Department of Mathematics Comsats Institute of Information Technology MTH- 486: Fluid Mechanics Instructor: Dr. Fahad Munir Abbasi Assistant Professor Department of Mathematics Comsats Institute of Information Technology Islamabad, Pakistan

Layout of lecture # 12 Highlights of the previous lecture “Mass” introduction and physical interpretation Law of conservation of mass Mathematical form of the law of conservation of mass: the Continuity equation Different forms of the continuity equation Rotational and irrotational flows Summary

Previously we studied about the… Perfect fluid Equations of streamlines Equations of streamtubes Equations of streamlines or streamtubes in cylindrical coordinates Equations of surface perpendicular to streamlines Examples

MTH-486, Fluid Mechanics Lec. # 12 Mass: A collection of incoherent particles, parts, or objects regarded as forming one body. Or Mass (m) is a dimensionless quantity representing the amount of matter in a particle or object. The standard unit of mass in the International System ( SI ) is the kilogram ( kg ).

MTH-486, Fluid Mechanics Lec. # 12 Mass is measured by determining the extent to which a particle or object resists a change in its direction or speed when a force is applied. Isaac Newton stated: A stationary mass remains stationary, and a mass in motion at a constant speed and in a constant direction maintains that state of motion, unless acted on by an outside force. For a given applied force, large masses are accelerated to a small extent, and small masses are accelerated to a large extent. The following formula applies: F = ma Note: Make it clear that there is a big difference in the mass and weight.

Law of conservation of mass: MTH-486, Fluid Mechanics Lec. # 12 Law of conservation of mass: The law of conservation of mass states that mass in an isolated system is neither created nor destroyed. Or! Conservation of mass implies that matter can be neither created nor destroyed—i.e., processes that change the physical or chemical properties of substances within an isolated system (such as conversion of a liquid to a gas) leave the total mass unchanged. Or! The law of conservation of mass states that mass in an isolated system is neither created nor destroyed by chemical reactions or physical transformations.

MTH-486, Fluid Mechanics Lec. # 12 The Law of Conservation of Mass states that matter can be changed from one form into another, mixtures can be separated or made, and pure substances can be decomposed, but the total amount of mass remains constant. We can state this important law in another way. The total mass of the universe is constant within measurable limits; whenever matter undergoes a change, the total mass of the products of the change is, within measurable limits, the same as the total mass of the reactants.

MTH-486, Fluid Mechanics Lec. # 12 Mathematical form of the law of conservation of mass: The Continuity equation For a control volume (CV) the principle of conservation of mass can be stated as: Rate at which mass enters the system = rate at which mass leaves the system + rate of accumulation of mass in the system Or rate of accumulation of mass in the CV + net rate of mass efflux from the CV = 0 The above equation can be expressed analytically in terms of velocity and density field of a flow and the resulting expression is known as the continuity equation.

MTH-486, Fluid Mechanics Lec. # 12 Let us consider a differential control volume with dimensions dx, dy and dz, which can be fixed in a given flow field with a rectangular coordinate system. Assume that the general flow directions of the fluid particles is such that the velocity and the density increase in the increasing coordinates directions. Assume that the center point of the control volume has the x-component of velocity “u” and density “ρ”. The outward drawn unit normal 𝒏 shows the direction of the specific area under consideration. The densities and velocities at the centers of the faces “abcd” and “oefg” (perpendicular to the x-axis) are determined (using the Taylor’s theorem) by considering only the first order derivatives.

MTH-486, Fluid Mechanics Lec. # 12 The x-component of the velocity at the center of the face “abcd” = 𝒖+ 𝝏𝒖 𝝏𝒙 𝒅𝒙 𝟐 . The density at the center of the face “abcd” = ρ+ 𝝏ρ 𝝏𝒙 𝒅𝒙 𝟐 .

