Viscoelastic flow of an Oldroyd-B fluid through a slowly varying contraction-expansion channel: pressure drop and elastic stress relaxation

Abstract

Viscoelastic fluid flows in narrow non-uniform geometries are ubiquitous in various engineering applications and physiological flow systems. For such flows, one of the key interests is understanding how fluid viscoelasticity affects the flow rate-pressure drop relation, which remains not fully understood. We analyze the flow of the Oldroyd-B fluid in slowly varying contraction-expansion channels, commonly referred to as constrictions. Unlike most previous theoretical studies focusing on contracting channels, we consider a constriction geometry and present a theory for calculating the elastic stresses and flow rate-pressure drop relation at low and high Deborah ($De$) numbers. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity approximates a parabolic and Newtonian profile. This results in a one-way coupling between the velocity and elastic stresses, allowing us to derive closed-form expressions for the elastic stresses and pressure drop for arbitrary values of $De$. We validate our theoretical predictions with numerical simulations, finding excellent agreement. We identify the physical mechanisms governing the pressure drop behavior and compare our results for the constriction with previous predictions for the contraction. At low $De$, the pressure drop in the constriction monotonically decreases with $De$, similar to the contraction. However, at high $De$, in contrast to a linear decrease for the contraction, the pressure drop across the constriction reaches a plateau due to the vanishing contribution of elastic normal stresses, leaving elastic shear stresses as the sole driver of the reduction. Furthermore, we elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following both constriction and contraction geometries, showing that the relaxation length is significantly shorter in the case of a constriction.

Figures (7)

• Figure 1: (a) Schematic illustration of the planar configuration consisting of a slowly varying and symmetric contraction-expansion channel of height $2h(x)$ and length $\ell$$(h\ll \ell)$, connected to two long straight channels of height $2h_0$ up- and downstream. The contraction-expansion channel, which we refer to as constriction, contains a viscoelastic fluid steadily driven by the imposed flow rate $q$. (b) Schematic illustration of the orthogonal curvilinear coordinates ($\xi,\eta$) used in the lubrication analysis of a slowly varying constriction. The coordinate $\xi$ is constant along vertical grid lines, and $\eta$, defined in (\ref{['curvilinear mapping']}), is constant along the curves going from left to right. We are interested in determining the pressure drop $\Delta p$ over the constriction region and the spatial relaxation of pressure and elastic stresses in the exit channel.
• Figure 2: Illustration of the smooth constriction and contraction geometries considered in this work. The black curve represents the constriction shape function \ref{['H_constriction']} and the gray curve represents the contraction shape function \ref{['H_contraction']}. All calculations were performed using $\delta=0.25$.
• Figure 3: The streamwise variation of leading-order elastic stresses on $\eta=0.5$ in a slowly varying constriction and exit channel in the ultra-dilute limit. (a--c) Scaled elastic stresses $\mathcal{A}_{11,0}/(18De^2\eta^2)$, $\mathcal{A}_{12,0}/(-3De\eta)$ and $\mathcal{A}_{22,0}$ in the constriction as a function of $X$ for (a) $De=0.001$, (b) $De=0.1$, and (c) $De=1$. (d--f) Scaled elastic stresses in the exit channel $\mathcal{A}_{11,0}/(18De^2\eta^2)$, $\mathcal{A}_{12,0}/(-3De\eta)$ and $\mathcal{A}_{22,0}$ in the exit channel as a function of $X_\ell$ for (d) $De=0.001$, (e) $De=0.1$, and (f) $De=1$. The insets in (c) and (f) show the zoom-in view. Solid lines represent the semi-analytical solutions \ref{['b22_Gen']}--\ref{['b11_Gen']} (constriction) and (\ref{['A_general_exit']}) (exit channel). Cyan dotted lines represent the low-$De$ asymptotic solutions (\ref{['A_ij_lowDe']}) (constriction) and (\ref{['A_general_exit']}) with (\ref{['Aij_ref_LowDe_OB_Ultra_pol4']}a--c) (exit channel). Red dashed lines represent the high-$De$ asymptotic solutions (\ref{['bij_ultradilute_OB_highDe_sol']}) (constriction) and (\ref{['A_general_exit']}) with (\ref{['Aij_ref_HighDe_OB_Ultra']}) (exit channel). All calculations were performed using $\delta=0.25$.
• Figure 4: Non-dimensional pressure drop at low and high Deborah numbers for the Oldroyd-B fluid in a constriction channel in the ultra-dilute limit. (a,b) Scaled pressure drop $\Delta P/ \Delta P_0$ as a function of $De=\lambda q/(2\ell h_0)$. Gray triangles represent the OpenFOAM simulation results. Black dots represent the semi-analytical solution \ref{['dP total OB low-beta']}. The cyan dotted line represents the low-$De$ asymptotic solution \ref{['dP total OB low-De']}. The purple solid line represents the low-$De$ Padé approximation \ref{['Pade-approximation OB low-De']}. The red dashed line represents the high-$De$ asymptotic solution \ref{['dP_nonuniform_HighDe constriction']}. All calculations were performed using $\beta_p=0.05$ and $\delta=0.25$.
• Figure 5: Comparison of non-dimensional pressure drop for the Oldroyd-B fluid in constriction and contraction channels in the ultra-dilute limit. Scaled pressure drop $\Delta P/ \Delta P_0$ as a function of $De=\lambda q/(2\ell h_0)$. Black dots and gray crosses represent the semi-analytical solutions \ref{['dP total OB low-beta']} for the constriction and contraction geometries. The red dashed line represents the high-$De$ asymptotic solution (\ref{['dP_nonuniform_HighDe constriction']}) for the constriction. The black dashed line represents the high-$De$ asymptotic solution (\ref{['dP_nonuniform_HighDe']}) for the contraction. All calculations were performed using $\beta_p=0.05$ and $\delta=0.25$.