STEADY VISCOUS FLOW THROUGH A VENTURI TUBE

CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 12, Number 2, Summer 2004 STEADY VISCOUS FLOW THROUGH A VENTURI TUBE K. B. RANGER ABSTRACT. Steady viscous flow through an axisymmetric convergent-divergent nozzle is discussed for a wide range of viscosities. It is shown that in general the Stokes flow represents a uniform approximation to the flow for arbitrary viscosity. Introduction The present article discusses the steady axisymmetric viscous flow through a convergent-divergent nozzle. The model for the nozzle is determined by a suitable conformal mapping and is represented by one family of coordinate curves depicted by β = constant. In the first section the Stokes flow is found to be a similarity solution of the coordinate β. This result was previously derived by Sampson [5], Roscoe [4] and is described in the book by Happel and Brenner [3]. The method of derivation given here is different from other authors. It is possible to calculate the pressure drop and wall stress over a section of the nozzle. The main thrust in this discussion is to show that even though the Stokes flow is symmetric it represents a uniform approximation to the motion for a wide range of viscosities. This is achieved by considering coordinate expansions for effectively small and large distances from the nozzle centre. The expansion for large distances is similar to that in [6]. The wall stress and pressure drop in a section of the channel are also valid for a wide range of viscosity and not just for Stokes flow. The applications of the Venturi tube have been discussed in [1]. 1 Axisymmetric Stokes flow through a constricted channel If (z, ρ, φ are cylindrical polar coordinates and ψ = ψ(z, ρ is the stream function for steady axisymmetric motion, the fluid velocity is described AMS subject classification: 34K20, 92D25. Copyright c Applied Mathematics Institute, University of Alberta. 199

200 K. B. RANGER by (1 q = curl { ψ } ρ φ = 1 ρ ψ ρ k + 1 ρ ψ z ρ, and for Stokes flow which plays a central role in this analysis the momentum equations can be written in the form (2 p z = µ ρ ρ (L 1(ψ, p ρ = µ ρ z (L 1(ψ, where p = p(z, ρ is the pressure field, and µ the coefficient of viscosity. The Stokes operator is defined by ( 2 (3 L 1 (ψ = z 2 + 2 ρ 2 1 ψ. ρ ρ If the pressure p is eliminated from equations (2, then the stream function ψ satisfies the Stokes or repeated operator equation (4 L 2 1(ψ = L 1 (L 1 (ψ = 0. There is a decomposition formula for the solution of iterated operator equations first given by Weinstein [2] and is expressible in the form (5 ψ = v ( 1 + v ( 3, where v ( k, k = 1, 3, satisfy the equation { (6 L k v ( k} = v zz ( k + v ρρ ( k k ρ v( k ρ = 0. v ( 1 and v ( 3 may be interpreted as potential flow stream functions in axisymmetric spaces of three and five dimensions, respectively. Now if z + iρ = F (α + iβ, F (α + iβ 0 is a conformal mapping of the meridional plane (z, ρ into a region of the (α, β plane, then v ( k satisfies the equation (7 ρ k {ρ α k v( k α } + ρ k {ρ β k v( k β } = 0,