Fig. A differential CV in rectangular coordinates fixed in MTH-486, Fluid Mechanics Lec. # 12 Fig. A differential CV in rectangular coordinates fixed in a given flow field

MTH-486, Fluid Mechanics Lec. # 12 The x-component of the velocity at the center of the face “oefg” = 𝒖− 𝝏𝒖 𝝏𝒙 𝒅𝒙 𝟐 . The density at the center of the face “oefg” = ρ− 𝝏ρ 𝝏𝒙 𝒅𝒙 𝟐 . The values of density and velocity at the center of the two faces of differential CV (for the x-direction flow) are uniformly distributed over the respective faces. The net mass efflux in the x-direction is 𝑵𝒆𝒕 𝒎𝒂𝒔𝒔 𝒆𝒇𝒇𝒍𝒖𝒙=𝑴𝒂𝒔𝒔 𝒆𝒇𝒇𝒍𝒖𝒙 −𝑴𝒂𝒔𝒔 𝒊𝒏𝒇𝒍𝒍𝒖𝒙 = ρ+ 𝝏ρ 𝝏𝒙 𝒅𝒙 𝟐 𝒖+ 𝝏𝒖 𝝏𝒙 𝒅𝒙 𝟐 𝒅𝒚𝒅𝒛− ρ− 𝝏ρ 𝝏𝒙 𝒅𝒙 𝟐 𝒖− 𝝏𝒖 𝝏𝒙 𝒅𝒙 𝟐 𝒅𝒚𝒅𝒛 = ρ 𝝏𝒖 𝝏𝒙 +𝒖 𝝏ρ 𝝏𝒙 𝒅𝒙𝒅𝒚𝒅𝒛 (1)

MTH-486, Fluid Mechanics Lec. # 12 Similar treatment in the y- and z-directions yields the expressions of net mass effluxes in these directions, respectively, as: 𝑵𝒆𝒕 𝒎𝒂𝒔𝒔 𝒆𝒇𝒇𝒍𝒖𝒙 𝒊𝒏 𝒕𝒉𝒆 𝒚−𝒅𝒊𝒓𝒆𝒄𝒕𝒊𝒐𝒏= ρ 𝝏𝒗 𝝏𝒚 +𝒗 𝝏ρ 𝝏𝒚 𝒅𝒙𝒅𝒚𝒅𝒛 𝑵𝒆𝒕 𝒎𝒂𝒔𝒔 𝒆𝒇𝒇𝒍𝒖𝒙 𝒊𝒏 𝒕𝒉𝒆 𝒛−𝒅𝒊𝒓𝒆𝒄𝒕𝒊𝒐𝒏= ρ 𝝏𝒘 𝝏𝒛 +𝒘 𝝏ρ 𝝏𝒛 𝒅𝒙𝒅𝒚𝒅𝒛 Adding (1)-(3), we get, 𝑵𝒆𝒕 𝒎𝒂𝒔𝒔 𝒆𝒇𝒇𝒍𝒖𝒙= 𝝆 𝝏𝒖 𝝏𝒙 + 𝝏𝒗 𝝏𝒚 + 𝝏𝒘 𝝏𝒛 +(𝒖 𝝏ρ 𝝏𝒙 +𝒗 𝝏ρ 𝝏𝒚 +𝒘 𝝏ρ 𝝏𝒛 ) 𝒅𝒙𝒅𝒚𝒅𝒛 (2) (3) (4)

MTH-486, Fluid Mechanics Lec. # 12 The total net mass efflux is causing reduction in the mass of the differential CV at a rate of 𝝏ρ 𝝏𝒕 per unit volume. The total mass reduction in the differential control volume is found by multiplying the rate by the volume 𝒅𝑽=𝒅𝒙𝒅𝒚𝒅𝒛, or 𝑴𝒂𝒔𝒔 𝒓𝒆𝒅𝒖𝒄𝒕𝒊𝒐𝒏 𝒊𝒏 𝒕𝒉𝒆 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒍 𝑪𝑽=− 𝝏ρ 𝝏𝒕 𝒅𝒙𝒅𝒚𝒅𝒛 The negative sign on the RHS of (5) is necessary as it expresses the reduction of mass in the differential CV dv. Therefore, total mass efflux out of dv is equal to the reduction of mass in dv; by equating (4) and (5), we obtain 𝝆 𝝏𝒖 𝝏𝒙 + 𝝏𝒗 𝝏𝒚 + 𝝏𝒘 𝝏𝒛 +(𝒖 𝝏ρ 𝝏𝒙 +𝒗 𝝏ρ 𝝏𝒚 +𝒘 𝝏ρ 𝝏𝒛 ) 𝒅𝒙𝒅𝒚𝒅𝒛=− 𝝏ρ 𝝏𝒕 𝒅𝒙𝒅𝒚𝒅𝒛 (5) (6)

