂x
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Continuity Equation• The net rate of mass leaving the control volume in the x direction• Mass leaving the control volume – Mass entering the control volume ∂ = ρ u + ( ρ u ) dx dy dz − ρ u dx dy dz ∂x ∂ = ( ρ u ) dx dy dz ∂x• In a similar manner, the net rate of mass leaving in the y direction ∂ρ ∂v = ρ + dy v + dy dx dz − ρ v dx dy dz ∂y ∂y ∂ = ( ρ v ) dx dy dz ∂y
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Continuity Equation• In a similar manner, the net rate of mass leaving in the z direction = ρ + ∂ρ dz w + ∂w dz dx dy − ρ w dx dy dz ∂z ∂z ∂ = ( ρ w) dx dy dz ∂z• The rate of accumulation of mass within the control volume is ∂ρ = dx dy dz ( dx dy dz ) ∂t {the volume is invariant with time}. From the law of conservation of mass for a control volume, it can be written that ∂ρ ∂ ∂ ∂ ∂t + ∂x ( ρ u ) + ∂y ( ρ v ) + ∂z ( ρ w) dx dy dz = 0 ∂u ∂v ∂w ∂x + ∂y + ∂z = 0For an steady, incompressible flow,
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Mass and Volume flow rates - Actual and averaged velocity profile as shown For simplification, we need Vavg instead of Vn - For the average velocity can be written as; 1 Vavg = Ac ∫ V dA Ac n c - For the flow with uniform density value across the cross section;Average Velocity m = ρ Vavg Ac &
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Volume Flow Rate & Mass ConservationVolume of fluid flow through a cross section per unittime is called “volume flow rate”-Volume flow rate:- Mass flow rate and volume flow rate can be correlated as; m = ρV & &
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Conservation of Mass PrincipleConservation of Mass Principle; conservation of mass in flow rate form; dmCV min − mout & & = dt
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Steady Mass Flow Conservation - For steady flow across CV, mass is CONSTANT - This means that; ∑m = ∑m & in & out - For incompressible flow, it is possible to write; ∑V A = ∑V A in n n out n nSteady Flow Conservation
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