STEADY VISCOUS FLOW THROUGH A VENTURI TUBE 201 and (8 L 1 (ψ 1 = F (α + iβf (α iβ { ρ α ρ ψ + ρ α β Also the momentum equations (2 transform in the form (9 p α = µ ρ β (L 1(ψ, Oblate ellipsoidal coordinates ρ p β = µ ρ α (L 1(ψ. The conformal mapping (10 z + iρ = F (α + iβ = c sinh(α + iβ produces the real equations (11 z = c sinh α cos β, ρ = c cosh α sin β and the surfaces β = constant, described by ( 2 ( 2 ρ z (12 = c 2 sin β cos β } ψ. β represent a system of confocal hyperboloids of one sheet with common focal ring located at z = 0, ρ = c. The surfaces α = constant, are the orthogonal trajectories and represent a confocal set of planetary or oblate ellipsoids also with common ring at z = 0, ρ = c. The range of the coordinates (α, β are given by 0 β < π/2, < α <. This coordinate system is appropriate to the description of flow through a convergent-divergent nozzle, and for the purposes of application describes a section of stenotic artery. A diagram of the coordinate system in the meridional plane together with a sketch of the flow model are given below in Figure 1 and Figure 2. 2 The boundary value problem for steady Stokes flow The stream function ψ satisfies equation (4 and for the present purposes an appropriate decomposition formula for the solution is described by (5. The boundary conditions of no seepage and no slip on the boundary together with finite flow on the axis are represented by (13 (14 ψ = M, ψ β = 0 at β = β 0, ψ = 0, at β = 0 for < α <.

202 K. B. RANGER ρ β=const. α=const. z FIGURE 1: Figure 1. Sketch of typical curves in the meridional plane x z y FIGURE 2: Figure 2. Sketch of the convergent-divergent nozzle

STEADY VISCOUS FLOW THROUGH A VENTURI TUBE 203 M is the constant volumetric flux rate through the nozzle. The equations satisfied by v ( 1, v ( 3 are; (15 (16 cosh α cosh α α cosh 3 α α cosh 3 α v ( 1 α v ( 3 α + sin β β + sin 3 β β sin β sin 3 β v ( 1 β v ( 3 β = 0, = 0. Suitable solutions for v ( 1, v ( 3 are forced by the boundary conditions, and in this case v ( 1 = v ( 1 (β, v ( 3 = v ( 3 (β so that (17 d dβ sin β dv ( 1 dβ The solutions are given by = 0, d dβ sin 3 β dv ( 3 dβ = 0. (18 v ( 1 (β = a cos β + b, v ( 3 (β = c cos β c 3 cos3 β + d, where a, b, c and d are constants. The stream function ψ can then be more conveniently written as; (19 ψ = A cos β + C cos 3 β + D where A, C andd are constants to be determined by the boundary conditions. These conditions require that A, C and D satisfy (20 A cos β 0 + C cos 3 β 0 + D = M, A + 3C cos 2 β 0 = 0, A + C + D = 0 and the complete Stokes solution is defined by (21 ψ = M [ cos 3 β 1 3 cos 2 β 0 (cos β 1 ] [3 cos 2 β 0 1 2 cos 3. β 0 ] The stream function is a similarity solution of the Stokes equations since the streamlines coincide with the curves β = constant. This flow was previously derived by Sampson, Roscoe and is also contained in the book by Happel and Brenner. The method of derivation in [3] is different from that given here, and the analysis presented in this section appears to be the first application of the decomposition formula (5 to this problem.

204 K. B. RANGER The components of fluid velocity u, v along and perpendicular to the symmetry axis respectively are described by (22 (23 u = 1 ρ ψ ρ = 1 ρ β ρψ β = 3M cos β ( cos 2 β 0 cos 2 β c 2 ( sinh 2 α + cos 2 β (3 cos 2 β 0 1 2 cos 3 β 0, v = 1 ρ ψ z = 1 ρ β zψ β = 3M sinh α ( cos 2 β cos 2 β 0 sin β c 2 cosh α ( sinh 2 α + cos 2 β [3 cos 2 β 0 1 2 cos 3 β 0 ]. The velocity on the axis at the origin α = β = 0 is given by 3M ( 1 cos 2 β 0 (24 u = c 2 [3 cos 2 β 0 1 2 cos 3 β 0 ] and decays like e 2 α or 1/r 2 as α. defined by the formula The fluid vorticity ω is (25 ω = L 1(ψ ρ 1 = c 3 cosh α sin β(sinh 2 α + cos 2 β { cosh ψ α cosh α α = sin β β sin β } ψ β 6M cos β sin β c 3 cosh α(sinh 2 α + cos 2 β [3 cos 2 β 0 1 2 cos 3 β 0 ]. In particular the vorticity on the boundary β = β 0 is expressed by (26 ω = 6M sin β 0 cos β 0 c 3 cosh α ( sinh 2 α + cos 2 β 0 [3 cos2 β 0 1 2 cos 3 β 0 ], and is a maximum at α = 0, and vanishes like O(e 3 α as α. The maximum is given by (27 ω = 6M tan β 0 c 3 [3 cos 2 β 0 1 2 cos 3 β 0 ],

