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THEORY OF ELASTICITY
A. T. LISTROV, A. D. CHERNYSHOV
ON THE STEADY FLOW OF A VISCO-PLASTIC MEDIUM WITH NONLINEAR VISCOSITY
(Presented by Academician A. Yu. Ishlinskii, 3 II 1964)
Numerous investigations have been devoted to the motion of a visco-plastic medium; among them we note \((^{1-3})\).
In the present work an exact solution is constructed for the problem of the motion of a visco-plastic medium with nonlinear viscosity. The problem of the steady motion of a rigid cylindrical body in a visco-plastic medium is considered (similar problems for linear viscosity were considered by another method in \((^4)\)). The problem investigated belongs to the class of problems of antiplane deformation.
Let us consider the steady flow of an unbounded isotropic visco-plastic medium caused by the motion of an infinitely long rigid cylinder with constant velocity \(u_0\) in the direction of its generator. The contour \(S_1\) of the transverse section of the cylinder is convex, smooth, and has no straight segments (Fig. 1).
Fig. 1 Fig. 2
Introduce a moving coordinate system rigidly connected with the cylinder. Direct the \(z\)-axis along the axis of the cylinder opposite to the direction of its motion, and place the \(x\)- and \(y\)-axes in the plane of its transverse section. The velocity \(u(x,y)\) of each particle of the medium will be directed along the \(z\)-axis. We write the relation between the stress and the shear rate in the form
\[ \gamma = a(\tau-k)^\mu,\qquad \mu>0, \tag{1} \]
where \(\tau\) is the shear stress; \(\gamma\) is the shear rate; \(k\) is the yield limit; \(\mu, a\) are constants.
Keeping the former notation, pass to dimensionless quantities. Refer the velocity \(u(x,y)\) to the quantity \(u_0\), the shear stress \(\tau\) to the yield limit \(k\); quantities having the dimension of length to the constant \(u_0/ak^\mu\), and the shear rate \(\gamma\) to the quantity \(ak^\mu\). The original relation (1) is written in the form \(\gamma = F(\tau) = (\tau-1)^\mu\).
Following the ideas of the works \((^{5,6})\), pass to the orthogonal coordinate net \(u, v\), formed by the lines of constant deformation velocity \(u=\mathrm{const}\) and the stress lines \(v=\mathrm{const}\). The vector \(\vec{\tau}\) is directed tangent to the lines \(v=\mathrm{const}\).
To solve the problem we shall use the relations \((^5)\)
\[ \frac{\partial x}{\partial u}=h_u\cos\varphi,\qquad \frac{\partial x}{\partial v}=-h_v\sin\varphi,\qquad \frac{\partial y}{\partial u}=h_u\sin\varphi,\qquad \frac{\partial y}{\partial v}=h_v\cos\varphi, \]
\[ h_u=\frac{1}{F(\tau)},\qquad h_v=\frac{1}{\tau},\qquad \frac{F}{\tau^2}\frac{\partial v}{\partial\varphi}=-\frac{\partial u}{\partial\tau},\qquad \frac{\tau F'}{F^2}\frac{\partial u}{\partial\varphi}=\frac{\partial v}{\partial\tau}. \tag{2} \]
The equation for the function \(u(\tau,\varphi)\) has the form \((^5)\)
\[ \frac{F^2}{\tau F'}\frac{\partial}{\partial\tau} \left(\frac{\tau^2}{F}\frac{\partial u}{\partial\tau}\right) +\frac{\partial^2 u}{\partial\varphi^2}=0, \tag{3} \]
where \(h_u, h_v\) are the Lamé parameters, and \(\varphi\) is the angle of inclination of the vector \(\vec{\tau}\) to the \(x\)-axis.
The boundary conditions for the function \(u\) are determined from the conditions of adhesion of the medium to the cylinder being flowed around and from the constancy of the deformation rate at the boundary of the rigid zone of the medium, whence
\[ u\big|_{S_1}=0,\qquad u\big|_{\tau=1}=1. \tag{4} \]
To find \(u(\tau,\varphi)\), it is necessary to solve equation (3) under the boundary conditions (4). Separating the variables in this equation and using the second of conditions (4), we obtain
\[ u(\tau,\varphi)=1+A_0\int_1^\tau \frac{F}{\tau^2}\,d\tau +(\tau-1)^{1+\mu}\sum_{n=1}^{\infty}\Phi_n \left(A_n\cos\lambda_n+B_n\sin\lambda_n\varphi\right), \tag{5} \]
where \(\lambda_n\) are the eigenvalues of the problem; \(\Phi_n(2-\alpha_n,\,2-\beta_n,\,2+\mu,\,1-\tau)\) are hypergeometric functions;
\[ \alpha_n=\frac{1}{2}(1-\mu)+\left[\frac{1}{4}(1-\mu)^2+\mu n\lambda_n^2\right]^{1/2}, \]
\[ \beta_n=\frac{1}{2}(1-\mu)-\left[\frac{1}{4}(1-\mu)^2+\mu n\lambda_n^2\right]^{1/2}. \]
The constants \(A_0,\ A_n,\ B_n\) \((n=1,2,\ldots)\) are determined from conditions (4) on the contour \(S_1\).