MTH-486, Fluid Mechanics Lec. # 12 Dividing this equation by 𝒅𝑽=𝒅𝒙𝒅𝒚𝒅𝒛 throughout and letting 𝒅𝑽→𝟎 (i.e. letting dv shrink dv to a point volume), we get 𝝏ρ 𝝏𝒕 +𝝆 𝝏𝒖 𝝏𝒙 + 𝝏𝒗 𝝏𝒚 + 𝝏𝒘 𝝏𝒛 +𝒖 𝝏ρ 𝝏𝒙 +𝒗 𝝏ρ 𝝏𝒚 +𝒘 𝝏ρ 𝝏𝒛 =𝟎 Or 𝝏ρ 𝝏𝒕 +𝝆 𝛁 . 𝑽 + 𝑽 . 𝛁 𝝆=𝟎 𝒅𝒊𝒗 𝝆 𝑽 + 𝝏ρ 𝝏𝒕 =𝟎 Where 𝒅𝒊𝒗 𝝆 𝑽 = 𝛁 .(𝝆 𝑽 )=𝝆 𝛁 . 𝑽 + 𝑽 . 𝛁 𝝆. (7) (7a) (8)

MTH-486, Fluid Mechanics Lec. # 12 Or 𝝏 𝝏𝒕 + 𝑽 . 𝛁 ρ+𝝆 𝛁 . 𝑽 =𝟎 𝟏 𝝆 𝒅ρ 𝒅𝒕 + 𝛁 . 𝑽 =𝟎 𝒅 𝒅𝒕 ( log ρ )+ 𝛁 . 𝑽 =𝟎 Equations (7)-(9) are different forms of the equation of continuity in the differential forms applicable to unsteady compressible flow field. Where, 𝑽 =𝒖 𝒊 +𝒗 𝒋 +𝒘 𝒌 . (9)

Different forms of the continuity equation MTH-486, Fluid Mechanics Lec. # 12 Different forms of the continuity equation The continuity equation for incompressible flows: Recall that, for incompressible flows, (ρ=𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕), therefore the flow is steady or unsteady, equation (7) reduces to 𝝏𝒖 𝝏𝒙 + 𝝏𝒗 𝝏𝒚 + 𝝏𝒘 𝝏𝒛 =𝟎 In vector notation (10) can be written as 𝛁 . 𝑽 =𝟎 (10) (11)

In cylindrical coordinates, (10) is written as MTH-486, Fluid Mechanics Lec. # 12 In cylindrical coordinates, (10) is written as 𝟏 𝒓 𝝏 𝝏𝒓 (𝒓 𝒗 𝒓 )+ 𝟏 𝒓 𝝏 𝒗 θ 𝝏θ + 𝝏 𝒗 𝒛 𝝏𝒛 =𝟎 For a steady flow, all fluid properties are be definition, independent of time. Thus, for steady compressible flow the equation of continuity can be written as 𝛁 . 𝝆The continuity equation for incompressible flows: =𝟎 (12) (*)

MTH-486, Fluid Mechanics Lec. # 12 Special case: For two dimensional incompressible flow fields, 𝒘= 𝝏𝒘 𝝏𝒛 =𝟎. Hence (10) reduces to 𝝏𝒖 𝝏𝒙 + 𝝏𝒗 𝝏𝒚 =𝟎 (13)

Irrotational flow: Rotational flow: MTH-486, Fluid Mechanics Lec. # 12 Irrotational flow: If 𝑪𝒖𝒓𝒍 𝑽 =𝟎, then the given flow field is irrotational. Rotational flow: If 𝑪𝒖𝒓𝒍 𝑽 ≠𝟎, then the given flow field is rotational. Note: There is a vector identity which states that the curl of a gradient of a scalar function is always zero. i.e. 𝛁 × − 𝛁 𝝋 =𝟎 If 𝑽 =− 𝛁 𝝋, then from (i) 𝛁 × 𝑽 =𝟎 →𝒊𝒓𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝒇𝒍𝒐𝒘 Thus for irrotational flow, 𝑽 =− 𝛁 𝝋. (i)

MTH-486, Fluid Mechanics Lec. # 12 Equation of continuity for an incompressible, irrotational flow: 𝒅𝒊𝒗 𝑽 =𝟎 Being irrotational, 𝑽 =− 𝛁 𝝋 Using (15) in (14), − 𝛁 ∙ 𝛁 𝝋=𝟎 Or 𝛁 𝟐 𝝋=𝟎 (14) (15) (16)

Summary “Mass” introduction and physical interpretation Law of conservation of mass Mathematical form of the law of conservation of mass: the Continuity equation Different forms of the continuity equation Rotational and irrotational flows

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