STEADY VISCOUS FLOW THROUGH A VENTURI TUBE 205 which is an increasing function of β 0 in the range 0 < β 0 < π/2. Since the fluid velocity vanishes at β = β 0, it follows ψ ββ ω = ρc 2 (sinh 2, α + cos 2 β β=β 0 so that the boundary vorticity is related to the shear stress on the boundary in a simple way. The shear stress is negative for all α on β = β 0, so that there is no possibility of streamline separation attached to the boundary. Finally it is noted that the numerical comparison can be substantiated by observing that the pressure force, A p = 1 (sec β 0 + 1 2. 3 The pressure field It has been shown that in (α, β coordinates the momentum equations are equivalent to (28 where (29 L 1 (ψ = p α = µ c cosh α sin β β L 1(ψ, p β = µ c cosh α sin β α L 1(ψ 6M cos β sin 2 β c 2 (sinh 2 α + cos 2 β[3 cos 2 β 0 1 2 cos 3 β 0 ]. By elimination of L 1 (ψ from these equations, the pressure field satisfies the Laplace equation and in (α, β coordinates ( 1 (30 cosh α p + 1 ( sin β p = 0. cosh α α α sin β β β Now if s = cos β, then from (28 it is found that p (31 s = 6Mµ ( s 1 c 3 cosh α α sinh 2 α + s 2 (3 cos 2 β 0 1 2 cos 3 β 0. Integration shows that (32 p = 3Mµ c 3 cosh α[3 cos 2 β 0 1 2 cos 3 β 0 ] { ln(sinh 2 α + s 2 } + A(α α 6Mµ sinh α = c 3 [3 cos 2 β 0 1 2 cos 3 β 0 ][sinh 2 α + cos 2 β] + A(α,

206 K. B. RANGER where A(α can be taken as a constant since it is readily verified that { cosh α + 1 ( sin β } (33 cosh α α α sin β β β sinh α (sinh 2 α + cos 2 β = 0. The axial pressure drop in a section of nozzle from α = +α 0 to α = α 0, z 0 = c sinh α 0, is given by (34 p = p α=α0 p α= α0 = 12Mµz 0 c 2 (z 2 0 + c2 [2 cos 3 β 0 + 1 3 cos 2 β 0 ]. In particular the pressure drops in the channel along the axis for β 0 = π/6, β 0 = π/4, β 0 = π/3 are given by (35 p π/6 = k 0.049, p π/4 = k 0.207, p π/3 = k 0.5, where the constant k = 12Mz 0 µ/(c 2 (z 2 0 + c2. The ratio of these drops are represented by (36 p π/6 p π/4 = 4.224, p π/6 p π/3 = 10.206. The minimum area of the cross-sections corresponding to these drops are: (37 A π/6 = 1 4 πc2, A π/4 = 1 2 πc2, A π/3 = 3 4 πc2 and the ratios are given by (38 A π/3 A π/6 = 3, A π/4 A π/6 = 2. These results indicate that the ratio of the pressure drops are slightly greater than the square of the cross-sectional ratios. Again the volume of liquid in the channel bounded by z 0 z z 0, the axis β = 0 and the meridian curve β = β 0, is represented by z0 (39 V = π ρ 2 dz, ρ 2 = sin 2 β 0 (c 2 + z2 z 0 cos 2 β 0 = 2π sin 2 β 0 (z 0 c 2 z0 3 + 3 cos 2 β 0

