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Module 3: Lecture 10 to 13 Continuity Equation, Angular Strain Rate and Euler's Equation of Motion

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Module 3: Lecture 10 to 13 Continuity Equation, Angular Strain Rate and Euler's Equation of Motion Module 3: Lecture 10 to 13 Continuity Equation, Angular Strain Rate and Euler's Equation of Motion Dr. Raj Nandkeolyar *** Topics Covered: 1. Continuity Equation (Conservation of mass). • Control System Approach. • Control Volume Approach. 2. Angular Strain Rate (Angular Deformation of Fluid Particle). • Different Cases: When Flow is 2D, Incompressible and Irrotational. 3. Dynamics of Inviscid Fluid. • Euler's Equation of Motion. • Bernoulli's Equation of Motion. 1 Continuity Equation (Conservation of Mass) Here in this section, we will do some gymnastics for driving an equation which enforces con- servation of mass i.e. the continuity equation. bContinuity Equationb In fluid dynamics, the continuity equation states that the rate at which mass enters a system is equal to the rate at which mass leaves the system plus the accumulation of mass within the system. bDifferential Form of the Continuity Equationb @ρ + r:(ρ~q) = 0: @t here, ρ is the fluid density, t is the time and ~q is the flow velocity vector field. .............................................................................................................................. Note: The Navier-Stoke's equation form a vector continuity equation describing the conservation of linear momentum (which wiil be discussed in the module 4). 1 This is not strictly a description of material behavior, but the resulting equation is often used as an identity to algebraically manipulate constitutive models describing material behavior. So it is worth reviewing. It is also central to the analysis of fluid flow because classical fluids analyses cannot be Lagrangian since the positions of all the fluid particles at t = 0 are unknown. 1.1 Control System Approach Let us consider a control system with mass m, Then; m = ρV =) log m = log ρ + log V Taking time derivative (actually the material derivative1) to the both sides of above equation (keeping in mind that the mass is treated as constant due to control system), we get; 0 1 Dm 1 Dρ 1 DV = + m Dt ρ Dt V Dt 1 Dρ 1 DV =) + = 0 (1) ρ Dt V Dt Note: The second term in equation (1) is nothing but the rate of change of volume per unit time, per unit volume i.e. volumetric strain rate. And that's the reason why we replaced @u @v @w it with the term @x + @y + @z in next step (for more details please refer to the module 2). " # " # 1 @ρ @ρ @ρ @ρ @u @v @w =) + u + v + w + + + = 0 ρ @t @x @y @z @x @y @z " # " # @ρ @ρ @ρ @ρ @u @v @w =) + u + v + w + ρ + + = 0 @t @x @y @z @x @y @z @ρ @(ρu) @(ρv) @(ρw) =) + + + = 0 @t @x @y @z @ρ =) + r:(ρ~q) = 0 (2) @t This is the required differential form of the continuity equation. The time derivative in equation (2) can be understood as the accumulation (or loss) of mass in the system, while the divergence term represents the difference in flow in versus flow out. In this context, this equation is also one of the Euler's equations (fluid dynamics). If the flow is incompressible: Flow incompressible means the volumetric strain rate is 1 DV 1 Dρ Dρ zero, i.e. V Dt = 0 in equation (1) =) ρ Dt = 0 =) Dt = 0: 1The material derivative is defined for any tensor field Q that is macroscopic, with the sense that it depends only on position and time coordinates, Q = Q(x; t): DQ @Q ≡ + ~q:rQ Dt @t where rQ is the covariant derivative of the tensor, and ~q is the flow velocity. 2 1.2 Control Volume Approach The equation is developed by adding up the rate at which mass is flowing in and out of a control volume, and setting the net in-flow equal to the rate of change of mass within it. This is demonstrated in the figure on right side and mathematically we can write it as; @ m_ − m_ = (m ): (3) in out @t CV wherem _ in is the mass flow rate that flows in,m _ out is the mass flow rate that flows out and mCV is the rate of change of mass in control volume. Figure 1: depicting the mass flow Now, let us do some gymnastics with equation (3); rates in control volume. @ [_m − m_ ] + [m _ − m_ ] + [m _ − m_ ] = (m ): (4) x x+∆x y y+∆y z z+∆z @t CV Now, using the Taylor's series expansion; ! @m_ m_ − m_ =m _ − m_ + x ∆x + ::: x x+∆x x x @x @m_ = − x ∆x − ::: @x Similarly, we can write others; @m_ m_ − m_ = − y ∆y − ::: y y+∆y @y @m_ m_ − m_ = − z ∆z − ::: z z+∆z @z Putting all these (ignoring the higher order terms) in equation (4), we get; " # @m_ @m_ @m_ @ − x ∆x + y ∆y + z ∆z = (m ): (5) @x @y @z @t CV also we have; m_ x = ρ∆y:∆z:u m_ y = ρ∆z:∆x:v m_ z = ρ∆x:∆y:w mCV = ρ∆x:∆y:∆z Therefore from equation (5), we get; " # @(ρu) @(ρv) @(ρw) @ − + + (∆x∆y∆z)= ρ(∆x∆y∆z) @x @y @z @t " # @(ρu) @(ρv) @(ρw) @ρ =) − + + = @x @y @z @t 3 @ρ @(ρu) @(ρv) @(ρw) =) + + + = 0 @t @x @y @z @ρ =) + r(ρ~q) = 0: @t And thus, again we get the differential form of the continuity equation. @ρ If the flow is incompressible: That means, if density is not variable x; y; z; t =) @t = 0 and that further =) r:~q = 0: Example 1.1: Check whether the flow is possible or not for the given velocity field ~q = x^i−y^j. Explanation: As r:~q = 0 for the given velocity field so the flow is possible or ~q is the possible velocity field. Further, note that if the equation of continuity is not satisfied then we say flow is not possible (for an example take ~q = x^i − y^j + zk^). 2 Angular Strain Rate (Angular Deformation of Fluid Particle) Consider a fluid element ABCD as shown in the figure with dimensions ∆x and ∆y. Let after ∆t time the fluid element deforms and occupies the shape of PQRS. We have OP = v:∆t and QN 0 = v(x + ∆x):∆t. By using Taylor's series expansion we have; @v QN 0 = v + :∆x + :::∆t @x now, QN = QN 0 − NN 0 or Figure 2: depicting the angular defor- QN = QN 0 − OP mation of fluid particle. Then by using values of QN 0 and OP we will get, @v QN = ∆x∆t + ::: @x also we have PN = ∆x + ::: ) in 4PNQ we have, QN tan(∆α) = PN @v ∆x∆t + ::: =) tan(∆α) = @x (6) ∆x + ::: If ∆t ! 0 then ∆α ! 0, in that case tan(∆α) ≈ ∆α and therefore from equation (6) we get; ∆α lim = @v ∆t!0 ∆t @x @v =) α_ = @x 4 similarly, doing whole process along y − coordinate we can get, @u β_ = @y whereα _ = lim ∆α and β_ = lim ∆β . u and v velocity components in x and y directions ∆t!0 ∆t ∆t!0 ∆t respectively. Angular Strain Rate: The angular strain rate is define as rate of change of angle between the line elements which were initially perpendicular to each other. In our case, angular strain rate is basically the rate of change of \SP Q with respect to time. π π Change in angle = − − ∆α − ∆β = ∆α + ∆β: ) 2 2 ∆α + ∆β Rate of shear strain or rate of angular strain = ) ∆t =α _ + β_ @v @u = + : @x @y Angular Velocity: We define the angular velocity of the fluid element as; 1 ! = (_α − β_) z 2 1@v @u ! = − : z 2 @x @y Vorticity: Vorticity is a pseudovector (a vector under a proper angular rotation), which is denoted by Ω~ and defined by; Ω~ = curl ~q = r × ~q ^i ^j k^ @ @ @ =) Ω~ = @x @y @z u v w @w @v @u @w @v @u =) Ω~ = ^i − + ^j − + k^ − @y @z @z @x @x @y ~ ^ =) Ω = ^i(2!x) + ^j(2!y) + k(2!z) =) Ω~ = 2~! ) vorticity = 2 × angular velocity where ~! = (!x;!y;!z) is the angular velocity vector. .............................................................................................................................. Note: Vorticity vector is just effect of angular velocity. Moreover, flow vorticity= 0 if and only if flow is irrotational or irrotational flow () curl~q = ~0. 5 2.1 Flow Analysis @u @v • Let the flow is 2-D and incompressible: The equation of continuity is @x + @y = 0. Let us define = (x; y) such that; @ @ u = and v = − : @y @x Then the function is called stream function. Consider = constant (represents a family of curve), then; =) @ = 0 @ @ =) dx + dy = 0 @x @y =) −vdx + udy = 0 dx dy =) = u v Therefore = constant, represents streamlines of the flow. • Let the flow is irrotational: Let φ(x; y; z) be a scalar function; ) curl(rφ) = ~0: Comparing both of the above equations, we can say that 9 a scalar function φ(x; y; z) such that; ~q = rφ ! @φ @φ @φ =) (u; v; w) = ; ; @x @y @z @φ @φ @φ =) u = ; v = ; w = : @x @y @z Note: The scalar function φ(x; y; z) is called velocity potential. If the fluid flow is irro- tational and inviscid then flow will always remain irrotational and such type of flow is called potential flow. • Let the flow be 2D, incompressible and irrotational: That is stream function = (x; y) exist and = constant. =) d = 0 dx dy =) = u v dy v =) = dx u v =) slopej = : =c u 6 Also φ = constant =) dφ = 0: @φ @φ =) dx + dy = 0 @x @y =) udx + vdy = 0 dy u =) = − dx v u =) slopej = − : φ=k v At the point of intersection of = c and φ = k, we have; (slopej =c):(slopejφ=k) = −1: =) = c and φ = k are orthogonal to each other.
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