If the contour \(S_1\) has \(r\) axes of symmetry, then \(\lambda_n=rn,\ B_n=0\). If the contour \(S_1\) consists of \(m\) identical arcs, so that any two neighboring arcs can be superposed by rotating one of them through the angle \(\varphi_0=2\pi/m\), then \(\lambda_n=mn\). If the contour is of arbitrary shape, then \(\lambda_n=n\).
The dependences \(x(\tau,\varphi), y(\tau,\varphi)\), and \(v(\tau,\varphi)\) are found by using the relation
\[ \frac{\partial x}{\partial\tau} =\frac{\partial x}{\partial u}\frac{\partial u}{\partial\tau} +\frac{\partial x}{\partial v}\frac{\partial v}{\partial\tau}, \]
the analogous relations for \(\dfrac{\partial x}{\partial\varphi},\ \dfrac{\partial y}{\partial\tau},\ \dfrac{\partial y}{\partial\varphi}\), and also expressions (2) and (5), whence
\[ x(\tau,\varphi)=A_0\left(1-\frac{\cos\varphi}{\tau}\right)+ \]
\[ +\sum_{n=1}^{\infty} \bigl[ A_n(K_n-F_n)\cos\varphi\cos\lambda_n\varphi -B_nL_n\sin\varphi\cos\lambda_n\varphi + \]
\[ +A_nL_n\sin\varphi\sin\lambda_n\varphi +B_nK_n\cos\varphi\sin\lambda_n\varphi +A_nF_n \bigr]+x_0, \]
\[ y(\tau,\varphi)=-A_0\frac{\sin\varphi}{\tau} +\sum_{n=1}^{\infty} \bigl[ A_n(K_n-F_n)\sin\varphi\cos\lambda_n\varphi +B_nL_n\cos\varphi\cos\lambda_n\varphi+ \]
\[ +B_n(K_n-F_n)\sin\varphi\sin\lambda_n\varphi -A_nL_n\cos\varphi\sin\lambda_n\varphi +\lambda_nB_nF_n \bigr]+y_0, \tag{6} \]
\[ v(\tau,\varphi)=-A_0\varphi+\sum_{n=1}^{\infty} \left\{ \frac{\mu(\mu+1)\lambda_n}{(1-\alpha_n)(1-\beta_n)} \left[ 1-\tau\Phi^*+\frac{\mu}{\alpha_n\beta_n}(1-\Phi_n^{**}) \right] +\right. \]
\[ \left. +\frac{1+\mu}{\mu} \right\} \{B_n\cos\lambda_n\varphi-A_n\sin\lambda_n\varphi\}+v_0, \]
where the following notation has been introduced:
\[ F_n=\frac{\mu+1}{1-\lambda_n^2},\qquad K_n=\frac{\mu(\mu+1)}{(1-\alpha_n)(1-\beta_n)}(1-\Phi_n^*)-(1-\tau)\Phi_n, \]
\[ L_n=\lambda_n\left[ \frac{\mu(\mu+1)}{(1-\alpha_n)(1-\beta_n)}(1-\Phi_n^*)F_n \right]. \]
The functions \(\Phi_n^*(1-\alpha_n, 1-\beta_n, 1+\mu, 1-\tau)\), \(\Phi_n^{**}(-\alpha_n, -\beta_n, \mu, 1-\tau)\) are solutions of hypergeometric equations.
Specifying the equation of the contour \(S_1\) in parametric form \(X(t)\), \(Y(t)\), the dependence \(t=t(\varphi)\) is found from the equation
\[ \frac{dX}{dt}\cos\varphi+\frac{dY}{dt}\sin\varphi=0. \tag{7} \]
From condition (4) on the contour \(S_1\) we obtain a system of equations for determining \(x_0, y_0, A_0, A_n, B_n\):
\[ X(t)=x(\tau,\varphi),\quad u(\tau,\varphi)=0,\quad t=t(\varphi),\quad Y(t_1)=y(\tau,\varphi_1),\quad t_1=t(\varphi_1), \tag{8} \]
where \(\varphi_1\) is a fixed value of the angle \(\varphi\).
Putting \(\tau=1\) in expressions (6), we find the equation of the boundary \(S_2\) in parametric form
\[ \begin{aligned} x(1,\varphi)={}&A_0(1-\cos\varphi)+\sum_{n=1}^{\infty}F_n\,[\lambda_n B_n\sin\varphi\cos\lambda_n\varphi -A_n\cos\varphi\cos\lambda_n\varphi\\ &\qquad\qquad\qquad{}-A_n\lambda_n\sin\varphi\sin\lambda_n\varphi+A_n]+x_0, \end{aligned} \tag{9} \]
\[ \begin{aligned} y(1,\varphi)={}&-A_0\sin\varphi+\sum_{n=1}^{\infty}F_n\,[\lambda_n A_n\cos\varphi\sin\lambda_n\varphi -A_n\sin\varphi\cos\lambda_n\varphi\\ &\qquad\qquad\qquad{}-B_n\lambda_n\cos\varphi\cos\lambda_n\varphi -B_n\sin\varphi\sin\lambda_n\varphi+\lambda_n B_n]+y_0. \end{aligned} \]
The determination of solutions of specific problems reduces to finding the constants \(x_0, y_0, A_0, A_n, B_n\) from (8) and (9).
We note that for linear viscosity \((\mu=1)\) the functions \(\Phi_n, \Phi_n^*, \Phi_n^{**}\) are polynomials; in this case finding the constants \(A_n, B_n\) is substantially simplified.
Voronezh State
University
Received
31 I 1964
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