STEADY VISCOUS FLOW THROUGH A VENTURI TUBE 207 ψ=μ Fig. 1 z=-z 0 ψ=0 z=z 0 β 0 =π/6 Fig. 2 z=-z 0 ψ=0 z=z 0 β 0 =π/4 Fig. 3 z=-z 0 ψ=0 z=z 0 β 0 =π/3

208 K. B. RANGER and the pressure drop along the axis from z = z 0 to z = z 0, is given by (40 p = p z=z0 p z= z0 = 12Mz 0 µ c 2 (z 2 0 + c2 [2 cos 3 β 0 + 1 3 cos 2 β 0 ]. This result can be contrasted with the corresponding values determined from steady Poiseulle flow through a straight pipe where the volumetric flux rate is the same and is applied to a section of the pipe with the same volume of liquid. If the radius of the pipe is R, then (41 V = 2πR 2 z 0, R 2 = sin 2 β 0 (c 2 z 2 0 + 3 cos 2 β 0 and from the known results for Poiseulle flow available in the standard texts (42 M = πp R4 9µ, P = dp dz where the pressure gradient is constant. The pressure drop is then 16Mz 0 µ (43 p = 2P z 0 = 2, π sin 4 β 0 (c 2 + z2 0 3 cos 2 β 0 and may be compared with the motion in a constricted channel by the ratio (44 S = 12Mπz 0 µ sin 4 β 0 (c 2 + z 2 0 /3 cos2 β 0 2 c 2 (c 2 + z 2 0 16Mµz 0(2 cos 3 β 0 + 1 3 cos 2 β 0, which is an increasing function of β 0 in the range 0 < β 0 < π/2. In fact as β 0 π/2, S. 4 Steady state Navier-Stokes equations For axisymmetric flow the fluid velocity field is defined by (45 q = curl { ψ } ρ φ = 1 ρ ψ ρ k + 1 ρ ψ z ρ

STEADY VISCOUS FLOW THROUGH A VENTURI TUBE 209 and the momentum equations can be written in the form (46 (47 1 ρ 2 ψ zl 1 (ψ = B z ν ρ 1 ρ 2 ψ ρl 1 (ψ = B ρ + ν ρ ρ L 1(ψ, z L 1(ψ where B = p/ρ 0 + (1/2 q 2 is the Bernoulli function or total head of pressure, and ν is the kinematic viscosity. Elimination of B results in the vorticity equation (48 ρ { ψ, L 1 (ψ/ρ 2} (z, ρ = νl 2 1 (ψ. In oblate ellipsoidal coordinates z + iρ = c sinh(α + iβ equations (46 (47 transform into: (49 (50 1 ρ 2 ψ αl 1 (ψ = B α ν ρ 1 ρ 2 ψ βl 1 (ψ = B β + ν ρ β (L 1(ψ, α (L 1(ψ and also there is corresponding mapping for the vorticity equation (51 ρ {ψ, L 1(ψ/ρ} (α, β = ν { cosh α α cosh α α + sin β ( } 1 L 1(ψ β sin β β where (52 { 1 L 1 (ψ = c 2 (sinh 2 α + cos 2 β cosh α α cosh α ψ α + sin β ( } 1 ψ. β sin β β

210 K. B. RANGER The coordinate expansion for large α In this situation the stream function ψ is expanded in the asymptotic series (53 ψ F n (βe nα, α, with boundary conditions (54 n=0 F 0 (β 0 = M, F n (β 0 = 0, n 1, F n(β 0 = 0, n 0 and F n (0 = 0 for n 0. For large α the leading order approximations show that (55 ρ { ψ, L 1 (ψ/ρ 2} = O(e 3α, α (α, β and (56 [ cosh α α cosh α + sin β α β sin β ] L 1 (ψ β = O(e 2α, α. Similar results are also determined as α, and in both cases the leading term in the expansion is the Stokes flow which satisfies the equation (57 [ sin β β sin β ] + 6 sin β ( 1 β β sin β F 0 (β = 0. The solution which produces finite flow on the axis is given by (58 F 0 (β = M[cos3 β 1 3 cos 2 β 0 (cos β 1] (3 cos 2 β 0 1 2 cos 3. β 0 Higher order approximations satisfy a system of inhomogenous linear partial differential equations of fourth order which in principle can be constructed successively. It is noted from equations (49, (50 that for consistency the components of convective-acceleration show that 1 ρ 2 ψ αl 1 (ψ + 1 [ (ψα 2 + ψ β 2 ] (59 2 α ρ 2 c 2 (sinh 2 = O(e 4α, α + cos 2 β (60 [ 1 ρ 2 ψ βl 1 (ψ + 1 2 β ψ 2 α + ψ2 β ρ 2 c 2 (sinh 2 α + cos 2 β ] = O(e 4α

STEADY VISCOUS FLOW THROUGH A VENTURI TUBE 211 and the components of viscous diffusion: (61 ν ρ β (L 1(ψ = O(e 3α, ν ρ α (L 1(ψ = O(e 3α for large α. This demonstrates that the ratio convection/diffusion = O(e α, α and the Stokes approximation is valid at large distances from the nozzle. Similar results apply to higher order approximations since the motion is finite in the flow region. A similar result can be obtained as α. It is not clear whether the leading order term is a good approximation for small α. However it is observed from the exact vorticity equation that on the boundary β = β 0, convectiveacceleration is absent and the governing equation is viscous diffusion, that is L 2 1 (ψ = 0. It then appears that the complete Stokes solution is a leading order approximation to the flow in a neighbourhood of the nozzle wall and is independent of viscosity. A second point is that if ψ = F 0 (β, then the convective-acceleration term from (48 is given by (62 ρ [ψ, L 1(ψ/ρ 2 ] (α, β Also the diffusion term [ (63 cosh α α cosh α = 0 at α = 0. + sin β α β sin β [ F ] β 0 (β cot βf 0(β sinh 2 α + cos 2 β vanishes independent of viscosity. It follows from the vorticity equation that the Stokes solution represents the leading order term in a coordinate expansion for small α described by (64 ψ F 0 (β + where α n G m (β, α 0 m=1 (65 G m (β 0 = G m (β 0 = G m (0 = 0 for m 1. Finally it is reasonable to assume that although the flow is asymmetric about the plane α = 0, that the Stokes solution is a uniformily valid leading order approximation for the flow independent of α and valid for arbitrary viscosity. ]

212 K. B. RANGER 5 Conclusions The results for the pressure drop and wall-stress over a section of the flow through a convergent-divergent nozzle appear to be new as the analysis suggest they are valid over a wide range of viscosity. The possibility of application to the flow of blood through a large artery is also included but since the motion is pulsatile the results would have to be interpreted in some sense as mean values over a suitable time period. Since the leading order approximation is symmetric about the plane of symmetry, but the complete flow is asymmetric there is the possibility of reverse flow occuring on a region of the wall downstream of the nozzle. However this effect is likely to be small compared with the main forcing flow. REFERENCES 1. G. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, p. 484, 1967. 2. J. Happel and H. Brenner, Low Reynolds number hydrodynamics, Martenus Hijoff Publishers, 138 141, 1983. 3. J. Happel and and H. Brenner, Low Reynolds number hydrodynamics, Martenus Hijhoff Publishers, p. 153, 1983. 4. R. Roscoe, Phil. Mag. 40 (1949, 338. 5. R. A. Sampson, Phil Trans. Roy. Soc. A182 (1891, 449. 6. A. Weinstein, Annali Di Mathematics Pure Et Applicata 1959 iv., 39. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3 E-mail address: ranger@math.utoronto.ca